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Interest Rate Modelling: A discussion on
theoretical construction and practical
implementation
Mohamed Reza Ismail (ISMREZ001)
A research dissertation submitted to the Faculty of Commerce, Univer-sity of the Cape Town, in partial fulfilment of the requirements for thedegree of Master of Business Administration.
November 12, 2014
Master of Business Administration,University of the Cape Town,
Cape Town.
Declaration
I declare that this dissertation is my own, unaided work. It is being submitted
for the Degree of Master of Business Administration in the University of the Cape
Town. It has not been submitted before for any degree or examination in any other
University.
November 12, 2014
Abstract
Since the seminal work by Black and Scholes in 1973, the theory and practice
around derivative instruments in general, and interest-rate modelling in particular,
have evolved considerably. The market for over-the-counter interest-rate dependent
derivatives, including interest-rate caps and floors, is today among the largest and
most liquid option markets. According to Rebonato (2002), because of this rapid
development, the academic and practitioner communities have not always commu-
nicated as productively as might be desirable.
What this thesis tries to explain is how prices of derivatives are, in general,
conceived of by academic theoreticians by incorporating the ideas of risk-neutral
pricing, martingales and partial differential equation theory in order to formulate
prices. In the view of the academic theoretician, the abstract theory paves the way
to indicative pricing which, in their view, should ultimately be the prevailing market
prices for derivatives.
However, for the practicing traders in interest-rate derivatives, the notions of risk-
neutral pricing as abstract mathematical constructions actually mean very little. I
provide a discussion of how practicing derivatives traders engage with interest-rate
derivatives, and I show why traders actually need to take the market prices of
instruments such as caps and floors as being given, and then calibrate their various
models accordingly. In the view of the trader then, demand-supply forces in the
market create equilibria, and the market participants effective define what is known
as the “market price of risk”, and this ultimately results in indicative pricing, which
traders hold as the standard to which the theoretical models must be calibrated.
This thesis looks at some of the mathematical reasons for why we rely on the
“market price of risk” in modern interest rate markets, and then examines the
calibration of the LIBOR market model to interest rate caps as an illustrative point
in case.
Acknowledgements
My thanks and gratitude goes to God, without whom we are nought. My sincere
gratitude and appreciation goes to my wife and family who have endured with me
through this work, and many of my other struggles to broaden my knowledge and
expand my horizons. This is for all of you.
We raise in degrees whom We will, but over every possessor of knowledge
is one [more] knowing. -Qur’an 12:76
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. Definitions and Notation . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 The Bank Account and the Short Rate . . . . . . . . . . . . . . . . . 52.2 Zero-Coupon Bonds and Spot Interest Rates . . . . . . . . . . . . . . 62.3 Fundamental Interest-Rate Curves . . . . . . . . . . . . . . . . . . . 92.4 Forward Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.5 Interest-Rate Caps/Floors and Swaptions . . . . . . . . . . . . . . . 12
3. No-Arbitrage Pricing and Numeraire Change . . . . . . . . . . . . 153.1 No-Arbitrage in Continuous Time . . . . . . . . . . . . . . . . . . . . 163.2 The Change-of-Numeraire Technique . . . . . . . . . . . . . . . . . . 183.3 A Change of Numeraire Toolkit - Adapted from Brigo and Mercurio
[2007] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.4 The Choice of a Convenient Numeraire . . . . . . . . . . . . . . . . . 223.5 The Forward Measure . . . . . . . . . . . . . . . . . . . . . . . . . . 223.6 The Fundamental Pricing Formulas . . . . . . . . . . . . . . . . . . . 24
4. One-factor short-rate models . . . . . . . . . . . . . . . . . . . . . . . 264.1 Introduction and Guided Tour . . . . . . . . . . . . . . . . . . . . . 264.2 Classical Time-Homogeneous Short-Rate Models . . . . . . . . . . . 284.3 The Vasicek Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.3.1 Affine Term-Structure Models . . . . . . . . . . . . . . . . . . 304.4 The Hull-White Extended Vasicek Model . . . . . . . . . . . . . . . 314.5 The Short-Rate Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 32
4.5.1 Bond and Option Pricing . . . . . . . . . . . . . . . . . . . . 33
5. Two-Factor Short-Rate Models . . . . . . . . . . . . . . . . . . . . . 355.1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . 355.2 The Two-Additive-Factor Gaussian Model G2++ . . . . . . . . . . . 37
5.2.1 The Short-Rate Dynamics . . . . . . . . . . . . . . . . . . . . 375.2.2 The Pricing of a Zero-Coupon Bond . . . . . . . . . . . . . . 39
6. “The Heath-Jarrow-Morton” framework . . . . . . . . . . . . . . . 436.1 The HJM Forward-Rate Dynamics . . . . . . . . . . . . . . . . . . . 45
7. The Log-Normal Forward-LIBOR model . . . . . . . . . . . . . . . 477.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477.2 Market Models: a Guided Tour . . . . . . . . . . . . . . . . . . . . . 477.3 Derivation of Black’s formula - adapted from Brigo and Mercurio [2007] 507.4 The Lognormal Forward-LIBOR Model (LFM) . . . . . . . . . . . . 54
8. Discussion: Market Completeness and Risk-Neutral Pricing: Whatthe relationship means for Derivative Traders . . . . . . . . . . . . 568.1 Completeness and Hedging . . . . . . . . . . . . . . . . . . . . . . . 56
8.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 568.1.2 Completeness in the BlackScholes Model . . . . . . . . . . . . 588.1.3 Completeness-Absence of Arbitrage Adapted from Bjork [2009] 62
8.2 Incomplete Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . 648.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 648.2.2 A Scalar Nonpriced Underlying Asset . . . . . . . . . . . . . 648.2.3 The Martingale Approach . . . . . . . . . . . . . . . . . . . . 738.2.4 Summing Up . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
9. Working example: Calibrating the LIBOR market model (LFM)to interest rate caps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 789.1 The role of the market . . . . . . . . . . . . . . . . . . . . . . . . . . 81
10.Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
A. Technical Appendix: Some Tools from Stochastic Calculus Adaptedfrom Brigo and Mercurio [2007] . . . . . . . . . . . . . . . . . . . . . 84A.1 From Deterministic to Stochastic Differential Equations . . . . . . . 84
A.1.1 Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . 85A.1.2 Stochastic Integrals . . . . . . . . . . . . . . . . . . . . . . . 86A.1.3 Martingales, Driftless SDEs and Semimartingales . . . . . . . 87A.1.4 Quadratic Variation . . . . . . . . . . . . . . . . . . . . . . . 88A.1.5 Quadratic Covariation . . . . . . . . . . . . . . . . . . . . . . 89A.1.6 Solution to a General SDE . . . . . . . . . . . . . . . . . . . 89A.1.7 Interpretation of the Coefficients of the SDE . . . . . . . . . 90
A.2 Ito’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90A.2.1 Stochastic Leibnitz Rule . . . . . . . . . . . . . . . . . . . . . 91A.2.2 Linear SDEs with Deterministic Diffusion Coefficient . . . . . 92A.2.3 Lognormal Linear SDEs . . . . . . . . . . . . . . . . . . . . . 92A.2.4 Geometric Brownian Motion . . . . . . . . . . . . . . . . . . 93A.2.5 Square-Root Processes . . . . . . . . . . . . . . . . . . . . . . 93
A.3 Two Important Theorems . . . . . . . . . . . . . . . . . . . . . . . . 93A.3.1 The Feynman-Kac Theorem . . . . . . . . . . . . . . . . . . . 94A.3.2 The Girsanov Theorem . . . . . . . . . . . . . . . . . . . . . 94
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
v
Chapter 1
Introduction
Rebonato (2002) divides the period of the evolution of interest-rate modelling into
four distinct periods, beginning with the Black and Scholes paper in 1973, then
moving to first generation yield-curve models with Vasicek (1977) et al, second
generation yield-curve models such as Hull and White (1990) et al, and culminating
with Heath et al. [1992]. In this thesis, there will be certain arguments, explanations
and mathematical constructions which draw from each of these four eras and, as
such, I will make brief mention of some of the key results from each of these eras,
somewhat chronologically in the paper.
In the first stage, the approaches by Black and Scholes (1973), Black [1976] and
Merton (1973), the same distributional assumption for the underlying, i.e. that of
log-normality was made, and the resulting expiry-time distribution was integrated
over the terminal pay-off of a European option.
The quantities that were typically considered to be log-normal were, for instance,
bond prices. As for the products priced using these modelling approaches, they
generally belonged to one of two markets and yield curves; the Repo market for
the pricing of plain-vanilla bond options and then the LIBOR (London Interbank
Offered Rate) environments for the pricing of caps, floors and swaptions. (Rebonato,
2002)
Rebonato (2002) further explains that, with regards early interest rate derivative
modelling, the most straightforward approach was to use the Black-Scholes formula
with the spot bond price as the underlying. This approach was popular, but it
immediately came under criticism because of the pull-to-par phenomenon i.e. the
Black and Scholes formula requires a constant percentage volatility of the underlying,
but for a coupon or discount bond, the volatility is not constant because it ultimately
has to converge to par at maturity.
A simple, work-around to this was to consider the bond-yield as the underlying
log-normal variable. The obvious advantage here was the absence of the pull-to-par
phenomenon, but more seriously, the drawback was the fact that a bond-yield was
Chapter 1. Introduction 2
not a traded asset, and, therefore, the Black and Scholes reasoning behind the self-
financing dynamic trading strategy that reproduces the final payoff of the option
could not easily be adapted. This is a key point, and one of great consequence for
the entire spirit of the thesis - we will come back to this all important notion of a
“traded asset” later on.
Despite this lack of academic justification, this early approach was widely used
by traders, and provides some early evidence of disconnect between the differing
arenas of academic abstraction and real world practical relevance. It illustrates that
unlike many other fields of practice in which there is a scientific underpin for key
principles, the financial world of interest rate derivatives trading does not necessarily
need a sound and complete academic edifice for its survival the market itself is a
powerful price-making machine.
Nevertheless, the log-normal yield model was eventually abandoned by traders
but it was not for lack for sound academic bedrock. It was replaced by what came
to be known as the standard “market formulae” for caps based on the Black [1976]
formula. This formula looks deceptively similar to the log-normal yield approach
with the Black-Scholes formula, but with all important nuanced differences. The
Black 76 formula uses the forward as opposed to spot price and the volatility of
the forward as inputs. No pull-to-par effect applies to a forward bond price which
remains of the same residual maturity throughout its life. (Rebonato, 2002)
Having made mention of all of the above, what we will therefore try to show in
the thesis is how this historical evolution in interest-rate modelling had unfolded,
but using the mathematical language of options theory. As I alluded to before, the
idea of the underlying being a “traded asset” is an all important one for derivatives
pricing, and for interest-rate derivatives in particular, as this is critically tied to the
mathematical idea of the “market price of risk”.
This thesis is not primarily a quantitative investigation. Rather it is a discursive
work, which uses the language and syntax of mathematics to explain what is other-
wise a philosophical digression about the nature and pricing of modern interest-rate
derivatives. Much of the paper is devoted to laying the mathematical groundwork,
which provides the tools for discussions later on.
The key chapter of this work, after most of the mathematical preliminaries have
been covered, is a discussion on market completeness and its implications for deriva-
tive traders. It is here that we will try to give some richer mathematical meaning
to the notion of the “market price of risk” and hopefully explain why the trading
practice of interest rate derivatives - which relies on market-forces like supply and
demand to define how much return compensation the market requires per unit of
risk assumed is different to the pricing other derivative instruments.
Chapter 1. Introduction 3
Chapter 1 covers foundational material such as the definitions of the money/bank
account, the zero-coupon bond, forward-rate agreements and the descriptions of
interest rate caps and floors.
Chapter 2 begins to describe some of the early ideas about risk-neutral pricing
in continuous-time, as detailed by the work of Harrison, Kreps and Pliska. Some of
this material is revisited in greater depth in later chapters. The numeraire, defined
as any non-dividend paying asset, is introduced. Some of the material here will rely
on concepts and terms which are detailed in a Technical Appendix at the end of the
paper, for the benefit of the reader with limited prior exposure.
Chapter 3 describes some of the first and second-generation one-factor yield-
curve models, the Vasicek (1977) and the Hull-White (1990) model are given special
attention. Time-homogeneity is essentially the key factor in separating the first and
second generation models.
Chapter 4 describes the Gaussian Vasicek Two-Factor model (G2++). This is
important for showing the limitations of prior one-factor models, but more impor-
tantly, the G2++ model will be referenced in the detailed derivation of the Black
[1976] formula for caplets in chapter 5.
Chapter 5 introduces formally the Heath-Jarrow-Morton framework. Here it
is shown that, in the case of one-factor short-rate models one is free to choose the
drift and instantaneous volatility co-efficient in the related diffusion dynamics as one
deems fit, with no general restrictions. In the HJM framework we choose the instan-
taneous forward rates as fundamental quantities to model, and thereafter derive an
arbitrage-free framework for the stochastic evolution of the entire yield curve, where
the forward-rates dynamics are fully specified through their instantaneous volatility
structures. This is a major difference with arbitrage free one-factor short-rate dy-
namics, where the volatility of the short rate alone does not suffice to characterize
the relevant interest-rate model.
Chapter 6 introduces the Lognormal Forward-LIBOR Model (LFM). Before mar-
ket models such as the LFM were introduced, there was no interest-rate dynamics
compatible with either Black’s 1976 formula for caps or Black’s formula for swap-
tions. I show here a detailed derivation of the Black [1976] formula for caplets,
as suggested by Brigo and Mercurio (2006). The formal definition and structure
of the LFM is shown. This chapter has a link-in to chapter 8 which discusses the
calibration of interest rate caps.
Chapter 7 is the key discussion chapter. Here we revisit some of the risk-neutral
pricing results introduced in chapter 2, with more mathematical rigour. More im-
portantly, the key demonstration of this chapter is that in an incomplete market the
requirement of no arbitrage is no longer sufficient to determine a unique price for the
Chapter 1. Introduction 4
derivative. We will have several possible martingale measures, and several market
prices of risk. The reason that there are several possible martingale measures simply
means that there are several different price systems for the derivatives, all of which
are consistent with absence of arbitrage. Furthermore, in an incomplete market the
price is also partly determined, in a nontrivial way, by aggregate supply and demand
on the market. Supply and demand for a specific derivative are in turn determined
by the aggregate risk aversion on the market, as well as by liquidity considerations
and other factors.
Chapter 8 provides a concrete instance to the abstract discussion on calibration,
which was discussed in the closing paragraphs of the previous chapter. More specif-
ically, I describe here the calibration of the LIBOR model (LFM) to interest rate
caps quoted in the market. The intention of this chapter is to show how the LIBOR
market model can be calibrated to market quoted cap volatilities. The purpose of
the calibration is to capture information from market data with a view to fitting
parameters in such a way as to make the model prices of interest rate caps match the
market prices as close as possible. The key point here, is that the market with its
embedded risk tolerances is the primary pricing machine. The market data is taken
as given, and the calibration procedure basically attempts to recover the market
prices.
Chapter 2
Definitions and Notation
2.1 The Bank Account and the Short Rate
The first definition we consider is the definition of a bank account, or money-market
account. A money-market account represents a (locally) riskless investments, where
profit is accrued continuously at the risk-free rate prevailing in the market at every
instant.
Definition 2.1 (Bank account (Money-market account)). We define B(t) to be the
value of a bank account at time t ≥ 0. We assume B(0) = 1 and that the bank
account evolves according to the following differential equation:
dB(t) = rtB(t)dt, B(0) = 1, (2.1)
where rt is a positive function of time. As a consequence,
B(t) = exp
(∫ t
0rsds
). (2.2)
The above definition tells us that investing a unit amount at time 0 yields at
time t the value in (2.2), and rt is the instantaneous rate at which the bank account
accrues. This instantaneous rate is usually referred to as instantaneous spot rate,
or briefly as short rate. In fact, a first order expansion in ∆t gives
B(t+ ∆t) = B(t)(1 + r(t)∆t), (2.3)
which amounts to say that, in any arbitrarily small time interval [t, t+ ∆t),
B(t+ ∆t)−B(t)
B(t)= r(t)∆t. (2.4)
It is then clear that the bank account grows at each time instant t at a rate r(t).
The bank-account numeraire B is important for relating amounts of currencies
available at different times. To this end, consider the following fundamental question:
What is the value at time t of one unit of currency available at time T?
2.2 Zero-Coupon Bonds and Spot Interest Rates 6
Assume for simplicity that the interest rate process r, and hence B, are deter-
ministic. We know that if we deposit A units of currency in the bank account at
time 0, at time t > 0 we have A × B(t) units of currency. Similarly, at time T > t
we have A×B(T ) units. If we wish to have exactly one unit of currency at time T ,
i.e., if we wish that
AB(t) = 1,
we have to initially invest the amount A = 1/B(T ), which is known since the process
B is deterministic. Hence, the value at time t of the amount A invested at the initial
time is
AB(t) =B(t)
B(T ).
We have thus seen that the value of one unit of currency payable at time T , as
seen from time t, is B(t)/B(T ). Coming back to the initial assumption of a general
(stochastic) interest-rate process, this leads to the following.
Definition 2.2 (Stochastic discount factor). The (stochastic) discount factor D(t, T )
between two time instants t and T is the amount at time t that is “equivalent” to
one unit of currency payable at time T , and is given by
D(t, T ) =B(t)
B(T )= exp
(−∫ T
trsds
). (2.5)
The probabilistic nature of rt is important since it affects the nature of the basic
asset of our discussion, the bank-account numeraire B. In many pricing applications,
especially when applying the Black and Scholes formula in equity or foreign-exchange
(FX) markets, r is assumed to be a deterministic function of time, so that both the
bank account (2.2) and the discount factors (2.5) at any future time are deterministic
functions of time. This is usually motivated by assuming that variability of interest
rates contributes to the price of equity or FX options by a smaller order of magnitude
with respect to the underlying’s movements.
However, when dealing with interest-rate products, the main variability that
matters is clearly that of the interest rates themselves. It is therefore necessary
to drop the deterministic setup and to start modeling the evolution of r in time
through a stochastic process. As a consequence, the bank account (2.2) and the
discount factors (2.5) will be stochastic processes, too.
2.2 Zero-Coupon Bonds and Spot Interest Rates
Definition 2.3 (Zero-coupon bond). T -maturity zero-coupon bond (pure discount
bond) is a contract that guarantees its holder the payment of one unit of currency at
2.2 Zero-Coupon Bonds and Spot Interest Rates 7
time T , with no intermediate payments. The contract value at time t < T is denoted
by P (t, T ). Clearly, P (T, T ) = 1 for all T .
If we are now at time t, a zero-coupon bond for the maturity T is a contract
that establishes the present value of one unit of currency to be paid at time T (the
maturity of the contract).
A natural question arising now is: What is the relationship between the discount
factor D(t, T ) and the zero-coupon-bond price P (t, T )? The difference lies in the
two objects being respectively an “equivalent amount of currency” and a “value of
a contract”.
If rates r are deterministic, then D is deterministic as well and necessarily
D(t, T ) = P (t, T ) for each pair (t, T ). However, if rates are stochastic, D(t, T )
is a random quantity at time t depending on the future evolution of rates r between
t and T . Instead, the zero-coupon-bond price P (t, T ), being the time t-value of a
contract with payoff at time T , has to be known (deterministic) at time t.
Definition 2.4 (Time to maturity). The time to maturity T − t is the amount of
time (in years) from the present time t to the maturity time T > t.
The definition “T − t” makes sense as long as t and T are real numbers asso-
ciated to two time instants. However, if t and T denote two dates expressed as
day/month/year, say D1 = (d1,m1, y1) and D2 = (d2,m2, y2), we need to define the
amount of time between the two dates in terms of the number of days between them.
This choice, however, is not unique, and the market evaluates the time between t
and T in different ways.
Definition 2.5 (Year fraction, Day-count convention). We denote by τ(t, T ) the
chosen time measure between t and T , which is usually referred to as year fraction
between the dates t and T . When t and T are less than one-day distant (typically
when dealing with limit quantities involving time to maturities tending to zero),
τ(t, T ) is to be interpreted as the time difference T − t (in years). The particular
choice that is made to measure the time between two dates reflects what is known as
the day-count convention.
In moving from zero-coupon-bond prices to interest rates, and vice versa, we
need to know two fundamental features of the rates themselves: The compounding
type and the day-count convention to be applied in the rate definition. What we
mean by “compounding type” will be clear from the definitions below, while the
day-count convention has been discussed earlier.
Definition 2.6 (Continuously-compounded spot interest rate). The continuously-
compounded spot interest rate prevailing at time t for the maturity T is denoted by
2.2 Zero-Coupon Bonds and Spot Interest Rates 8
R(t, T ) and is the constant rate at which an investment of P (t, T ) units of currency
at time t accrues continuously to yield a unit amount of currency at maturity T . In
formulas:
R(t, T ) := − lnP (t, T )
τ(t, T ). (2.6)
The continuously-compounded interest rate is therefore a constant rate that is
consistent with the zero-coupon-bond prices in that
eR(t,T )τ(t,T )P (t, T ) = 1, (2.7)
from which we can express the bond price in terms of the continuously-compounded
rate R:
P (t, T ) = e−R(t,T )τ(t,T ). (2.8)
The year fraction involved in continuous compounding is usually τ(t, T ) = T − t,the time difference expressed in years.
An alternative to continuous compounding is simple compounding, which applies
when accruing occurs proportionally to the time of the investment. We indeed have
the following.
Definition 2.7 (Simply-compounded spot interest rate). The simply-compounded
spot interest rate prevailing at time t for the maturity T is denoted by L(t, T ) and
is the constant rate at which an investment has to be made to produce an amount of
one unit of currency at maturity, starting from P (t, T ) units of currency at time t,
when accruing occurs proportionally to the investment time. In formulas:
L(t, T ) :=1− P (t, T )
τ(t, T )P (t, T ). (2.9)
The market LIBOR rates are simply-compounded rates, which motivates why we
denote by L such rates. LIBOR rates are typically linked to zero-coupon-bond prices
by the Actual/360 day-count convention for computing τ(t, T ). Definition (2.9)
immediately leads to the simple-compounding counterpart of (2.7), i.e.,
P (t, T )(1 + L(t, T )τ(t, T )) = 1 (2.10)
so that a bond price can be expressed in terms of L as:
P (t, T ) =1
1 + L(t, T )τ(t, T ).
A further compounding method that is considered is annual compounding. Annual
compounding is obtained as follows. If we invest today a unit of currency at the
simply-compounded rate Y , in one year we will obtain the amount A = 1(1 + Y ).
2.3 Fundamental Interest-Rate Curves 9
Suppose that, after this year, we invest such an amount for one more year at the
same rate Y , so that we will obtain A(1 + Y ) = (1 + Y )2 in two years. If we keep
on reinvesting for n years, the final amount we obtain is (1 + Y )n. Based on this
reasoning, we have the following.
Definition 2.8 (Annually-compounded spot interest rate). The annually-compounded
spot interest rate prevailing at time t for the maturity T is denoted by Y (t, T ) and
is the constant rate at which an investment has to be made to produce an amount of
one unit of currency at maturity, starting from P (t, T ) units of currency at time t,
when reinvesting the obtained amounts once a year. In formulas
Y (t, T ) :=1
[P (t, T )]1/τ(t,T )− 1. (2.11)
Analogously to (2.7) and (2.10), we then have
P (t, T )(1 + Y (t, T ))τ(t,T ) = 1, (2.12)
which implies that bond prices can be expressed in terms of annually-compounded
rates as
P (t, T ) =1
(1 + Y (t, T ))τ(t,T ). (2.13)
A year fraction τ that can be associated to annual compounding is for example
ACT/365.
2.3 Fundamental Interest-Rate Curves
A fundamental curve that can be obtained from the market data of interest rates
is the zero-coupon curve at a given date t. This curve is the graph of the function
mapping maturities into rates at times t. Precisely, we have the following.
Definition 2.9. The zero-coupon curve (sometimes also referred to as “yield curve”)
at time t is the graph of the function
T 7→
L(t, T ) t < T ≤ t+ 1 (years),
Y (t, T ) T > t+ 1 (years).(2.14)
Such a zero-coupon curve is also called the term structure of interest rates at
time t. It is a plot at time t of simply-compounded interest rates for all maturities
T up to one year and of annually compounded rates for maturities T larger than
one year. An example of such a curve is shown in Figure 1.1. In this example, the
curve is not monotonic, and its initially-inverted behaviour resurfaces periodically
in the market. The Italian curve, for example, has been inverted for several years
2.4 Forward Rates 10
before becoming monotonic. The Euro curve has often shown a monotonic pattern,
although in our figure we see an example of the opposite situation.
The term “yield curve” is often used to denote several different curves deduced
from the interest-rate-market quotes, and is in fact slightly ambiguous.
Finally, at times, we may consider the same plot for rates with different com-
pounding conventions, such as for example
T 7→ R(t, T ), T > t.
The term zero-coupon curve will be used for all such curves, no matter the com-
pounding convention being used.
Definition 2.10 (Zero-bond curve). The zero-bond curve at time t is the graph of
the function
T 7→ P (t, T ), T > t,
which, because of the positivity of interest rates, is a T -decreasing function starting
from P (t, t) = 1. Such a curve is also referred to as term structure of discount
factors.
2.4 Forward Rates
We now move to the definition of forward rates. Forward rates are characterized by
three time instants, namely the time t at which the rate is considered, its expiry T
and its maturity S, with t ≤ T ≤ S. Forward rates are interest rates that can be
locked in today for an investment in a future time period, and are set consistently
with the current term structure of discount factors.
We can define a forward rate through a prototypical forward-rate agreement
(FRA). A FRA is a contract involving three time instants: The current time t, the
expiry time T > t, and the maturity time S > T . The contract gives its holder an
interest-rate payment for the period between T and S. At the maturity S, a fixed
payment based on a fixed rate K is exchanged against a floating payment based on
the spot rate L(T, S) resetting in T and with maturity S. Basically, this contract
allows one to lock-in the interest rate between times T and S at a desired value K,
with the rates in the contract that are simply compounded. Formally, at time S one
receives τ(T, S)KN units of currency and pays the amount τ(T, S)L(T, S)N , where
N is the contract nominal value. The value of the contract in S is therefore
Nτ(T, S)(K − L(T, S)), (2.15)
2.4 Forward Rates 11
where we assume that both rates have the same day-count convention.1 Clearly, if
L is larger than K at time T , the contract value is negative, whereas in the other
case it is positive. Recalling the expression (1.8) for L, we can rewrite this value as
N
[τ(T, S)K − 1
P (T, S)+ 1
]. (2.16)
Now consider the term A = 1/P (T, S) as an amount of currency held at time S. Its
value at time T is obtained by multiplying this amount A for the zero coupon price
P (T, S):
P (T, S)A = P (T, S)1
P (T, S)= 1,
so that this term is equivalent to holding one unit of currency at time T . In turn, one
unit of currency at time T is worth P (t, T ) units of currency at time t. Therefore,
the amount 1/P (T, S) in S is equivalent to an amount of P (t, T ) in t.
Then consider the other two terms in the contract value (2.16). The amount
B = τ(T, S)K + 1 at time S is worth
P (t, S)B = P (t, S)τ(T, S)K + P (t, S)
at time t. The total value of the contract at time t is therefore
FRA(t, T, S, τ(T, S), N,K) = N [P (t, S)τ(T, S)K − P (t, T ) + P (t, S)]. (2.17)
There is just one value of K that renders the contract fair at time t, i.e. such that
the contract value is 0 in t. This value is of course obtained by equating to zero the
FRA value. The resulting rate defines the (simply-compounded) forward rate.
Definition 2.11 (Simply-compounded forward interest rate). The simply-compounded
forward interest rate prevailing at time t for the expiry T > t and maturity S > T
is denoted by F (t;T, S) and is defined by
F (t;T, S) :=1
τ(T, S)
(P (t, T )
P (t, S)− 1
). (2.18)
It is that value of the fixed rate in a prototypical FRA with expiry T and maturity
S that renders the FRA a fair contract at time t.
Notice that we can rewrite the value of the above FRA (2.17) in terms of the
just-defined simply compounded forward interest rate:
FRA(t, T, S, τ(T, S), N,K) = NP (t, S)τ(T, S)(K − F (t;T, S)). (2.19)
1 It is actually market practice to apply different day-count conventions to fixed and floating
rates. Here, however, we assume the same convention for simplicity, without altering the treatment
of the whole subject. The generalization to different day-count conventions for the fixed and floating
rates is indeed straight-forward.
2.5 Interest-Rate Caps/Floors and Swaptions 12
Therefore, to value a FRA, we just have to replace the LIBOR rate L(T, S) in
the payoff (2.15) with the corresponding forward rate F (t;T, S), and then take the
present value of the resulting (deterministic) quantity.
The forward rate F (t;T, S) may thus be viewed as an estimate of the future spot
rate L(T, S), which is random at time t, based on market conditions at time t. In
particular, we will see later on that F (t;T, S) is the expectation of L(T, S) at time
t under a suitable probability measure.
When the maturity of the forward rate collapses towards its expiry, we have the
notion of instantaneous forward rate. Indeed, let us consider the limit
limS→T+
= − limS→T+
1
P (t, S)
P (t, S)− P (t, T )
S − T
= −frac1P (t, T )∂P (t, T )
∂T(2.20)
= −∂ lnP (t, T )
∂T,
where we use our convention that τ(T, S) = S − T when S is extremely close to T .
This leads to the following.
Definition 2.12 (Instantaneous forward interest rate). The instantaneous forward
interest rate prevailing at time t for the maturity T > t is denoted by f(t, T ) and is
defined as
f(t, T ) := limS→T+
F (t;T, S) = −∂ lnP (t, T )
∂T, (2.21)
so that we have
P (t, T ) = exp
(−∫ T
tf(t, u)du
).
Clearly, for this notion to make sense, we need to assume smoothness of the
zero-coupon-price function T 7→ P (t, T ) for all T ’s.
Intuitively, the instantaneous forward interest rate f(t, T ) is a forward interest
rate at time t whose maturity is very close to its expiry T , say f(t, T ) ≈ F (t;T, T +
∆T ) with ∆T small.
Instantaneous forward rates are fundamental quantities in the theory of interest
rates. Indeed, it turns out that one of the most general ways to express “fairness”
of an interest-rate model is to relate certain quantities in the expression for the
evolution of f .
2.5 Interest-Rate Caps/Floors and Swaptions
We conclude this chapter by introducing the two main derivative products of the
interest-rates market, namely caps and swaptions.
2.5 Interest-Rate Caps/Floors and Swaptions 13
Interest-Rate Caps/Floors
A cap is a contract that can be viewed as a payer IRS where each exchange payment
is executed only if it has positive value. The cap discounted payoff is therefore given
byβ∑
i=α+1
D(t, Ti)Nτi(L(Ti−1, Ti)−K)+.
Analogously, a floor is equivalent to a receiver IRS where each exchange payment is
executed only if it has positive value. The floor discounted payoff is therefore given
byβ∑
i=α+1
D(t, Ti)Nτi(K − L(Ti−1, Ti))+.
Where do the terms “cap” and “floor” originate from? Consider the cap case.
Suppose a company is LIBOR indebted and has to pay at certain times Tα+1, . . . , Tβ
the LIBOR rates resetting at times Tα, . . . , Tβ−1, with associated year fractions
τ = τα+1, . . . , τβ, and assume that the debt notional amount is one. Set T =
Tα, . . . , Tβ. The company is afraid that LIBOR rates will increase in the future,
and wishes to protect itself by locking the payment at a maximum “cap” rate K. In
order to do this, the company enters a cap with the payoff described above, pays its
debt in terms of the LIBOR rate L and receives (L −K)+ from the cap contract.
The difference gives what is paid when considering both contracts:
L− (L−K)+ = min(L,K).
This implies that the company pays at most K at each payment date, since its
variable (L-indexed) payments have been capped to the fixed rateK, which is termed
the strike of the cap contract or, more briefly, cap rate. The cap, therefore, can be
seen as a contract that can be used to prevent losses from large movements of interest
rates when indebted at a variable (LIBOR) rate.
A cap contract can be decomposed additively. Indeed, its discounted payoff is a
sum of terms such as
D(t, Ti)Nτi(L(Ti−1, Ti)−K)+.
Each such term defines a contract that is termed caplet. The floorlet contracts are
defined in an analogous way.
It is market practice to price a cap with the following sum of Black’s formulas
(at time zero)
CapBlack(0, T , τ,N,K, σα,β) = N
β∑i=α+1
P (0, Ti)τiBl(K,F (0, Ti−1, Ti), vi, 1), (2.22)
2.5 Interest-Rate Caps/Floors and Swaptions 14
where, denoting by Φ the standard Gaussian cumulative distribution function,
Bl(K,F, v, w) = FwΦ(wd1(K,F, v))−KwΦ(wd2(K,F, v)),
d1(K,F, v) =ln(F/K) + v2/2
v,
d2(K,F, v) =ln(F/K)− v2/2
v,
vi = σα,β√Ti−1,
with the common volatility parameter σα,β that is retrieved from market quotes.
Analogously, the corresponding floor is priced according to the formula
FlrBlack(0, T , τ,N,K, σα,β) = N
β∑α+1
P (0, Ti)τiBl(K,F (0, Ti−1, Ti), vi,−1). (2.23)
There are several caps (floors) whose implied volatilities are quoted by the market.
For example we have quotes with α = 0, T0 equal to three months, and all other
Ti’s equally three-month spaced. Another example concerns quotes with T0 equal
to three months, the next Ti’s up to one year equally three-month spaced, and all
other Ti’s equally six-month spaced.
Definition 2.13. Consider a cap (floor) with payment times Tα+1, . . . , Tβ, associ-
ated year fractions τα+1, . . . , τβ and strike K. The cap (floor) is said to be at-the-
money (ATM) if and only if
K = KATM := Sα,β(0) =P (0, Tα)− P (0, Tβ)∑β
i=α+1 τiP (0, Ti).
The cap is instead said to be in-the-money (ITM) if K < KATM, and out-of-the-
money (OTM) if K > KATM, with the converse holding for a floor.
Using the equality
(L−K)+ − (K − L)+ = L−K,
we can note that the difference between a cap and the corresponding floor is equiv-
alent to a forward-start swap. It is, therefore, easy to prove that a cap (floor) is
ATM if and only if its price equals that of the corresponding floor (cap).
Notice also that in case the cap has only one payment date (α + 1 = β), the
cap collapses to a single caplet. In such a case, the at-the-money caplet strike is
KATM = F (0, Tα, Tα+1), and the caplet is ITM if K < F (0, Tα, Tα+1). The reason
for these terms is particularly clear in the caplet case. In fact, if the contract
terminal payoff is evaluated now with the current value of the underlying replacing
the corresponding terminal value, we have a positive amount in the ITM case (so
that we are “in the money”), whereas we have a non-positive amount in the other
case (we are “out of the money”).
Chapter 3
No-Arbitrage Pricing and
Numeraire Change
The fundamental economic assumption in the seminal paper by Black and Scholes
[1973] is the absence of arbitrage opportunities in the considered financial market.
Roughly speaking, absence of arbitrage is equivalent to the impossibility to invest
zero today and receive tomorrow a nonnegative amount that is positive with positive
probability. In other words, two portfolios having the same payoff at a given future
date must have the same price today. By constructing a suitable portfolio having
the same instantaneous return as that of a riskless investment, Black and Scholes
could then conclude that the portfolio instantaneous return was indeed equal to
the instantaneous risk-free rate, which immediately led to their celebrated partial
differential equation and, through its solution, to their option-pricing formula.
The basic Black and Scholes argument was subsequently used by Vasicek [1977]
to develop a model for the evolution of the term structure of interest rates and for
the pricing of interest-rate derivatives (see also next Chapter 3). However, such
argument deals with infinitesimal quantities in a way that only later works have
justified. Indeed, the first rigorous and mathematically sound approach for the
arbitrage-free pricing of general contingent claims was developed by Harrison and
Kreps [1979] and Harrison and Pliska [1981, 1983]. Their general theory has then
inspired the work by Heath et al. [1992] who developed a general framework for
interest-rates dynamics.
In this chapter, we start by reviewing the main results of Harrison and Pliska
[1981, 1983]. We then describe the change-of-numeraire technique as developed
by Geman et al. [1995] and provide a useful toolkit explaining how the various
dynamics change when changing the numeraire.
3.1 No-Arbitrage in Continuous Time 16
3.1 No-Arbitrage in Continuous Time
We here briefly consider the case of the continuous-time economy analyzed by Harri-
son and Kreps [1979] and Harrison and Pliska [1981, 1983]. We do this for historical
reasons, but also for introducing a terminology that will be helpful later on.
We consider a time horizon T > 0, a probability space (Ω,F , Q0) and a right-
continuous filtration F = Ft : 0 ≤ t ≤ T . In the given economy, K + 1 non dividend
paying securities are traded continuously from time 0 until time T . Their prices are
modeled by a K+1 dimensional adapted semi-martingale S = St : 0 ≤ t ≤ T , whose
components S0, S1, . . . , SK are positive. The asset indexed by 0 is a bank account.
Its price then evolves according to
dS0t = rtS
0t dt,
with S00 = 1 and where rt is the instantaneous short-term rate at time t. Notation
(see previous chapter), St0 = B(t) and 1/S0t = D(0, t) for each t.
Definition 3.1. A trading strategy is a (K + 1 dimensional) process φ = φt : 0 ≤t ≤ T, whose components φ0, φ1, . . . , φK are locally bounded and predictable. The
value process associated with a strategy φ is defined by
Vt(φ) = φtSt =K∑k=0
φkt Skt , 0 ≤ t ≤ T,
and the gains process associated with a strategy φ by
Gt(φ) =
∫ t
0φudSu =
K∑k=0
∫ t
0φkudS
ku, 0 ≤ t ≤ T.
The k-th component φkt of the strategy φt at time t, for each t, is interpreted as
the number of units of security k held by an investor at time t. The predictability
condition on each φk means that the value φkt is known immediately before time t.
This is done to reduce the investor’s freedom at any jump time. This issue, however,
is not relevant if S has continuous paths (as happens for example in case S follows
a diffusion process). Moreover, Vt(φ) and Gt(φ) are respectively interpreted as the
market value of the portfolio φt and the cumulative gains realized by the investor
until time t by adopting the strategy φ.
Definition 3.2. A trading strategy φ is self-financing if V (φ) ≥ 0 and
Vt(φ) = V0(φ) +Gt(φ), 0 ≤ t < T. (3.1)
3.1 No-Arbitrage in Continuous Time 17
Intuitively, a strategy is self-financing if its value changes only due to changes in
the asset prices. In other words, no additional cash inflows or outflows occur after
the initial time.
A relation similar to 3.1 holds also when asset prices are all expressed in terms
of the bank-account value. Indeed, Harrison and Pliska [1981] proved the following.
Proposition 3.3. Let φ be a trading strategy. Then, φ is self-financing if and only
if D(0, t)Vt(φ) = V0(φ) +∫ t
0 φud(D(0, u)Su).
A key result in Harrison and Kreps [1979] and Harrison and Pliska [1981, 1983]
is the established connection between the economic concept of absence of arbitrage
and the mathematical property of existence of a probability measure, the equiva-
lent martingale measure (or risk-neutral measure, or risk-adjusted measure), whose
definition is given in the following.
Definition 3.4. An equivalent martingale measure Q is a probability measure on
the space (Ω,F) such that
1. Q0 and Q are equivalent measures, that is Q0(A) = 0 if and only if Q(A) = 0,
for every A ∈ F ;
2. the Radon-Nikodym derivative dQ/dQ0 belongs to L2(Ω,F , Q0) (i.e. it is
square integrable with respect to Q0).
3. the “discounted asset price” process D(0, )S is an (F, Q)-martingale, that is
E(D(0, t)Skt |Fu) = D(0, u)Sku, for all k = 0, 1, . . . ,K and all 0 ≤ u ≤ t ≤ T ,
with E denoting expectation under Q.
An arbitrage opportunity is defined, in mathematical terms, as a self-financing
strategy φ such that V0(φ) = 0 but Q0VT (φ) > 0 > 0.Harrison and Pliska [1981,
1983] then proved the fundamental result that the existence of an equivalent mar-
tingale measure implies the absence of arbitrage opportunities.
Definition 3.5. A contingent claim is a square-integrable and positive random
variable on Ω,F , Q0. A contingent claim H is attainable if there exists some self-
financing φ such that VT (φ) = H. Such a φ is said to generate H, and φt = Vt(φ)
is the price at time t associated with H.
The following proposition, proved by Harrison and Pliska [1981] provides the
mathematical characterization of the unique no-arbitrage price associated with any
attainable contingent claim.
3.2 The Change-of-Numeraire Technique 18
Proposition 3.6. Assume there exists an equivalent martingale measure Q and let
H be an attainable contingent claim. Then, for each time t, 0 ≤ t ≤ T , there exists
a unique price πt associated with H, i.e.,
πt = E(D(t, T )H|Ft). (3.2)
When the set of all equivalent martingale measures is nonempty, it is then pos-
sible to derive a unique no-arbitrage price associated to any attainable contingent
claim. Such a price is given by the expectation of the discounted claim payoff under
the measure Q equivalent to Q0. This result generalizes that of Black and Scholes
[1973] to the pricing of any claim, which, in particular, may be path-dependent. Also
the underlying-asset-price dynamics are quite general, which makes formula (3.2)
applicable in quite different situations.
Definition 3.7. A financial market is complete if and only if every contingent claim
is attainable.
Harrison and Pliska [1983] proved the following fundamental result. A financial
market is (arbitrage free and) complete if and only if there exists a unique equiva-
lent martingale measure. The existence of a unique equivalent martingale measure,
therefore, not only makes the markets arbitrage free, but also allows the derivation
of a unique price associated with any contingent claim.
We can summarize what stated above by the following well known stylized char-
acterization of no-arbitrage theory via martingales.
• The market is free of arbitrage if (and only if) there exists a martingale mea-
sure;
• The market is complete if and only if the martingale measure is unique;
• In an arbitrage-free market, not necessarily complete, the price of any attain-
able claim is uniquely given, either by the value of the associated replicating
strategy, or by the risk neutral expectation of the discounted claim payoff
under any of the equivalent (risk-neutral) martingale measures.
3.2 The Change-of-Numeraire Technique
Formula (3.2) gives the unique no-arbitrage price of an attainable contingent claim
H in terms of the expectation of the claim payoff under the selected martingale mea-
sure Q.Geman et al. [1995] noted, however, that an equivalent martingale measure
Q, as in Definition 3.5, is not necessarily the most natural and convenient mea-
sure for pricing the claim H. Indeed, under stochastic interest rates, for example,
3.2 The Change-of-Numeraire Technique 19
the presence of the stochastic discount factor D(t, T ) complicates considerably the
calculation of the expectation. In such cases, a change of measure can be quite
helpful, and this is the approach followed, for instance, by Jamshidian [1989] in the
calculation of a bond-option price under the Vasicek [1977] model.
Geman et al. [1995] introduced the following.
Definition 3.8. A numeraire is any positive non-dividend-paying asset.
In general, a numeraire Z is identifiable with a self-financing strategy φ in that
Zt = Vt(φ) for each t. Intuitively, a numeraire is a reference asset that is chosen so
as to normalize all other asset prices with respect to it. Choosing a numeraire Z
then implies that the relative prices Sk/Z, k = 0, 1, . . . ,K, are considered instead of
the securities prices themselves. The value of the numeraire will be often used to
denote the numeraire itself.
As proven by Geman et al. [1995], Proposition 3.3 can be extended to any
numeraire, in that self-financing strategies remain self-financing after a numeraire
change. Indeed, written in differential form, the self-financing condition
dVt(φ) =
K∑k=0
φkt dSkt
implies that
d
(Vt(φ)
Zt
)=
K∑k=0
φkt d
(SktZt
).
Therefore, an attainable claim is also attainable under any numeraire.
In the definition of equivalent martingale measure of the previous section, it has
been implicitly assumed the bank account S0 as numeraire. However, as already
pointed out, this is just one of all possible choices and it turns out, in fact, that
there can be more convenient numeraires as far as the calculation of claim prices is
concerned. The following proposition by Geman et al. [1995] provides a fundamental
tool for the pricing of derivatives and is the natural generalization of Proposition 3.6
to any numeraire.
Proposition 3.9. Assume there exists a numeraire N and a probability measure
QN , equivalent to the initial Q0, such that the price of any traded asset X (without
intermediate payments) relative to N is a martingale under QN , i.e.,
Xt
Nt= EN
XT
NT|Ft
0 ≤ t ≤ T. (3.3)
Let U be an arbitrary numeraire. Then there exists a probability measure QU , equiv-
alent to the initial Q0, such that the price of any attainable claim Y normalized by
3.3 A Change of Numeraire Toolkit - Adapted from Brigo and Mercurio [2007] 20
U is a martingale under QU , i.e.,
YtUt
= EUYTUT|Ft
0 ≤ t ≤ T. (3.4)
Moreover, the Radon-Nikodym derivative defining the measure QU is given by
dQU
dQN=UTN0
U0NT. (3.5)
The derivation of (3.5) is outlined as follows. By definition of QN , we know that
for any tradable asset price Z,
EN[ZTNT
]= EU
[U0
N0
ZTUT
](3.6)
(both being equal to Z0/N0). By definition of Radon-Nikodym derivative, we know
also that for all Z
EN[ZTNT
]= EU
[ZTNT
dQN
dQU
].
By comparing the right-hand sides of the last two equalities, from the arbi-
trariness of Z we obtain (3.5). The general formula (3.4) follows from immediate
application of the Bayes rule for conditional expectations.
3.3 A Change of Numeraire Toolkit - Adapted
from Brigo and Mercurio [2007]
In this section, we present some useful considerations and formulas on the change-
of-numeraire technique developed in the previous section.
There are basically three facts on the change of numeraire technique one should
consider in practice. Before addressing the third one, namely the change of numeraire
toolkit, we give a useful summary of the first two relevant change-of-numeraire facts
we have encountered earlier.
FACT ONE The price of any asset divided by a reference positive non-dividend-
paying asset (called numeraire) is a martingale (no drift) under the measure
associated with that numeraire. This first fact follows from Proposition 3.9 As
a first fundamental example, we may consider any (say non-dividend paying)
asset S divided by the bank account B(t). This process St/B(t), recalling that
dB(t) = rtB(t)dt and assuming for simplicity B(0) = 1, reads
StB(t)
= e−∫ t0 rsdsSt.
3.3 A Change of Numeraire Toolkit - Adapted from Brigo and Mercurio [2007] 21
Fact One requires this process to be a martingale under the measure QB
associated to the bank account numeraire B, i.e. the risk neutral measure.
If for a second we call b(t, St) the risk neutral drift of S, we have 1
d(e−
∫ t0 rsdsSt.
)= St
(−rte−
∫ t0 rsds
)dt+ e−
∫ t0 rsdsdSt
= −rtSte−∫ t0 rsdsdt+ e−
∫ t0 rsdsb(t, St)dt+ (. . . )dWt
= e−∫ t0 rsds(−rtSt + b(t, St))dt+ (. . . )dWt.
Fact One tells us that the drift of this last equation should be zero, since the
ratio has to be a martingale. By zeroing the drift we obtain
−rtSt + b(t, St) = 0⇒ b(t, St) = rtSt.
We see that Fact One implies the risk neutral drift of an asset S dynamics to
be the risk free interest rate rt times the asset St itself, consistently with what
we have seen earlier.
As a second fundamental example, the forward LIBOR rate F2 between expiry
T1 and maturity T2:
F2(t) =(P (t, T1)− P (t, T2))/(T2 − T1)
P (t, T2)
This is a portfolio of two zero coupon bonds divided by the zero coupon bond
P (,T2). If we take the measure Q2 associated with the numeraire P (,T2), by
Fact One F2 will be a martingale (no drift) under that measure.
FACT TWO The time-t risk neutral price
Pricet = EBt
B(t)Payoff(T )
B(T )
is invariant by change of numeraire: If S is any other numeraire, we have
Pricet = ESt
[St
Payoff(T )
ST
].
In other terms, if we substitute the three occurrences inside the boxes of the
original numeraire with a new numeraire the price does not change. This
second fact is a rephrasing of Formula(3.6) seen before.
1 No need to use the stochastic Leibnitz rule of Appendix A here, since dStdt = 0
3.4 The Choice of a Convenient Numeraire 22
3.4 The Choice of a Convenient Numeraire
As far as pricing derivatives is concerned, the change-of-numeraire technique is typi-
cally employed as follows. A payoff h(XT ) is given, which depends on an underlying
variable X (an interest rate, an exchange rate, a commodity price, etc.) at time T .
Pricing such a payoff amounts to compute the risk-neutral expectation
E0D(0, T )h(XT ).
The risk-neutral numeraire is the money-market account
B(t) = D(0, t)−1 = exp
(∫ t
0rsds
).
By using formula (3.6) for pricing under a new numeraire S, we obtain
E0D(0, T )h(XT ) = S0EQSh(XT )
ST. (3.7)
Motivated by the above formula, we look for a numeraire S with the following two
properties.
1. XtSt is (the price of) a tradable asset (in such a case S is sometimes termed
the natural payoff of X).
2. The quantity h(XT )/ST is conveniently simple.
Why are these properties desirable?
The first condition ensures that (XtSt)/St = Xt is a martingale under QS , so
that one can assume for example a lognormal martingale dynamics for X:
dXt = σ(t)XtdWt, QS
with the consequent ease in computing expected values of functions of X. Indeed,
in this case, the distribution of X under QS is known, and we have
lnXt = N(
lnX0 −1
2
∫ t
0σ(s)2ds,
∫ t
0σ(s)2ds
).
The second condition ensures that the new numeraire renders the computation of
the right hand side of (3.7) simpler, instead of complicating it.
3.5 The Forward Measure
In many concrete situations, a useful numeraire is the zero-coupon bond whose
maturity T coincides with that of the derivative to price. In such a case, in fact,
3.5 The Forward Measure 23
ST = P (T, T ) = 1, so that pricing the derivative can be achieved by calculating
an expectation of its payoff (divided by one). The measure associated with the
bond maturing at time T is referred to as T -forward risk-adjusted measure, or more
briefly as T -forward measure, and will be denoted by QT . The related expectation
is denoted by ET .
Denoting by πt the price of the derivative at time t, formula (3.4) applied to the
above numeraire yields
πt = P (t, T )ET HT |Ft, (3.8)
for 0 ≤ t ≤ T , and where HT is the claim payoff at time T .
The reason why the measure QT is called forward measure is justified by the
following.
Proposition 3.10. Any simply-compounded forward rate spanning a time interval
ending in T is a martingale under the T -forward measure, i.e.,
ET F (t;S, T )|Fu = F (u;S, T ),
for each 0 = u = t = S ¡ T. In particular, the forward rate spanning the interval
[S, T ] is the QT -expectation of the future simply-compounded spot rate at time S for
the maturity T , i.e,
ET L(S, T )|Ft = F (t;S, T ), (3.9)
for each 0 ≤ t ≤ S < T .
Proof. From the definition of a simply-compounded forward rate
F (t;S, T ) =1
τ(S, T )
[P (t, S)
P (t, T )− 1
],
with τ(S, T ) the year fraction from S to T , we have that
F (t;S, T )P (t, T ) =1
τ(S, T )[P (t, S)− P (t, T )]
is the price at time t of a traded asset, since it is a multiple of the difference of two
bonds. Therefore, by definition of the T -forward measure
F (t;S, T )P (t, T )
P (t, T )= F (t;S, T )
is a martingale under such a measure. The relation (3.9) then immediately follows
from the equality F (S;S, T ) = L(S, T ).
The equality (3.9) can be extended to instantaneous rates as well. This is done
in the following.
3.6 The Fundamental Pricing Formulas 24
Proposition 3.11. The expected value of any future instantaneous spot interest rate,
under the corresponding forward measure, is equal to related instantaneous forward
rate, i.e.,
ET rT |Ft = f(t, T ),
for each 0 ≤ t ≤ T .
Proof. Let us apply (3.8) with HT = rT and remember the risk-neutral valuation
formula (3.2). We then have
ET rT |Ft =1
P (t, T )ErT e−
∫ Tt rsds|Ft
= − 1
P (t, T )E
∂
∂Te−
∫ Tt rsds|Ft
= − 1
P (t, T )
∂P (t, T )
∂T
= f(t, T ).
3.6 The Fundamental Pricing Formulas
The results of the previous sections are developed in case of a finite number of
basic market securities. A (theoretical) bond market, however, has a continuum of
basic assets (the zero-coupon bonds), one for each possible maturity until a given
time horizon. The no-arbitrage theory in such a situation is more complicated even
though quite similar in spirit. What we will do is simply to assume the existence of a
risk-neutral measure Q under which the price at time t of any attainable contingent
claim with payoff HT at time T > t is given by
πt = E(e−
∫ Tt rsdsHT |Ft
), (3.10)
consistently with formula (3.2), where a claim is now meant to be attainable if it can
be replicated by a self-financing strategy involving a finite number of basic assets at
every trading time.
Remark 1 (Attainability Assumption). For all contingent claims that will be priced
we will assume attainability to hold. Therefore, when a claim is priced, we are
assuming the existence of a suitable self-financing strategy that replicates the claim.
The particular case of a European call option with maturity T , strike X and
written on a unit-principal zero-coupon bond with maturity S > T leads to the
pricing formula
ZBC(t, T, S,X) = E(e−
∫ Tt rsds(P (T, S)−X)+|Ft
). (3.11)
3.6 The Fundamental Pricing Formulas 25
Moreover, the change-of-numeraire technique can be applied exactly as before, since
it is just based on changing the underlying probability measure. For example, the
forward measure QT is defined by the Radon-Nikodym derivative
dQT
Q=P (T, T )B(0)
P (0, T )B(T )=e−
∫ T0 rsds
P (0, T )=D(0, T )
P (0, T ),
so that the price at time t of the above claim is also given by (3.8), that is
πt = P (t, T )ET (HT |Ft) . (3.12)
In the case of the above call option, we then have
zbc(t, T, S,X) = P (t, T )ET((P (T, S)−X)+|Ft
)(3.13)
Of course, formula (3.12), and hence formula (3.13), turn out to be useful when
the quantities defining the claim payoff have known dynamics (or distributions)
under the forward measure QT . Typically, if P (T, S) has a log-normal distribution
conditional on Ft under the T -forward measure, the above expectation reduces to a
Black-like formula with a suitable volatility input.
Chapter 4
One-factor short-rate models
4.1 Introduction and Guided Tour
The theory of interest-rate modeling was originally based on the assumption of spe-
cific one-dimensional dynamics for the instantaneous spot rate process r. Modeling
directly such dynamics is very convenient since all fundamental quantities (rates
and bonds) are readily defined, by no-arbitrage arguments, as the expectation of a
functional of the process r. Indeed, the existence of a risk-neutral measure implies
that the arbitrage-free price at time t of a contingent claim with payoff HT at time
T is given by
Ht = EtD(t, T )HT = Et
e−
∫ Tt r(s)dsHT
, (4.1)
with Et denoting the time t-conditional expectation under that measure. In partic-
ular, the zero-coupon-bond price at time t for the maturity T is characterized by a
unit amount of currency available at time T , so that HT = 1 and we obtain
P (t, T ) = Et
e−
∫ Tt r(s)ds
. (4.2)
From this last expression it is clear that whenever we can characterize the distribu-
tion of e−∫ Tt r(s)ds in terms of a chosen dynamics for r, conditional on the information
available at time t, we are able to compute bond prices P . As we have seen earlier
in Chapter 3, from bond prices all kind of rates are available, so that indeed the
whole zero-coupon curve is characterized in terms of distributional properties of r.
The pioneering approach proposed by Vasicek (1977) was based on defining the
instantaneous-spot-rate dynamics under the real-world measure. His derivation of
an arbitrage-free price for any interest-rate derivative followed from using the basic
Black and Scholes (1973) arguments, while taking into account the non-tradable
feature of interest rates.
The construction of a suitable locally-riskless portfolio, as in Black and Scholes
(1973), leads to the existence of a stochastic process that only depends on the current
time and instantaneous spot rate and not on the maturities of the claims constituting
4.1 Introduction and Guided Tour 27
the portfolio. Such process, which is commonly referred to as market price of risk,
defines a Girsanov change of measure from the real-world measure to the risk-neutral
one also in case of more general dynamics than Vasicek’s. Precisely, let us assume
that the instantaneous spot rate evolves under the real-world measure Q0 according
to
dr(t) = µ(t, r(t))dt+ σ(t, r(t))dW 0(t),
where µ and σ are well-behaved functions and W0 is a Q0-Brownian motion. It is
possible to show the existence of a stochastic process λ such that if
dP (t, T ) = µT (t, r(t))dt+ σT (t, r(t))dW 0(t),
thenµT (t, r(t))− r(t)P (t, T )
σT (t, r(t))= λ(t)
for each maturity T , with λ that may depend on r but not on T . Moreover, there
exists a measure Q that is equivalent to Q0 and is defined by the Radon-Nikodym
derivativedQ
dQ0
∣∣∣∣Ft
= exp
(−1
2
∫ t
0λ2(s)ds−
∫ t
0λ(s)dW 0(s)
),
where Ft is the σ-field generated by r up to time t. As a consequence, the process
r evolves under Q according to
dr(t) = [µ(t, r(t))− λ(t)σ(t, r(t))]dt+ σ(t, r(t))dW (t),
where W (t) = W 0(t) +∫ t
0 λ(s)ds is a Brownian motion under Q.1
Let us comment briefly on this setup. The above equation for dP actually ex-
presses the bond-price dynamics in terms of the short rate r. It expresses how the
bond price P evolves over time. Now recall that r is the instantaneous-return rate
of a risk-free investment, so that the difference µ − r represents a difference in re-
turns. It tells us how much better we are doing with respect to the risk-free case,
i.e. with respect to putting our money in a riskless bank account. When we divide
this quantity by σT , we are dividing by the amount of risk we are subject to, as
measured by the bond-price volatility σT . This is why λ is referred to as “market
price of risk”. An alternative term could be “excess return with respect to a risk-
free investment per unit of risk”. The crucial observation is that in order to specify
completely the model, we have to provide λ. In effect, the market price of risk λ
connects the real-world measure to the risk-neutral measure as the main ingredient
in the mathematical object dQ/dQ0 expressing the connection between these two
1 The Radon-Nikodym derivative and the Girsanov change of measure are briefly reviewed in
Appendix C. For a formal treatment see Musiela and Rutkowski (1998).
4.2 Classical Time-Homogeneous Short-Rate Models 28
“worlds”. The way of moving from one world to the other is characterized by our
choice of λ. However, if we are just concerned with the pricing of (interestrate)
derivatives, we can directly model the rate dynamics under the measure Q, so that
λ will be implicit in our dynamics. We put ourselves in the world Q and we do not
bother about the way of moving to the world Q0. Then we would be in troubles
only if we needed to move under the objective measure, but for pricing derivatives,
the objective measure is not necessary, so that we can safely ignore it.
4.2 Classical Time-Homogeneous Short-Rate Models
The first instantaneous short rate models being proposed in the financial literature
were time-homogeneous, meaning that the assumed short rate dynamics depended
only on constant coefficients. The success of models like that of Vasicek (1977) and
that of Cox, Ingersoll and Ross (1985) was mainly due to their possibility of pricing
analytically bonds and bond options. However, as observed in the introduction,
these models produce an endogenous term structure of interest rates, in that the
initial term structure of (e.g. continuously-compounded) rates T 7→ R(0, T ) =
−(lnP (0, T ))/T does not necessarily match that observed in the market, no matter
how the model parameters are chosen.
4.3 The Vasicek Model
Vasicek (1977) assumed that the instantaneous spot rate under the real-world mea-
sure evolves as an Ornstein-Uhlenbeck process with constant coefficients. For a suit-
able choice of the market price of risk (more on this later, see equation (4.7), this
is equivalent to assume that r follows an Ornstein-Uhlenbeck process with constant
coefficients under the risk-neutral measure as well, that is
dr(t) = k[θ − r(t)]dt+ σdW (t), r(0) = r0, (4.3)
where r0, k, θ and σ are positive constants.
Integrating equation (4.3), we obtain, for each s = t,
r(t) = r(s)e−k(t−s) + θ(
1− e−k(t−s))
+ σ
∫ t
se−k(t−u)dW (u), (4.4)
so that r(t) conditional on Fs is normally distributed with mean and variance given
respectively by
Er(t)|Fs = r(s)e−k(t−s) + θ(
1− e−k(t−s))
Varr(t)|Fs =σ2
2k[1− e−2k(t−s)].
(4.5)
4.3 The Vasicek Model 29
This implies that, for each time t, the rate r(t) can be negative with positive prob-
ability. The possibility of negative rates is indeed a major drawback of the Vasicek
model. However, the analytical tractability that is implied by a Gaussian density is
hardly achieved when assuming other distributions for the process r.
As a consequence of (4.5), the short rate r is mean reverting, since the expected
rate tends, for t going to infinity, to the value θ. The fact that θ can be regarded
as a long term average rate could be also inferred from the dynamics (4.3) itself.
Notice, indeed, that the drift of the process r is positive whenever the short rate is
below θ and negative otherwise, so that r is pushed, at every time, to be closer on
average to the level θ.
The price of a pure-discount bond can be derived by computing the expecta-
tion (4.2). We obtain
P (t, T ) = A(t, T )e−B(t,T )r(t), (4.6)
where
A(t, T ) = exp
(θ − σ2
2k2
)[B(t, T )− T + t]− σ2
4kB(t, T )2
B(t, T ) =
1
k
[1− e−k(T−t)
].
Objective measure dynamics and historical estimation. We can consider the
objective measure dynamics of the Vasicek model as a process of the form
dr(t) = [kθ − (k + λσ)r(t)]dt+ σdW 0(t), r(0) = r0, (4.7)
where lambda is a new parameter, contributing to the market price of risk. Compare
this Q0 dynamics to the Q-dynamics (3.5). Notice that for λ = 0 the two dynamics
coincide, i.e. there is no difference between the risk neutral world and the objective
world. More generally, the above Q0-dynamics is expressed again as a linear Gaus-
sian stochastic differential equation, although it depends on the new parameter λ.
This is a tacit assumption on the form of the market price of risk process. Indeed,
requiring that the dynamics be of the same nature under the two measures, imposes
a Girsanov change of measure of the following kind to go from (4.3) to (4.7):
dQ
dQ0
∣∣∣∣Fs
= exp
(−1
2
∫ t
0λ2r(s)2ds+
∫ t
0λr(s)dW 0(s)
)(see Appendix A for an introduction to Girsanov’s theorem).
In other terms, we are assuming that the market price of risk process λ(t) has
the functional form
λ(t) = λr(t)
4.3 The Vasicek Model 30
in the short rate. Of course, in general there is no reason why this should be the
case. However, under this choice we obtain a short rate process that is tractable
under both measures.2
4.3.1 Affine Term-Structure Models
Affine term-structure models are interest-rate models where the continuously-compounded
spot rate R(t, T ) is an affine function in the short rate r(t), i.e.
R(t, T ) = α(t, T ) + β(t, T )r(t)
, where α and β are deterministic functions of time. If this happens, the model is
said to possess an affine term structure. This relationship is always satisfied when
the zerocoupon bond price can be written in the form
P (t, T ) = A(t, T )e−B(t,T )r(t),
since then clearly it suffices to set
α(t, T ) = −(lnA(t, T ))/(T − t), β(t, T ) = B(t, T )/(T − t).
Both the Vasicek and CIR models we have seen earlier are affine models, since the
bond price has an expression of the above form in both cases. The Dothan model is
not an affine model.
A computation that will be helpful in the following is the instantaneous absolute
volatility of instantaneous forward rates in affine models. Since in general
f(t, T ) = −∂ lnP (t, T )
∂T,
for affine models we have
f(t, T ) = −∂ lnA(t, T )
∂T+∂B(t, T )
∂Tr(t),
so that clearly
f(t, T ) = (. . . )dt+∂B(t, T )
∂Tσ(t, r(t))dW (t)
where σ(t, r(t)) is the diffusion coefficient in the short rate dynamics. It follows that
the absolute volatility of the instantaneous forward rate f(t, T ) at time t in a short
rate model with an affine term structure is
σf (t, T ) =∂B(t, T )
∂Tσ(t, r(t)). (4.8)
2 Indeed, the market price of risk under the Vasicek model is usually chosen to be constant, i.e.,
λ(t) = λ. However, this is just another possible formulation.
4.4 The Hull-White Extended Vasicek Model 31
In particular, for the Vasicek and CIR models we can obtain explicit expressions for
this quantity by using the known expressions for B. We will see later on why this
volatility function is important.
Given that by inspection one sees that the Vasicek and CIR models are affine
models whereas Dothan is not, one may wonder whether there is a relationship
between the coefficients in the risk-neutral dynamics of the short rate and affinity
of the term structure in the above sense. Assume we have a general risk-neutral
dynamics for the short rate,
dr(t) = b(t, r(t))dt+ σ(t, r(t))dW (t).
We may wonder whether there exist conditions on b and σ such that the resulting
model displays an affine term structure. The answer is simply that the coefficients b
and σ2 need be affine functions themselves (see for example Bjork (1997) or Duffie
(1996)). If the coefficients b and σ2 are of the form
b(t, x) = λ(t)x+ η(t), σ2(t, x) = γ(t)x+ δ(t)
for suitable deterministic time functions λ, η, γ, delta then the model has an affine
term structure, with α and β (or A and B) above depending on the chosen func-
tions λ, η, γ, delta. The functions A and B can be obtained from the coefficients
λ, η, γ, delta by solving the following differential equations:
∂
∂tB(t, T ) + λ(t)B(t, T )− 1
2γ(t)B(t, T )2 + 1 = 0, B(T, T ) = 0,
∂
∂t[lnA(t, T )]− η(t)B(t, T ) +
1
2δ(t)B(t, T )2 = 0, A(T, T ) = 1.
4.4 The Hull-White Extended Vasicek Model
The poor fitting of the initial term structure of interest rates implied by the Vasicek
model has been addressed by Hull and White in their 1990 and subsequent papers.
Ho and Lee (1986) have been the first to propose an exogenous term-structure model
as opposed to models that endogenously produce the current term structure of rates.
However, their model was based on the assumption of a binomial tree governing the
evolution of the entire term structure of rates, and even its continuous-time limit,
as derived by Dybvig (1988) and Jamshidian (1988), cannot be regarded as a proper
extension of the Vasicek model because of the lack of mean reversion in the short-rate
dynamics.
The need for an exact fit to the currently-observed yield curve, led Hull and
White to the introduction of a time-varying parameter in the Vasicek model. No-
tice indeed that matching the model and the market term structures of rates at the
4.5 The Short-Rate Dynamics 32
current time is equivalent to solving a system with an infinite number of equations,
one for each possible maturity. Such a system can be solved in general only after in-
troducing an infinite number of parameters, or equivalently a deterministic function
of time.
By considering a further time-varying parameter, Hull and White (1990b) pro-
posed an even more general model that is also able to fit a given term structure of
volatilities. Such a model, however, may be somehow dangerous when applied to
concrete market situations as we will hint at below. This is the main reason why in
this section we stick to the extension where only one parameter, corresponding to
the Vasicek θ, is chosen to be a deterministic function of time.
4.5 The Short-Rate Dynamics
Hull and White (1990) assumed that the instantaneous short-rate process evolves
under the risk-neutral measure according to
dr(t) = [ϑ(t)− a(t)r(t)]dt+ σ(t)dW (t), (4.9)
where ϑ, a, σ are deterministic functions of time.
Such a model can be fitted to the term structure of interest rates and the term
structure of spot or forward-rate volatilities. However, if an exact calibration to
the current yield curve is a desirable feature, the perfect fitting to a volatility term
structure can be rather dangerous and must be carefully dealt with. The reason is
two-fold. First, not all the volatilities that are quoted in the market are significant:
some market sectors are less liquid, with the associated quotes that may be neither
informative nor reliable. Second, the future volatility structures implied by (4.9) are
likely to be unrealistic in that they do not conform to typical market shapes, as was
remarked by Hull and White (1995b) themselves.
We therefore concentrate on the following extension of the Vasicek model being
analyzed by Hull and White (1994a)
dr(t) = [ϑ(t)− ar(t)]dt+ σdW (t), (4.10)
where a and σ are now positive constants and ϑ is chosen so as to exactly fit the
term structure of interest rates being currently observed in the market. It can be
shown that, denoting by fM (0, T ) the market instantaneous forward rate at time 0
for the maturity T , i.e.,
fM (0, T ) = −∂ lnPM (0, T )
∂T,
4.5 The Short-Rate Dynamics 33
with PM (0, T ) the market discount factor for the maturity T , we must have
v(t) =∂fM (0, t)
∂T+ afM (0, t) +
σ2
2a(1− e−2at), (4.11)
where ∂fM
∂T denotes partial derivative of fM with respect to its second argument.
Equation (4.10) can be easily integrated so as to yield
r(t) = r(s)e−a(t−s) +
∫ t
se−a(t−u)ϑ(u)du+ σ
∫ t
se−a(t−u)dW (u)
= r(s)e−a(t−s) + α(t)− α(s)e−a(t−s) + σ
∫ t
se−a(t−u)dW (u),
(4.12)
where
α(t) = fM (0, t) +σ2
2a2(1− e−at)2. (4.13)
Therefore, r(t) conditional on Fs is normally distributed with mean and variance
given respectively by
Er(t)|Fs = r(s)e−a(t−s) + α(t)− α(s)e−a(t−s)Varr(t)|Fs =σ2
2a
[1− e−2a(t−s)
].
(4.14)
Notice that defining the process x by
dx(t) = −ax(t)dt+ σdW (t), x(0) = 0, (4.15)
we immediately have that, for each s ¡ t,
x(t) = x(s)e−a(t−s) + σ
∫ t
se−a(t−u)dW (u),
so that we can write r(t) = x(t) + α(t) for each t.
4.5.1 Bond and Option Pricing
The price at time t of a pure discount bond paying off 1 at time T is given by the
expectation (4.2). Such expectation is relatively easy to compute under the dynam-
ics (4.10). Notice indeed that, due to the Gaussian distribution of r(T ) conditional
on Ft, t ≤ T,∫ Tt r(u)du is itself normally distributed. Precisely we can show that∫ T
tr(u)du|Ft
∼ N(B(t, T )[r(t)− α(t)] + ln
PM (0, t)
PM (0, T )+
1
2[V (0, T )− V (0, t)], V (t, T )
),
4.5 The Short-Rate Dynamics 34
where
B(t, T ) =1
a
[1− e−a(T−t)
],
V (t, T ) =σ2
a2
[T − t+
2
ae−a(T−t) − 1
2ae−2a(T−t) − 3
2a
],
so that we obtain
P (t, T ) = A(t, T )e−B(t,T )r(t), (4.16)
where
A(t, T ) =PM (0, T )
PM (0, t)exp
B(t, T )fM (0, t)− σ2
4a(1− e−2at)B(t, T )2
Derivations shown for above will be shown in detail in the two-factor case to
follow in the next chapter.
Chapter 5
Two-Factor Short-Rate Models
5.1 Introduction and Motivation
We have seen previously that the short rate rt may constitute the fundamental
coordinate with which the whole yield curve can be characterized. Indeed, knowledge
of the short rate and of its distributional properties leads to knowledge of bond prices
as from the usual relationship
P (t, T ) = E
[exp
(−∫ T
trsds
)].
From all bond prices P (t, T ) at a given time t one can reconstruct the whole zero-
coupon interest-rate curve at the same time t, so that indeed the evolution of the
whole curve is characterized by the evolution of the single quantity r. However,
choosing a poor model for the evolution of r will result in a poor model for the
evolution of the yield curve. In order to clarify this rather important point, let us
consider for a moment the Vasicek model (4.3).
Recall from the previous chapter that the Vasicek model assumes the evolution
of the short-rate process r to be given by the linear-Gaussian SDE
drt = k(θ − rt)dt+ σdWt.
Recall also the bond price formula P (t, T ) = A(t, T ) exp(−B(t, T )rt), from which all
rates can be computed in terms of r. In particular, continuously-compounded spot
rates are given by the following affine transformation of the fundamental quantity r
R(t, T ) = − lnA(t, T )
T − t+B(t, T )
T − trt =: a(t, T ) + b(t, T )rt.
Consider now a payoff depending on the joint distribution of two such rates at time
t. For example, we may set T1 = t+1 years and T2 = t+10 years. The payoff would
then depend on the joint distribution of the one-year and ten-year continuously-
compounded spot interest rates at “terminal” time t. In particular, since the joint
5.1 Introduction and Motivation 36
distribution is involved, the correlation between the two rates plays a crucial role.
With the Vasicek model such terminal correlation is easily computed as
Corr(R(t, T1), R(t, T2)) = Corr(a(t, T1) + b(t, T1)rt, a(t, T2) + b(t, T2)rt) = 1,
so that at every time instant rates for all maturities in the curve are perfectly
correlated. For example, the thirty-year interest rate at a given instant is perfectly
correlated with the three-month rate at the same instant. This means that a shock
to the interest rate curve at time t is transmitted equally through all maturities,
and the curve, when its initial point (the short rate rt) is shocked, moves almost
rigidly in the same direction. Clearly, it is hard to accept this perfect-correlation
feature of the model. Truly, interest rates are known to exhibit some decorrelation
(i.e. non-perfect correlation), so that a more satisfactory model of curve evolution
has to be found.
One-factor models such as HW may still prove useful when the product to be
priced does not depend on the correlations of different rates but depends at every
instant on a single rate of the whole interest-rate curve (say for example the six-
month rate).
But in general, whenever the correlation plays a more relevant role, or when a
higher precision is needed anyway, we need to move to a model allowing for more
realistic correlation patterns. This can be achieved with multi-factor models, and
in particular with two-factor models. Indeed, suppose for a moment that we replace
the Gaussian Vasicek model with its hypothetical two-factor version (G2):
rt = xt + yt,
dxt = kx(θx − xt)dt+ σxdW1(t),
dyt = ky(θy − yt)dt+ σydW2(t),
(5.1)
with instantaneously-correlated sources of randomness, dW1dW2 = ρdt. Again, we
will see later on in the chapter that also for this kind of models the bond price is an
affine function, this time of the two factors x and y,
P (t, T ) = A(t, T ) exp(−Bx(t, T )xt −By(t, T )yt),
where quantities with the superscripts “x” or “y” denote the analogous quantities
for the one-factor model where the short rate is given by x or y, respectively. Taking
this for granted at the moment, we can see easily that now
Corr(R(t, T1), R(t, T2)) = Corr(bx(t, T1)xt + by(t, T1)yt, bx(t, T2)xt + by(t, T2)yt),
and this quantity is not identically equal to one, but depends crucially on the cor-
relation between the two factors x and y, which in turn depends, among other
5.2 The Two-Additive-Factor Gaussian Model G2++ 37
quantities, on the instantaneous correlation ρ in their joint dynamics. How much
flexibility is gained in the correlation structure and whether this is sufficient for
practical purposes will be debated in the chapter. It is however clear that the choice
of a multi-factor model is a step forth in that correlation between different rates of
the curve at a given instant is not necessarily equal to one.
What we learn from these analyses is that, in the objective world, a two- or
three-dimensional process is needed to provide a realistic evolution of the whole
zero-coupon curve. Since the instantaneous-covariance structure of the same process
when moving from the objective probability measure to the risk-neutral probability
measure does not change, we may guess that also in the risk-neutral world a two-
or three-dimensional process may be needed in order to obtain satisfactory results.
This is a further motivation for introducing a two- or three-factor model for the
short rate. In particular, we will consider additive models of the form
rt = xt + yt + ϕ(t), (5.2)
where ϕ is a deterministic shift which is added in order to fit exactly the initial
zero-coupon curve. This formulation encompasses the classical Hull and White two-
factor model as a deterministically-shifted two-factor Vasicek (G2++), we will focus
especially on the two-factor additive Gaussian model G2++.
5.2 The Two-Additive-Factor Gaussian Model G2++
In this section we consider an interest-rate model where the instantaneous-short-rate
process is given by the sum of two correlated Gaussian factors plus a deterministic
function that is properly chosen so as to exactly fit the current term structure of
discount factors. The model is quite analytically tractable in that explicit formulas
for discount bonds, European options on pure discount bonds, hence caps and floors,
can be readily derived.
5.2.1 The Short-Rate Dynamics
We assume that the dynamics of the instantaneous-short-rate process under the
risk-adjusted measure Q is given by
r(t) = x(t) + y(t) + ϕ(t), r(0) = r0, (5.3)
where the processes x(t) : t ≥ 0 and y(t) : t ≥ 0 satisfy
dx(t) = −ax(t)dt+ σdW1(t), x(0) = 0,
dy(t) = −by(t)dt+ ηdW2(t), y(0) = 0,(5.4)
5.2 The Two-Additive-Factor Gaussian Model G2++ 38
where (W1,W2) is a two-dimensional Brownian motion with instantaneous correla-
tion ρ as from
dW1(t)dW2(t) = ρdt,
where r0, a, b, σ, η are positive constants, and where −1 ≤ ρ ≤ 1. The function ϕ
is deterministic and well defined in the time interval [0, T ∗], with T ∗ a given time
horizon, typically 10, 30 or 50 (years). In particular, ϕ(0) = r0. We denote by Ftthe sigma-field generated by the pair (x, y) up to time t.
Simple integration of equations (5.4) implies that for each s < t
r(t) = x(s)e−a(t−s) + y(s)e−b(t−s)
+ σ
∫ t
se−a(t−u)dW1(u) + η
∫ t
se−b(t−u)dW2(u) + ϕ(t),
meaning that r(t) conditional on Fs is normally distributed with mean and variance
given respectively by
Er(t)|Fs = x(s)e−a(t−s) + y(s)e−b(t−s) + ϕ(t),
Varr(t)|Fs =σ2
2a
[1− e−2a(t−s)
]+η2
2b
[1− e−2b(t−s)
]+ 2ρ
ση
a+ b
[1− e−(a+b)(t−s)
].
(5.5)
In particular
r(t) = σ
∫ t
0e−a(t−u)dW1(u) + η
∫ t
0e−b(t−u)dW2(u) + ϕ(t). (5.6)
The dynamics of the processes x and y can be also expressed in terms of two inde-
pendent Brownian motions W1 and W2 as follows: 1
dx(t) = −ax(t)dt+ σdW1(t),
dy(t) = −by(t)dt+ ηρdW1(t)η√
1− ρ2dW2(t),(5.7)
where
dW1(t) = dW1(t), dW2(t) = ρdW1(t) +√
1− ρ2dW2(t)
so that we can also write
r(t) = x(s)e−a(t−s) + y(s)e−b(t−s) + σ
∫ t
se−a(t−u)dW1(u)
+ ηρ
∫ t
se−b(t−u)dW1(u) + η
√1− ρ2
∫ t
se−b(t−u)dW2(u) + ϕ(t)
1 This is equivalent to performing a Cholesky decomposition on the variance-covariance matrix
of the pair (W1(t),W2(t)).
5.2 The Two-Additive-Factor Gaussian Model G2++ 39
5.2.2 The Pricing of a Zero-Coupon Bond
We denote by P (t, T ) the price at time t of a zero-coupon bond maturing at T and
with unit face value, so that
P (t, T ) = Ee−
∫ Tt rsds|Ft
,
where E denotes the expectation under the risk-adjusted measure Q. In order to
explicitly compute this expectation, we need the following.
Lemma 5.1. For each t, T the random variable
I(t, T ) :=
∫ T
t[x(u) + y(u)]du
conditional to the sigma-field Ft is normally distributed with mean M(t, T ) and
variance V (t, T ), respectively given by
M(t, T ) =1− e−a(T−s)
ax(t) +
1− eb(T−t)
by(t) (5.8)
and
V (t, T ) =σ2
a2
[T − t+
2
ae−a(T−t) − 1
2ae−2a(T−t) − 3
2a
]+η2
b2
[T − t+
2
be−b(T−t) − 1
2be−2b(T−t) − 3
2b
]+ 2ρ
ση
ab
[T − t+
e−a(T−t) − 1
a+e−b(T−t) − 1
b− e−(a+b)(T−t) − 1
a+ b
].
(5.9)
Proof. Stochastic integration by parts implies that∫ T
tx(u)du = Tx(T )− tx(t)−
∫ T
tudx(u) =
∫ T
t(T −u)dx(u) + (T − t)x(t). (5.10)
By definition of x, the integral in the right-hand side can be written as∫ T
t(T − u)dx(u) = −a
∫ T
t(T − u)x(u)du+ σ
∫ T
t(T − u)dW1(u)
by substituting the expression for dx(u), and∫ T
t(T − u)x(u)du = x(t)
∫ T
t(T − u)e−a(u−t)du
+ σ
∫ T
t(T − u)
∫ u
te−a(u−s)dW1(s)du.
5.2 The Two-Additive-Factor Gaussian Model G2++ 40
by substituting the expression for x(u). Calculating separately the last two integrals
(multiplied by −a), we have
−ax(t)
∫ T
t(T − u)e−a(u−t)du = −x(t)(T − t)− e−a(T−t) − 1
ax(t)
and, again by integration by parts,
− aσ∫ T
t(T − u)
∫ u
te−a(u−s)dW1(s)du
= −aσ∫ T
t
(∫ u
teasdW1(s)
)du
(∫ u
t(T − v)e−avdv
)= −aσ
[(∫ T
teaudW1(u)
)(∫ T
t(T − v)e−avdv
)−∫ T
t
(∫ u
t(T − v)e−avdv
)eaudW1(u)
]= −aσ
∫ T
t
(∫ T
u(T − v)e−avdv
)eaudW1(u)
= −σ∫ T
t
[(T − u) +
e−a(T−u) − 1
a
]dW1(u),
where in the last step we have used the fact that∫ T
u(T − v)e−avdv =
(T − u)e−au
a+e−aT + e−au
a2.
Adding up the previous terms, we obtain∫ T
tx(u)du =
1− e−a(T−t)
ax(t) +
σ
a
∫ T
t
[1− e−a(T−u)
]dW1(u).
The analogous expression for y is obtained by replacing a, σandW1 respectively with
b, η and W2, i.e.,∫ T
ty(u)du =
1− e−b(T−t)
by(t) +
η
b
∫ T
t
[1− e−b(T−u)
]dW2(u),
so that (5.8) is immediately verified. As to the calculation of the conditional variance,
we have
VarI(t, T )|Ft = Var
σ
a
∫ T
t
[1− e−a(T−u)
]dW1(u)
+η
b
∫ T
t
[1− e−b(T−u)
]dW2(u)
∣∣∣∣Ft=σ2
a2
∫ T
t
[1− e−a(T−u)
]2du+
η2
b2
∫ T
t
[1− e−b(T−u)
]2du
+ 2ρση
ab
∫ T
t
[1− e−a(T−u)
] [1− e−b(T−u)
]du.
Simple integration then leads to (5.9).
5.2 The Two-Additive-Factor Gaussian Model G2++ 41
Theorem 5.2. The price at time t of a zero-coupon bond maturing at time T and
with unit face value is
P (t, T ) = exp
−∫ T
tϕ(u)du− 1− e−a(T−t)
ax(t)
− 1− e−b(T−t)
by(t) +
1
2V (t, T )
. (5.11)
Proof. Being ϕ a deterministic function, the theorem follows from straight forward
application of Lemma 5.1 and the fact that if Z is a normal random variable with
mean mZ and variance σ2Z , then Eexp(Z) = exp(mZ + 1
2σ2Z).
Let us now assume that the term structure of discount factors that is currently
observed in the market is given by the sufficiently smooth function T 7→ PM (0, T ).
If we denote by fM (0, T ) the instantaneous forward rate at time 0 for a maturity
T implied by the term structure T 7→ PM (0, T ), i.e.,
fM (0, T ) = −∂ lnPM (0, T )
∂T,
we then have the following.
Corollary 5.3. The model (5.3) fits the currently-observed term structure of dis-
count factors if and only if, for each T ,
ϕ(T ) = fM (0, T ) +σ2
2a2(1− e−aT )2
+η2
2b2(1− e−bT )2 + ρ
ση
ab(1− e−aT )(1− e−bT ), (5.12)
i.e., if and only if
exp
−∫ T
tϕ(u)du
=PM (0, T )
PM (0, t)exp
−1
2[V (0, T )− V (0, t)]
, (5.13)
so that the corresponding zero-coupon-bond prices at time t are given by
P (t, T ) =PM (0, T )
PM (0, t)expA(t, T )
A(t, T ) =1
2[V (t, T )− V (0, T ) + V (0, t)]
− 1− e−a(T−t)
ax(t)− 1− e−b(T−t)
by(t).
(5.14)
Proof. The model (5.3) fits the currently-observed term structure of discount factors
if and only if for each maturity T ≤ T ∗ the discount factor P (0, T ) produced by the
model (5.3) coincides with the one observed in the market, i.e., if and only if
PM (0, T ) = exp
−∫ T
0ϕ(u)du+
1
2V (0, T )
.
5.2 The Two-Additive-Factor Gaussian Model G2++ 42
Now let us take logs of both sides and differentiate with respect to T , so as to obtain
(4.12) by noting that
V (t, T ) =σ2
a2
∫ T
t
[1− e−a(T−u)
]2du+
η2
b2
∫ T
t
[1− e−b(T−u)
]2du
+ 2ρση
ab
∫ T
t
[1− e−a(T−u)
] [1− e−b(T−u)
]du
Equality (5.13) follows from noting that, under the specification (5.12),
exp
−∫ T
tϕ(u)du
= exp
−∫ T
0ϕ(u)du
exp
∫ t
0ϕ(u)du
=PM (0, T ) exp−1
2V (0, T )PM (0, t) exp−1
2V (0, t).
Equality (5.14) immediately follows from (5.11) and (5.13).
Remark 2 (Is it really necessary to derive the market instantaneous forward curve?).
Notice that, at a first sight, one may have the impression that in order to implement
the G2++ model we need to derive the whole ϕ curve, and therefore the market
instantaneous forward curve T 7→ fM (0, T ). Now, this curve involves differentiating
the market discount curve T 7→ PM (0, T ), which is usually obtained from a finite
set of maturities via interpolation. Interpolation and differentiation may induce a
certain degree of approximation, since the particular interpolation technique being
used has a certain impact on (first) derivatives.
However, it turns out that one does not really need the whole ϕ curve. Indeed,
what matters is the integral of ϕ between two given instants. This integral has
been computed in (5.13). From this expression, we see that the only curve needed
is the market discount curve, which need not be differentiated, and only at times
corresponding to the maturities of the bond prices and rates desired, thus limiting
also the need for interpolation.
Chapter 6
“The Heath-Jarrow-Morton”
framework
Modeling the interest-rate evolution through the instantaneous short rate has some
advantages, mostly the large liberty one has in choosing the related dynamics. For
example, for one-factor short-rate models one is free to choose the drift and instan-
taneous volatility coefficient in the related diffusion dynamics as one deems fit, with
no general restrictions. We have seen several examples of possible choices in Chap-
ter 3. However, short-rate models have also some clear drawbacks. For example, an
exact calibration to the initial curve of discount factors and a clear understanding
of the covariance structure of forward rates are both difficult to achieve, especially
for models that are not analytically tractable.
The first historically important alternative to short-rate models has been pro-
posed by Ho and LEE [1986], who modeled the evolution of the entire yield curve in
a binomial-tree setting. Their basic intuition was then translated in continuous time
by Heath et al. [1992] who developed a quite general framework for the modeling
of interest-rate dynamics. Precisely, by choosing the instantaneous forward rates
as fundamental quantities to model, they derived an arbitrage-free framework for
the stochastic evolution of the entire yield curve, where the forward-rates dynamics
are fully specified through their instantaneous volatility structures. This is a major
difference with arbitrage free one-factor short-rate dynamics, where the volatility of
the short rate alone does not suffice to characterize the relevant interest-rate model.
But in order to clarify the matter, let us consider the Merton (1973) toy short-rate
model.
Assume we take the following equation for the short rate under the risk-neutral
measure:
drt = θdt+ σdWt, r0.
If you wish, this is a very particular toy version of the Hull-White model (4.9) seen
in Chapter 4 with constant coefficient θ. Now, for this (affine) model, one can easily
Chapter 6. “The Heath-Jarrow-Morton” framework 44
compute the bond price,
P (t, T ) = exp
[σ2
6(T − t)3 − θ
2(T − t)2 − (T − t)rt
],
and the instantaneous forward rate
f(t, T ) = −∂ lnP (t, T )
∂T= −σ
2
2(T − t)2 + θ(T − t) + rt.
Differentiate this and substitute the short-rate dynamics to obtain
df(t, T ) = (σ2(T − t)− θ)dt+ θdt+ σdWt,
or
df(t, T ) = σ2(T − t)dt+ σdWt.
Look at this last equation. The drift σ2(T − t) in the f ’s dynamics is determined as
a suitable transformation of the diffusion coefficient σ in the same dynamics. This
is no mere coincidence due to the simplicity of our toy model, but a general fact
that Heath et al. [1992] expressed in full generality.
Clearly, if one wishes to directly model this instantaneous forward rate, there is
no liberty in selecting the drift of its process, as it is completely determined by the
chosen volatility coefficient. This is essentially due to the fact that we are modeling
a derived quantity f and not the fundamental quantity r. Indeed, f is expressed in
terms of the more fundamental r by
f(t, T ) = −∂ lnEt
[exp(−
∫ Tt r(s)ds)
]∂T
,
and, as you see, an expectation has already acted in the definition of f , adding
structure and taking away freedom, so to say.
More generally, under the HJM framework, one assumes that, for each T , the
forward rate f(t, T ) evolves according to
df(t, T ) = α(t, T )dt+ σ(t, T )dW (t),
where W is a (possibly multi-dimensional) Brownian motion. As we have just seen
in the above example, contrary to the short-rate modeling case, where one is free
to specify the drift of the considered diffusion, here the function α is completely
determined by the choice of the (vector) diffusion coefficient σ.
The importance of the HJM theory lies in the fact that virtually any (exogenous
term-structure) interest-rate model can be derived within such a framework. How-
ever, only a restricted class of volatilities is known to imply a Markovian short-rate
process. This means that, in general, burdensome procedures, like those based on
6.1 The HJM Forward-Rate Dynamics 45
non-recombining lattices, are needed to price interest-rate derivatives. Substantially,
the problem remains of defining a suitable volatility function for practical purposes.
This is the reason why in this work we do not devote too much attention to the
HJM theory, preferring to deal with explicitly formulated models.
6.1 The HJM Forward-Rate Dynamics
Heath et al. [1992] assumed that, for a fixed a maturity T , the instantaneous forward
rate f(t, T ) evolves, under a given measure, according the following diffusion process:
df(t, T ) = α(t, T )dt+ σ(t, T )dW (t),
f(0, T ) = fM (0, T ),(6.1)
with T 7→ fM (0, T ) the market instantaneous-forward curve at time t = 0, and where
W = (W1, . . . ,WN ) is anN -dimensional Brownian motion, σ(t, T ) = (σ1(t, T ), . . . , σN (t, T ))
is a vector of adapted processes and α(t, T ) is itself an adapted process. The product
σ(t, T )dW (t) is intended to be the scalar product between the two vectors σ(t, T )
and dW (t).
The advantage of modeling forward rates as in (6.1) is that the current term
structure of rates is, by construction, an input of the selected model. Remember, in
fact, that the following relations between zero-bond prices and forward rates hold:
f(t, T ) = −∂ lnP (t, T )
∂T,
P (t, T ) = e−∫ Tt f(t,u)du.
The dynamics in (6.1) is not necessarily arbitrage-free. Following an approach simi-
lar to that of Harrison and Pliska [1981]. Heath, Jarrow and Morton proved that, in
order for a unique equivalent martingale measure to exist, the function a cannot be
arbitrarily chosen, but it must equal a quantity depending on the vector volatility
s and on the drift rates in the dynamics of N selected zero-coupon bond prices. In
particular, if the dynamics (6.1) are under the risk-neutral measure, then we must
have
α(t, T ) = σ(t, T )
∫ T
tσ(t, s)ds =
N∑i=1
σi(t, T )
∫ T
tσi(t, s)ds, (6.2)
so that the integrated dynamics of f(t, T ) under the risk-neutral measure are
f(t, T ) = f(0, T ) +
∫ t
0σ(u, T )
∫ T
uσ(u, s)dsdu+
∫ t
0σ(s, T )dW (s)
= f(0, T ) +N∑i=1
∫ t
0σi(u, T )
∫ T
uσi(u, s)dsdu+
N∑i=1
∫ t
0σi(s, T )dWi(s)
6.1 The HJM Forward-Rate Dynamics 46
and are fully specified once the vector volatility function σ is provided. Given this
dynamics of the instantaneous forward rate f(t, T ), application of Ito’s lemma gives
the following dynamics of the zero-coupon bond price P (t, T ):
dP (t, T ) = P (t, T )
[r(t)dt−
(∫ T
tσ(t, s)ds
)dW (t)
],
where r(t) is the instantaneous short term interest rate at time t, that is
r(t) = f(t, t) = f(0, t) +
∫ t
0σ(u, t)
∫ t
uσ(u, s)dsdu+
∫ t
0σ(s, t)dW (s)
= f(0, t) +N∑i=1
∫ t
0σi(u, t)
∫ t
uσi(u, s)dsdu+
N∑i=1
∫ t
0σi(s, t)dWi(s). (6.3)
Chapter 7
The Log-Normal
Forward-LIBOR model
7.1 Introduction
In this chapter we consider one of the most popular and promising families of
interest-rate models: The market models.
Indeed, the log-normal forward-LIBOR model (LFM) prices caps with Black’s
cap formula, which is the standard formula employed in the cap market.
Before market models were introduced, there was no interest-rate dynamics com-
patible with either Black’s formula for caps or Black’s formula for swaptions. These
formulas were actually based on mimicking the Black and Scholes model for stock
options under some simplifying and inexact assumptions on the interest-rates dis-
tributions. The introduction of market models provided a new derivation of Black’s
formulas based on rigorous interest-rate dynamics.
7.2 Market Models: a Guided Tour
Before market models were introduced, short-rate models used to be the main choice
for pricing and hedging interest-rate derivatives. Short-rate models are still chosen
for many applications and are based on modeling the instantaneous spot interest rate
(“short rate”) via a (possibly multidimensional) diffusion process. This diffusion
process characterizes the evolution of the complete yield curve in time.
To fix ideas, let us consider the time-0 price of a T2-maturity caplet resetting at
time T1(0 < T1 < T2) with strike X and a notional amount of 1. Let τ denote the
year fraction between T1 and T2. Such a contract pays out the amount
τ(L(T1, T2)−X)+
at time T2, where in general L(u, s) is the LIBOR rate at time u for maturity s.
7.2 Market Models: a Guided Tour 48
Again to fix ideas, let us choose a specific short-rate model and assume we are us-
ing the shifted two-factor Vasicek model G2++ given in (5.3). The parameters of this
two-factor Gaussian additive short-rate model are here denoted by ? = (a, σ, b, η, ρ).
Then the short rate rt is obtained as the sum of two linear diffusion processes xt
and yt, plus a deterministic shift ϕ that is used for fitting the initially observed yield
curve at time 0:
rt = xt + yt + ϕ(t; θ).
Such model allows for an analytical formula for forward LIBOR rates F ,
F (t;T1, T2) = F (t;T1, T2;xt, yt, θ),
L(T1, T2) = F (T1;T2, T2;xT1 , yT1 , θ)
At this point one can try and price a caplet. To this end, one can compute the
risk-neutral expectation of the payoff discounted with respect account numeraire
exp(∫ T2
0 rsds) so that one has to
E
[exp
(−∫ T2
0rsds
)τ(F (T1;T1, T2, xT1 , yT1 , θ)−X)+
].
This too turns out to be feasible, and leads to a function
UC(0, T1, T2, X, θ).
On the other hand, the market has been pricing caplets (actually caps) with Black’s
formula for years. One possible derivation of Black’s formula for caplets is based on
the following approximation. When pricing the discounted payoff
E
[exp
(−∫ T2
0rsds
)τ(L(T1, T2)−X)+
]= . . .
one first assumes the discount factor exp(−∫ T2
0 rsds) to be deterministic and identi-
fies it with the corresponding bond price P (t, T2). Then one factors out the discount
factor to obtain:
· · · ≈ P (0, T2)τE[(L(T1, T2)−X)+] = P (0, T2)τE[(F (T1;T1, T2)−X)+].
Now, inconsistently with the previous approximation, one goes back to assuming
rates to be stochastic, and models the forward LIBOR rate F (t;T1, T2) as in the
classical Black and Scholes option pricing setup, i.e as a (driftless) geometric Brow-
nian motion:
dF (t;T1, T2) = vF (t;T1, T2)dWt, (7.1)
where v is the instantaneous volatility, assumed here to be constant for simplicity,
and W is a standard Brownian motion under the risk-neutral measure Q.
7.2 Market Models: a Guided Tour 49
Then the expectation
E[(F (T1;T1, T2)−X)+
]can be viewed simply as a T1-maturity call-option price with strike X and whose
underlying asset has volatility v, in a market with zero risk-free rate.
We therefore obtain:
Cpl(0, T1, T2, X) := P (0, T2)τE(F (T1;T1, T2)−X)+
= P (0, T2)τ [F (0;T1, T2)Φ(d1(X,F (0;T1, T2), v√T1))
−XΦ(d2(X,F (0;T1, T2), v√T1))],
d1(X,F, u) =ln(F/X) + u2/2
u, d2(X,F, u) =
ln(F/X)− u2/2
u,
where Φ is the standard Gaussian cumulative distribution function.
From the way we just introduced it, this formula seems to be partly based on
inconsistencies. However, within the change-of-numeraire setup, the formula can
be given full mathematical rigor as follows. Denote by Q2 the measure associ-
ated with the T2-bond-price numeraire P (·, T2) (T2-forward measure) and by E2
the corresponding expectation. Then, by the change-of numeraire approach, we can
switch from the bank-account numeraire B(t) = B0 exp(∫ t
0 rsds) associated with the
risk-neutral measure Q = QB to the bond-price numeraire P (t, T2) and obtain (set
F2 = F (·, T1, T2) for brevity, so that L(T1, T2) = F (T1;T1, T2) = F2(T1))
E
[exp
(−∫ T2
0rsds
)τ(L(T1, T2)−X)+
]=
= E
[exp
(−∫ T2
0rsds
)τ(F2(T1)−X)+
]=
= E B
B(0)
B(T2)τ(F2(T1)−X)+
= E 2
P (0, T2)
P (T2, T2)τ(F2(T1)−X)+
= . . .
where we have used Fact Two on the change of numeraire technique (Section 3.3),
obtaining an invariant quantity by changing numeraire in the three boxes. Take out
P (0, T2) and recall that P (T2, T2) = 1. We have
E
[exp
(−∫ T2
0rsds
)τ(L(T1, T2)−X)+
]= P (0, T2)τE2[(L(T1, T2)−X)+].
What has just been done, rather than assuming deterministic discount factors, is
a change of measure. We have “factored out” the stochastic discount factor and
replaced it with the related bond price, but in order to do so we had to change the
probability measure under which the expectation is taken. Now the last expectation
7.3 Derivation of Black’s formula - adapted from Brigo and Mercurio [2007] 50
is no longer taken under the risk-neutral measure but rather under the T2-forward
measure. Since by definition F (t;T1, T2) can be written as the price of a tradable
asset divided by P (t, T2), it needs follow a martingale under the measure associated
with the numeraire P (t, T2), i.e. under Q2 (Fact One on the change of numeraire,
again Section 3.3). Martingale means “driftless” when dealing with diffusion pro-
cesses. Therefore, the dynamics of F (t;T1, T2) under Q2 is driftless, so that the
dynamics
dF (t;T1, T2) = vF (t;T1, T2)dWt (7.2)
is correct under the measure Q2, where W is a standard Brownian motion under
Q2. Notice that the driftless (log-normal) dynamics above is precisely the dynamics
we need in order to recover exactly Black’s formula, without approximation. We
can say that the choice of the numeraire P (·, T2) is based on this fact: It makes the
dynamics (7.2) of F driftless under the related Q2 measure, thus replacing rigorously
the earlier arbitrary assumption on the F dynamics (7.1) under the risk-neutral
measure Q. Following this rigorous approach we indeed obtain Black’s formula,
since the process F has the same distribution as in the approximated case above,
and hence the expected value has the same value as before. Nonetheless, it can be
instructive to derive Black’s formula in detail at least once in the work. We do so
now.
7.3 Derivation of Black’s formula - adapted from Brigo
and Mercurio [2007]
We change slightly the notation to derive a formula for time-varying volatilities. As
seen above, by Fact One on the change of numeraire F2 is a martingale (no drift)
under Q2. Take a geometric Brownian motion
dF (t;T1, T2) = σ2(t)F (t;T1, T2)dW (t),
where σ2 is the instantaneous volatility, and W is a standard Brownian motion under
the measure Q2.
Let us solve this equation and compute the caplet price term EQ2[(F2(T1)−X)+].
7.3 Derivation of Black’s formula - adapted from Brigo and Mercurio [2007] 51
By Ito’s formula:
d ln(F2(t)) = ln′(F2)dF2 +1
2ln′′(F2)dF2dF2
=1
F2dF2 +
1
2
(− 1
(F2)2
)dF2dF2
=1
F2σ2F2dW −
1
2(F2)2(σ2F2dW )(σ2F2dW )
= σ2dW −1
2(F2)2σ2
2F22 dWdW
= σ2(t)dW (t)− 1
2σ2
2(t)dt
(where ′ and ′′ denote here the first and second derivative and where we used
dWdW = dt). So we have
d ln(F2(t)) = σ2(t)dW (t)− 1
2σ2
2(t)dt.
Integrate both sides:∫ T
0d ln(F2(t)) =
∫ T
0σ2(t)dW (t)− 1
2
∫ T
0σ2
2(t)dt
ln(F2(t))− ln(F2(0)) =
∫ T
0σ2(t)dW (t)− 1
2
∫ T
0σ2
2(t)dt
lnF2(t)
F2(0)=
∫ T
0σ2(t)dW (t)− 1
2
∫ T
0σ2
2(t)dt
F2(t)
F2(0)= exp
(∫ T
0σ2(t)dW (t)− 1
2
∫ T
0σ2
2(t)dt
)F2(T ) = F2(0) exp
(∫ T
0σ2(t)dW (t)− 1
2
∫ T
0σ2
2(t)dt
).
Compute the distribution of the random variable in the exponent.
It is Gaussian, since it is a stochastic integral of a deterministic function times
a Brownian motion (roughly, sum of independent Gaussians is Gaussian).
Compute the expectation:
E
[∫ T
0σ2(t)dW (t)− 1
2
∫ T
0σ2
2(t)dt
]= 0− 1
2
∫ T
0σ2
2(t)dt
and the variance
Var
[∫ T
0σ2(t)dW (t)− 1
2
∫ T
0σ2
2(t)dt
]= Var
[∫ T
0σ2(t)dW (t)
]= E
[(
∫ T
0σ2(t)dW (t))2
]− 02 =
∫ T
0(σ2(t))2dt
7.3 Derivation of Black’s formula - adapted from Brigo and Mercurio [2007] 52
where we have used Ito’s isometry in the last step. We thus have
I(T ) :=
∫ T
0σ2(t)dW (t)− 1
2
∫ T
0σ2
2(t)dt ∼
∼ m+ VN (0, 1), m = −1
2
∫ T
0σ2(t)2dt,
V 2 =
∫ T
0σ2(t)2dt.
Recall that we have
F2(T ) = F2(0) exp(I(T )) = F2(0)em+VN (′,∞).
Compute now the option price term
EQ2[(F2(T1)−X)+] = EQ
2[(F2(0)em+VN (0,1) −X)+]
=
∫ +∞
−∞(F2(0)em+V y −X)+pN (0,1)(y)dy = . . .
Note that F2(0) exp(m+ V y)−X > 0 if and only if
y >− ln(F2(0)
X )−mV
=: y
so that
· · · =∫ +∞
y(F2(0) exp(m+ V y)−X)pN (0,1)(y)dy
= F2(0)
∫ +∞
yem+V ypN (0,1)(y)dy −X
∫ +∞
ypN (0,1)(y)dy
= F2(0)1√2π
∫ +∞
ye−
12y2+V y+mdy −X(1− Φ(y))
= F2(0)1√2π
∫ +∞
ye−
12
(y−V )2+m− 12V 2dy −X(1− Φ(y))
= F2(0)em−12V 2 1√
2π
∫ +∞
ye−
12
(y−V )2dy −X(1− Φ(y))
= F2(0)em−12V 2 1√
2π
∫ +∞
y−Ve−
12z2dz −X(1− Φ(y))
= F2(0)em−12V 2
(1− Φ(y − V ))−X(1− Φ(y))
= F2(0)em−12V 2
Φ(−y + V )−XΦ(−y)
= F2(0)Φ(d1)−XΦ(d2), d1,2 =ln F2(0)
X ± 12
∫ T10 σ2
2(t)dt√∫ T10 σ2
2(t)dt.
7.3 Derivation of Black’s formula - adapted from Brigo and Mercurio [2007] 53
When including the initial discount factor P (0, T2) and the year fraction τ this is
exactly the classic market Black’s formula for the T1 − T2 caplet.
The example just introduced is a simple case of what is known as “log-normal
forward-LIBOR model”. It is known also as Brace-Gatarek-Musiela (1997) model,
from the name of the authors of one of the first papers where it was introduced
rigorously.
Let us now go back to our short-rate model formula UC and ask ourselves whether
this formula can be compatible with the above reported Black’s market formula.
It is well known that the two formulas are not compatible. Indeed, by the two-
dimensional version of Itos formula we may derive the Q2-dynamics of the forward
LIBOR rate between T1 and T2 under the short-rate model,
dF (t;T1, T2;xt, yt, θ) =∂F
∂(t, x, y)d[t, xt, yt]
′ +1
2d[xt, yt]
∂2F
∂2(x, y)d[xt, yt]
′, Q2, (7.3)
where the Jacobian vector and the Hessian matrix have been denoted by their partial
derivative notation. This dynamics clearly depends on the linear-Gaussian dynamics
of x and y under the T2-forward measure. The thus obtained dynamics is easily seen
to be incompatible with the log-normal dynamics leading to Black’s formula. More
specifically, for no choice of the parameters θ does the distribution of the forward
rate F in (7.3) produced by the short-rate model coincide with the distribution of
the “Black”-like forward rate F following (7.2). In general, no known short-rate
model can lead to Black’s formula for caplets (and more generally for caps).
What is then done with short-rate models is the following. After setting the
deterministic shift ϕ so as to obtain a perfect fit of the initial term structure, one
looks for the parameters θ that produce caplet (actually cap) prices UC that are
closest to a number of observed market cap prices. The model is thus calibrated to
(part of) the cap market and should reproduce well the observed prices. Still, the
prices UC are complicated nonlinear functions of the parameters θ, and this renders
the parameters themselves difficult to interpret. On the contrary, the parameter
v in the above “market model” for F has a clear interpretation as a log-normal
percentage (instantaneous) volatility, and traders feel confident in handling such
kind of parameters.
When dealing with several caplets involving different forward rates, different
structures of instantaneous volatilities can be employed. One can select a different v
for each forward rate by assuming each forward rate to have a constant instantaneous
volatility. Alternatively, one can select piecewise-constant instantaneous volatilities
for each forward rate. Moreover, different forward rates can be modeled as each
having different random sources W that are instantaneously correlated.
7.4 The Lognormal Forward-LIBOR Model (LFM) 54
7.4 The Lognormal Forward-LIBOR Model (LFM)
Let t = 0 be the current time. Consider a set E = T0, . . . , TM from which expiry-
maturity pairs of dates (Ti−1, Ti) for a family of spanning forward rates are taken.
We shall denote by τ0, ..., τM the corresponding year fractions, meaning that τi is
the year fraction associated with the expiry-maturity pair (Ti−1, Ti) for i > 0, and
τ0 is the year fraction from settlement to T0. Times Ti will be usually expressed in
years from the current time. We set T−1 := 0.
Consider the generic forward rate Fk(t) = F (t;Tk−1, Tk), k = 1, . . . ,M , which is
“alive” up to time Tk−1, where it coincides with the simply compounded spot rate
Fk(Tk−1) = L(Tk−1, Tk). In general L(S, T ) is the simply compounded spot rate
prevailing at time S for the maturity T .
Consider now the probability measure Qk associated with the numeraire P (·, Tk),i.e. to the price of the bond whose maturity coincides with the maturity of the
forward rate. Qk is often called the forward (adjusted) measure for the maturity Tk.
Under simple compounding, it follows immediately by definition that
Fk(t)P (t, Tk) = [P (t, Tk−1)− P (t, Tk)]/τk.
Therefore, Fk(t)P (t, Tk) is the price of a tradable asset (difference between two dis-
count bonds with notional amounts 1/τk). As such, when its price is expressed with
respect to the numeraire P (·, Tk), it has to be a martingale under the measure Qk as-
sociated with that numeraire (by definition of measure associated with a numeraire).
But the price Fk(t)P (t, Tk) of our tradable asset divided by this numeraire is simply
Fk(t) itself. Therefore, Fk follows a martingale under Qk. It follows that if Fk is
modeled according to a diffusion process, it needs to be driftless under Qk.
We assume the following driftless dynamics for Fk under Qk:
dFk(t) = σk(t)Fk(t)dZk(t), t ≤ Tk−1, (7.4)
where Zk(t) is an M -dimensional column-vector Brownian motion (under Qk) with
instantaneous covariance ρ = (ρi,j)i,j=1,...,M ,
dZk(t)dZk(t)′ = ρdt,
and where σk(t) is the horizontal M -vector volatility coefficient for the forward rate
Fk(t). Unless differently stated, from now on we will assume that
σj(t) = [00 . . . σj(t) . . . 00],
with the only non-zero entry σj(t) occurring at the j-th position in the vector σj(t).
We hope the reader is not disoriented by our notation. The point of switching from
7.4 The Lognormal Forward-LIBOR Model (LFM) 55
vector to scalar volatility is that in some computations the former will be more
convenient, whereas in other cases the latter will be useful. Just notice that the
above equation for Fk can be rewritten under Qk as
dFk(t) = σk(t)Fk(t)dZkk (t), t ≤ Tk−1,
where Zkk (t) is the k-th component of the vector Brownian motion Zk and is thus a
standard Brownian motion. Lower indices indicate the component we are considering
in a vector, whereas upper indices show under which measure we are working. Upper
indices are usually omitted when the context is clear, so that we write
dFk(t) = σk(t)Fk(t)dZk(t), t ≤ Tk−1. (7.5)
With this scalar notation, σk(t) now bears the usual interpretation of instantaneous
volatility at time t for the forward LIBOR rate Fk.
Notice that in case σ is bounded (as in all of our volatility parameterizations
below), this last equation has a unique strong solution, since it describes a geometric
Brownian motion. This can be checked immediately also by writing, through Ito’s
formula,
d lnFk(t) = −σk(t)2
2dt+ σk(t)dZk(t), t ≤ Tk−1,
and by observing that the coefficients of this last equation are bounded, so that there
exists trivially a unique strong solution. Indeed, we have immediately the unique
solution
lnFk(T ) = lnFk(0)−∫ T
0
σk(t)2
2dt+
∫ T
0σk(t)dZk(t).
We will often consider piecewise-constant instantaneous volatilities:
σk(t) = σk,β(t), t > T−1 = 0,
where in general β(t) = m if Tm−2 < t ≤ Tm−1,m ≥ 1, so that
t ∈ (Tβ(t)−2, Tβ−1].
In the particular case t = T−1 = 0 we set σk(0) = σk,1.
Notice that β(t) is the index of the first forward rate that has not expired by
t : Fβ(t). All preceding forward rates Fβ(t)−1, Fβ(t)−2·· have expired by t.
At times we will use the notation Zt = Z(t).
As is clear from our assumption on the vector “noise” dZ above, the single
“noises” in the dynamics of different forward rates are assumed to be instantaneously
correlated according to
dZi(t)dZj(t) = d〈Zi, Zj〉tρi, jdt.
Chapter 8
Discussion: Market
Completeness and Risk-Neutral
Pricing: What the relationship
means for Derivative Traders
As mentioned in the introduction, this chapter represents the heart of the argument
matter. Here we revisit some of the risk-neutral pricing results introduced in chapter
two, with more mathematical rigor.
More importantly, the key demonstration of this chapter is that in an incomplete
market (such as the fixed interest market) the requirement of no arbitrage is no longer
sufficient to determine a unique price for the derivative. We eventually show that in
an incomplete market, we will have several possible martingale measures, and several
market prices of risk. The reason that there are several possible martingale measures
simply means that there are several different price systems for the derivatives, all of
which are consistent with absence of arbitrage.
Furthermore, in an incomplete market the price is also partly determined, in a
nontrivial way, by aggregate supply and demand on the market. Supply and demand
for a specific derivative are in turn determined by the aggregate risk aversion on the
market, as well as by liquidity considerations and other factors.
8.1 Completeness and Hedging
8.1.1 Introduction
We start with a fairly general situation by considering a financial market with a price
vector process S = S1, . . . , SN , governed by an objective probability measure P .
The process S is as usual interpreted as the price process of the exogenously given
8.1 Completeness and Hedging 57
underlying assets and we now want to price a contingent T -claim X . We assume
that all the underlying assets are traded on the market, but we do not assume that
there exists an a priori market (or a price process) for the derivative. To avoid
trivialities we also assume that the underlying market is arbitrage free.
Definition 8.1. We say that a T -claim X can be replicated, alternatively that it
is reachable or hedgeable, if there exists a self-financing portfolio h such that
V h(T ) = X , P − a.s. (8.1)
In this case we say that h is a hedge against X . Alternatively, h is called a repli-
cating or hedging portfolio. If every contingent claim is reachable we say that the
market is complete.
Let us now consider a fixed T -claim X and let us assume that X can be replicated
by a portfolio h. Then we can make the following mental experiment:
1. Fix a point in time t with t ≤ T .
2. Suppose that we, at time t, possess V h(t) ZAR
3. We can then use this money to buy the portfolio h(t). If furthermore we follow
the portfolio strategy h on the time interval [t, T ] this will cost us nothing, since
h is self-financing. At time T the value of our portfolio will then be V h(T )
ZAR.
4. By the replication assumption the value, at time T , of our portfolio will thus be
exactly X ZAR, regardless of the stochastic price movements over the interval
[t, T ].
5. From a purely financial point of view, holding the portfolio h is thus equivalent
to the holding of the contract X .
6. The “correct” price of X at time t is thus given by Π(t;X ) = V h(t).
For a hedgeable claim we thus have a natural price process, Π(t;X ) = V h(t), and
we may now ask if this has anything to do with absence of arbitrage.
Proposition 8.2. Suppose that the claim X can be hedged using the portfolio h.
Then the only price process Π(t;X ) which is consistent with no arbitrage is given
by Π(t;X ) = V h(t). Furthermore, if X can be hedged by g as well as by h then
V g(t) = V h(t) holds for all t with probability 1.
Proof. If at some time t we have Π(t;X ) < V h(t) then we can make an arbitrage by
selling the portfolio short and buying the claim, and vice versa if Π(t;X ) > V h(t).
A similar argument shows that we must have V g(t) = V h(t).
8.1 Completeness and Hedging 58
8.1.2 Completeness in the BlackScholes Model
We will now investigate completeness for the generalized BlackScholes model given
by
dB(t) = rB(t)dt, (8.2)
dS(t) = S(t)α(t, S(t))dt+ S(t)σ(t, S(t))dW (t), (8.3)
where we assume that σ(t, s) > 0 for all (t, s). The main result is the following.
Theorem 8.3. The model (8.2)-(8.3) is complete. The proof of this theorem requires
some fairly deep results from probability theory and is thus outside the scope of this
work. We will prove a weaker version of the theorem, namely that every simple claim
can be hedged. This is often quite sufficient for practical purposes, and our proof
of the restricted completeness also has the advantage that it gives the replicating
portfolio in explicit form. We will use the notational convention h(t) = [h0(t), h∗(t)]
where h0 is the number of bonds in the portfolio, whereas h∗ denotes the number
of shares in the underlying stock. We thus fix a simple T -claim of the form X =
Φ(S(T )) and we now want to show that this claim can be hedged.
Lemma 8.4. Suppose that there exists an adapted process V and an adapted process
u = [u0, u∗] with
u0(t) + u∗(t) = 1, (8.4)
such thatdV (t) = V (t)u0(t)r + u∗(t)α(t, S(t))dt+ V (t)u∗(t)σ(t, S(t))dW (t),
V (T ) = Φ(S(T )).(8.5)
Then the claim X = Φ(S(T )) can be replicated using u as the relative portfolio. The
corresponding value process is given by the process V and the absolute portfolio h is
given by
h0(t) =u0(t)V (t)
B(t), (8.6)
h∗(t) =u∗(t)V (t)
S(t). (8.7)
Our strategy now is to look for a process V and a process u satisfying the conditions
above.
Begin Heuristics Adapted from Bjork [2009]
We assume what we want to prove, namely that X = Φ(S(T )) is indeed replicable,
and then we ponder on what the hedging strategy u might look like. Since the
8.1 Completeness and Hedging 59
S-process and (trivially) the B-process are Markov processes it seems reasonable
to assume that the hedging portfolio is of the form h(t) = h(t, S(t)) where, with a
slight misuse of notation, the h in the right member of the equality is a deterministic
function. Since, furthermore, the value process V (we suppress the superscript h) is
defined as V (t) = h0(t)B(t) + h∗(t)S(t) it will also be a function of time and stock
price as
V (t) = F (t, S(t)), (8.8)
where F is some real valued deterministic function which we would like to know
more about.
Assume therefore that (8.8) actually holds. Then we may apply the Ito formula
to V in order to obtain the V -dynamics as
dV =
Ft + αSFs +
1
2σ2S2Fss
dt+ σSFsdW , (8.9)
where we have suppressed the fact that V and S are to be evaluated at time t,
whereas α, σ and F are to be evaluated at (t, S(t)). Now, in order to make (8.9)
look more like (8.5) we rewrite (8.9) as
dV = V
Ft + αSFs + 1
2σ2S2Fss
V
dt+ V
SFsV
σdW . (8.10)
Since we have assumed that X is replicated by V we see from (8.10) and (8.5) that
u∗ must be given by
u∗(t) =S(t)Fs(t, S(t))
F (t, S(t)), (8.11)
(remember that we have assumed that V (t) = F (t, S(t)), and if we substitute (8.11)
into (8.10) we get
dV = V
Ft + 1
2σ2S2Fss
rFr + u∗α
dt+ V u∗σdW . (8.12)
Comparing this expression to (8.5) we see that the natural choice for u0 is given by
u0 =Ft + 1
2σ2S2Fss
rF, (8.13)
but we also have to satisfy the requirement u0 + u∗ = 1 of (8.4). Using (8.11)
and (8.13) this gives us the relation
Ft + 12σ
2S2Fss
rF=F − SFs
F, (8.14)
which, after some manipulation, turns out to be the familiar BlackScholes equation
Ft + rSFs +1
2σ2S2Fss − rF = 0. (8.15)
8.1 Completeness and Hedging 60
Furthermore, in order to satisfy the relation F (T, S(T )) = Φ(S(T )) of (8.5) (remem-
ber that we assume that V (t) = F (t, S(t))) we must have the boundary value
F (T, s) = Φ(s), for all s ∈ R+. (8.16)
End Heuristics
Since at this point the reader may well be somewhat confused as to the logic of the
reasoning, let us try to straighten things out. The logic of the reasoning above is
basically as follows.
• We assumed that the claim X was replicable.
• Using this and some further (reasonable) assumptions we showed that they
implied that the value process of the replicating portfolio was given as V (t) =
F (t, S(t)) where F is a solution of the BlackScholes equation. This is of course not
at all what we wish to achieve. What we want to do is to prove that X really can
be replicated. In order to do this we put the entire argument above within a logical
parenthesis and formally disregard it. We then have the following result.
Theorem 8.5. Consider the market (8.2)-(8.3), and a contingent claim of the form
X = F (S(T )). Define F as the solution to the boundary value problemFt + rsFs +1
2σ2s2Fss − rF = 0
F (T, s) = Φ(s)(8.17)
Then X can be replicated by the relative portfolio
u0(t) =F (t, S(t))− S(t)Fs(t, S(t))
F (t, S(t)), (8.18)
u∗(t) =S(t)Fs(t, S(t))
F (t, S(t)). (8.19)
The corresponding absolute portfolio is given by
h0(t) =F (t, S(t))− S(t)Fs(t, S(t))
B(t), (8.20)
h∗(t) = Fs(t, S(t)), (8.21)
and the process V h is given by
V h(t) = F (t, S(t)). (8.22)
Proof. Applying the Ito formula to the process V (t) defined by (8.22) and performing
exactly the same calculations as in the heuristic argument above, will show that we
can apply Lemma 8.4.
8.1 Completeness and Hedging 61
The result above gives us an explanation of the surprising facts that there actually
exists a unique price for a derivative asset in the BlackScholes model and that this
price does not depend on any particular assumptions about individual preferences.
The arbitrage free price of a derivative asset is uniquely determined simply because in
this model the derivative is superfluous. It can always be replaced by a corresponding
“synthetic” derivative in terms of a replicating portfolio.
Since the replication is done with P -probability 1, we also see that if a contingent
claim X is replicated under P by a portfolio h and if P ∗ is some other probability
measure such that P and P ∗ assign probability 1 to exactly the same events (such
measures P and P ∗ are said to be equivalent), then h will replicate X also under the
measure P ∗. Thus the pricing formula for a certain claim will be exactly the same
for all measures which are equivalent to P . It is a well known fact (the Girsanov
theorem) in the theory of SDEs that if we change the measure from P to some other
equivalent measure, this will change the drift in the SDE, but the diffusion term
will be unaffected. Thus the drift will play no part in the pricing equation, which
explains why α does not appear in the BlackScholes equation.
Proposition 8.6. Consider the model
dB(t) = rB(t)dt, (8.23)
dS(t) = S(t)α(t, S(t))dt+ S(t)σ(t, S(t))dW (t), (8.24)
and let X be a T -claim of the form
X = Φ(S(T ), Z(T )), (8.25)
where the process Z is defined by
Z(t) =
∫ t
0g(u, S(u))du, (8.26)
for some choice of the deterministic function g. Then X can be replicated using a
relative portfolio given by
u0(t) =F (t, S(t), Z(t))− S(t)Fs(t, S(t), Z(t))
F (t, S(t), Z(t)), (8.27)
u∗(t) =S(t)Fs(t, S(t), Z(t))
F (t, S(t), Z(t)), (8.28)
where F is the solution to the boundary value problemFt + srFs +1
2s2σ2Fss + gFz − rF = 0,
F (T, s, z = Φ(s, z)).(8.29)
8.1 Completeness and Hedging 62
The corresponding value process V is given by V (t) = F (t, S(t), Z(t)), and F has
the stochastic representation
F (t, s, z) = e−r(T−t)EQt,s,z[Φ(S(T ), Z(T ))], (8.30)
where the Q-dynamics are given by
dS(u) = rS(u)du+ S(u)σ(u, S(u))dW (u), (8.31)
S(t) = s, (8.32)
dZ(u) = g(u, S(u))du, (8.33)
Z(t) = z, (8.34)
Again we see that the arbitrage free price of a contingent claim is given as the
expected value of the claim discounted to the present time. Here, as before, the
expected value is to be calculated using the martingale measure Q instead of the
objective probability measure P . As we have said before, this general structure of
arbitrage free pricing holds in much more general situations, and as a rule of thumb
one can view the martingale measure Q as being defined by the property that all
traded underlying assets have r as the rate of return under Q. It is important to
stress that it is only traded assets which will have sr as the rate of return under
Q. For models with nontraded underlying objects we have a completely different
situation, which we will encounter below.
8.1.3 Completeness-Absence of Arbitrage Adapted from Bjork
[2009]
In this section we will give some general rules of thumb for quickly determining
whether a certain model is complete and/or free of arbitrage. The arguments will
be purely heuristic.
Let us consider a model with M traded underlying assets plus the risk free asset
(i.e. totally M + 1 assets). We assume that the price processes of the underlying
assets are driven by R “random sources”. We cannot give a precise definition of what
constitutes a “random source” here, but the typical example is a driving Wiener
process. If, for example, we have five independent Wiener processes driving our
prices, then R = 5. Another example of a random source would be a counting
process such as a Poisson process. In this context it is important to note that if the
prices are driven by a point process with different jump sizes then the appropriate
number of random sources equals the number of different jump sizes.
When discussing completeness and absence of arbitrage it is important to realize
that these concepts work in opposite directions. Let the number of random sources
8.1 Completeness and Hedging 63
R be fixed. Then every new underlying asset added to the model (without increasing
R) will of course give us a potential opportunity of creating an arbitrage portfolio,
so in order to have an arbitrage free market the number M of underlying assets must
be small in comparison to the number of random sources R.
On the other hand we see that every new underlying asset added to the model
gives us new possibilities of replicating a given contingent claim, so completeness
requires M to be great in comparison to R.
We cannot formulate and prove a precise result here, but the following rule of
thumb, or “meta-theorem”, is nevertheless extremely useful. In concrete cases it
can in fact be given a precise formulation and a precise proof. See Chapters 10 and
14. We will later use the meta-theorem when dealing with problems connected with
nontraded underlying assets in general and interest rate theory in particular.
Theorem 8.7. Let M denote the number of underlying traded assets in the model
excluding the risk free asset, and let R denote the number of random sources. Gener-
ically we then have the following relations:
1. The model is arbitrage free if and only if M = R.
2. The model is complete if and only if M = R.
3. The model is complete and arbitrage free if and only if M = R.
As an example we take the BlackScholes model, where we have one underlying asset
S plus the risk free asset so M = 1. We have one driving Wiener process, giving
us R = 1, so in fact M = R. Using the meta-theorem above we thus expect the
BlackScholes model to be arbitrage free as well as complete and this is indeed the
case.
8.2 Incomplete Markets 64
8.2 Incomplete Markets
8.2.1 Introduction
We know from the meta-theorem that markets generically are incomplete when
there are more random sources than there are traded assets, and this can occur in
an infinite number of ways, so there is no “canonical” way of writing down a model
of an incomplete market. We will confine ourselves to study a particular type of
incomplete market, namely a “factor model”, i.e. a market where there are some
nontraded underlying objects. Before we go on to the formal description of the
models let us briefly recall what we may expect in an incomplete model.
• Since, by assumption, the market is incomplete we will not be able to hedge a
generic contingent claim.
• In particular there will not be a unique price for a generic derivative.
8.2.2 A Scalar Nonpriced Underlying Asset
We will start by studying the simplest possible incomplete market, namely a market
where the only randomness comes from a scalar stochastic process which is not the
price of a traded asset. We will then discuss the problems which arise when we want
to price derivatives which are written in terms of the underlying object. The model
is as follows.
Assumption 8.8. The only objects which are a priori given are the following.
• An empirically observable stochastic process X, which is not assumed to be the
price process of a traded asset, with P -dynamics given by
dX(t) = µ(t,X(t))dt+ σ(t,X(t))dW (t). (8.35)
Here W is a standard scalar P -Wiener process.
• A risk free asset (money account) with the dynamics
dB(t) = rB(t)dt, (8.36)
where r as usual is the deterministic short rate of interest.
We now consider a given contingent claim, written in terms of the process X.
More specifically we define the T -claim Y by
Y = Φ(X(T )), (8.37)
8.2 Incomplete Markets 65
where Φ is some given deterministic function, and our main problem is that of
studying the price process Π(t;Y) for this claim.
In order to give some substance to the discussion, and to understand the differ-
ence between the present setting and that of the previous chapters, let us consider
a specific concrete, interpretation of the model adapted from Bjork [2009] We may,
for example, interpret the process X as the temperature at some specific point on
the earth, say the end of the Palace Pier in Brighton. Thus X(t) is the temperature
(in centigrade) at time t at the Palace Pier. Suppose now that you want to go to
Brighton for a holiday, but that you fear that it will be unpleasantly cold at the
particular time T when you visit Brighton. Then it may be wise to buy “holiday
insurance”, i.e. a contract which pays you a certain amount of money if the weather
is unpleasant at a prespecified time in a prespecified place. If the monetary unit is
the pound sterling, the contract function Φ above may have the form
Φ(x) =
100,ifx ≤ 20,
0,ifx > 20.(8.38)
In other words, if the temperature at time T is below 20C(degrees centigrade) you
will obtain 100 pounds from the insurance company, whereas you will get nothing if
the temperature exceeds 20C.
The problem is now that of finding a “reasonable” price for the contract above,
and as usual we interpret the word “reasonable” in the sense that there should be
no arbitrage possibilities if we are allowed to trade the contract. This last sentence
contains a hidden assumption which we now formalize.
Assumption 8.9. There is a liquid market for every contingent claim
If we compare this model with the standard BlackScholes model, we see many
similarities. In both cases we have the money account B, and in both cases we have
an a priori given underlying process. For the BlackScholes model the underlying
process is the stock price S, whereas we now have the underlying process X, and in
both models the claim to be priced is a deterministic function Φ of the underlying
process, evaluated at time T .
In view of these similarities it is now natural to assume that the results from the
BlackScholes analysis will carry over to the present case, i.e. we are (perhaps) led
to believe that the price process for the claim Y is uniquely determined by the P -
dynamics of the underlying process X. It is, however, very important to understand
that this is, most emphatically, not the case, and the reasons are as follows:
1. If we consider the a priori given market, which only consists of the money
account B, we see that the number R of random sources in this case equals
8.2 Incomplete Markets 66
one (one driving Wiener process), while the number M of traded assets (always
excluding the money account B) equals zero. From the meta-theorem it now
follows that the market is incomplete. The incompleteness can also be seen
from the obvious fact that in the a priori given market there are no interesting
ways of forming self-financing portfolios. The only strategy that is allowed is to
invest all our money in the bank, and then we can only sit down and passively
watch our money grow at the rate r. In particular we have no possibility to
replicate any interesting derivative of the form Φ(X(T )). We thus conclude
that, since we cannot replicate our claim, we cannot expect to obtain a unique
arbitrage free price process.
2. One natural strategy to follow in order to obtain a unique price for the claim
X is of course to imitate the scheme for the BlackScholes model. We would
assume that the price process Π(t;Y) is of the form Π(t;Y) = F (t,X(t)), and
then we would form a portfolio based on the derivative F and the underlyingX.
Choosing the portfolio weights such that the portfolio has no driving Wiener
process would give us a riskless asset, the rate of return on which would have
to equal the short rate r, and this last equality would finally have the form
of a PDE for the pricing function F . This approach is, however, completely
nonsensical, since in the present setting the process X is (by assumption) not
the price of a traded asset, and thus it is meaningless to talk about a “portfolio
based on X”. In our concrete interpretation this is eminently clear. Obviously
you can buy any number of insurance contracts and put them in your portfolio,
but it is also obvious that you cannot meaningfully add, for example, 15Cto
that portfolio.
We can summarize the situation as follows:
• The price of a particular derivative will not be completely determined by the
specification (8.35) of the X-dynamics and the requirement that the market
[B(t),Π(t;Y)] is free of arbitrage.
• The reason for this fact is that arbitrage pricing is always a case of pricing a
derivative in terms of the price of some underlying assets. In our market we
do not have sufficiently many underlying assets.
Thus we will not obtain a unique price of a particular derivative. This fact does
not mean, however, that prices of various derivatives can take any form whatsoever.
From the discussion above we see that the reason for the incompleteness is that we
do not have enough underlying assets, so if we adjoin one more asset to the market,
8.2 Incomplete Markets 67
without introducing any new Wiener processes, then we expect the market to be
complete. This idea can be expressed in the following ways.
Idea
• We cannot say anything about the price of any particular derivative.
• The requirement of an arbitrage free derivative market implies that prices of
different derivatives (i.e. claims with different contract functions or different
times of expiration) will have to satisfy certain internal consistency relations
in order to avoid arbitrage possibilities on the bond market. In terms of our
concrete interpretation this means that even if we are unable to produce a
unique price for a fixed weather insurance contract, say the “20Ccontract”
Φ(x) =
100, if x ≤ 20,
0, if x > 20.(8.39)
for the fixed date T , there must be internal consistency requirements between
the price of this contract and the price of the following “25Ccontract”
Γ(x) =
100, if x ≤ 25,
0, if x > 25.(8.40)
with some expiration date T (where of course we may have T = T ′).
• In particular, if we take the price of one particular “benchmark” derivative as a
priori given, then the prices of all other derivatives will be uniquely determined
by the price of the benchmark. This fact is in complete agreement with the
meta-theorem, since in an a priori given market consisting of one benchmark
derivative plus the risk free asset we will have R = M = 1, thus guaranteeing
completeness. In our concrete interpretation we expect that the prices of all
insurance contracts should be determined by the price of any fixed benchmark
contract. If, for example, we choose the 20Ccontract above as our benchmark,
and take its price as given, then we expect the 25Ccontract to be priced
uniquely in terms of the benchmark price.
To put these ideas into action we now take as given two fixed T -claims, Y and Z,
of the form
Y = Φ(X(T )),
Z = Γ(X(T )),
where Φ and Γ are given deterministic real valued functions. The project is to find
out how the prices of these two derivatives must be related to each other in order to
8.2 Incomplete Markets 68
avoid arbitrage possibilities on the derivative market. As above we assume that the
contracts are traded on a frictionless market, and as in the BlackScholes analysis we
make an assumption about the structure of the price processes.
Assumption 8.10. We assume that
• There is a liquid, frictionless market for each of the contingent claims Y and
Z.
• The market prices of the claims are of the form
Π(t;Y) = F (t,X(t)),
Π(t;Z) = G(t,X(t)),
where F and G are smooth real valued functions.
We now proceed exactly as in the BlackScholes case. We form a portfolio based
on F and G, and choose the weights so as to make the portfolio locally riskless. The
rate of return of this riskless portfolio has to equal the short rate of interest, and
this relation will give us some kind of equation, which we then have to analyze in
detail.
From the assumption above, the Ito formula, and the X-dynamics we obtain the
following price dynamics for the price processes F (t,X(t)) and G(t,X(t)).
dF = αFFdt+ σFFdW , (8.41)
dG = αGGdt+ σGGdW . (8.42)
Here the processes aF and sF are given by
αF =Ft + µFx + 1
2σ2Fxx
F,
αF =σFxF
,
and correspondingly for αG and σG. As usual we have suppressed most arguments,
so in more detail we have, for example,
µFx = µ(t,X(t))∂F
∂x(t,X(t)).
We now form a self-financing portfolio based on F and G, with portfolio weights
denoted by uF and uG respectively. The portfolio dynamics are given by
dV = V
uF ·
dF
F+ uG ·
dG
G
,
8.2 Incomplete Markets 69
and using the expressions above we get
dV = V uF · αF + uG · αGdt+ V uF · σF + uG · σGdW . (8.43)
In order to make this portfolio locally riskless we must choose uF and uG such that
uF · σF + uG · σG = 0, and we must also remember that they must add to unity.
Thus we define uF and uG as the solution to the following system of equations.uF + uG = 1,
uF · σF + uG · σG = 0.(8.44)
The solution to this system is given byuF =
−σGσF − σG
,
uG =σF
σF − σG,
(8.45)
and inserting this into the portfolio dynamics equation (8.43) gives us
dV = V ·αG · σF − αF · σG
σF − σG
dt.
We have thus created a locally riskless asset, and the absence of arbitrage must
imply the equationαG · σF − αF · σG
σF − σG= r.
After some reshuffling we can rewrite this equation as
αF − rσF
=αG − rσG
.
The important fact to notice about this equation is that the left-hand side does not
depend on the choice of G, while the right-hand side does not depend upon the
choice of F . The common quotient will thus depend neither on the choice of F nor
on the choice of G, and we have proved the following central result.
Proposition 8.11. Assume that the market for derivatives is free of arbitrage. Then
there exists a universal process λ(t) such that, with probability 1, and for all t, we
haveαF (t)− rσF (t)
= λ(t), (8.46)
regardless of the specific choice of the derivative F .
There is a natural economic interpretation of this result, and of the process λ. In
eqn (8.46) the numerator is given by αF − r, and from (8.41) we recognize aF as the
local mean rate of return on the derivative F . The numerator αF−r is thus the local
8.2 Incomplete Markets 70
mean excess return on the derivative F over the riskless rate of return r, i.e. the risk
premium of F . In the denominator we find the volatility σF of the F process, so we
see that λ has the dimension “risk premium per unit of volatility”. This is a concept
well known from CAPM theory, so λ is commonly called “the market price of risk”.
Proposition 8.11 can now be formulated in the following, slightly more flashy, form:
• In a no arbitrage market all derivatives will, regardless of the specific choice
of contract function, have the same market price of risk.
We can obtain more explicit information from eqn (8.46) by substituting our earlier
formulas for aF and sF into it. After some algebraic manipulations we then end up
with the following PDE.
ft + µ− λσ+1
2σ2Fxx − rF = 0.
This is really shorthand notation for an equation which must hold with probability
1, for each t, when all terms are evaluated at the point (t,X(t)). Assuming for
the moment that the support of X is the entire real line, we can then draw the
conclusion that the equation must also hold identically when we evaluate it at an
arbitrary deterministic point (t, x). Furthermore it is clear that we must have the
boundary condition
F (T, x) = Φ(x), ∀x ∈ R,
so we finally end up with the following result.
Proposition 8.12. Pricing equation Assuming absence of arbitrage, the pricing
function F (t, x) of the T -claim Φ(X(T )) solves the following boundary value problem.
Ft(t, x) +AF (t, x)− rF (t, x) = 0, (t, x) ∈ (0, T )∏
R, (8.47)
F (T, x) = Φ(x), x ∈ R, (8.48)
where
AF (t, x) = µ(t, x)− λ(t, x)σ(t, x)Fx(t, x) +1
2σ2(t, x)Fxx(t, x).
At first glance this result may seem to contradict the moral presented above.
We have stressed earlier the fact that, because of the incompleteness of the market,
there will be no unique arbitrage free price for a particular derivative. In Proposi-
tion 8.12, on the other hand, we seem to have arrived at a PDE which, when solved,
will give us precisely the unique pricing function for any simple claim. The solution
to this conundrum is that the pricing equation above is indeed very nice, but in
order to solve it we have to know the short rate of interest r, as well as the func-
tions µ(t, x), σ(t, x),Φ(x) and λ(t, x). Of these, only r, µ(t, x), σ(t, x) and Φ(x) are
8.2 Incomplete Markets 71
specified exogenously. The market price of risk λ, on the contrary, is not specified
within the model. We can now make the Idea above more precise.
Firstly we see that, even though we cannot determine a unique price for a par-
ticular derivative, prices of different derivatives must satisfy internal consistency
requirements. This requirement is formulated precisely in Proposition 8.11, which
says that all derivatives must have the same market price of risk. Thus, if we con-
sider two different derivative assets with price processes F and G, these may have
completely different local mean rates of return (αF and αG), and they may also have
completely different volatilities (σF and σG). The consistency relation which has to
be satisfied in order to avoid arbitrage between the derivatives is that, at all times,
the quotientαF − rσF
must equalαG − rσG
.
Secondly, let us assume that we take the price process of one particular derivative
as given. To be concrete let us fix the “benchmark” claim Γ(X(T )) above and assume
that the pricing function G(t, x), for Γ(X(T )), is specified exogenously. Then we
can compute the market price of risk by the formula
λ(t, x) =αG(t, x)− rαG(t, x)
. (8.49)
Let us then consider an arbitrary pricing function, say the function F for the claim
Φ(X(T )). Since the market price of risk is the same for all derivatives we may
now take the expression for λ obtained from (8.49) and insert this into the pricing
equation (8.47)-(8.48) for the function F . Now everything in this equation is well
specified, so we can (in principle) solve it, in order to obtain F . Thus we see that
the price F of an arbitrary claim is indeed uniquely determined by the price G of
any exogenously specified benchmark claim.
We can obtain more information from the pricing equation by applying the Feyn-
manKac representation. The result can be read off immediately and is as follows.
Proposition 8.13. Risk neutral valuation Assuming absence of arbitrage, the pric-
ing function F (t, x) of the T -claim Φ(X(T )) is given by the formula
F (t, x) = e−r(T−t)EQt,x[Φ(X(T ))], (8.50)
where the dynamics of X under the martingale measure Q are given by
dX(t) = µ(t,X(t))− λ(t,X(t))σ(t,X(t))dt+ σ(t,X(t))dW (t).
Here W is a Q-Wiener process, and the subscripts t, x indicate as usual that X(t) =
x.
8.2 Incomplete Markets 72
Again we have an “explicit” risk neutral valuation formula for the pricing func-
tion. The arbitrage free price of the claim is given as the discounted value of the
mathematical expectation of the future claim. As before the expectation is, how-
ever, not to be taken under the objective measure P , but under the martingale (“risk
adjusted”) measure Q. Note that there is a one-to-one correspondence between the
martingale measure and the market price of risk. Thus, choosing a particular λ is
equivalent to choosing a particular Q
Proposition 8.14. The martingale measure Q is characterized by any of the fol-
lowing equivalent facts.
• The local mean rate of return of any derivative price process Π(t) equals the
short rate of interest, i.e. the Π(t)-dynamics have the following structural form
under Q
dΠ(t) = rΠ(t)dt+ σΠΠ(t)(t)dW,
where W is a Q-Wiener process, and σΠ is the same under Q as under P .
• With Π as above, the process Π(t)/B(t) is a Q-martingale, i.e. it has a zero
drift term.
As for the pricing equation above, we have to know λ in order to compute
the expected value in the risk neutral valuation formula. Thus, in the present
context, and in contrast to the BlackScholes case, the martingale measure Q is
not (generically) determined within the model.
It is instructive to see how the standard BlackScholes model can be interpreted
within the present model. Let us therefore assume, in contrast to our earlier as-
sumptions, that X is in fact the price of a stock in a BlackScholes model. This of
course means that the P -dynamics of X are given by
dX = αGdt+ σGdW ,
for some constants α and σ. Thus, in terms of the notation in (8.35), µ(t, x) = x·αand σ(t, x) = x · σ. Our project is still that of finding the arbitrage free price for
the claim Φ(X(T )), and to this end we follow the arguments given above, which of
course are still valid, and try to find a suitable benchmark claim Γ. In the present
setting we can make a particularly clever choice of Γ(X(T )), and in fact we define
the claim by
Γ(X(T )) = X(T ).
This is the claim which at time T gives us exactly one share of the underlying stock,
and the point is that the price process G(t,X(t)) for this claim is particularly simple.
8.2 Incomplete Markets 73
We have in fact G(t,X(t)) = X(t), the reason being that the claim can be trivially
replicated using a buy-and-hold strategy which consists of one unit of the stock
itself. Note that for this argument to hold we use critically our new assumption that
X really is the price of a traded asset. Using the identity dG = dX we now trivially
obtain the G-dynamics as
dG = αXdt+ σXdW .
Thus, in terms of our earlier discussion we have αG(t, x) = α and σG(t, x) = σ.
We can now determine the market price of risk as
λ(t, x) =αG − rσG
=α− rσ
,
and the point to note is that, because of our tradability assumption, λ is now
determined within the model. We now go on to the pricing PDE, into which we
substitute λ given as above. The critical part, AF , of the PDE now becomes
AF (t, x) = µ(t, x)− λ(t, x)σ(t, x)Fx(t, x) +1
2σ2(t, x)Fxx(t, x)
=
αx− α− r
σ· σx
Fx +
1
2x2σ2Fxx
= rxFx +1
2x2σ2Fxx,
so we end up (as expected) with the BlackScholes equationFt + rxFx +1
2x2σ2Fxx − rF = 0,
F (T, x) = Φ(x),
and in the risk neutral valuation formula we will get the old BlackScholesQ-dynamics
dX = rXdt+ σXdW.
8.2.3 The Martingale Approach
Let us, briefly recollect some main points.
• Since we have assumed that the only asset given a priori is B and that X is
not the price vector of traded assets, the normalized price process Z is one
dimensional with Z = Z0 = B/B = 1. Now: the constant process Z0(t) = 1 is
a martingale regardless of the choice of measure, so every equivalent measure
Q will be a martingale measure
• We always have the risk neutral valuation formula
Π(t;Y) = EQ[e−∫ Tt r(s)ds · Y|Ft] (8.51)
8.2 Incomplete Markets 74
for any T -claim Y and any specification of the short rate process r. In partic-
ular, r does not have to be of the form r(t,X(t)).
• Denoting a fixed choice of martingale measure by Q, where ϕ is the correspond-
ing Girsanov kernel and λ = −ϕ the market price of risk, the Q dynamics of
X are
dX(t) = µ(t,X(t))− δ(t,X(t))λ(t)dt+ δ(t,X(t))dW (t) (8.52)
• In general, the market price of risk process λ is just an adapted (possibly path
dependent) process, so generically X will not be a Markov process under Q.
• If we assume that the market prices of risk is of the form λ(t,X(t)), then X
will be Markovian under the corresponding Q. In this special case we can use
the PDE methods discussed earlier in the chapter for pricing.
8.2.4 Summing Up
We may now sum up our experiences from the preceding sections.
• In an arbitrage free market, regardless of whether the market is complete
or incomplete, there will exist a market price of risk process, λ(t), which is
common to all assets in the market. More precisely, let Π(t) be any price
process in the market, with P -dynamics
dΠ(t) = Π(t)αΠ(t)dt+ Π(t)σΠ(t)dW (t).
Then the following holds, for all t, and P -a.s.
αΠ(t)− r = σΠ(t)λ(t).
• In a complete market the price of any derivative will be uniquely determined
by the requirement of absence of arbitrage. In hedging terms this means that
the price is unique because the derivative can equally well be replaced by
its replicating portfolio. Phrased in terms of pricing PDEs and risk neutral
valuation formulas, the price is unique because a complete market has the
property that the martingale measure Q, or equivalently the market price of
risk λ, is uniquely determined within the model.
• In an incomplete market the requirement of no arbitrage is no longer sufficient
to determine a unique price for the derivative. We have several possible mar-
tingale measures, and several market prices of risk. The reason that there are
8.2 Incomplete Markets 75
several possible martingale measures simply means that there are several dif-
ferent price systems for the derivatives, all of which are consistent with absence
of arbitrage.
Schematically speaking the price of a derivative is thus determined by two major
factors.
• We require that the derivative should be priced in such a way so as to not
introduce arbitrage possibilities into the market. This requirement is reflected
by the fact that all derivatives must be priced by formula (8.50) where the same
Q is used for all derivatives, or equivalently by the pricing PDE (8.47)-(8.48),
where the same λ is used for all derivatives.
• In an incomplete market the price is also partly determined, in a nontrivial
way, by aggregate supply and demand on the market. Supply and demand
for a specific derivative are in turn determined by the aggregate risk aversion
on the market, as well as by liquidity considerations and other factors. All
these aspects are aggregated into the particular martingale measure used by
the market.
When dealing with derivative pricing in an incomplete market we thus have to fix
a specific martingale measure Q, or equivalently a λ, and the question arises as to
how this is to be done.
Question:Who chooses the martingale measure?
From the discussions above the answer should by now be fairly clear.
Answer:The market!
The main implication of this message is that, within our framework, it is not the job
of the theorist to determine the “correct” market price of risk. The market price of
risk is determined on the market, by the agents in the market, and in particular this
means that if we assume a particular structure of the market price of risk, then we
have implicitly made an assumption about the preferences on the market. To take
a simple example, suppose that in our computations we assume that λ = 0. This
means in fact that we have assumed that the market is risk neutral (why?).
From this it immediately follows that if we have a concrete model, and we want
to obtain information about the prevailing market price of risk, then we must go to
the concrete market and get that information using empirical methods. It would of
course be nice if we were able to go to the market and ask the question “What is
today’s market price of risk?”, or alternatively “Which martingale measure are you
using?”, but for obvious reasons this cannot be done in real life. The information
that can be obtained from the market is price data, so a natural idea is to obtain
8.2 Incomplete Markets 76
implicit information about the market price of risk using the existing prices on the
market. This method, sometimes called “calibrating the model to market data”,
“backing out the parameters”, or “computing the implied parameter values”, can
schematically be described as follows.
Let us take a concrete model as given, say the one defined by eqn (8.35). We
assume that we know the exact form of µ and σ. Our problem is that of pricing a
fixed claim Φ(X(T )), and in order to do this we need to know the market price of
risk λ(t, x), and we have to consider two possible scenarios for the derivative market,
which we are trying to model.
The first case occurs if there is no existing market for any derivative product
with X as its underlying object, i.e. no weather insurance contracts for Brighton
are traded. Then we are stuck. Since the market price of risk is determined by the
market, then if there is no market, there is no market price of risk.
The second case occurs if some contracts are already traded. Let us thus assume
that there exists a market for the claims Φi(X(T )), i = 1, . . . , n. Let us furthermore
assume that we want to choose our market price of risk from a parameterized family
of functions, i.e. we assume a priori that λ is of the form
λ = λ(t, x;β), β ∈ Rk.
We have thus a priori specified the functional form of λ but we do not know
which parameter vector β we should use. We then carry out the following scheme.
We are standing at time t = 0.
• Compute the theoretical pricing functions F i(t, x) for the claims Φ1, . . . ,Φn.
This is done by solving the pricing PDE for each contract, and the result will
of course depend on the parameter β, so the pricing functions will be of the
form
F i = F i(t, x;β), i = 1, . . . , n.
• In particular, by observing today’s value of the underlying process, say X(0) =
x0, we can compute today’s theoretical prices of the contracts as
Πi(0;β) = F i(0, x0;β).
• We now go to the concrete market and observe the actually traded prices for
the contracts, thus obtaining the observed prices
Πi∗(0),
where the superscript ∗ indicates an observed value.
8.2 Incomplete Markets 77
• We now choose the “implied” parameter vector β∗ in such a way that the
theoretical prices are “as close as possible” to the observed prices, i.e. such
that
Πi∗(0) ≈ Πi(0;β∗), i = 1, . . . , n.
One way, out of many, of formalizing this step is to determine β∗ by solving
the least squares minimization problem
minβ∈Rk
[n∑i=1
Πi(0;β)−Πi∗(0)2].
In the theory of interest rates, this procedure is widely used under the name of “the
inversion of the yield curve” and in that context we will study it in some detail.
Chapter 9
Working example: Calibrating
the LIBOR market model
(LFM) to interest rate caps
To reiterate, the purpose of calibration is essentially to capture information from
market data, and to fit various model parameters such that the model prices of
interest rate caps match the market prices as close as possible.
Interest rate caps and floors can be priced using a variety of different models but
one model in particular (the LIBOR market model) has gained popularity due to an
ability to encompass well-established market formulas Brigo and Mercurio [2007].
Classic distributional assumptions of interest-rate models include the normal
(Gaussian) distribution and the log-normal distribution. The Hull-White model and
the LIBOR market model represent the assumption of normality and log-normality,
respectively, and are commonly known and applied in practice Brigo and Mercurio
[2007].
Both models can be initially fitted to market prices of interest rate caps but an
exact fit is somewhat more difficult to obtain with the Hull-White model Rebonato
[2004]
A distinct advantage of the LIBOR market model is that it provides a closed
form solution to pricing interest rate caplets which is equivalent to Blacks formula.
Moreover, the equivalent pricing formula implies a link between the implied volatil-
ities of Blacks formula for caplets and the instantaneous volatilities of forward
rates in the LIBOR market model. The link between the volatilities enables the
model to be exactly calibrated to market observable prices of interest rate caps and
floors. Bloch Mikkelsen [2010]
However, as we have seen, the LIBOR market model depends on the specification
of instantaneous volatility (and correlation) of the forward rates, i.e. the volatility
functions influence the models evolution of future interest rates and, hence, the
Chapter 9. Working example: Calibrating the LIBOR market model (LFM) to interest rate caps 79
pricing of derivatives within the model. In principle this means that the user of the
model has to specify the instantaneous volatility. Given different specifications of
the instantaneous volatility the user will obtain different models. Bloch Mikkelsen
[2010]
The use of Black’s formula for pricing caplets and thus interest rate caps has been
standard market practice for years. As I mentioned in the introduction, even before
the development of a consistent no arbitrage framework to support the log-normal
assumption of Blacks formula, traders used Black’s formula for pricing caps.
The market quotes interest rate caps in terms of the implied Black volatility
and not in terms of the nominal value of the cap. The volatility parameter, entered
into the Black cap formula account for how the correct market cap price is obtained
Different interpretations of the volatility parameter apply depending on whether it
is the volatility of a cap or of a single caplet. Brigo and Mercurio [2007].
According to Bjork [2009], implied Black volatilities are referred to as either flat
volatilities or as spot volatilities. In case of an interest rate cap the implied Black
volatility is referred to as implied flat volatility, that is the volatility implied by
the Black cap formula assuming that the same volatility is applied for each caplet
in the cap. In case of a single caplet the implied Black volatility is referred to as
implied spot volatility which is the implied volatility of a single caplet corresponding
to the volatility that has to be entered into Blacks caplet formula in order to obtain
the market price of the caplet. The difference between the flat volatility and the
spot volatility is that the flat volatility imposes the same volatility for all caplets
in the cap, implicitly assuming that all caplets share the same volatility. Brigo and
Mercurio [2007].
A cap volatility structure (flat volatilities) can be illustrated by plotting the cap
volatility for e.g. at-the-money (ATM) caps across different maturities. This is
illustrated in Brigo and Mercurio [2007].
If we recall the forward measure and forward rate dynamics of the LIBOR market
model, we recall that each of the simple LIBOR forward rates is assumed to be log-
normally distributed.
Using the forward measure the dynamics of the LIBOR forward rates can be
described by a system of stochastic differential equations. The forward measure is
denoted as Qi. The forward measure is the measure associated with the numeraire
P (·, Ti) which mean that Qi is associated to the price of a zero-coupon bond with
the same maturity date Ti as that numeraire at maturity is 1. The forward measure
is also referred to as the settlement linked measure. Under the forward measure
Qi the settlement-linked forward rate Fi(t) can be shown to be a martingale which
means that the dynamics of the forward rate can be described by a driftless process
Chapter 9. Working example: Calibrating the LIBOR market model (LFM) to interest rate caps 80
under Qi:
dFi(t) = σi(t)Fi(t)dWii (t), t ≤ Ti−1 (9.1)
where W ii (t) is a standard Brownian motion under the associated forward measure
corresponding to the ith component of the vector of Brownian motions and σi(t) is
the instantaneous volatility of the forward rate Fi(t) at time t Brigo and Mercurio
[2007].
Hence, each forward rate can be expressed as a martingale under the settlement-
linked measure which means that a model in which the LIBOR forward rates evolve
after driftless processes can be formulated. i.e. at any given point in time the
probability distribution associated with the numeraire asset implies that the forward
LIBOR rate is equal to the expected value of the future LIBOR spot rate, i.e.:
Fi(t) = EQi
t [L(Ti−1, Ti)], t < Ti−1 (9.2)
The LIBOR spot rate is eventually the rate that will be used for settlement of
the caplet contract. Thus, given that the forward rate (i.e. the expected value of the
future LIBOR spot rate) is log-normally distributed under the associated forward
measure, log(Fi(Ti−1)) ∼ N(Fi(t), σ2Ti−1,caplet · (Ti−1− t)) (the volatility corresponds
to equation 9.3 below), the price of a caplet at current time t would be given by
Black’s formula with the only difference being the volatility entering the caplet
pricing formula which is defined as:
σ2Ti−1,caplet =
1
Ti−1
∫ Ti−1
0σ2i (s)ds (9.3)
That is the Black volatility should equal the average variance of the forward
rates also referred to as the root-mean-square volatility Rebonato [2004].
Sometimes the pricing of an instrument requires that all forward rates evolve
under the same forward measure. In general, the dynamics of the forward rate Fi(t)
under a measure Qj different from the settlement-linked measure Qi will not be a
driftless process. Therefore, in order to obtain the dynamics of the forward rates
under one common measure the dynamics must be appropriately adjusted. This can
be done by applying proposition 6.3.1 from Brigo and Mercurio [2007] which states
that:
The dynamics of forward rate Fi(t) under the forward adjusted measure Qj in
the three cases, j < i, j = i and j > i are,
9.1 The role of the market 81
j < i, t ≤ Tj :
dFi(t) = σi(t)Fi(t)i∑
k=j+1
ρi,kτkσk(t)Fk(t)
1 + τkFk(t)dt+ σi(t)Fi(t)dWi(t)
j = i, t ≤ Ti−1 :
dFi(t) = σi(t)Fi(t)dWi(t)
j > i, t ≤ Ti−1 :
dFi(t) = −σi(t)Fi(t)j∑
k=i+1
ρi,kτkσk(t)Fk(t)
1 + τkFk(t)dt+ σi(t)Fi(t)dWi(t)
(9.4)
Where W = W j is a standard Brownian motion under the forward measure Qj . In
addition, the terminal measure corresponds to the pricing under a single common
measure.
The instantaneous volatility is commonly represented using the approach of
piecewise constant volatilities or by using an approach in which the volatilities are
represented by a parametric form. Piecewise-constant volatilities imply that for
short intervals of time the volatility of a forward rate is constant, the instantaneous
volatilities can thus be organized in a matrix.
It is important to distinguish between the instantaneous volatilities of the for-
ward rates and the corresponding Black implied volatilities (the spot volatilities).
As mentioned previously in equation (9.3) the relationship between Black caplet
volatilities and instantaneous forward rate volatilities is given (as before) by:
σ2Ti−1,caplet =
1
Ti−1
∫ Ti−1
0σ2i (s)ds
The recovery of Black caplet prices within the LIBOR market model requires the
instantaneous volatilities to be deterministic functions of time. Limited only by
the relation in the above equation this gives us infinity of possible specifications
to choose from. Notice that the Black implied volatility is the root-mean-square
instantaneous volatility and that the term structure of volatilities can be represented
using instantaneous volatilities. Rebonato [2002]
9.1 The role of the market
The market behaviour of interest rate cap volatilities change over time. The features
of the market behaviour can be studied by observing the term structure of volatilities.
According to Rebonato [2002] and figure 3.2 in section 3.4.2 previous observations
9.1 The role of the market 82
suggests that the Black implied volatilities of caplets plotted with maturity is either
a:
• humped shape or
• monotonically decreasing.
According to Rebonato [2002] the humped shape can be explained by the trading
patterns of different forward and futures contracts together with the influence of the
central banks on the short rate. Given “normal” market conditions Rebonato argues
that a higher level of transparency exists as central banks usually indicate their views
on the rate before making changes. Thus actions taken by the central banks tend
to be expected by the market meaning a lower level of volatility for the very short
end of the curve.
Getting back to the calibration, Equation (9.3) defines the connection between
the caplet volatilities of Black’s formula and the instantaneous volatilities of the for-
ward rates in the LIBOR market model. The link between implied and instantaneous
volatilities enables the calibration of the volatility parameterizations to market ob-
served volatilities and thus calibration to market prices. To ensure an exact match
between market and model prices a two-step procedure can be applied. Brigo and
Mercurio [2007] collect the two-step procedure into one expression:
σTi−1,capletmarket· Ti−1 = ϕ2
iλ2(Ti−1 − t, θ)
where λ(·) defines the volatility parameterization and θ denotes the set of parame-
ters used in the volatility specification. The first step of the calibration procedure
involves fitting the parameterization i.e. the “lambda” part of equation which also
is referred to as the time - homogeneous fit. This is done by applying a numeri-
cal iterative procedure, for instance least-squares, where the parameters theta are
adjusted until the best fit is achieved. The second step then involves choosing ϕ
accordingly such that an exact fit is achieved. ϕ is given by the expression:
ϕ2i =
σ2Ti−1,capletmarket
· Ti−1
λ2i (Ti−1 − t, θ)
(9.5)
The parameters θ are given by the first calibration step such that λ2i (Ti−1− t, θ) rep-
resents the caplet volatility implied implied by the parameterization Brigo and Mer-
curio [2007]. Equation (9.5) ensures that differences between the time-homogeneous
parameterization and the market volatilities are leveled out by ϕ.
Chapter 10
Conclusion
I remarked at the outset that this work was not intended to yield numerical results
in response to a quantitative investigation. Rather, it is intended as a discursive
work that explains the structural reasons for why the risk-neutral pricing arguments
developed by academic theoreticians are not an entirely sufficient apparatus for the
practical purposes of interest-rate derivatives trading what it needs in addition is
to incorporate the views of market participants.
The key observation, which I have tried to demonstrate mathematically, and,
occasionally, heuristically, was that in the BlackScholes model the underlying asset
was the stock S, and at first glance this would correspond to the short rate of interest
“r” in the fixed-interest context. However, we have the major difference between the
BlackScholes model and the fixed interest situation. The short rate of interest “r”
is not the price of a traded asset, i.e. there is no asset on the market whose price
process is given by “r”. Thus it is meaningless to form a portfolio “based on r”.
Stated formally, the fixed-interest market is therefore an incomplete market, and
following on from this, a key result is that for the theoretical interest-rate models
to have any practical use, they must incorporate the buy-sell attitudes and risk
preferences of market participants. Market participants effectively shape and define
what is known as the market price of risk.
In the illustrated case of interest-rate caps, this risk preference of the market
is captured, for instance, by the market’s quoted implied volatilities. The formal
process of calibration is merely a mathematical mechanism by which the parameters
of the theoretical model are chosen so as to recover market quoted prices and in so
doing formally incorporate the views of the market at a point in time.
Appendix A
Technical Appendix: SomeTools from Stochastic CalculusAdapted from Brigo andMercurio [2007]
In order to have stochastic differential equations defined we will illustrate the Brow-nian motion process, a fundamental continuous-paths process. Given its importancein default modeling, we also introduce the Poisson process, to some extent the purelyjump analogous of Brownian motion. Brownian motions and Poisson processes areamong the most important random processes of probability.
We note that the understanding, and subsequent implementation, of most of theessential and important issues in interest rate modeling do not require excessively-exotic tools of stochastic calculus. The basic paradigms, risk neutral valuation andchange of numeraire, in fact, essentially involve Ito’s formula and the Girsanovtheorem.
A.1 From Deterministic to Stochastic DifferentialEquations
We consider a probability space (Ω,F , (Ft)t,P). The usual interpretation of thisspace as an experiment can help intuition. The generic experiment result is denotedby ω ∈ Ω; Ω represents the set of all possible outcomes of the random experiment,and the σ-field F represents the set of events A ⊂ Ω with which we shall work. Theσ-field Ft represents the information available up to time t. We have Ft ⊆ Fu ⊆ Ffor all t ≤ u, meaning that “the information increases in time”, never exceeding thewhole set of events F . The family of σ-fields (Ft)t≥0 is called filtration.
If the experiment result is ω and ω ∈ A ∈ F , we say that the event A occurred.If ω ∈ A ∈ Ft, we say that the event A occurred at a time smaller or equal to t.
We use the symbol E to denote expectation, and E[|F ] denotes expectationconditional on the information contained in F .
We begin by a simple example. Consider a population growth model. Letx(t) = xt ∈ R, xt ≥ 0, be the population at time t ≥ 0. The simplest model
A.1 From Deterministic to Stochastic Differential Equations 85
for the population growth is obtained by assuming that the growth rate dxt/dt isproportional to the current population. This can be translated into the differentialequation:
dxt = Kxtdt, x0,
where K is a real constant. Now suppose that, due to some complications, it isno longer realistic to assume the initial condition x0 to be a deterministic constant.Then we may decide to let x0 be a random variable X0(ω), and to model the popu-lation growth by the differential equation:
dXt(ω) = KXt(ω)dt,X0(ω).
The solution of this last equation is Xt(ω) = X0(ω) exp[Kt]. Note that Xt(ω) is arandom variable, but all its randomness comes from the initial condition X0(ω). Foreach experiment result ω, the map t 7→ Xt(ω) is called the path of X associated toω.
As a further step, suppose that not even K is known for certain, but that alsoour knowledge of K is perturbed by some randomness, which we model as the“increment” of a stochastic process Wt(ω), t ≥ 0, so that
dXt(ω) = (Kdt+ dWt(ω))Xt(ω), X0(ω),K ≥ 0. (A.1)
Here, dWt(ω) represents a noise process that adds randomness to K. Equation (A.1)is an example of stochastic differential equation (SDE).
More generally, a SDE is written as
dXt(ω) = ft(Xt(ω))dt+ σt(Xt(ω))dWt(ω), X0(ω). (A.2)
The function f , corresponding to the deterministic part of the SDE, is called thedrift. The function σt (or sometimes its square at at := σ2
t ) is called the diffusioncoefficient. Note that the randomness enters the differential equation from twosources: The “noise term” σt()dWt(ω) and the initial condition X0(ω).
Usually, the solution X of the SDE is called also a diffusion process, because ofthe fact that some particular SDE’s can be used to arrive at a model of physicaldiffusion. In general the paths t 7→ Xt(ω) of a diffusion process X are continuous.
A.1.1 Brownian Motion
The process whose “increments” dWt(ω) are candidate for representing the noiseprocess in (A.2) is the Brownian motion. This process has important properties: Ithas stationary and independent Gaussian increments “dWt(ω)”, or more preciselyfor any 0 < s < t < u and any h > 0:
Wu(ω)−Wt(ω) independent of Wt(ω)−Ws(ω) (indep. increments) (A.3)
Wt+h(ω)−Ws+h(ω) Wt(ω)−Ws(ω) (stationary increments) (A.4)
Wt(ω)−Ws(ω) ∼ N (0, t− s) (Gaussian increments). (A.5)
The above assumptions imply intuitively that, for example, Wt+h(ω) − Wt(ω) isindependent of the history of W up to time t. Therefore, Wt+h(ω) − Wt(ω) can
A.1 From Deterministic to Stochastic Differential Equations 86
assume any value independently of Ws(ω), s ≤ t. The definition of Brownianmotion requires also the paths t 7→ Wt(ω) to be continuous. It turns out thatthe properties listed above imply that although the path be continuous, they are(almost surely) nowhere differentiable. In fact, the paths have unbounded variation,and hence
Wt(ω) =d
dtWt(ω)
does not exist.
A.1.2 Stochastic Integrals
Since Wt(ω) does not exist, what meaning can we give to equation (A.2)? Theanswer relies on rewriting (A.2) in integral form:
Xt(ω) = X0(ω) =
∫ t
0fs(Xs(ω))ds+
∫ t
0σs(Xs(ω))dWs(ω), (A.6)
so that, from now on, all differential equations involving terms like dW are meantas integral equations, in the same way as (A.2) will be an abbreviation for (A.6).
However, we are not done yet, since we have to deal with the new problem ofdefining an integral like
∫ t0 σs(Xs(ω))dWs(ω). A priori it is not possible to define it as
a Stieltjes integral on the paths, since they have unbounded variation. Nonetheless,under some “reasonable” assumptions that we do not mention, it is still possible todefine such integrals a la Stieltjes. The price to be paid is that the resulting integralwill depend on the chosen points of the sub-partitions (whose mesh tends to zero)used in the limit that defines the integral. More specifically, consider the followingdefinition. Take an interval [0, T ] and consider the following dyadic partition of [0, T ]depending on an integer n,
Tni = min
(T,
i
2n
), i = 1, . . . ,∞.
Notice that from a certain i on all terms collapse to T , i.e. Tni = T for all i > 2nT .For each n we have such a partition, and when n increases the partition containsmore elements, giving a better discrete approximation of the continuous interval[0, T ]. Then define the integral as∫ T
0φs(ω)dWs(ω) = lim
n→∞
∞∑i=0
φtni (ω)[WTni+1
(ω)−WTni
(ω)]
where tni is any point in the interval [Tni , Tni+1). Now, by choosing tni := Tni (initial
point of the subinterval) we have the definition of the Ito integral, whereas by takingtni := (Tni + Tni+1)/2 (middle point) we obtain a different result, the Stratonovichintegral.
The Ito integral has interesting probabilistic properties (for example, it is amartingale, an important type of stochastic process that will be briefly definedbelow), but leads to a calculus where the standard chain rule is not preserved sincethere is a non-zero contribution of the second order terms. On the contrary, although
A.1 From Deterministic to Stochastic Differential Equations 87
probabilistically less interesting, the Stratonovich integral does preserve the ordinarychain rule, and is preferable from the viewpoint of properties of the paths.
To better understand the difference between these two definitions, we can resortto the following classical example of stochastic integral computed both with the Itocalculus and the Stratonovich calculus:
Ito⇒ d(Wt(ω)2) = dt+ 2Wt(ω)dWt(ω),
Stratonovich⇒ d(Wt(ω)2) = 2Wt(ω) dWt(ω),
In the Ito version, the “dt” term originates from second order effects, which arenot negligible like in ordinary calculus. Note that the first integral is a martingale(so that, for example, it has constant expected value equal to zero, which is animportant probabilistic property), but does not satisfy formal rules of calculus, asinstead does the second one (which is not a martingale).
In general, stochastic integrals are defined a la Lebesgue rather than a la Riemann-Stieltjes. One defines the stochastic integral for increasingly more sophisticated in-tegrands (indicators, simple functions...), and then takes the limit in some sense.The reader interested in the deeper mathematical aspects of stochastic integrationin connection with SDEs may consult books such as, for example, Øksendal [1992].
A.1.3 Martingales, Driftless SDEs and Semimartingales
In our discussion above, we have mentioned the concept of martingale. To give aquick idea, consider a process X satisfying the following measurability and integra-bility conditions.
Measurability: Ft includes all the information on X up to time t, usually ex-pressed in the literature by saying that (Xt)t is adapted to (Ft)t; Integrability: therelevant expected values exist.
A martingale is a process satisfying these two conditions and such that thefollowing property holds for each t ≤ T :
E[XT |Ft] = Xt.
This definition states that, if we consider t as the present time, the expected valueat a future time T given the current information is equal to the current value. Thisis, among other things, a picture of a “fair game”, where it is not possible to gainor lose on average. It turns out that the martingale property is also suited to modelthe absence of arbitrage in mathematical finance. To avoid arbitrage, one requiresthat certain fundamental processes of the economy be martingales, so that there areno “safe” ways to make money from nothing out of them.
Consider an SDE admitting a unique solution (resulting in a diffusion process,as stated above). This solution is a martingale when the equation has zero drift. Inother terms, the solution of the SDE (A.2) is a martingale when ft() = 0 for all t:
dXt(ω) = σt(Xt(ω))dWt(ω), X0(ω).
Therefore, in diffusion-processes language, martingale means driftless diffusion pro-cess.
A.1 From Deterministic to Stochastic Differential Equations 88
A submartingale is a similar process X satisfying instead
E[XT |Ft] ≥ Xt.
This means that the expected value of the process grows in time, and that averagesof future values of the process given the current information always exceed (or atleast are equal to) the current value.
Similarly, a supermartingale satisfies
E[XT |Ft] ≤ Xt
and the expected value of the process decreases in time, so that averages of futurevalues of the process given the current information are always smaller than (or atmost are equal to) the current value.
A process X that is either a supermartingale or a submartingale is usually termedsemimartingale.
A.1.4 Quadratic Variation
The quadratic variation of a stochastic process Yt with continuous paths t 7→ Yt(ω)is defined as follows:
〈Y 〉T = limn lim∞
∞∑i=1
(YTni
(ω)− YTni−1
(ω))2.
Intuitively this could be written as a “second order” integral:
〈Y 〉T = “
∫ T
0(dYs(ω))2”,
or, even more intuitively, in the differential form
“d〈Y 〉t = dYt(ω)dYt(ω)”.
It is easy to check that a process Y whose paths t 7→ Yt(ω) are differentiable foralmost all ω satisfies 〈Y 〉t = 0. In case Y is a Brownian motion, it can be proved,instead, that
〈W 〉T = T, for each T,
which can be written in a more informal way as
dWt(ω)dWt(ω) = dt .
Again, this comes from the fact that the Brownian motion moves so quickly thatsecond order effects are not negligible. Instead, a process whose trajectories aredifferentiable cannot move so quickly, and therefore its second order effects do notcontribute.
In case the process Y is equal to the deterministic process t 7→ t, so that dYt = dt,we immediately retrieve the classical result from (deterministic) calculus:
dtdt = 0 .
A.1 From Deterministic to Stochastic Differential Equations 89
A.1.5 Quadratic Covariation
One can also define the quadratic covariation of two processes Y and Z with con-tinuous paths as
〈Y,Z〉T = limn→∞
∞∑i=1
(YTni
(ω)− YTni−1
(ω))(ZTni
(ω)− ZTni−1
(ω)).
Intuitively this could be written as a “second order” integral:
〈Y, Z〉T = “
∫ T
0dYs(ω)dZs(ω)”,
or, in differential form,“d〈Y,Z〉t = dYt(ω)dZt(ω)”.
It is then easy to check that, denoting again by t the deterministic process t 7→ t,
〈W, t〉T = 0, for each T,
which can be informally written as
dWt(ω)dt = 0 .
A.1.6 Solution to a General SDE
Let us go back to our general SDE, and let us take time-homogeneous coefficientsfor simplicity:
dXt(ω) = f(Xt(ω))dt+ σ(Xt(ω))dWt(ω), X0(ω). (A.7)
Under which conditions does it admit a unique solution in the Ito sense? Standardtheory tells us that it is enough to have both the f and σ coefficients satisfyingLipschitz continuity (and linear growth, which does not follow automatically in thetime-inhomogeneous case or with local Lipschitz continuity only). These sufficientconditions are valid for deterministic differential equations as well, and can be weak-ened, especially in dimension one. Typical examples showing how, without Lipschitzcontinuity or linear growth, existence and uniqueness of solutions can fail are thefollowing:
dxt = x2tdt, x0 = 1⇒ xt =
1
1− t, t ∈ [0, 1),
(explosion in finite time) and
dxt = 3x2/3t dt, x0 = 0⇒ xt = 1(a,∞)(t)(t− a)3, t ∈ [0,+∞)
for any positive a (no uniqueness).The proof of the fact that the existence and uniqueness of a solution to a SDE
is guaranteed by Lipschitz continuity and linear growth of the coefficients, is similarin spirit to the proof for deterministic equations. See again Øksendal [1992] for thedetails.
A.2 Ito’s Formula 90
A.1.7 Interpretation of the Coefficients of the SDE
We conclude by presenting an interpretation of the drift and diffusion coefficient fora SDE. For deterministic differential equations such as dxt = f(xt)dt, with a smoothfunction f , one clearly has
limh7→0
xt+h − xth
∣∣∣∣xt=y
= f(y),
limh7→0
(xt+h − xt)2
h
∣∣∣∣xt=y
= 0,
The “analogous” relations for the SDE
dXt(ω) = f(Xt(ω))dt+ σ(Xt(ω))dWt(ω),
are the following:
limh7→0
EXt+h(ω)−Xt(ω)
h
∣∣∣∣Xt = y
= f(y)
limh7→0
E
[Xt+h(ω)−Xt(ω)]2
h
∣∣∣∣Xt = y
= σ2(y).
The second limit is non-zero because of the “infinite velocity” of Brownian motion,while the first limit is the analogous of the deterministic case.
A.2 Ito’s Formula
Now we are ready to introduce the famous Ito formula, which gives the “chain rule”for differentials in a stochastic context.
For deterministic differential equations such as
dxt = f(xt)dt,
given a smooth transformation φ(t, x), one can write the evolution of φ(t, xt) via thechain rule:
dφ(t, xt) =∂φ
∂t(t, xt)dt+
∂φ
∂x(t, xt)dxt.
We already observed in A.1.2 that whenever a Brownian motion is involved, sucha fundamental rule of calculus needs to be modified. The general formulation ofthe chain rule for stochastic differential equations is the following. Let φ(t, x) be asmooth function and let Xt(ω) be the unique solution of the stochastic differentialequation (A.7). Then, Ito’s formula reads as
dφ(t,Xt(ω)) =∂φ
∂t(t,Xt(ω)) +
∂φ
∂x(t,Xt(ω))dXt(ω)
+1
2
∂2φ
∂x2(t,Xt(ω))dXt(ω)dXt(ω), (A.8)
A.2 Ito’s Formula 91
or, in a more compact notation,
dφ(t,Xt) =∂φ
∂t(t,Xt)dt+
∂φ
∂x(t,Xt)dXt +
1
2
∂2φ
∂x2(t,Xt)d〈X〉t .
Comparing equation (A.8) with its “deterministic” counterpart, we notice that theextra term
1
2
∂2φ
∂x2(t,Xt(ω))dXt(ω)dXt(ω)
appears in our stochastic context, and this is the term due to the Ito integral.1 Theterm dXt(ω)dXt(ω) can be developed algebraically by taking into account the ruleson the quadratic variation and covariation seen above:
dWt(ω)dWt(ω) = dt, dWt(ω)dt = 0, dtdt = 0.
We thus obtain
dφ(t,Xt(ω)) =∂φ
∂t(t,Xt(ω))dt+
∂φ
∂x(t,Xt(ω))f(Xt(ω))dt
1
2
∂2φ
∂x2(t,Xt(ω))σ2(Xt)dt+
∂φ
∂x(t,Xt(ω))σ(Xt(ω))dWt(ω).
A.2.1 Stochastic Leibnitz Rule
Also the classical Leibnitz rule for differentiation of a product of functions is modi-fied, analogously to the chain rule. The related formula can be derived as a corollaryof Ito’s formula in two dimensions, and is reported below.
For deterministic and differentiable functions x and y we have the deterministicLeibnitz rule
d(xtyt) = xtdyt + ytdxt.
For two diffusion processes (and more generally semimartingales) Xt(ω) and Yt(ω)we have instead2
d(Xt(ω)dYt(ω)) = Xt(ω)dYt(ω) + Yt(ω)dXt(ω) + dXt(ω)dYt(ω)
or, in more compact notation,
d(XtYt) = XtdYt + YtdXt + d〈X,Y 〉t .
1 Notice that, when writing instead the equation for dφ(t,Xt(ω)) in the Stratonovich sence, wewould re-obtain
dφ(t,Xt(ω)) =∂φ
∂t(t,Xt(ω))dt+
∂φ
∂x(t,Xt(ω)) dXt(ω),
since formal rules of calculus are preserved.2 Clearly, using the Stratonovich calculus, we would still have d(XtYt) = Xt dYt + Yt dXt.
A.2 Ito’s Formula 92
A.2.2 Linear SDEs with Deterministic Diffusion Coefficient
A SDE is said to be linear if both its drift and diffusion coefficients are first or-der polynomials (or affine functions) in the state variable. We here consider theparticular case:
dXt(ω) = βtXt(ω) + αt)dt+ vtdWt(ω), X0(ω) = x0, (A.9)
where α, β, v are deterministic functions of time that are regular enough to ensureexistence and uniqueness of a solution.
It can be shown that a stochastic integral of a deterministic function is the sameboth in the Stratonovich and in the Ito sense. As a consequence, by writing (A.9)in integral form we see that the same equation holds in the Stratonovich sense:
dXt(ω) = (βtXt(ω) + αt)dt+ vt dWt(ω), X0(ω) = x0,
so that we can solve it by ordinary calculus for linear differential equations: Weobtain
Xt(ω) = e∫ t0 βsds
[x0 +
∫ t
0e−
∫ s0 βuduαsds+
∫ t
0e−
∫ s0 βuduvsdWs(ω)
]= x0e
∫ t0 βsds +
∫ t
0e∫ ts βuduαsds+
∫ t
0e∫ ts βuduvsdWs(ω).
A remarkable fact is that the distribution of the solution Xt is normal at each timet. Intuitively, this holds since the last stochastic integral is a limit of a sum ofindependent normal random variables. Indeed, we have
Xt ∼ N(x0e
∫ t0 βsds +
∫ t
0e∫ ts βuduαsds,
∫ t
0e2
∫ ts βuduv2
sds
).
The major examples of models based on a SDE like (A.9) are that of Vasicek (1978)and that of Hull and White (1990).
A.2.3 Lognormal Linear SDEs
Another interesting example of linear SDE is that where the diffusion coefficient isa first order homogeneous polynomial in the underlying variable. This SDE can beobtained as an exponential of a linear equation with deterministic diffusion coeffi-cient. Indeed, let us take Yt = exp(Xt), where Xt evolves according to (A.9), andwrite by Ito’s formula
dYt(ω) = deXt(ω) = eXt(ω)dXt(ω) +1
2eXt(ω)dXt(ω)dXt(ω)
= Yt(ω)[αt + βt lnYt(ω) +1
2v2t ]dt+ vtYt(ω)dWt(ω).
As a consequence, the process Y has a lognormal marginal density.
A.3 Two Important Theorems 93
A.2.4 Geometric Brownian Motion
The geometric Brownian motion is a particular case of a process satisfying a lognor-mal linear SDE. Its evolution is defined according to
dXt(ω) = µXt(ω)dt+ σXt(ω)dWt(ω),
where µ and σ are positive constants. To check that X is indeed a lognormal process,one can compute d ln(Xt)via Ito’s formula and obtain
Xt(ω) = X0 exp
(µ− 1
2σ2
)t+ σWt(ω)
.
From the seminal work of Black and Scholes (1973) on, processes of this type arefrequently used in option pricing theory to model general asset price dynamics.Notice that this process is a submartingale, in that clearly
E[XT |Ft] = eµ(T−t)Xt ≥ Xt.
Finally, notice also that by setting Yt(ω) = e−µt)Xt(ω), we obtain
dYt(ω) = σYt(ω)dWt(ω).
Therefore, since the drift of this last SDE is zero, e−µtXt is a martingale.
A.2.5 Square-Root Processes
An interesting case of non-linear SDE is given by
dXt(ω) = (βtXt(ω) + αt)dt+ vt√Xt(ω)dWt(ω), X0(ω) = x0. (A.10)
A process following such dynamics is commonly referred to as square-root pro-cess. Major examples of models based on this dynamics are the Cox, Ingerssolland Ross (1985) instantaneous interest rate model and a particular case of theconstant-elasticity of variance (CEV) model for stock prices:
dXt(ω) = µXt(w)dt+ σ√Xt(ω)dWt(ω).
In general, square-root processes are naturally linked to non-central χ-square distri-butions.3 In particular, there are simplified versions of (A.10) for which the resultingprocess X is strictly positive and analytically tractable, like in the case of the Cox,Ingerssoll and Ross (1985) model.
A.3 Two Important Theorems
We now introduce (informally) two important results. See Øksendal [1992] for moredetails.
3 They may also display exit boundaries, for example the CEV process features absorption in theorigin.
A.3 Two Important Theorems 94
A.3.1 The Feynman-Kac Theorem
The Feynman-Kac theorem, under certain assumptions, allows us to express thesolution of a given partial differential equation (PDE) as the expected value ofa function of a suitable diffusion process whose drift and diffusion coefficient aredefined in terms of the PDE coefficients.
Theorem A.1 (The Feynman-Kac Theorem). Given Lipschitz continuous f(x) andσ(x) and a smooth f , the solution of the PDE
∂V
∂t+∂V
∂x(t, x)f(x) +
1
2
∂2V
∂x2(t, x)σ2(x) = rV (t, x) (A.11)
with terminal boundary condition
V (T, x) = φ(x) (A.12)
can be expressed as the following expected value
V (t, x) = e−t(T−t)Eφ(XT )|Xt = x, (A.13)
where the diffusion process X has dynamics, starting from x at time t, given by
dXs(ω) = f(Xs(ω))ds+ σ(Xs(ω))dWs(ω), s ≥ t,Xt(ω) = x (A.14)
under the probability measure P under which the expectation E is taken. The
process W is a standard Brownian motion under P.
Notice that the terminal condition determines the function φ of the diffusionprocess whose expectation is relevant, whereas the PDE coefficients determine thedynamics of the diffusion process.
This theorem is important because it establishes a link between the PDE’s oftraditional analysis and physics and diffusion processes in stochastic calculus. So-lutions of PDE’s can be interpreted as expectations of suitable transformations ofsolutions of stochastic differential equations and vice versa.
A.3.2 The Girsanov Theorem
The Girsanov theorem shows how a SDE changes due to changes in the underlyingprobability measure. It is based on the fact that the SDE drift depends on theparticular probability measure P in our probability space (Ω,F , (Ft)t,P), and that,if we change the probability measure in a “regular” way, the drift of the equationchanges while the diffusion coefficient remains the same. The Girsanov theoremcan be thus useful when we want to modify the drift coefficient of a SDE. Indeed,suppose that we are given two measures P∗ and P on the space Ω,F , (Ft)t. Twosuch measures are said to be equivalent, written P∗ ∼ P, if they share the samesets of null probability (or of probability one, which is equivalent). Therefore twomeasures are equivalent when they agree on which events of F hold almost surely.Accordingly, a proposition holds almost surely under P if and only if it holds almost
A.3 Two Important Theorems 95
surely under P∗. Similar definitions apply also for the measures restriction to Ft,thus expressing equivalence of the two measures up to time t.
When two measures are equivalent, it is possible to express the first in terms ofthe second through the Radon-Nikodym derivative. Indeed, there exists a martingaleρt on (Ω,F , (Ft)t,P) such that
P∗ =
∫Aρt(ωdP(ω)), A ∈ Ft,
which can be written in a more concise form as
dP∗
dP
∣∣∣∣Ft
= ρt.
The process ρt is called the Radon-Nikodym derivative of P∗ with respect to Prestricted to Ft.
When in need of computing the expected value of an integrable random variableX, it may be useful to switch from one measure to another equivalent one. Indeed,it is possible to prove that the following equivalence holds:
E∗[X] =
∫ΩX(ω)dP∗(ω) =
∫ΩX(ω)
dP∗
dP(ω)dP(ω) = E
[XdP∗
dP
],
where E∗ and E denote expected values with respect to the probability measures P∗and P, respectively. More generally, when dealing with conditional expectations, wecan prove that
E∗[X|Ft] =E[X dP∗
dP |Ft]ρt
.
Theorem A.2 (The Girsanov theorem). Consider again the stochastic differentialequation, with Lipschitz coefficients,
dXt(ω) = f(Xt(ω))dt+ σ(Xt(ω))dWt(ω), x0,
under P. Let be given a new drift f∗(x) and assume (f∗(x) − f(x))/σ(x) to bebounded. Define the measure P∗ by
dP∗
dP(ω)
∣∣∣∣Ft
= exp
− 1
2
∫ t
0
(f∗(Xs(ω))− f(Xs(ω))
σ(Xs(ω))
)2
ds
+
∫ t
0
f∗(Xs(ω))− f(Xs(ω))
σ(Xs(ω))dWs(ω)
.
Then P∗ is equivalent to P. Moreover, the process W ∗ defined by
dW ∗t (ω) = −[f∗(Xt(ω))− f(Xt(ω))
σ(Xt(ω))
]dt+ dWt(ω)
is a Brownian motion under P∗, and
dXt(ω) = f∗(Xt(ω))dt+ σ(Xt(ω))dW ∗t (ω), x0.
A.3 Two Important Theorems 96
As already noticed, this theorem is fundamental when we wish to change thedrift of a SDE. It is now clear that we can do this by defining a new probabilitymeasure P∗, via a suitable Radon-Nikodym derivative, in terms of the difference“desired drift - given drift”.
In mathematical finance, a classical example of application of the Girsanov the-orem is when one moves from the “real-world” asset price dynamics
dSt(ω) = µSt(ω)dt+ σSt(ω)dWt(ω)
to the risk-neutral ones
dSt(ω) = rSt(ω)dt+ σSt(ω)dW ∗t (ω).
This is accomplished by setting
dP∗
dP(ω)
∣∣∣∣Ft
= exp
−1
2
(µ− rσ
)2
t− µ− rσ
Wt(ω)
. (A.15)
We finally stress that above we assumed boundedness for simplicity, but less stringentassumptions are possible for the theorem to hold. See for example Øksendal [1992].
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