Libor market model with stochastic volatility · 2020-06-14 · Interest rates are also a vital tool for monetary policy and within the nancial envi-ronment. However, managing interest

MSc THESIS STOCHASTICS AND FINANCIAL MATHEMATICS

Libor market model with stochastic volatility

Author:Seonmi Lee (10034153)

Supervisors:Ir. Bart Hoorens (ING)

Geert-Jan Bralten MSc. (ING)Dr. Peter Spreij (UvA)

August 2012

Acknowledgments

This thesis is the result of my master project carried out at the MRMB Trading de-partment of ING Bank Amsterdam. I would like to express my gratitude to those whohelped and encouraged me in accomplishing this thesis. I did this project in the Quan-titative Analytics team, under supervision of Bart Hoorens and Geert-Jan Bralten. Mysupervisor from the university was Peter Spreij, professor at the Korteweg-de Vries In-stitute (KdVI) for Mathematics of the University of Amsterdam. I would like to thankmy supervisors Bart Hoorens, Geert-Jan Bralten and Peter Spreij for all their guidancethroughout this project. I greatly benefited from their expertise and ideas in both afinancial working environment and university. I am further grateful to all my colleaguesat ING Bank, who supported me in accomplishing this project.

2

MSc THESIS STOCHASTICS AND FINANCIAL MATHEMATICS

Libor market model with stochastic volatility

Abstract:

In this thesis, we investigate the Libor Market Model (LMM) with Displaced Diffusionand Stochastic Volatility (LMM-DDSV) for pricing of interest rate derivatives. In partic-ular, we derive pricing formulas for caplets and swaptions under the LMM-DDSV modelwith constant parameters, and we extend the model with time-dependent parameters.The time-dependent LMM-DDSV model is approximated by a time-homogeneous modelwith averaged parameters, obtained by using the parameter averaging method. We showthat the skewness and curvature of the implied volatility curves can be controlled by theLMM-DDSV model parameters, which fits the market implied volatility curves bettercompared to the standard model without stochastic volatility, which implies a flat andunrealistic volatility curve. A well-known drawback of stochastic volatility models is thata so-called moment explosion can occur in some parameter settings, which we demon-strate by using an in-arrears swap. Despite of a potential moment explosion issue, weshow that the LMM-DDSV model with parameter averaging generally yields accurateprice approximations.

Keywords:

Interest rate derivatives, pricing of caplets and swaptions, LIBOR, Libor Market Model(LMM), LMM-DDSV model, displaced diffusion, stochastic volatility, implied volatil-ity, skewness and curvature, time-dependent parameters, parameter averaging method,moment explosion, in-arrears swaps.

4

Contents

1 Introduction 8

2 Background 102.1 Zero-coupon bonds and interest rates . . . . . . . . . . . . . . . . . . . . . 102.2 Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.1 T -Forward measure . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2.2 Spot measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Black formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4 Fixed income instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4.1 Forward rate agreements (FRA) . . . . . . . . . . . . . . . . . . . 172.4.2 Plain vanilla swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4.3 In-arrears swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.4.4 Interest rate caps and floors . . . . . . . . . . . . . . . . . . . . . . 222.4.5 European swaptions . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 The LIBOR Market Model (LMM) 243.1 LIBOR market model dynamics and measures . . . . . . . . . . . . . . . . 24

3.1.1 The choice of σn(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.1.2 Brace-Gatarek-Musiela (BGM) model . . . . . . . . . . . . . . . . 28

3.2 LMM with displaced diffusion and stochastic volatility (LMM-DDSV) . . 293.2.1 Rank reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2.2 The principal components analysis (PCA) . . . . . . . . . . . . . . 313.2.3 Construction of λk(t) . . . . . . . . . . . . . . . . . . . . . . . . . 33

4 Pricing IR Derivatives under the LMM-DDSV Model 374.1 The Heston model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2 Pricing of caplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.3 Pricing of swaptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.4 Parameter averaging method . . . . . . . . . . . . . . . . . . . . . . . . . 464.5 The impact of the parameters on implied volatility curves . . . . . . . . . 494.6 Simulation of the LMM-DDSV model . . . . . . . . . . . . . . . . . . . . 50

4.6.1 Simulation of the variance process . . . . . . . . . . . . . . . . . . 514.6.2 Simulation of the forward rates processes . . . . . . . . . . . . . . 54

5

CONTENTS

5 Numerical Results 565.1 General simulation settings . . . . . . . . . . . . . . . . . . . . . . . . . . 565.2 Time-homogeneous LMM-DDSV model . . . . . . . . . . . . . . . . . . . 57

5.2.1 Caplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.2.2 Swaptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.3 Time-dependent LMM-DDSV model and parameter averaging method . . 625.3.1 Caplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.3.2 Swaptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.4 Moment explosion for stochastic volatility models . . . . . . . . . . . . . . 65

6 Conclusion 68

A Theorems and Proofs 71A.1 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71A.2 Proof of Theorem 4.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72A.3 Proof of Proposition 4.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Bibliography 78

6

Chapter 1

Introduction

The concept of interest rates is widely known in everybody’s daily life when people savetheir money in a bank account, lend money to others, or when they borrow capital froma bank for their mortgage: an interest should be paid by the borrower to the lender.Interest rates are also a vital tool for monetary policy and within the financial envi-ronment. However, managing interest rate risk (i.e., the control of uncertain cash flowsdue to fluctuations in interest rates) is an issue with greater complexity. In particular,pricing and hedging of securities which depend on interest rates, create the necessity forfinancial mathematical interest rate models.

In this thesis we investigate mathematical models which model the forward rates, espe-cially the forward LIBOR rates (London Interbank Offered Rates), which are one of themost important interest rates quoted in the market; it is widely used as a reference ratefor many securities in financial markets. LIBOR is a rate at which banks are willingto lend to each other, and it is also used to determine a term structure for the interestrates, also known as a yield curve. The yield curve reflects the relation between the costof borrowing (interest rate) and the term (time-to-maturity) at which time the moneyshould be paid back to the lender. It is also an important variable in the interest rate(IR) derivatives market, which is the largest derivatives market in the world. For pricingIR derivatives, it is important to develop a model which is able to describe the evolu-tion of a yield curve realistically, and remains manageable from a practical point-of-view.

Despite the popularity of LIBOR, the research on interest rate models have been mainlyfocused on describing the short-rate r(t), which represents the interest rate valid for aninfinitesimally short period of time after timepoint t. Short-rate models have been exten-sively studied in the literature, see Vasicek [31], Ingersoll and Ross [14], Ho and Lee [18].A popular short-rate model is the Hull-White model [20], for which analytic formulasare available to price plain vanilla IR products such as bond options, caps and swaptions.

Models for forward rates have been proposed by Heath, Jarrow and Merton (HJM)[16], and Brace, Gatarek and Musiela (BGM) [9]. The HJM model is a general frame-

8

CHAPTER 1. INTRODUCTION

work to model instantaneous forward rates, which are not observable in the market, andcaptures the full dynamics of the entire forward rate curve, while the short-rate modelsonly capture the dynamics of a point on the curve. When the volatility and drift of theinstantaneous forward rate are assumed to be deterministic, this model is known as theGaussian HJM model of forward rates, see Musiela and Rutkowski [28].

The BGM model [9] is a mathematical model describing the evolution of LIBOR rates,which belongs to the class of Libor Market Models (LMM). The model assumes thatforward rates have a lognormal distribution and has served as a benchmark model forinterest rate derivatives. However, a limitation of the BGM model is that it implies flatimplied volatility curves, whereas the market implied volatility curves observed in theLIBOR markets exhibit some skew and curvature.

To overcome the shortcoming of the BGM model, in this thesis we investigate a LMMmodel with stochastic volatility. The most popular stochastic volatility model is theHeston model [17], well-known from equities and is used to incorporate skewness andcurvature in the implied volatility. The advantage of the Heston model is that theskewness and curvature can be controlled by the model parameters and the model isanalytically tractable for closed-form pricing of European style of derivatives. For IRderivatives, this model has been modified to specific characteristics of IR markets, seeAndersen and Andreasen [2], Andersen and Brotherton-Ratcliffe [3], Piterbarg [29], andWu and Zhang [32].

In this thesis we are particularly interested in pricing plain vanilla interest rate deriva-tives, like swaptions and caplets, since there exist closed-form pricing formulas for theseinstruments and they serve as calibration instruments to calibrate the model parameters.

The outline of this thesis is as follows: In Chapter 2, we give some background informa-tion on interest rate modelling and give an overview of fixed income instruments and themost common interest rate derivatives. In Chapter 3 we discuss the BGM model, andwe introduce the LMM model with stochastic volatility. In Chapter 4, we derive pricingformulas for caplets and swaptions under the LMM model with Displaced Diffusion andStochastic Volatility, and we extend our model with time-dependent parameters. Theparameter averaging method is used to approximate the dynamics of the time-dependentmodel by dynamics with time-homogenous parameters. In Chapter 5 we present numer-ical results and we investigate the so-called moment explosion, a well-known problemfor stochastic volatility models. Finally, in Chapter 6 we give the conclusion and discussdirections for further research.

9

Chapter 2

Background

In this chapter we introduce several basic concepts of financial instruments, which wewill use throughout this thesis. We assume an economy where non-dividend paying se-curities are traded continuously from time 0 to time T . We assume that the pricesof these securities are defined on a probability space (Ω,F ,P) where the filtrationF = Ft = σW (s) : 0 ≤ s ≤ t : 0 ≤ t ≤ T satisfies the usual conditions andW is a m-dimensional Brownian motion. The economy is also assumed to be free of ar-bitrage and complete. A random variable V is called a contingent claim or a T -maturityderivative security if V (T ) is FT -measurable. The T -maturity derivative security paysout an amount of V (T ) at time T and makes no payments before T .

In this thesis we use annualized time-scales, and this can possibly be extended withday-count conventions. There are different types of day-count conventions which mea-sure the difference between two dates such as Actual/365 and Actual/360. For moredetailed information about day-count conventions, we refer to Brigo and Mercurio [10].

2.1 Zero-coupon bonds and interest rates

One of the simplest contingent claims is a T -maturity zero-coupon bond, which is definedas follows.

Definition 2.1.1. (zero-coupon bond) A T -maturity zero-coupon bond is a securitypaying one unit of a given currency at time T , its value at time t is denoted by P (t, T ),t ∈ [0, T ]. Clearly, we have that P (T, T ) = 1.

We define a money-market account (or bank account) where the value at time tis denoted by β(t). In this account, profits are accrued continuously at the prevailingrisk-free rate in the market. Its dynamics are assumed to be given by

dβ(t) = r(t)β(t)dt, β(0) = 1,

10

CHAPTER 2. BACKGROUND

where r(t) is the short-rate or instantaneous rate (for the exact definition of short-rate,see Definition 2.1.5). Under this assumption, it can be easily verified that β(t) satisfies:

β(t) = e∫ t0 r(u)du.

We also note that any positive traded non-dividend paying security can be used as anumeraire. In general, for any numeraire N , there exists a probability measure PN suchthat the price of any contingent claim V normalized by N is a martingale under PN . Inthe following definition we introduce such a measure corresponding to β(t).

Definition 2.1.2. (risk-neutral measure) We define the risk-neutral measure Q tobe the measure such that V (t)/β(t) is a martingale under the measure Q. It holds that,for all t ≤ T ,

V (t) = β(t)EQt

[V (T )

β(T )

],

where V (t) denotes the price at time t of a derivative security making an FT -measurablepayment of V (T ) and EQ

t is the conditional expectation with respect to Ft under themeasure Q. The money-market account β(t) is called the numeraire corresponding tothe measure Q.

Under the risk-neutral measure, we can derive the value at time t of a T -maturityzero-coupon bond. This result is given in the following lemma.

Lemma 2.1.3. (zero-coupon bond price) The price of a zero-coupon bond with ma-turity T at time t is given by

P (t, T ) = EQt

[e−∫ Tt r(u)du

].

Proof. By definition of the risk-neutral measure and P (T, T ) = 1, we have that

P (t, T ) = β(t)EQt

[P (T, T )

β(T )

]= EQ

t

[e−∫ Tt r(u)du

].

Under the arbitrage-free assumption, the following relationship should hold:

P (t, T + τ) = P (t, T )EQt [P (T, T + τ)], τ > 0. (2.1)

Otherwise, if the above relationship does not hold, then there exists a self-financingtrading strategy that results in a risk-less profit. To see this, suppose that the inequalityP (t, T + τ) > P (t, T )EQ

t [P (T, T + τ)] holds for some t ≤ T ≤ T + τ . Then, at time t,we can sell P (t, T )EQ

t [P (T, T + τ)]/P (t, T + τ) time (T + τ)-maturity bonds and buyEQt [P (T, T + τ)] time T -maturity bonds. Furthermore, at maturity time T , we receive

EQt [P (T, T + τ)] units of currency and buy one time (T + τ)-maturity bond. Finally, at

time T + τ we pay P (t, T )EQt [P (T, T + τ)]/P (t, T + τ) units of currency and receive one

11


unit of currency. Following this strategy results in a profit of 1 − P (t, T )EQt [P (T, T +

τ)]/P (t, T + τ) at time T + τ . A similar trading strategy with a risk-less profit could beestablished in case the other inequality P (t, T + τ) < P (t, T )EQ

t [P (T, T + τ)] holds.

The time interval τ is often called a tenor. The financial interpretation of Equation(2.1) is that given a cash-flow of 1 unit of currency at T + τ , its corresponding value atanytime t < T + τ must be the same if we discount back in a single step from T + τ tot or in two steps. It is known as the no-arbitrage relation. Next, we define the forwardprice of the zero-coupon bond.

Definition 2.1.4. (zero-coupon bond forward price) For τ > 0, the time t forwardprice of the zero-coupon bond spanning [T, T + τ ], is given by

P (t, T, T + τ) := EQt [P (T, T + τ)] .

and from Equation (2.1), it follows that

P (t, T, T + τ) =P (t, T + τ)

P (t, T ). (2.2)

It is also important to define rates and yields related to a zero-coupon bond price.We give the following definitions:

• continuously compounded forward yields

• compounded forward rates

• instantaneous forward rates

• short rates

Definition 2.1.5. (continuously compounded forward yield) For a given tenorτ > 0, the continuously compounded forward yield y(t, T, T + τ) is defined by

e−y(t,T,T+τ)τ = P (t, T, T + τ), for any t < T.

Definition 2.1.6. (compounded forward rate) The simply compounded forwardrate L(t, T, T + τ) is given by

1 + τL(t, T, T + τ) =1

P (t, T, T + τ).

The simply compounded forward rate is a constant rate for the time interval [T, T+τ ]prevailing at time t. We assume that the spot rates L(T, T, T + τ) are the LondonInterbank Offered Rates (LIBOR) quoted in the interbank market. We also denoteL(t, T, T + τ) as the forward LIBOR rates.

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Note that by taking the limit τ → 0, we have the following equations:

f(t, T ) := limτ→0+

L(t, T, T + τ)

= limτ→0+

1

τ

(1

P (t, T, T + τ)− 1

)= lim

τ→0+

1

τ

P (t, T )− P (t, T + τ)

P (t, T )

= − 1

P (t, T )

∂P (t, T )

∂T

= −∂ lnP (t, T )

∂T.

Definition 2.1.7. (instantaneous forward rate) The quantity defined by f(t, T ) isthe time t instantaneous forward rate to time T , and it satisfies that

P (t, T ) = exp

(−∫ T

tf(t, u)du

).

Definition 2.1.8. (short-rate) The short-rate r(t) at time t is defined by

r(t) := f(t, t) = limT↓t

f(t, T ).

2.2 Measures

Previously, we introduced the risk-neutral measure correspoinding to β(t). By choos-ing different numeraires, there exist different measures. In this section, we introducenumeraires and measures used in pricing interest rates derivatives for later reference.

2.2.1 T -Forward measure

We introduce a measure which takes the T -maturity zero-coupon bond as the numeraire.

Definition 2.2.1. (T -forward measure) The T -forward measure QT is the measurewhich makes V (t)/P (t, T ) a martingale, that is,

V (t) = P (t, T )ETt[V (T )

P (T, T )

]= P (t, T )ETt [V (T )], for all t ≤ T,

where ETt is the conditional expectation w.r.t. Ft under the T -forward measure QT .

The forward LIBOR rate L(t, T, T + τ) with t ≤ T and τ > 0 is a martingale underthe (T + τ)-forward measure QT+τ . This result is proven by the following lemma.

13


Lemma 2.2.2. (martingale under forward measure) The forward LIBOR rateL(t, T, T + τ) is a martingale under the (T + τ)-forward measure QT+τ . Hence, we havethe following equation:

L(t, T, T + τ) = ET+τt [L(T, T, T + τ)], t ≤ T,

where τ > 0.

Proof. Recall the definition of a forward LIBOR rate L(t, T, T + τ),

1 + τL(t, T, T + τ) =1

P (t, T, T + τ).

From Equation (2.2), we have that

1 + τL(t, T, T + τ) =P (t, T )

P (t, T + τ), (2.3)

which is equivalent to

L(t, T, T + τ) =1

τ

[P (t, T )

P (t, T + τ)− 1

]. (2.4)

To show that L(t, T, T + τ) is a martingale under the (T + τ)-forward measure, we needto show that

L(t, T, T + τ) = ET+τt [L(s, T, T + τ)],

holds for any t ≤ s ≤ T . Equation (2.4) yields that

ET+τt (L(s, T, T + τ)) = ET+τ

t

[1

τ

(P (s, T )

P (s, T + τ)− 1

)]=

1

τ

ET+τt

[P (s, T )

P (s, T + τ)

]− 1

.

Since P (s, T ) is a traded asset and a traded asset normalized by the numeraire is amartingale, P (s, T )/P (s, T + τ) is a martingale under the (T + τ)-forward measure.Hence, we have that

ET+τt [L(s, T, T + τ)] =

1

τ

(P (t, T )

P (t, T + τ)− 1

)= L(t, T, T + τ).

2.2.2 Spot measure

The spot measure is defined in this subsection. We consider a set of dates

0 = T0 < T1 < · · · < TN , τn = Tn+1 − Tn.

and define an asset price process B(t) by

B(t) =

P (t, Ti+1)

∏in=0(1 + τnLn(Tn)), for t > 0,

1, for t = 0,(2.5)

14


where i is such that Ti < t ≤ Ti+1 and

Ln(Tn) := L(Tn, Tn, Tn+1). (2.6)

Note that Equation (2.5) is equivalent to

B(t) = P (t, Ti+1)

i∏n=0

1/P (Tn, Tn+1),

by the definition of compounded forward rate Ln(t). The asset price process B(t) rep-resents the time t value of an asset starting from one unit of currency at time 0 andreinvesting at each tenor date Tn in the zero-coupon bonds for the next tenor Tn+1. It isa discrete-time equivalent of the continuously compounded money market account β(t).

Definition 2.2.3. (spot measure) The spot measure QB is the measure which makesV (t)/B(t) a martingale, that is,

V (t) = B(t)EBt[V (T )

B(T )

].

where EBt is the conditional expectation w.r.t. Ft under the spot measure QB.

The advantage of using the spot measure over a forward measure in the simulationis that a possible bias coming from discretizing the drift is more evenly distributed overthe different rates. We will discuss this in further detail in Chapter 3.

2.3 Black formula

In this section we discuss the Black formula, which is a theoretical pricing model foroptions. First, we give a brief introduction of options.

Definition 2.3.1. (options) An option is a financial instrument which gives the holderthe right to buy or sell the underlying security F for a certain price K at a future time T .The price K is also known as the strike price and the time T is known as the maturity.A call (put) option is an option which gives the right to buy (sell) the underlying asset.

Given the strike price K and the underlying level F , options are said to be in-the-money (ITM), at-the-money (ATM), or out-of-the-money (OTM). In-the-money optionsare options which have intrinsic value if they were exercised. Out-the-money options havezero intrinsic value. In other words: if F > K, then the call option is ITM and its corre-sponding put option is OTM, and vice versa. The option is said to be ATM when F = K.

If the holder of an option can only exercise at maturity, then the option is called aEuropean option. For American options, the holder may exercise the option at any timebefore the maturity date. American options are generally computed numerically with

15


for example binomial tree methods, Monte Carlo methods, or finite-difference methods.

For European options, the analytical pricing formula is well-known under certain as-sumptions as derived by Black and Scholes [8] and Merton [27]. Here we will introducethe pricing formula derived by Black.

Definition 2.3.2. (Black formula) At maturity T , let F (T ) be the underlying priceand it is assumed to be log-normally distributed. We assume that the volatility of F (T )is given by σ and we denote µ(0) := E(F (T )). The call price (under the risk-neutralmeasure) at time 0, for a European call option with strike K and maturity T is givenby the formula:

c(0) = P (0, T )(µ(0)N (d1)−KN (d2)), (2.7)

where

d1 =log(µ(0)/K) + 1

2σ2T

σ√T

,

d2 = d1 − σ√T , and N (x) =

∫ x−∞

1√2πe−

12y2 dy is the standard normal cumulative dis-

tribution function.

Accordingly, the price of a European put option is given by

p(0) = P (0, T )(KN (−d2)− µ(0)N (−d1)). (2.8)

The Equations (2.7) and (2.8) are known as the Black formula, here it is denoted byBlack(0, F, T,K, σ).

Remark 2.3.3. (implied volatility) The Black formula depends on the input param-eter σ (and strike K and maturity T ). However, the volatility σ is typically extractedfrom market data, which gives rise to a notion of implied volatility. The implied volatil-ity of an option is the volatility that is implied by the market price of the option basedon a pricing model like the Black’s model. Given the current market price V of theoption price, the implied volatility σimp is the level of volatility such that the followingequation is satisfied:

Black(0, F, T,K, σimp)− V = 0.

Remark 2.3.4. (volatility surface) For different strikes and maturities, the impliedvolatilities can be calculated numerically by observing the market prices. The mappingfrom strike K and maturity T to implied volatility is known as the volatility surface.The implied volatilities for options which are not directly observed in the market aredetermined by using interpolation methods and using the information from the volatilitysurface. From this volatility surface, the price of options which may not be observed inthe market can be calculated by using the Black formula.

16


If we use the Black formula throughout this thesis, then we refer to Equation (2.7)or Equation (2.8) depending on the type of options.

2.4 Fixed income instruments

In this section, we will introduce the definitions and pricing formulas of fixed incomesecurities which are traded in the financial market. First, we introduce forward rateagreements (FRA), plain vanilla swaps, and in-arrears swaps. Later in this section, wediscuss interest rate caps, floors and swaptions.

2.4.1 Forward rate agreements (FRA)

A forward rate agreement is a fixed income instrument which allows one to lock-in theinterest rate for a future time interval at a certain fixed rate. The formal definition of aforward rate agreement is as follows.

Definition 2.4.1. (Forward Rate Agreement) A Forward Rate Agreement (FRA)for the period [T, T + τ ] is a contract to exchange a payment based on a fixed rate Kagainst a payment based on the time T spot Libor rate with tenor τ . The exchangehappens at time T + τ .

In other words, the fixed rate payer pays out the amount τK and receives τL(T, T, T+τ) at time T + τ . Hence, the value of a FRA at time T + τ from the perspective of thefixed rate payer is given by:

VFRA(T + τ) = τ(L(T, T, T + τ)−K).

Accordingly, the value of the contract at time t < T is given by

VFRA(t) = P (t, T + τ)ET+τt [τ(L(T, T, T + τ)−K)].

Since L(t, T, T + τ) is a martingale under the (T + τ)-forward measure QT+τ , we havethat

VFRA(t) = τP (t, T + τ)(L(t, T, T + τ)−K),

and Equation (2.4) implies the following relation:

VFRA(t) = P (t, T )− P (t, T + τ)− τKP (t, T + τ).

17


2.4.2 Plain vanilla swaps

We introduce the plain vanilla swap, which is a generalization of the FRA.

Definition 2.4.2. (swap) A swap is an agreement between two counterparties to ex-change cash at each pre-defined time date. A plain vanilla interest rate swap is a swapin which one leg is a stream of fixed rate payments and the other a stream of paymentsbased on a floating rate (which is often the LIBOR rate).

More formally, consider a vanilla interest rate swap with a fixed rate K and assumethat both legs follow the tenor structure (i.e, the set of pre-defined time dates for anexchange of cash):

0 ≤ T0 < T1 < T2 < · · · < TN−1 < TN , τn = Tn+1 − Tn. (2.9)

At time Tn+1, the net cash flow of the fixed rate payer (receiver) is given by (on a unitnotional)

τnw(Ln(Tn)−K),

where Ln(t) = L(t, Tn, Tn+1) for n = 0, · · · , N−1. Here, we introduce the payer/receiverindicator w which is defined by

w =

1, for the payer,−1, for the receiver.

Hence, for any t ∈ [0, T0], the price of a payer (receiver) swap is given by

Vswap(t) =N−1∑n=0

τnP (t, Tn+1)En+1t [w(Ln(Tn)−K)]

=N−1∑n=0

τnP (t, Tn+1)w(Ln(t)−K)

= w

(N−1∑n=0

τnP (t, Tn+1)

)(∑N−1n=0 τnP (t, Tn+1)Ln(t)∑N−1

n=0 τnP (t, Tn+1)−K

).

(2.10)

where we define En+1t := ETn+1

t . If we define A(t) and S(t) by

A(t) := A0,N (t) =

N−1∑n=0

τnP (t, Tn+1),

S(t) := S0,N (t) =

∑N−1n=0 τnP (t, Tn+1)Ln(t)∑N−1

n=0 τnP (t, Tn+1),

then we can simplify the formula of a payer (receiver) swap price to

Vswap(t) = wA(t)(S(t)−K). (2.11)

18


The quantity A(·) is called the annuity of the swap or Present Value of a Basis Point(PVBP), and the quantity S(t) is the forward swap rate. The forward swap rate is thefixed rate K which makes Vswap(t) equal to 0. Therefore, the swap rate S(t) is oftencalled the par or break-even rate of the swap.

Another definition of the forward swap rate can also be given with the result in thenext Remark.

Remark 2.4.3. Since Equation (2.4) can be rewritten as

τnLn(t) =P (t, Tn)

P (t, Tn+1)− 1,

we have that

S(t) =

∑N−1n=0 τnP (t, Tn+1)Ln(t)

A(t)

=

∑N−1n=0 P (t, Tn+1)

(P (t,Tn)P (t,Tn+1) − 1

)A(t)

=

∑N−1n=0 (P (t, Tn)− P (t, Tn+1))

A(t)

=P (t, T0)− P (t, TN )

A(t).

We introduce the generalized definition of the annuity A(t) and the forward swaprate S(t) for later reference.

Definition 2.4.4. (annuity rate and swap rate) For any integers k,m satisfying0 ≤ k < N , m > 0 and k +m ≤ N , the annuity rate Ak,m is defined by

Ak,m(t) =

k+m−1∑n=k

P (t, Tn+1)τn,

and the swap rate Sk,m(t) at time t ≤ Tk is defined by

Sk,m(t) =P (t, Tk)− P (t, Tk+m)

Ak,m(t).

Note that the annuity rate is also used as a numeraire because it is the linear com-bination of zero-coupon bonds. Therefore there exist a measure corresponding to theannuity rate, which is defined as follows.

Definition 2.4.5. (swap measure) The swap measure Qk,m is the measure, which isinduced by the annuity factor Ak,m(t) as a numeraire. It gives that

V (t) = Ak,m(t)Ek,mt[V (T )

Ak,m(T )

],

where Ek,mt is a conditional expectation w.r.t. Ft under the swap measure Qk,m.

19


An advantage of using the swap measure is that the swap rate is a martingale underthe swap measure. This is explained in the following lemma.

Lemma 2.4.6. The swap rate Sk,m(t) is a martingale under the measure Qk,m.

Proof. Since P (·, Tk) and P (·, Tk+m) are traded assets, the difference P (·, Tk)−P (·, Tk+m)is also a traded asset. It yields that (P (·, Tk)−P (·, Tk+m))/Ak,m(·) is a martingale underthe swap measure Qk,m. For any T ≤ Tk, we have that

Ek,mt [Sk,m(T )] = Ek,mt[P (T, Tk)− P (T, Tk+m)

Ak,m(T )

]=P (t, Tk)− P (t, Tk+m)

Ak,m(t)= Sk,m(t).

2.4.3 In-arrears swaps

In this subsection, we introduce the in-arrears swaps which are interest rate swaps thatsets and pays interest in arrears. These swaps are complex than the plain vanilla swaps(see Section 2.4.2). The formal definition is given below.

Definition 2.4.7. (in-arrears swap) An in-arrears swap is an interest rate swap inwhich the floating rate is observed and paid on the same day. Consider the tenorstructure:

0 ≤ T0 < T1 < · · · < TN , τn = Tn+1 − Tn,where n ∈ 0, · · · , N−1. At time Tn, the net cash flow of the fixed rate payer (receiver)is given by (on a unit notional):

τnw(Ln(Tn)−K),

where K is the fixed rate and w is the payer/receiver indicator.

At time t ≤ T0, the price of an in-arrears payer swap is given by

V (t) =N−1∑n=0

τnP (t, Tn)Ent [Ln(Tn)−K]

=N−1∑n=0

τnP (t, Tn)(Ent [Ln(Tn)]− τnP (t, Tn)K).

(2.12)

Note that Ln(Tn) is not a martingale under the Tn-forward measure. By using theChange of Numeraire theorem (see Theorem A.1.2 in Appendix A), we have that

Ent (Ln(Tn)) = En+1t

[Ln(Tn)

P (Tn, Tn)/P (0, Tn)

P (Tn, Tn+1)/P (0, Tn+1)

]= En+1

t

[Ln(Tn)

P (Tn, Tn)/P (Tn, Tn+1)

P (0, Tn)/P (0, Tn+1)

]= En+1

t

[Ln(Tn)

1 + τnLn(Tn)

1 + τnLn(0)

]=Ln(t) + τnEn+1

t [L2n(Tn)]

1 + τnLn(0),

(2.13)

20


where Equation (2.4) is used in the third equality. From Equation (2.12) and (2.13), weobtain the following relation:

V (t) =N−1∑n=0

τnP (t, Tn)Ln(t) + τnEn+1

t [L2n(Tn)]

1 + τnLn(0)− τnP (t, Tn)K

=N−1∑n=0

τnP (t, Tn)

(Ln(t) + τnEn+1

t [L2n(Tn)]

1 + τnLn(0)−K

).

(2.14)

Hence, the price of an in-arrears swap at time t = 0 is given by

V (0) =N−1∑n=0

τnP (0, Tn)

(Ln(0) + τnEn+1[L2

n(Tn)]

1 + τnLn(0)−K

). (2.15)

Remark 2.4.8. (convexity adjustment) To derive an analytical formula of En+1[L2n(Tn)],

we need to make an assumption about the distribution of Ln(t). We assume that Ln(t)is log-normally distributed, i.e., Ln(t) follows the SDE given by

dLn(t) = σn(t)Ln(t)dWn+1(t),

where σn(t) is a deterministic function and Wn+1(t) is a Brownian motion under theTn+1-forward measure.

The assumption is equivalent to

Ln(t) = Ln(0) exp

(−1

2

∫ t

0σ2n(s) ds+

∫ t

0σn(s) dWn+1(s)

).

One can show that:

En+1[L2n(Tn)] = En+1

[L2n(0) exp

(−∫ Tn

0σn(s)2 ds+ 2

∫ Tn

0σn(s) dWn+1(s)

)]= L2

n(0) exp

(−∫ Tn

0σ2n(s) ds

)En+1

[exp

(2

∫ Tn

0σn(s) dWn+1(s)

)].

Note that∫ Tn

0 σn(s) dWn+1(s) is normally distributed with zero mean and variance∫ Tn0 σ2

n(s) ds. For a random variable X having a Normal distribution with mean µand variance σ2, the moment generating function of X is well-known to be given by

E[etX ] = eµt+12σ2t2 . (2.16)

Together with Equation (2.16) this yields that:

En+1[L2n(Tn)] = L2

n(0) exp

(−∫ Tn

0σ2n(s) ds

)En+1

[exp

(2

∫ Tn

0σn(s) dWn+1(s)

)]= L2

n(0) exp

(−∫ Tn

0σ2n(s) ds

)exp

(2

∫ Tn

0σ2n(s) ds

)= L2

n(0) exp

(∫ Tn

0σ2n(s) ds

).

21


Hence, the price of an in-arrears swap at time t = 0 is given by the formula:

V (0) =

N−1∑n=0

τnP (0, Tn)

(Ln(0) + τnEn+1(L2

n(Tn))

1 + τnLn(0)−K

)

=N−1∑n=0

τnP (0, Tn)

(Ln(0) + τnL

2n(0) exp(

∫ Tn0 σ2

n(s) ds)

1 + τnLn(0)−K

)

=N−1∑n=0

τnP (0, Tn)

Ln(0) +τnL

2n(0)

1 + τnLn(0)

(exp

(∫ Tn

0σ2n(s) ds

)− 1

)︸︷︷︸

=Cadj

−K

.

(2.17)

Here, the term Cadj is called the convexity adjustment. For more information on con-vexity adjustments, we refer to Hunt and Kennedy [21].

2.4.4 Interest rate caps and floors

In this subsection we introduce a cap or floor which is a popular interest rate option.A cap or floor provides insurance against the interest rates rising above or falling belowa certain level. A cap/floor is a portfolio of call/put options which are also calledcaplets/floorlets.

Definition 2.4.9. (caplets and floorlets) A caplet (floorlet) is a European call (put)option on a LIBOR rate. A caplet (floorlet) on LIBOR rate L(t, T, T + τ) with strike Kpays out at time T + τ

τw(L(T, T, T + τ)−K)+,per unit notional where w is a caplet/floorlet indicator such that w = 1 for a caplet andw = −1 for a floorlet.

Since a caplet (floorlet) is a European call (put) option on a LIBOR rate, we canprice a caplet (floorlet) by the Black formula under the following assumptions. Weassume that L(T, T, T + τ) is log-normally distributed with the forward price µ(t) andvolatility σ. Note that µ(0) = L(0, T, T + τ) because L(t, T, T + τ) is a martingale underthe (T + τ)-forward measure. The price of a caplet (floorlet) at time 0 is given in thefollowing form:

Vcaplet/floorlet(0) = τP (0, T + τ)(wL(0, T, T + τ)N (wd1)− wKN (wd2)),

d1 =log(L(0, T, T + τ)/K) + σ2T/2

σ√T

,

d2 = d1 − σ√T .

For later reference, we call the above equation as the Black formula for caplets (floorlets).Now, we define caps and floors as follows.

22


Definition 2.4.10. A cap is a strip of caplets and a floor is a strip of floorlets.

The price of an N -period cap (floor) at time 0 is simply the sum of time 0-value of Ncaplets. Hence, using the Black formula, we price a cap(floor) with strike K at time 0by

Vcap/floor(0) =

N−1∑n=0

P (0, Tn+1)(wLn(0)N (wd1)− wKN (wd2))

d1 =log(Ln(0)/K) + σ2

nT/2

σn√T

,

d2 = d1 − σn√T .

where σn is volatility corresponding to Ln(t). To price a cap (floor), we consider twoapproaches. One approach is to use different volatility for each caplet (floorlet), whichis referred to as spot volatility. The other approach is to use the same volatility for allcaplets (floorlets) comprising any particular cap (floor) but to vary this volatility accord-ing to the life of the cap (floor). These volatilities are referred to as flat volatilities. Theflat volatilities are usually quoted in the market. The spot volatilities can be strippedfrom these quotes.

2.4.5 European swaptions

In this subsection, we introduce another popular interest option called the (European)swaptions. The definition of a European swaption is given below.

Definition 2.4.11. (swaptions) A European swaption gives the holder the right toenter into a swap at a future date at a given fixed rate K. A payer (receiver) swaptionis an option where the underlying is a payer (receiver) swap.

From Equation (2.11) and assuming that the underlying swap starts on the expirydate T0 of the option, the payoff of a swaption at time T0 is given by

(Vswap(T0))+ = (wA(T0)(S(T0)−K))+.

Under the swap measure Q0,m, the price of a swaption at time 0 can be calculated by

Vswaption(0) = A(0)E0,m

[(wA(T0)(S(T0)−K))+

A(T0)

]= A(0)E0,m[(w(S(T0)−K))+].

For a swaption, there exists a Black-like formula. More precisely, the price of a swaptionwith strike K at time 0 is given by

Vswaption(0) = A(0)(wS(0)N (wd1)− wKN (wd2)),

d1 =log(S(0)/K) + σ2T0/2

σ√T0

,

d2 = d1 − σ√T0.

For later reference, we call the above equation as the Black formula for swaptions.

23

Chapter 3

The LIBOR Market Model (LMM)

In this chapter, we introduce the LIBOR Market Model which is one of the most popularinterest rate models. We give an overview of the relevant chapters in Andersen andPiterbarg (see [5]). In Section 3.1 we establish notations and some preliminary results todescribe the LMM and briefly discuss the standard LMM or BGM model [9]. A drawbackof the BGM model is that is not able to capture skew and curvature in the impliedvolatility. Therefore, in Section 3.2, we introduce a LMM with stochastic volatility inorder to overcome the shortcoming of the standard BGM model.

3.1 LIBOR market model dynamics and measures

Before we study the LIBOR market model, we will define a tenor structure. For a fixedN ∈ N, we define the following tenor structure:

0 = T0 < T1 < · · · < TN , τn = Tn+1 − Tn, (3.1)

where n = 0, · · · , N − 1. The distance τn is usually set to 0.25 or 0.5 (correspondingto 3 or 6 months). We focus on a finite set of zero-coupon bonds P (t, Tn), for the setn : t < Tn ≤ TN. As t moves forward this set shrinks. It is useful to introduce theso-called index function q(t) which satisfies

Tq(t)−1 ≤ t < Tq(t).

The index function q(t) can be seen as the index of the first zero-coupon bond that hasnot expired by time t. For any Tn > t, the price of a zero-coupon bond is given by

P (t, Tn) = P (t, Tq(t))

n−1∏i=q(t)

1

(1 + τiLi(t)).

Note that we can also rewrite the asset price process B(t) as

B(t) = P (t, Tq(t))

q(t)−1∏n=0

(1 + τnLn(Tn)). (3.2)

24

CHAPTER 3. THE LIBOR MARKET MODEL (LMM)

Now, we consider the set of forward Libor rates Lq(t)(t), Lq(t)+1(t), · · · , LN−1(t). Recall

that for all n ≥ q(t), Ln(t) is a martingale under the Tn+1-forward measure QTn+1 .Hence, the dynamics have to be driftless. It leads us to assume that the dynamics offorward rates are of the following form:

dLn(t) = σn(t)>dWn+1(t), (3.3)

where Wn+1(t) = W Tn+1(t) is an m-dimensional standard Brownian motion under theTn+1-forward measure QTn+1 and σn is an m-dimensional process adapted to the filtra-tion generated by Wn+1(t). In Subsection 3.1.1, we will discuss the details of the processσn.

In the following Lemma, we establish the dynamics of the Libor forward rate Ln(t)under the neighbouring Tn-forward measure.

Lemma 3.1.1. Assume that Ln(t) is given by Equation (3.3). The dynamics of Ln(t)under the Tn-forward measure QTn are given by

dLn(t) = σn(t)>(

τnσn(t)

1 + τnLn(t)dt+ dWn(t)

), (3.4)

where Wn(t) is an m-dimensional standard Brownian motion under the Tn-forward mea-sure QTn

Proof. By the Change of Numeraire theorem (see Theorem A.1.2 in Appendix A), wehave that

ς(t) = E(n+1)t

(dQTn

dQTn+1

)=

P (t, Tn)/P (0, Tn)

P (t, Tn+1)/P (0, Tn+1)= (1 + τnLn(t))

P (0, Tn+1)

P (0, Tn),

where the last equality follows from Equation (2.4). Since P (0, Tn+1), P (0, Tn) and τnare known for a given n, we have that

dς(t) =P (0, Tn+1)

P (0, Tn)τndLn(t) =

P (0, Tn+1)

P (0, Tn)τnσn(t)>dWn+1(t). (3.5)

Dividing Equation (3.5) by ς(t), we obtain the relation

dς(t)

ς(t)=

τnσn(t)>

1 + τnLn(t)dWn+1(t),

and Girsanov’s theorem (see Theorem A.1.3 in Appendix A) implies that

dWn(t) = dWn+1(t)− τnσn(t)

1 + τnLn(t)dt, (3.6)

where dWn(t) is a Brownian motion under the Tn-forward measure QTn . Hence, togetherwith Equation (3.3) it yields that

dLn(t) = σn(t)>(

τnσn(t)

1 + τnLn(t)dt+ dWn(t)

).

25


Lemma 3.1.1 gives an idea how to connect the Tn+1-forward measure QTn+1 to theTn-forward measure QTn . Using Lemma 3.1.1 iteratively, we can represent the dynamicsof Ln(t) under the terminal measure QTN which is the TN -forward measure.

Lemma 3.1.2. Assume that the dynamics of Ln(t) are given by Equation (3.3). Underthe terminal measure QTN , the dynamics of Ln(t) are given by

dLn(t) = σn(t)>

− N−1∑j=n+1

τjσj(t)

1 + τjLj(t)+ dWN (t)

. (3.7)

Proof. Since Equation (3.6) holds for all n = q(t), · · · , N−1, we can apply this recursivelysuch that

dWN (t) = dWN−1(t) +τN−1σN−1(t)

1 + τN−1LN−1(t)dt

= dWN−2(t) +

N−1∑j=N−2

τjσj(t)

1 + τjLj(t)dt

...

= dWn+1(t) +N−1∑j=n+1

τjσj(t)

1 + τjLj(t)dt,

which proves Equation (3.7).

The dynamics of Ln(t) can also be written under the spot measure (see Section 2.2.2).The result is stated in the following lemma.

Lemma 3.1.3. Under the spot measure QB, the process Ln(t) is given by

dLn(t) = σn(t)>

n∑j=q(t)

τjσj(t)

1 + τjLj(t)dt+ dWB(t)

, (3.8)

where WB(t) is an m-dimensional Brownian motion under the spot measure QB.

Proof. For convenience, recall that the asset price process B(t) is given by (see Equation(3.2))

B(t) = P (t, Tq(t))

q(t)−1∏n=0

(1 + τnLn(Tn)).

At any time t, the random part of B(t) is P (t, Tq(t)). Hence, we need to derive the

dynamics under the Tq(t)-forward measure QTq(t) . Applying Lemma 3.1.1 recursively, weobtain that

dWn+1(t) =

n∑j=q(t)

τjσj(t)>

1 + τjLj(t)dt+ dW q(t)(t).

For the more rigorous proof, we refer to Jamshidian [23].

26


Remark 3.1.4. (Terminal measure and spot measure) For the simulation of Ln(t),we can consider the dynamics under the terminal measure or under the spot measure.Under the terminal measure, the number of terms in the drift summation of Ln(t) isalways N − n − 1 for all t ≤ Tn. This means that for m < n, Lm(t) is possibly morebiased than Ln(t). However, the number of terms in the drift summation of Ln(t) underthe spot measure is given by n − q(t) + 1. It means that the number of terms in thedrift summation of Ln(t) decreases as t increases. Hence, the possible bias from thediscretization of the drift is more evenly distributed among the forward rates. Thismotivates us to consider the dynamics of Ln(t) under the spot measure instead of theterminal measure.

3.1.1 The choice of σn(t)

We assume that the volatility process σn(t) is separable in a deterministic part and anon-deterministic part. The reason why we assume the separable form is that it makesthe model analytically tractable, which allows for the derivation of closed-form pricingformulas for plain vanilla options. The existence of closed-form formulas allows for anefficient calibration of the model to plain vanilla derivatives.

Under the separability assumption, the process σn(t) is given by:

σn(t) = λn(t)ϕ(Ln(t)), (3.9)

where λn(t) is a bounded vector-valued deterministic function and ϕ : R → R is atime-homogeneous local volatility function.

Proposition 3.1.5. Assume that Equation (3.9) holds with ϕ(0) = 0 and that Ln(0) ≥ 0for all n. Furthermore, assume that ϕ is locally Lipschitz continuous and satisfies thegrowth condition

ϕ(x)2 ≤ C(1 + x2), x > 0,

or equivalently,|ϕ(x)| ≤ C ′(1 + x), x > 0,

where C and C ′ are some positive constants. Then, non-explosive, pathwise uniquesolutions of the no-arbitrage SDEs for Ln(t), q(t) ≤ n ≤ N −1, exist under all measuresQTi, q(t) ≤ i ≤ N . If Ln(0) > 0, then Ln(t) stays positive at all t.

Proof. We refer the reader to Piterbarg [5].

There are some standard parameterizations of ϕ. The most popular one is ϕ(x) = xwhich implies a log-normal formulation of the Libor market model and is also known asthe BGM model. We will briefly discuss this model in the next subsection. The otherpossible choice of ϕ(x) is bx + a with a 6= 0 and 0 < b ≤ 1. It is called the displacedlog-normal formulation. We will assume the displaced log-normal formulation for theLIBOR market model with stochastic volatility. The stability properties related to thedisplaced log-normal specification are stated in the following lemma.

27


Lemma 3.1.6. Consider a local volatility Libor market model with local volatility func-tion ϕ(x) = bx + a, where 0 < b ≤ 1 and a 6= 0. Assume that bLn(0) + a > 0 anda/b < τ−1

n for all n = 1, 2, · · · , N − 1. Then non-explosive, pathwise unique solutionsof the no-arbitrage SDEs for Ln(t), q(t) ≤ n ≤ N − 1, exist under all measure QTi,q(t) ≤ i ≤ N . All Ln(t) are bounded from below by −a/b.


More specifically, if we assume b > 0, a = (1 − b)Ln(0) and (1 − b)Ln(0)/b < τ−1n ,

then we have non-explosive, pathwise unique solutions of the SDEs for Ln(t).

3.1.2 Brace-Gatarek-Musiela (BGM) model

As we briefly discussed in the previous section, we call the Libor market model with localvolatility function ϕ(x) = x the “log-normal forward LIBOR model”. It is also known asthe Brace-Gatarek-Musiela (BGM) model [9]. The dynamics of the log-normal forwardrate are given by

dLn(t) = Ln(t)λn(t)> dWn+1(t), (3.10)

where λn : R → Rm is a bounded deterministic function and Wn+1 is a m-dimensionalstandard Brownian motion under the Tn+1-forward measure. For the BGM model, thepricing formula for a capet is known as follows.

Theorem 3.1.7. The time 0 price of a caplet on forward rate Ln(t) is given by

Vcaplet(0) = τnP (0, Tn+1)(Ln(0)N (d1)−KN (d2)), (3.11)

where

d1 =log(Ln(0)/K) + v2/2

v,

d2 =log(Ln(0)/K)− v2/2

v,

and

v2 =

∫ Tn

0||λn(t)||2 dt.

Proof. For the proof, we refer to Brace, Gatarek and Musiela [9].

Note that Equation (3.11) corresponds to the Black formula for caplets with σ2Tn =v2(t).

28


3.2 LMM with displaced diffusion and stochastic volatility (LMM-DDSV)

A drawback of the BGM model is that it is less flexible in recovering essential charac-teristics of interest rate markets, in particular, the volatility smile. More precisely, itimplies a flat implied volatility curve (see Figure 3.1a), but in many markets we observethat the volatility smile has a skewed shape such that the volatility curve is downwardsloping for smaller strikes and has some curvature for larger strikes (see Figure 3.1b1).Capturing the volatility skew is important in pricing derivatives, especially for exoticproducts. To overcome the shortcomings of the BGM model, we introduce the Libormarket model with stochastic volatility.

−200 −150 −100 −50 0 50 100 150 2000.15

0.16

0.17

0.18

0.19

0.2

0.21

0.22

0.23

0.24

0.25

ATM + x(bps)

Impl

ied

Vol

atili

ty

(a) Implied volatility from the BGM model

−200 −150 −100 −50 0 50 100 150 200

0.16

0.17

0.18

0.19

0.2

0.21

0.22

0.23

0.24

0.25

ATM + x(bps)

Impl

ied

Vol

atili

ty

(b) Market implied volatilities (April 2011)

Figure 3.1: Flat versus skewed implied volatility curves

In order to generate curvature in the implied volatilities we introduce a mean-revertingscalar process z(t), with dynamics of the form

dz(t) = θ(z0 − z(t))dt+ η√z(t) dZB(t), z(0) = z0 = 1, (3.12)

where θ, z0, and η are positive constants, and ZB is a Brownian motion under the spotmeasure QB. This process is also known as the Cox-Ingersoll-Ross(CIR) process. Theprocess z has the role of a scaling factor so typically z(0) = z0 = 1. Given the varianceprocess, we assume that the forward rate processes Ln(t) in QB follows, for all n ≥ q(t),

dLn(t) =√z(t)ϕ(Ln(t))λn(t)>(

√z(t)µn(t) dt+ dWB(t)),

µn(t) =

n∑j=q(t)

τjϕ(Lj(t))λj(t)

1 + τjLj(t).

(3.13)

Note that the above formulation can be seen as Equation (3.8) with

σn =√z(t)ϕ(Ln(t))λn(t).

1Source: ING

29


In order to derive pricing formulas, it is convenient to consider the dynamics of theforward rates under the Tn+1-forward measure. Since Ln(t) is a martingale (driftless)under the Tn+1-forward measure QTn+1 , we should have

dLn(t) =√z(t)ϕ(Ln(t))λn(t)>dWn+1(t).

It means that under the Tn+1-forward measure, the dynamics of Ln(t) are given by:

dWn+1(t) =√z(t)µn(t) dt+ dWB(t).

where Wn+1(t) is a m-dimensional Brownian motion under the Tn+1-forward measure.

By defining the m-dimensional vector

a(t) = d〈Z,WB〉(t)/dt,

we can writedZ(t) = a(t)>dWB(t) +

√1− ||a(t)||2 dW (t),

where W (t) is a scalar Brownian motion independent of WB(t). Under the Tn+1-forwardmeasure QTn+1 , we have that

dZ(t) = a(t)>(dWn+1(t)−√z(t)µn(t)dt) +

√1− ||a(t)||2 dW (t)

= dZn+1(t)− a(t)>√z(t)µn(t) dt,

where dZn+1(t) = a(t)>dWn+1(t) +√

1− ||a(t)||2dW (t).

The process z(t) under Tn+1-forward measure QTn+1 , n ≥ q(t)− 1, is given by:

dz(t) = θ(z0 − z(t)) dt+ η√z(t)(−

√z(t)µn(t)>d〈Z,WB〉(t) + dZn+1(t)), (3.14)

where Zn+1(t) is a Brownian motion in measure QTn+1 . Note that if Z(t) and WB(t)are independent, that is, d〈Z,WB〉(t) = 0, then the term

√z(t)µn(t)>d〈Z,WB〉(t) in

Equation (3.14) is equal to 0. It simplifies the dynamics of z(t) :

dz(t) = θ(z0 − z(t)) dt+ η√z(t) dZn+1(t),

since dZ(t) = dZn+1(t). We assume that Z(t) and WB(t) are independent, and we usethe displaced log-normal formulation for the local volatility function ϕ(x) (specifically,ϕ(Ln(t)) = bLn(t) + (1 − b)Ln(0) for 0 < b ≤ 1) because it is tractable and it givesthe possibility to control the slope of the implied volatility skew with the parameter b.Now we turn to pricing of interest rate derivatives under the following model (called theLMM-DDSV: LMM with Displaced Diffusion and Stochastic Volatility).

Model: The LMM-DDSV

dLn(t) =√z(t)(bLn(t) + (1− b)Ln(0))λn(t)>dWn+1(t),

dz(t) = θ(z0 − z(t)) dt+ η√z(t) dZn+1(t), z0 = z(0) = 1

d〈Zn+1,Wn+1〉(t) = 0

(3.15)

30


3.2.1 Rank reduction

Short rate models for interest rates imply the perfectly correlated movement of forwardrates. However, in practice, various points on the forward rates curve do not moveco-monotonically with each other. To incorporate this observation, the LIBOR marketmodel uses a multi-dimensional Brownian motion where the components drives eachforward rate. In general, three or four dimensions are sufficient to capture the most ofvariation of the points on the forward curve (see Andersen and Piterbarg [5]). It gives usthe motivation to choose the dimension m of Brownian motion of forward rate dynamics(see Equation (3.3)) smaller than the total number N − 1 of forward rates in the tenorstructure.

Note that from the dynamics given by (3.13), we have

d〈Lk, Lj〉(t) = z(t)ϕ(Lk(t))ϕ(Lj(t))λk(t)>λj(t) dt.

This implies that the instantaneous correlation between Lk(t) and Lj(t) is given by

ρk,j(t) =d〈Lk, Lj〉(t)√

d〈Lk, Lk〉(t)d〈Lj , Lj〉(t)=

λk(t)>λj(t)

||λk(t)|| ||λj(t)||. (3.16)

When m = 1, we can see that ρk,j(t) = 1, that is, Lk(t) and Lj(t) are fully correlated.As the number of Brownian motion increases, we are able to capture more complicatedcorrelation structures, but it also increases the complexity of the model. Once we get asample variance-covariance matrix from empirical data, the smallest m which is sufficientto duplicate correlation can be found by the tools of principal components analysis(PCA). In the next section we present the PCA briefly.

3.2.2 The principal components analysis (PCA)

In this section we present a general discussion on PCA. For more details, we refer toAndersen and Piterbarg [4]. Consider a p-dimensional Gaussian variable Z with a givencovariance matrix Σ. Without loss of generality, we assume that the mean of Z is 0 andthat Σ has full rank (positive definite). Our aim is to find a (p×m)-dimensional matrixD such that

Zd≈DX, (3.17)

where X is an m-dimensional vector of independent standard Gaussian variables, m ≤ p.Define

D∗ := arg min f(D) = tr(

(Σ−DD>)(Σ−DD>)>).

where tr(A) is the trace of a matrix A. It is shown that [4]:

D∗ = Vm√Em,

31


where Em is an m ×m diagonal matrix containing the m largest eigenvalues of Σ, andVm is a p×m matrix of m p-dimensional eigenvectors corresponding to the eigenvaluesin Em. Using D∗ in Equation (3.17), we have

Z ≈ Z := Vm√EmX =

√e1v1X1 +

√e2v2X2 + · · ·+

√emvmXm,

where vi denotes the i-th column of Vm and the ej ’s are the eigenvalues, sorted indecreasing order of magnitude. The vi is called the i-th principal component of Z, andthe variable

√eiXi as the i-th principal factor.

A sample variance-covariance matrix

In this subsection, we introduce some ways to obtain the empirical instantaneous corre-lations between forward rates from market data, described in Andersen and Piterbarg[5]. For a fixed τ , we define “sliding” forward rates l(t, x) with tenor x as

l(t, x) = L(t, t+ x, t+ x+ τ).

For a given set of tenors x1, · · · , xNx and a given set of calendar times t0, t1, · · · , tNt , wedefine the Nx ×Nt matrix O with

Oi,j =l(tj , xi)− l(tj−1, xi)√

tj − tj−1, i = 1, · · · , Nx, j = 1, · · · , Nt

where√tj − tj−1 is the annualized factor. Assuming time-homogeneity and ignoring

small drift terms, a sample variance-covariance matrix C can be obtained by

C =OO>

Nt.

Introducing the diagonal matrix c such that

ci,j =

√Ci,j , if i = j,0, if i 6= j,

the empirical forward rate correlation matrix R is given by: R = c−1Cc−1. In general,we expect that the correlation matrix R = [rij ], i, j = 1, · · · , Nx satisfies the following:

• The matrix R is real and symmetric

• rii = 1

• The matrix R is positive semi-definite

• i 7→ rij is decreasing

• i 7→ ri+p,i is increasing for fixed p ∈ 1, · · · , Nx − 1

32


The empirical correlation matrix R is often relatively noisy in a sense that it might notsatisfy one of these conditions. Hence, it is convenient to work with a parametric form.A possible parametric form (see Rebonato [30]) is given by

ρj,k(t) = q(Tk − t, Tj − t),

where

q(x, y) = ρ∞ + (1− ρ∞) exp(−|x− y|(β exp(α(min(x, y)))). (3.18)

where −1 ≤ ρ∞ ≤ 1, β > 0 and α ∈ R. We can use the parameters ξ∗ = (α∗, β∗, ρ∗∞)which satisfies

ξ∗ = arg minξ

(tr(

(R−R1(ξ))(R−R1(ξ))>))

,

where R is the empirical correlation matrix, R1(ξ) is the correlation matrix generated byξ, and the solution for ξ∗ minimizes the Frobenius norm. Variations of this parametricform are also possible. An other alternative to obtain the matrix R is to calibrate R tothe prices of spread options in the market (see [5]).

3.2.3 Construction of λk(t)

As we have seen in Equation (3.16), λk(t) is closely related to correlation and volatility||λk(t)|| of forward rates. It implies that we need to find λk(t) which determine theoverall correlation and volatility structure of forward rates in the model. We assumethat

λk(t) = h(t, Tk − t), ||λk(t)|| = g(t, Tk − t),

where h : R2+ → Rm and g : R2

+ → R+. For the function g, we use

g(t, x) = g(x) = (a+ bx)e−cx + d, a, b, c, d ∈ R+. (3.19)

Usually, the function ||λk(t)|| is considered to be piecewise constant in t, with disconti-nuities at Tn, n = 1, · · · , N − 1,

||λk(t)|| =k∑

n=1

1Tn−1≤t<Tn||λn,k|| =k∑

n=1

1q(t)=n||λn,k||,

To find ||λn,k||, we use a bilinear interpolation method. Given a grid of times and tenorstj × xj, i = 1, · · · , Nt, j = 1, · · · , Nx, we introduce a matrix G such that

Gi,j = (a+ bxj)e−cxj + d.

Then, ||λn,k|| is given by

||λn,k|| = w++Gi,j + w+−Gi,j−1 + w−+Gi−1,j + w−−Gi−1,j−1,

33


and by denoting τn,k = Tk − Tn−1, with

i = mina : ta ≥ Tn−1, j = minb : xb ≥ τn,k,

w++ =(Tn−1 − ti−1)(τn,k − xj−1)

(ti − ti−1)(xj − xj−1), w+− =

(Tn−1 − ti−1)(xj − τn,k)(ti − ti−1)(xj − xj−1)

,

w−+ =(ti − Tn−1)(τn,k − xj−1)

(ti − ti−1)(xj − xj−1), w−− =

(ti − Tn−1)(xj − τn,k)(ti − ti−1)(xj − xj−1)

.

For each Tn, let the matrix R(Tn) be an (N − n) × (N − n) instantaneous correlationmatrix such that

(R(Tn))i,j = ρi,j(Tn−), i, j = n, · · · , N − 1,

and define a diagonal volatility matrix c(Tn) with elements

(c(Tn))i,j =

||λn,n+j−1||, for i = j,

0, for i 6= j,

where j = 1, · · · , N − n. Given R(Tn) and c(Tn), an instantaneous covariance matrixC(Tn) for forward rates is given by

C(Tn) = c(Tn)R(Tn)c(Tn). (3.20)

Suppose that the (N − n)×m matrix H(Tn) consists of elements

(H(Tn))j,i = hi(Tn, Tn+j−1 − Tn),

for j = 1, · · · , N − n and i = 1, · · · ,m. Then, it follows that

C(Tn) = H(Tn)H(Tn)>.

Together with Equation (3.20), it yields

H(Tn)H(Tn)> = c(Tn)R(Tn)c(Tn).

Applying the PCA decomposition, the matrix R(Tn) can be written as

R(Tn) = D(Tn)D(Tn)>. (3.21)

Hence, we getH(Tn) = c(Tn)D(Tn).

Assuming piecewise constant interpolation of λk(t) for t, the full set of factor volatilitiesλk(t) can be constructed for all t and Tk.

34


Remark 3.2.1. When we apply PCA decomposition on a correlation matrix, the ap-proximated matrix is in general not a valid correlation matrix anymore, meaning thatthe diagonal elements are not equal to 1. Hence, we need to rescale the approximatedmatrix to make sure that it has diagonal elements equal to 1. More precisely, the matrixD(Tn) in Equation (3.21) is

D(Tn) =

v11 v21 · · · vm1

v12 v22 · · · vm2...

.... . .

...v1N−n v2N−n · · · vmN−n

√e1 0 · · · 00

√e2 · · · 0

......

. . ....

0 0 · · · √em

=

v11√e1 v21

√e2 · · · vm1

√em

v12√e1 v22

√e2 · · · vm2

√em

......

. . ....

v1N−n√e1 v2N−n

√e2 · · · vmN−n

√em

where vi = (vi1, vi2, · · · , viN−n)> is an eigenvector corresponding to the eigenvalue ei.

Then, the approximating matrix is given by

R(Tn) = D(Tn)D(Tn)>

=

v11√e1 v21

√e2 · · · vm1

√em

v12√e1 v22

√e2 · · · vm2

√em

......

. . ....

v1N−n√e1 v2N−n

√e2 · · · vmN−n

√em

v11√e1 v12

√e1 · · · v1N−n

√e1

v21√e2 v22

√e2 · · · v2N−n

√e2

......

. . ....

vm1√em vm2

√em · · · vmN−n

√em

=

∑m

k=1 v2k1ek

∑mk=1 vk1vk2ei · · ·

∑mk=1 vk1vkN−nek∑m

k=1 vk2vk1ek∑m

k=1 v2k2ek · · ·

∑mk=1 vk2vkN−nek

......

. . ....∑m

k=1 vkN−nvk1ek∑m

k=1 vkN−nvk2ek · · ·∑m

k=1 v2kN−nek

.

To make the diagonal elements equal to 1, we consider a rescaled correlation matrixR(Tn) with elements

(R(Tn))ij = rij =rijrii

=

∑mk=1 vkivkjek∑mk=1 v

2kiek

,

where rij is a element of the matrix R(Tn) or equivalently, the rescaled decompositionmatrix D(Tn) has elements

dij =vji√ej√∑m

k=1 v2kiek

.

35


The matrix form of forward rate dynamics

From the definition of λn(t), we have

λn(t) =

||λn(t)||v1n

√e1

||λn(t)||v2n√e2

...||λn(t)||vmn

√em

,

and this implies that

λ>n (t)λj(t) =(||λn(t)||v1n

√e1 ||λn(t)||v2n

√e2 · · · ||λn(t)||vmn

√em)||λj(t)||v1j

√e1

||λj(t)||v2j√e2

...||λj(t)||vmj

√em

= ||λn(t)||ρn,j ||λj(t)||,

where ρn,j is the correlation between Ln and Lj . The dynamics of Ln(t) are equivalentto

dLn(t) =√z(t)(bLn(t) + (1− b)Ln(0))||λn(t)||

· (√z(t)µ′n(t) dt+

(v1n√e1 v2n

√e2 · · · vmn

√em)dWB(t)),

where

µ′n(t) =n∑

j=q(t)

τj(bLj(t) + (1− b)Lj(0))ρn,j ||λj(t)||1 + τjLj(t)

.

We can also consider the set of forward rates dynamics. The matrix form of the dynamicsare given by dL1(t)

...dLN−1(t)

=√z(t)

ϕ(L1(t))||λ1(t)|| · · · 0...

. . ....

0 · · · ϕ(LN−1(t))||λN−1(t)||

·

√z(t) µ′1(t)

...µ′N−1(t)

dt+

v11 · · · vm1...

. . ....

v1N−1 · · · vmN−1

√e1 · · · 0...

. . ....

0 · · · √em

dW

B1 (t)...

dWBm (t)

,

(3.22)

whereϕ(Ln(t)) = bLn(t) + (1− b)Ln(0),

and WBk is the k-th component of an m-dimensional Brownian motion WB.

36

Chapter 4

Pricing IR Derivatives under theLMM-DDSV Model

In this chapter, we derive pricing formulas under the Libor Market Model with DisplacedDiffusion and Stochastic Volatility (see Andersen and Piterbarg [5]). The LMM-DDSVmodel is closely related to the Heston model [17], which is one of the most well-knownstochastic volatility models for equities. A benefit of using the Heston model is that incase of constant parameters there exist analytical pricing formulas for European options.Before we present the pricing formula for caplets and swaptions, we first introduce theHeston model in Section 4.1. In Sections 4.2 and 4.3 we derive pricing formulas for capletsand swaptions under the LMM-DDSV model with constant (time-homogeneous) param-eters, and in Section 4.4 we extend the LMM-DDSV model with time-dependent param-eters. The time-dependent LMM-DDSV model is approximated by a time-homogeneousmodel with averaged parameters, obtained by a parameter averaging method. Finally,in Section 4.5 we investigate the impact of parameters on the implied volatility curve,and in Section 4.6 we discuss simulation methods for the LMM-DDSV model.

4.1 The Heston model

We start with the dynamics of an asset price process X(t) under the Heston model.They are given by:

dX(t) = µX(t)dt+√z(t)X(t)dW (t),

dz(t) = θ(z0 − z(t))dt+ η√z(t)dZ(t), z(0) = z0 = 1,

(4.1)

where Z(t) and W (t) are Brownian motions under a probability measure P and withd〈Z,W 〉(t) = ρ dt. This model has been extensively studied in the literature and Heston[17] suggested a technique to derive a closed-form solution to price of a European (call)option under the stochastic volatility model, where the asset price process is driven by

37

CHAPTER 4. PRICING IR DERIVATIVES UNDER THE LMM-DDSV MODEL

the dynamics of Equation (4.1). We can match the LMM-DDSV model given by Equa-tion (3.15) to the Heston model after some transformations. Transformation techniquesare described in later sections (Section 4.2 and Section 4.3) and allow us to use theseresults to derive pricing formulas for caplets and swaptions. In this subsection, we willinvestigate the Heston model before we derive analytical pricing formulas for a capletand a swaption.

Following Piterbarg [29], we consider the following modified dynamics (set µ = 0 inEquation (4.1):

dX(t) =√z(t)λ(t)X(t) dW (t), X(0) = 1,

dz(t) = θ(z0 − z(t)) dt+ η(t)√z(t) dZ(t), z(0) = z0 = 1,

(4.2)

where λ(t) and η(t) are deterministic functions and d〈Z,W 〉(t) = 0 to avoid undesirableeffects of common measure changes on the stochastic variance process when correlationis non-zero. For λ(t), it is assumed to be continuous and bounded on time interval [0, T ]for tractability. The expectation part of the pricing formula of a T -maturity call optionwith strike K, EP(X(T ) −K)+, can be obtained by using the Fundamental Transformtechnique (see Lewi [26]).

Before we present the result, we define the moment generating function of logX(t)by:

ΨX(u, t) := EP(eu logX(t)

).

Since dX(t)/X(t) =√z(t)λ(t) dW (t), we have that

lnX(t) =

∫ t

0λ(s)

√z(s) dW (s)− 1

2

∫ t

0λ(s)2z(s) ds,


ΨX(u, t) = EP(eu logX(t)

)= EP

(eu(∫ t0 λ(s)

√z(s) dW (s)− 1

2

∫ t0 λ(s)2z(s) ds)

)= EP

(eu∫ t0 λ(s)

√z(s) dW (s)− 1

2u2∫ t0 λ(s)2z(s) dse

12u(u−1)

∫ t0 λ(s)2z(s) ds

)= EP

(ς(t)e

12u(u−1)


),

whereς(t) = eu

∫ t0 λ(s)

√z(s) dW (s)− 1

2u2∫ t0 λ(s)2z(s) ds.

The function ς(t) is in fact an exponential martingale. From Girsanov’s theorem (seeA.1.3 in Appendix A), we have

ΨX(u, t) = EP(ς(t)e

12u(u−1)


)= Ψλ2z

(1

2u(u− 1), t

),

38


whereΨλ2z(v, t) := EP

(ev∫ t0 λ(s)2z(s) ds

). (4.3)

Let us defineG(t, z) := EP

(ev∫ T0 λ(s)2z(s) ds|z(t) = z

),

then, by applying the Feynman-Kac formula, we can derive the PDE which is affine inz (see [4]). The solution of the PDE can be made such that

G(t, z) = exp(A(t, T ) + z(t)B(t, T )),

where A(t, T ) and B(t, T ) satisfy the following relationship:

d

dtA(t, T ) + z

d

dtB(t, T ) + (θz0 − θz)B(t, T ) +

η2

2zB(t, T )2 + vλ(t)2z = 0.

Since G(0, z0) = Ψλ2z(v, u;T ), we have that

Ψλ2z(v, T ) = exp(A(0, T ) + z0B(0, T ))

where

d

dtA(t, T ) + θz0B(t, T ) = 0,

d

dtB(t, T )− θB(t, T ) +

η2(t)

2B(t, T )2 + vλ(t)2 = 0,

with the terminal condition (A(T, T ), B(T, T )) = (0, 0). Under the assumption thatλ(t) = λ and η(t) = η are both constant, the Riccati ODE’s can be solved analytically.It will be convenient to address the result here.

Lemma 4.1.1. Assuming that λ(t) = λ and η(t) = η, Ψλ2z(v, T ) is given by

Ψλ2z(v, T ) = Ψz(vλ2, T ),

where Ψz(v, T ) := EP(ev∫ T0 z(s) ds) = exp(A(v, T ) + z0B(v, T )), with

A(v, Tn) =θz0

η2

[2 log

(2γ

θ + γ − e−γTn(θ − γ)

)+ (θ − γ)Tn

],

B(v, Tn) =2v(1− e−γTn)

(θ + γ)(1− e−γTn) + 2γe−γTn,

and γ = γ(v) =√θ2 − 2η2v.

Proof. We refer to Andersen and Piterbarg [4].

39


Note that the z(t) is an affine process. The process X = Xt : 0 ≤ t ≤ T is calledaffine if the Ft conditional characteristic function of X(T ) is exponential affine in X(t),for all t ≤ T . That is, there exist functions A(u, t) and B(u, t) satisfying

E[euX(T )|Ft] = eA(u,T−t)+B(u,T−t)X(t).

If the dynamics of the process X(t) are assumed to be

dX(t) = (b(t) + β(t)X(t))dt+√a(t) + α(t)X(t) dW (t),

where a(t), b(t), α(t) and β(t) are deterministic continuous functions, then the functionsA and B satisfy the Riccati ODE’s,

d

dtA(u, T ) =

1

2a(t)B2(t, T )− b(t)B(t, T )

d

dtB(t, T ) = −1

2α(t)B2(t, T ) + β(t)B(t, T )− 1

with the terminal conditions (A(T, T ), B(T, T )) = (0, 0).

Now, we state the analytical pricing formula of the European call option under theHeston model in the following Theorem. This theorem will be used in deriving pricingformula of caplets and swaptions because caplets and swaptions are European call optiontypes.

Theorem 4.1.2. Let X(t) be the asset price process defined by Equation (4.2). Thetheoretical fair price for a European call option with strike K and maturity T under theHeston model is given by

EP(X(T )−K)+ = Black(0, X, T,K, σ)

− K

π

∫ ∞0

Re

(e−(iw+α) log(K)

(α+ iw)(1− α− iw)q(α+ iw)

)dw,

where α is usually set to 1/2 and with

q(u) = ΨX(u, T )−Ψ0X(u, T ),

Ψ0X(u, T ) = exp

(1

2z0u(u− 1)

∫ T

0λ(s)2 ds

).

The Black formula is given by

Black(0, X, T,K, σ) := X(0)N (d1)−KN (d2),

with ν2 = z0

∫ T0 λ2(t) dt, d1 =

log(X(0)/K)+ 12ν2

ν , d2 = d1 − ν, and σ = ν√T

.

Proof. For the proof, we refer the reader to Appendix A.2.

40


Remark 4.1.3. By setting α = 1/2, the indefinite integral in Theorem 4.1.2 can besimplified as follows. Note that:

q

(1

2+ iw

)= ΨX

(1

2+ iw, T

)−Ψ0

X

(1

2+ iw, T

)= Ψλ2z

(1

2

(1

2+ iw

)(1

2+ iw − 1

), T

)− e

12z0( 1

2+iw)( 1

2+iw−1)

∫ T0 λ2(s) ds

= Ψλ2z

(−1

2

(1

4+ w2

), T

)− e−

12z0( 1

4+w2)

∫ T0 λ2(s) ds.

Hence, q(1/2 + iw) is a real-valued function, and it implies that

K

π

∫ ∞0Re

(e−(iw+1/2) log(K)

(w2 + 1/4)q(1/2 + iw)

)dw

=K

π

∫ ∞0

√1

K

cos(w log (1/K))

(w2 + 1/4)q(1/2 + iw) dw.

According to Attari [7], the simplified equation has the benefit in terms of computationaltime.

Remark 4.1.4. (Numerical implementation) In order to compute the improperintegral, we can approximate the improper integral by a proper integral, such that theabsolute difference is smaller than a given tolerance ε. From the result by Hoorens [19],we know that if we set

wmax =1

2tan

(π2− ε

4

),

then we have that ∣∣∣∣∫ ∞wmax

cos(w log (1/K))

(w2 + 1/4)q(1/2 + iw) dw

∣∣∣∣ ≤ ε.Hence, we can approximate the improper integral by a proper integral:

K

π

∫ ∞0

√1

K

cos(w log (1/K))

(w2 + 1/4)q(1/2 + iw) dw

≈ K

π

∫ wmax

0

√1

K

cos(w log (1/K))

(w2 + 1/4)q(1/2 + iw) dw.

In our implementation, we use the MATLAB function quadgk, also known as the adap-tive Gauss-Kronrod quadrature, because it is an efficient method for high accuracies andoscillatory integrands. It truncates the region of integration by an internal criteria.

41


4.2 Pricing of caplets

In this subsection, we derive the pricing formula for caplets under the LMM-DDSVmodel. For convenience, we repeat the dynamics of forward rate Ln(t) here:

dLn(t) =√z(t)(bLn(t) + (1− b)Ln(0))λn(t)>dWn+1(t), (4.4a)

dz(t) = θ(z0 − z(t)) dt+ η√z(t) dZn+1(t), z0 = z(0) = 1, (4.4b)

where Wn+1(t) and Zn+1(t) is an m-dimensional Brownian motion and one-dimensionalBrownian motion under Tn+1-forward measure, respectively. Equation (4.4a) is equiva-lent to

dLn(t) =√z(t)(bLn(t) + (1− b)Ln(0))||λn(t)|| λn(t)>

||λn(t)||dWn+1(t).

In order to derive the pricing formula, we want to use the results from Section 4.1.Therefore, we have to map the LMM-DDSV model onto the Heston model. Since theLMM-DDSV model assumes an m-dimensional Brownian motion Wn+1(t), the first stepis to replace this by one-dimensional Brownian motion. Hence, we define Y n+1(t) by

Y n+1(t) =

∫ t

0

λn(s)>

||λn(s)||dWn+1(s).

The differential equation of Y n+1(t) is given by

dY n+1(t) =λn(t)>

||λn(t)||dWn+1(t).

Since λn(t) is bounded, the Ito integral Y n+1 is martingale with mean 0 and variancet. By the Levy’s characterization of Brownian motions (see Karatzas and Shreve [25]),Y n+1 can be identified as a Brownian motion. Substituting λn(t)>/||λn(t)||dWn+1(t) bydY n+1(t), the dynamics are rewritten as

dLn(t) =√z(t)(bLn(t) + (1− b)Ln(0))||λn(t)||dY n+1(t), (4.5a)

dz(t) = θ(z0 − z(t))dt+ η√z(t)dZn+1(t) z0 = z(0) = 1, (4.5b)

where Y n+1(t) and Zn+1(t) are independent Brownian motions under the Tn+1-forwardmeasure QTn+1 . The next step in order to map the LMM-DDSV model onto the Hestonmodel is to define a coordinate transformation. For a given strike K, we define theshifted forward process Ln and the shifted strike K by

Ln(t) = bLn(t) + (1− b)Ln(0),

K = bK + (1− b)Ln(0),

and the dynamics of Ln(t) are given by:

dLn(t) = b√z(t)Ln(t)||λn(t)||dY n+1(t), Ln(0) = Ln(0),

dz(t) = θ(z0 − z(t))dt+ η√z(t)dZn+1(t), z0 = z(0) = 1.

(4.6)

42


Note that the initial value Ln(0) may not be equal to 1. As the last step to mapthe LMM-DDSV model onto the Heston model defined by Equation (4.2), we need tointroduce a process X(t) = Ln(t)/Ln(0) with X(0) = 1.

Theorem 4.2.1. Assuming that ||λn(t)|| is constant, the price of a caplet under theLMM-DDSV model at time t = 0 is given by

Vcaplet(0) =1

bP (0, Tn+1)τnEn+1(Ln(Tn)− K)+. (4.7)

where

En+1((Ln(Tn)− K)+) = Black(0, Ln, Tn, K, ||λn||b)

− K

π

∫ ∞0

Re

(e−(iw+1/2) log(K/Ln(0))

(w2 + 1/4)q(1/2 + iw)

)dw,

(4.8)

with the function q(u) defined by

q(u) = Ψz

(1

2(||λn||b)2u(u− 1), Tn

)− e

12||λn||2b2z0Tnu(u−1),

and also Ψz(u, t) is defined in Lemma 4.1.1 and Black(0, X, T,K, σ) is given by Theorem4.1.2.

Proof. Note that

En+1(Ln(Tn)− K)+ = Ln(0)En+1

(Ln(Tn)

Ln(0)− K

Ln(0)

)+

.

Define X(t) = Ln(t)/Ln(0), the dynamics of X(t) given by

dX(t) = b√z(t)X(t)||λn||dY n+1(t), X(0) = 1,

dz(t) = θ(z0 − z(t))dt+ η√z(t)dZ(t), z0 = z(0) = 1,

which are the dynamics considered in Theorem 4.1.2. Hence, it implies that

Ln(0)En+1

(Ln(Tn)

Ln(0)− K

Ln(0)

)+

= Black(0, Ln, T, c, σ)− K

π

∫ ∞0

Re

(e−(iw+1/2) log(K/Ln(0))

1/4 + w2q(1/2 + iw)

)dw,

where σ2 = z0(b||λn||)2 = (b||λn||)2 and

q(u) = ΨX(u, Tn)−Ψ0X(u, Tn)

= Ψz

(1

2u(u− 1)(b||λn||)2, Tn

)− e

12u(u−1)z0b2||λn||2Tn .

43


4.3 Pricing of swaptions

Consider a swaption and the tenor structure given by (3.1). Recall that the price of theswaption with maturity Tj at time t is given by

Vswaption(t) = A(t)Ej,k−j(S(Tj −K)+,

where

S(t) = Sj,k−j(t) =P (t, Tj)− P (t, Tk)

A(t),

and

A(t) = Aj,k−j(t) =

k−1∑n=j

P (t, Tn+1)τn,

with the underlying security being a swap, making payments at time Tj+1, Tj+2, · · · , Tk,j < k ≤ N . It can be seen as a European call option on the swap rate, so that it isconvenient to derive the dynamics of the forward swap rate S(t).

Proposition 4.3.1. Assume that the forward rate dynamics are given by Equation(3.15). Under the swap measure Qj,k−j, the dynamics of the swap rate are given by

dS(t) =√z(t)(bS(t) + (1− b)S(0))

k−1∑n=j

wn(t)λn(t)>dW j,k−j(t), (4.9)

where W j,k−j is a Brownian motion under the swap measure Qj,k−j and the stochasticweights wn are given by

wn(t) =bLn(t) + (1− b)Ln(0)

bS(t) + (1− b)S(0)× ∂S(t)

∂Ln(t)

=bLn(t) + (1− b)Ln(0)

bS(t) + (1− b)S(0)×

αn(t) +τn

1 + τnLn(t)

n−1∑i=j

αi(t)(Li(t)− S(t))

,

where

αn(t) =τnP (t, Tn+1)∑k−1r=j τrP (t, Tr+1)

.

Proof. For the proof, we refer the reader to Appendix A.3.

In order to derive the pricing formula for swaptions, we map the dynamics of theswap rate given by Equation (4.9) onto the Heston model. However, it can immediatelybe seen that the dynamics of the swap rate given in Equation (4.9) are not analyticallytractable, because the weights are stochastic. One possible way to overcome this prob-lem is to freeze the weights at their time 0 values, which is accurate when ∂S(t)/∂Ln(t)is a near-constant, see Andersen and Piterbarg [5].

44


Define

λS(t) =

k−1∑n=j

wn(0)λn(t),

where wn(0) is the value of the stochastic weights wn(t) at time 0. The swap ratedynamics can be approximated by

dS(t) =√z(t)(bS(t) + (1− b)S(0))||λS(t)||dY j,k−j(t),

dz(t) = θ(z0 − z(t))dt+ η√z(t)dZ(t),

(4.10)

where Y j,k−j and Z(t) are independent scalar Brownian motions under the swap measureQj,k−j , and

||λS(t)||dY j,k−j(t) =k−1∑n=j

wn(0)λn(t)>dW j,k−j(t),

Analogous to Section 4.2, we can map the dynamics of the swap rate given by Equation(4.10) onto Heston model by using a coordinate transformation. The pricing formula ofa swaption is stated in the following theorem.

Theorem 4.3.2. Assume that ||λS(t)|| = ||λS ||, for all t. If we define S(t) and K by

S(t) = bS(t) + (1− b)S(0),

K = bK + (1− b)S(0),

then the time 0 price of the payer swaption is given by

Vswaption(0) = A(0)EA((S(Tj)−K)+) =1

bA(0)EA((S(Tj)− K)+), (4.11)

and where the expectation is given by

EA((S(Tj)− K)+) = Black(0, S, Tj , K, ||λS ||b)

− K

π

∫ ∞0

Re

(e−(iw+1/2) log(K/S(0))

(w2 + 1/4)q(1/2 + iw)

)dw.

The function for q(u) is given by

q(u) = Ψz

(1

2(||λS ||b)2u(u− 1), Tj

)− e

12||λS ||2b2z0Tju(u−1),

where Ψz(u, t) is defined in Lemma 4.1.1 and Black(0, X, T,K, σ) is given in Theorem4.1.2.

Proof. Analogous to Theorem 4.2.1.

45


4.4 Parameter averaging method

The dynamics given by Equation (4.9) are less flexible to capture the variability involatility skews across maturities. Therefore, we consider the forward rate dynamicswith time-dependent parameters, which are given by

Model: LMM-DDSV with time-dependent parameters

dLn(t) =√z(t)(bn(t)Ln(t) + (1− bn(t))Ln(0))λn(t)>dWn+1(t),

dz(t) = θ(z0 − z(t))dt+ η(t)√z(t)dZn+1(t).

(4.12)

where Wn+1(t) is an m-dimensional Brownian motion under the Tn+1-forward measureQTn+1 . We call these dynamics the LMM-DDSV model with time-dependent parameters.Piterbarg [29] derived the dynamics of the swap rate for the above model and the resultis given in the following theorem.

Theorem 4.4.1. The approximate dynamics of the swap rate S(t) under the swap mea-sure Qj,k−j(t) are given by

dS(t) ≈√z(t)(bS(t)S(t) + (1− bS(t))S(0))λ>S dW

j,k−j(t),

where

λS(t) =

k−1∑n=j

∂S(0)

∂Ln(0)

Ln(0)

S(0)λn(t),

bS(t) =

k−1∑n=j

pn(t)bn(t),

and for n = j, j + 1, · · · , k − 1, we have

pn(t) =λn(t)>λS(t)

(k − j)||λS(t)||2.


Note that as we discussed in the Section 4.3, we can derive the dynamics of the swaprate which is driven by a scalar Brownian motion. The dynamics are given by:

dS(t) =√z(t)(bS(t)S(t) + (1− bS(t))S(0))||λS(t)|| dY j,k−j(t),

dz(t) = θ(z0 − z(t)) dt+ η(t)√z(t) dZ(t).

(4.13)

Unless otherwise stated we will use the dynamics from Equation 4.13. A benefit ofconsidering time-dependent parameters is that it can describe the market volatility skews

46


for different maturities more accurately. Therefore, we want to replace the dynamics,given in Equation (4.13), by

dS(t) =√z(t)(bSS(t) + (1− bS)S(0))||λS || dY j,k−j(t),

dz(t) = θ(z0 − z(t)) dt+ η√z(t) dZ(t).

or similarly, we want to replace the dynamics of the forward rates given in Equation(4.12) by

dLn(t) =√z(t)(bnLn(t) + (1− bn)Ln(0))||λn|| dY n+1(t),

dz(t) = θ(z0 − z(t)) dt+ η√z(t) dZ(t).

These can be done by three steps, as suggested in Andersen and Piterbarg [4]. Theprocedure is presented below for the swap rate dynamics. The procedure is similar forthe forward rate dynamics.

Parameter averaging method:Step 1: We approximate the stochastic variance process in Equation (4.13) by a processwith constant parameters:

dz(t) = θ(z0 − z(t)) dt+ η√z(t) dZ(t), z0 = z0 = 1. (4.14)

Since the curvature of the implied volatility is related to the kurtosis of the distributionof S(Tj) and controlled by the variance of∫ Tj

0||λS(t)||2z(t) dt,

we choose the parameter η such that the first and the second moments of∫ Tj

0||λS(t)||2z(t) dt, and

∫ Tj

0||λS(t)||2z(t) dt,

are equal. The constant parameter η such that the process given by Equation (4.14)satisfies these conditions, is given by

η2 =

∫ Tj0 e2θrη2(r)ρTj (r) dr∫ Tj

0 e2θrρTj (r) dr,

where

ρTj (r) =

∫ Tj

re−θs||λS(s)||2

∫ Tj

s||λS(t)||2e−θt dt ds.

Step 2: In this step, we replace the time-dependent skew parameter bS(t) by a constant

47


parameter bS , that is, the dynamics of the swap rate, given by Equation (4.13), areapproximated by

dS(t) =√z(t)(bSS(t) + (1− bS)S(0))||λS(t)|| dY j,k−j(t), S(0) = S(0),

in a way that the parameter bS minimizes

E(S(Tj)− S(Tj))2.

From Andersen and Piterbarg [4], it is known that the constant parameter bS is givenby

bS =

∫ Tj

0bS(t)w(t) dt,

and

w(t) =ν2(t)||λS(t)||2∫ Tj

0 ν2(t)||λS(t)||2 dt,

ν2(t) = z20(t)

∫ t

0||λS(s)||2 ds+ z0η

2e−θt∫ t

0||λS(s)||2 e

θs − e−θs

2θds.

Step 3: The last step to make the model time-homogeneous is to approximate the

dynamics of S(t) defined by

dS(t) =√z(t)(bSS(t) + (1− bS)S(0))||λS(t)|| dY j,k−j(t), S(0) = S(0)

with a process S(t) which dynamics are given by

dS(t) =√z(t)(bSS(t) + (1− bS)S(0))||λS || dY j,k−j(t), S(0) = S(0).

This is done in a way such that

Ej,k−j(S(Tj − S(0))+ = Ej,k−j(S(Tj)− S(0))+.

The effective volatility parameter λS which satisfies this condition is obtained by solvingthe equation:

Ψ||λS ||2z(c, Tj) = Ψz(cλ2S , Tj),

where Ψλ2z(u, t) is defined in Equation (4.3) and Ψz(u, t) is defined in Lemma 4.1.1.

48


4.5 The impact of the parameters on implied volatility curves

In this section, we investigate the impact of the parameters b, η and ||λn(t)|| of theLMM-DDSV model on the implied volatility curve. We consider caplets starting in 3year and maturing in 3.5 year (3 year caplets for short) and the analytical approximatingformula for caplets is available for the LMM-DDSV model. For swaptions the impactof the parameters on implied volatility curves are expected to be similar to caplet case.This is because the dynamics of swap rates are equivalent to the Heston model.

The basic parameter settings for the simulations are:

b = 0.2, ||λn(t)|| = 0.15, θ = 0.4, η = 0.8.

In each test case, we vary the value of only one parameter, while the other parametersare kept constant.

Firstly, the implied volatility curves of 3 year caplets with different skew parametersb = 0.1, 0.2, 0.5, 1 are depicted in Figure 4.1. As it can be seen in Figure 4.1, thesmaller the skew parameter b the steeper the implied volatility curves. The result can beinterpreted as follows. As b becomes close to 1, the dynamics of forward rates becomesimilar to the log-normal dynamics, which imply a flat volatility curve. On the otherhand, for the dynamics of forward rates with b close to 0, the implied distributions be-come close to a normal distribution, which implies that we can control the skew/slopeof the implied volatility curve by adjusting the value of b.

0.025 0.03 0.035 0.04 0.045 0.05 0.0550.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

Strike

Impl

ied

Vol

atili

ty

b=0.1b=0.2b=0.5b=1

Figure 4.1: Implied volatility curves for different skew parameters b ∈ 0.1, 0.2, 0.5, 1

Secondly, Figure 4.2 presents implied volatilities of 3 year caplets. The volatility-of-volatility parameter is varied between η = 0.2, 0.8, 1.3 and 2. The smaller the value ofη the flatter implied volatility curve is. The parameter η controls the variance of thevariance for the forward rate (see Section 4.4) and it results in controlling the kurtosisof the distribution of forward rates, which is related to the curvature of the implied

49


0.025 0.03 0.035 0.04 0.045 0.05 0.0550.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

0.21

Strike

Impl

ied

Vol

atili

ty

η=0.2η=0.8η=1.3η=2

Figure 4.2: Implied volatility curves for curvature parameter η ∈ 0.2, 0.8, 1.3, 2

volatility curve. Note that we can also control the curvature of the implied volatilitycurve by the parameter θ in the variance process, but it works in the opposite way asthe parameter η, that is, the larger the value for θ gives the flatter implied volatilitycurve. This can be explained roughly as follows. The long-term variance of the varianceprocess is given by

z0η2

2θ,

and as θ increases the long-term variance of the variance process decreases.

Finally, we vary the volatility parameter ||λn(t)||. The variance of the dynamics attime T is given by

b

∫ T

0z(t)||λn(t)||2 dt.

As we increase the value of ||λn(t)||, the terminal volatility of the forward rate increases.Hence, the larger ||λn(t)|| results in a larger level of the implied volatility curve. Thiscan be seen in Figure 4.3 which shows implied volatilities for 3 year caplets with differentvolatility parameters.

4.6 Simulation of the LMM-DDSV model

In this section, we discuss efficient Monte Carlo methods for simulating the LMM-DDSVmodel. To get a certain acceptable accuracy, Monte Carlo methods generally need a largenumber of sample paths which mean that they are typically computational intensive.Hence, an efficient implementation is desirable. We will consider simulations methodsfor the variance process and the forward rates processes in the following subsections.

50


0.025 0.03 0.035 0.04 0.045 0.05 0.0550.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Strike

Impl

ied

Vol

atili

ty

||λ

n||=0.1

||λn||=0.15

||λn||=0.2

||λn||=0.4

Figure 4.3: Implied volatility curves for volatility parameter ||λn(t)|| ∈0.1, 0.15, 0.2, 0.4

4.6.1 Simulation of the variance process

In this subsection, methods for simulating the variance process given by Equation (3.12)will be discussed. The most straightforward and well-known way to simulate the varianceprocess is by applying the Euler scheme which (equally) discretizes the time interval [0, T ]into N sub-intervals with time increment ∆. Then, at each time i∆, i ∈ [0, · · · , N − 1],zi+1 is obtained by

zi+1 = zi + θ(z0 − zi)∆ + η√zi√

∆Wi, z0 = z0,

where Wi is a standard normal random variable. This scheme is simple to implement,but a drawback is that the value z(t) can become negative, since W follows the standardnormal distribution. This is an undesired property because z(t) is a variance processand by definition of the variance it should remain positive. To overcome this problem,we introduce the log-Euler scheme.

1. The log-Euler scheme: Although the log-Euler scheme is intuitive and easy toimplement, discretization introduces a bias to the simulation results and a largenumber of time steps may be needed to reduce the discretization bias. It is analternative scheme of the Euler scheme by introducing an invertible transformationz(t) = f(y(t)), with f : R → R+, and then applying the Euler scheme to y ateach step recovering z as f(y). More precisely, set z(t) := exp(y(t)), such thaty(t) = log(z(t)). By Ito’s lemma, it follows that:

dy(t) =

(θ(z0 − z(t))

z(t)− 1

2

η2z(t)

z2(t)

)dt+

η√z(t)

z(t)dZ(t).

By applying the Euler scheme on the above SDE and using the definition of y(t),we obtain

zi+1 = zi exp

((θ(z0 − zi)

zi− 1

2

η2ziz2i

)dt+

η√zi

zi

√∆W

), (4.15)

51


which gives a positive value.

2. Sampling from a χ2 distribution: This method proposed by Broadie and Kaya[11] samples from the exact distribution of z(t) given z(u) for u < t. The transitionlaw of z(t) is given by:

z(t) =η2(1− e−θ(t−u)

)4θ

χ2d

(4θe−θ(t−u)

η2(1− e−θ(t−u))z(u)

),

where χ2d(λ) denotes the non-central chi-squared random variable with d = 4z0θ/η

2

degrees of freedom, and non-centrality parameter

λ =4θe−θ(t−u)

η2(1− e−θ(t−u))z(u).

Sampling from a non-central chi-squared distribution can be computationally ex-pensive (see [1]), especially when 0 < d < 1.

3. The Quadratic-Exponential(QE) Scheme: It is a moment-matching schemeproposed by Andersen [1], and the procedure is as follows:

(a) Compute the first and the second moments of z(t + ∆) given z(t). Sincez(t + ∆) is distributed as e−θ∆/λ time a non-central chi-square distributionwith d degrees of freedom and non-centrality parameter λ defined above, thefirst and the second moments of z(t+ ∆) given z(t) are given by

m := E[z(t+ ∆)|z(t)] = z0 + (z(t)− z0)e−θ∆,

s2 := V ar[z(t+ ∆)|z(t)] =z(t)η2e−θ∆

θ

(1− e−θ∆

)+z0η

2

2θ

(1− e−θ∆

)2.

(b) Compute ψ = s2

m2 .

(c) Pick ψc ∈ [1, 2].

(d) If ψ ≤ ψc, thenz(t+ ∆) ≈ d(c+ Z)2, Z ∼ N (0, 1)

where

c2 =2

ψ− 1 +

√2

ψ

√2

ψ− 1,

d =m

1 + c2.

52


(e) Otherwise,z(t+ ∆) ≈ ψ−1(U ; p; q)

where

ψ−1(u; p; q) =

0, 0 ≤ u ≤ p

1q log

(1−p1−u

), p < u ≤ 1

with p = ψ−1ψ+1 , q = 2

m(ψ+1) and for U ∼ Uniform(0, 1).

0.06 0.07 0.08 0.09 0.1 0.11 0.120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

F(x

)

z(t)=0.09;∆=0.01;θ=0.5;η=0.25

ExactValueThe Q−E Methodlog−Euler method

(a) ∆ = 0.01

0 0.05 0.1 0.15 0.2 0.25 0.30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

F(x

)

z(t)=0.09;∆=0.1;θ=0.5;η=0.25


(b) ∆ = 0.1

Figure 4.4: Distribution of the variance process when the Feller condition is satisfied

0 0.05 0.1 0.15 0.2 0.25 0.30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

F(x

)

z(t)=0.09;∆=0.01;θ=0.5;η=1.6


(a) ∆ = 0.01

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

F(x

)

z(t)=0.09;∆=0.1;θ=0.5;η=1.6


(b) ∆ = 0.1

Figure 4.5: Distribution of the variance process when the Feller condition fails

Now, we will test the performance and show the test results here. Computationally,the QE method allows one to use a larger time-step, which is computationally more

53


efficient. It is known that the process z(t) remains positive if

2θz0 ≤ η2,

which is called the Feller condition. If the Feller condition is violated, then the origin isaccessible and strongly reflecting (see Proposition 2 in Andersen [1]). Figures 4.4a and4.4b show the distribution of z(t+ ∆) given z(t) = 0.09 with different ∆ when the Fellercondition is satisfied. In both cases, the distribution of z(t + ∆) from the QE schemefits the exact distribution better than the one from the log-Euler scheme.

Figures 4.5a and 4.5b also represent the distribution of z(t + ∆) given z(t) = 0.09with different ∆ when the Feller condition is violated. Same as in Figure 4.4, the QEscheme gives a better match to the exact distribution than the log-Euler scheme. Inaddition, we can see that the process z(t) has a strong affinity of the area around theorigin. However, once z(t) becomes zero, log-Euler scheme cannot be used because z(t)is placed in the denominator, see Equation (4.15). In practice, the Feller condition canbe violated under certain market conditions. Hence, the QE scheme is more suitable forour simulations.

4.6.2 Simulation of the forward rates processes

Once we simulate the variance process z(t), we can simulate the forward rates processes.For the simulation of forward rates, we can use the Euler method. More precisely, forn = q(t+ ∆), q(t+ ∆) + 1, · · · , N − 1, the forward rates are simulated by

Ln(t+ ∆) = Ln(t) +√z(t)(bLn(t) + (1− b)Ln(0))λn(t)>

√z(t)µn(t)∆ +√

∆

Z1...Zm

,

µn(t) =

n∑j=q(t)

τj(bLj(t) + (1− b)Lj(0))λj(t)

1 + τjLj(t),

where q(t) is the index function. However, instead of simulating forward rates one byone, we can simulate the vector of forward rates at once by using the matrix form,defined in Equation (3.22). The step of the Euler method from t to t+ ∆ is given by L1(t+ ∆)

...LN−1(t+ ∆)

=

L1(t)...

LN−1(t)

+√z(t)

ϕ(L1(t))||λ1(t)|| · · · 0...

. . ....

0 · · · ϕ(LN−1(t))||λN−1(t)||

·

√z(t) µ′1(t)

...µ′N−1(t)

∆ +

v11 · · · vm1...

. . ....

v1N−1 · · · vmN−1

√e1 · · · 0...

. . ....

0 · · · √em

√∆

Z1...Zm

.

It turns out that the simulation of the vector of forward rates is faster than the simulationof forward rates one by one. The other remark is that for the simulation of the forward

54


rates process we can reduce the computational time by using the recursive relation ofthe drift term µn(t). To see this, note that

µn(t) =n∑

j=q(t)

τjϕ(Lj(t))λj(t)

1 + τjLj(t)

=n−1∑j=q(t)

τjϕ(Lj(t))λj(t)

1 + τjLj(t)+τnϕ(Ln(t))λn(t)

1 + τnLn(t)

= µn−1(t) +τnϕ(Ln(t))λn(t)

1 + τnLn(t).

There are other simulation methods for the BGM model proposed in the literature, suchas the lagging predictor-corrector method. In this method the terminal values of theforward rates are first predicted, and then these values are used to correct the approxi-mation in the drift coefficient, see Hunter, Jackel, and Joshi [22] for more information.

55

Chapter 5

Numerical Results

The goal of this chapter is to discuss the numerical results of the LMM-DDSV modelby pricing IR derivatives. In Chapter 4, we considered the pricing formulas of capletsand swaptions under the LMM-DDSV model. Therefore, the simulation will be mainlydone for caplets and swaptions, so that we are able to compare the analytical prices tothe Monte Carlo simulation results. In Section 5.1, we describe the general simulationsettings. The results will be shown for two different cases in Section 5.2 and Section 5.3,respectively:

• The LMM-DDSV model with time-homogeneous parameters: see Equation (3.15)

• The LMM-DDSV model with time-dependent parameters: see Equation (4.12)

For the LMM-DDSV model with time-dependent parameters, we use the parameteraveraging method as an approximation. In Section 5.4, we will also discuss a shortcomingof the stochastic volatility models which is the possibility of a so-called moment explosion,see [6]. This can be seen for interest rate derivatives for which prices are not onlydependent on the first moment, but also on higher moments of the forward rate process.Typically, LMM is used for products that depend on higher moments. An in-arrearsswap is such a derivative, therefore we investigate the moment explosion problem usingthis product.

5.1 General simulation settings

We consider a semi-annual tenor structure up to 8 years (i.e., TN = 8). Since thelast forward rate fixes at 7.5 years, there are 15 forward rates to be considered. Thecorrelation between forward rates follows a parametric form given by Equation (3.18)with parameters: ρ∞ = 0.3, α = 0.05 and β = 0.05. As mentioned in Section 3.2.1, weassume that there are three Brownian motions which drive the forward rates (instead of15). We choose the following constant parameters in the time-homogeneous model:

b = 0.2, Ln(0) = 0.4, θ = 0.4, η = 0.8, ||λn(t)|| = 0.15.

56

CHAPTER 5. NUMERICAL RESULTS

To see the performance of the parameter averaging method in the time-dependentmodel, we consider piecewise constant parameters (see Figure 5.1):

• η(t) =∑15

j=1 1[Tj−1,Tj)(t)(2.2− 0.5j)

• bn(t) =∑15

j=1 1[Tj−1,Tj)(t)(0.6− j/50 + 0.01(n− 1))

and for ||λn(t)|| we use the parametric form in Equation (3.19) with a = 0.1, b = 0, c =0.3 and d = 0.1. As an initial condition we take Ln(0) = 0.04. These parameters werechosen to keep the level of implied volatility around realistic levels 15% ∼ 20%.

0 1 2 3 4 5 6 7 81.5

1.6

1.7

1.8

1.9

2

2.1

2.2

2.3

time

η(t)

(a) η(t)

0 2 4 6 80.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

time

b n(t)

n=1n=2n=3n=4n=5n=6n=7n=8n=9n=10n=11n=12n=13n=14n=15

(b) bn(t)

Figure 5.1: (Piece-wise constant) time-dependent parameters

For the Monte Carlo simulations, we use 106 sample paths and time increments ∆ =0.01. The Euler method is used for the simulation of forward rates and the QuadraticExponential is used to simulate the stochastic variance process (see Section 4.6). Thestrikes we consider are equidistant around the ATM level, i.e., we take ATM ± d, ford ∈ 0, 0.005, 0.01, 0.015.

5.2 Time-homogeneous LMM-DDSV model

In this section, we consider the LMM-DDSV model with time-homogeneous parameters.Under this model, we price caplets with different maturities and swaptions with differentmaturities and different tenors. We denote a swaption with a x-year option maturitythat gives the right to enter into a z-year swap as a xy × zy swaption.

5.2.1 Caplets

First, we consider the caplet starting in 1-year and maturing in 1.5-year (the 1-yearcaplet for short). Figure 5.2a shows the prices and implied (caplet) volatilities.

57


0.02 0.04 0.060

20

40

60

80

Strike

pric

e(bp

s)

NrSims=1000000,dt =0.01,Tn=1,σ=0.15

MC capletUpper CILower CIAnalytical

0.02 0.04 0.06−0.02

0

0.02

0.04

0.06

StrikeA

naly

tic−

MC

(bp

s)

L0=0.04,b=0.2,θ=0.4,η=0.8

0.02 0.04 0.0614

16

18

20

22

strike

Impl

ied

Vol

(%)

MCAnalytic

0.02 0.04 0.06−60

−40

−20

0

20

strike

Ana

lytic

−M

C Im

plie

d V

ol (

bps)

(a) 1-year caplet prices and implied volatilities

0 5 10 15 20 2570.6

70.7

70.8

70.9

71

71.1

71.2

Volatility(%)

Cap

let p

rice(

bps)

from

Bla

ck fo

rmul

a

black priceMC priceAnalytical Priceupper CIlower CI

ATM ITMOTM

(b) Black price as function of implied volatility

Figure 5.2: Caplet prices and its implied volatility

For the 1-year caplet, the top right graph in Figure 5.2a shows the difference (in basis-points or bps) between the analytical price and the MC price. The bottom right graphin Figure 5.2a shows the difference in terms of implied volatilities. Although we see asmall difference (0.04 bps) for the ITM caplets, we see a large difference of 43.75 bps interms of implied volatility.

0.02 0.04 0.060

20

40

60

80

Strike

pric

e(bp

s)

NrSims=1000000,dt =0.01,Tn=3,σ=0.15


0.02 0.04 0.06−0.02

−0.01

0

0.01

0.02

Strike

Ana

lytic

−M

C (

bps)

L0=0.04,b=0.2,θ=0.4,η=0.8

0.02 0.04 0.0612

14

16

18

20

strike

Impl

ied

Vol

(%)

MCAnalytic

0.02 0.04 0.06−10

−5

0

5

strike

Ana

lytic

−M

C Im

plie

d V

ol (

bps)

(a) 3-year caplet price

0 5 10 15 20 2565

65.5

66

66.5

67

67.5

68

68.5

69

69.5

Volatility(%)

Cap

let p

rice(

bps)

from

Bla

ck fo

rmul

a

black priceMC priceAnalytical Priceupper CIlower CI

ATMOTM ITM

(b) Black price as function of implied volatility

Figure 5.3: Caplet prices and its implied volatility

For comparison purposes, we consider the caplet starting in 3-year and maturing in 3.5-year (the 3-year caplet for short). Figure 5.3a shows also small difference in terms ofprice similarly to the 1-year caplet case. Since there is no approximation involved toderive the analytical formula for caplets, this error comes from the Monte Carlo sim-

58


ulation (and/or from the numerical integration errors). However, we observe that forthe 3-year caplet the error of the ITM implied volatility is much smaller than the onefor the 1-year caplet. This can be explained by the fact that a caplet with a short ma-turity is less sensitive to changes in volatility (i.e., the vega sensitivity ∂V

∂σ is smaller).This implies that a small difference in price results in a big difference in implied volatility.

To understand these results better, we show the ITM caplet prices (strike K = 0.025), asfunction of the implied volatility, in Figures 5.2b and 5.3b. The ATM implied volatilityis shown in the Figures with a vertical dashed line. For the OTM region of the 1-yearcaplet, we can observe in Figure 5.2b that the Black price is “nearly flat” as a functionof the implied volatility, i.e., ∂V

∂σ ≈ 0. On the other hand, for the OTM of the 3-yearcaplet we can see in Figure 5.3b that the Black price is not “fully flat” as a function ofthe implied volatility, i.e., ∂V∂σ 6= 0. Hence, the large errors in terms of implied volatilitiesfor the 1-year caplet can be explained by the small vega sensitivity.

5.2.2 Swaptions

For caplets there exists an analytical formula, however we have to approximate the swaprate dynamics in order to derive closed-form pricing formulas for swaptions. Hence, thedifference between the analytical prices and Monte Carlo prices comes not only from theMonte Carlo simulation error, but also from the assumptions made for the approxima-tion method.

In Table 5.1, we present the performance of the approximation of swaptions for dif-ferent maturities and tenors. “MC” refers to Monte Carlo simulation results and “Ap-prox” refers to the results obtained by Equation (4.11). Overall, we can see that theapproximation performs reasonably well (maximum difference in price is −0.65 bps andin implied volatility it is −147.19 bps for the smallest strikes). The large difference inimplied volatility for the 1y × 7y swaption can be explained similarly as for the 1-yearcaplet (see Figure 5.4), and see section 5.2.1 (small vega for options with short maturity).

0 5 10 15 20 25872

873

874

875

876

877

878

879

Volatility(%)

Sw

aptio

n pr

ice(

bps)

from

Bla

ck fo

rmul

a

black priceMC priceupper CIlower CIApprox

Figure 5.4: Swaption (Black) price as function of implied volatility

59


Table 5.1: The swaption prices and their implied volatilities obtained by the MonteCarlo simulation and pricing formula

Price(bps) Implied vol(%)Swap. Strike MC(Error) Approx Diff MC Approx Diff(bps)

1y × 1y 0.025 140.15(0.05) 140.12 -0.03 20.02 19.61 -41.320.030 94.47(0.05) 94.46 -0.01 17.43 17.40 -3.070.035 52.70(0.05) 52.72 0.03 15.73 15.76 2.680.040 21.83(0.03) 21.89 0.05 14.67 14.71 3.680.045 6.39(0.02) 6.45 0.06 14.19 14.13 6.570.050 1.40(0.01) 1.43 0.03 14.06 14.13 6.570.055 0.25(0.00) 0.26 0.01 14.07 14.17 9.88

1y × 2y 0.025 274.86(0.11) 274.80 -0.07 20.05 19.57 -48.040.030 185.26(0.10) 185.22 -0.04 17.42 17.36 -6.120.035 103.32(0.09) 103.33 0.00 15.72 15.72 0.170.040 42.78(0.06) 42.82 0.04 14.66 14.68 1.380.045 12.51(0.04) 12.57 0.07 14.18 14.21 2.970.050 2.74(0.02) 2.78 0.04 14.05 14.10 4.570.055 0.49(0.01) 0.51 0.02 14.06 14.14 7.59

1y × 7y 0.025 874.15(0.32) 873.50 -0.65 20.56 19.09 -147.190.030 588.60(0.31) 588.00 -0.60 17.22 16.91 -30.650.035 326.37(0.28) 325.91 -0.46 15.36 15.29 -7.720.040 132.50(0.20) 132.26 -0.24 14.28 14.26 -2.640.045 37.30(0.11) 37.25 -0.05 13.82 13.81 -0.750.050 7.74(0.05) 7.79 0.05 13.71 13.73 1.800.055 1.31(0.02) 1.34 0.01 13.76 13.79 2.98

2y × 1y 0.025 135.91(0.07) 135.80 -0.11 19.44 19.14 -29.740.030 94.04(0.07) 93.95 -0.09 17.18 17.08 -10.150.035 57.17(0.06) 57.12 -0.05 15.60 15.56 -3.360.040 29.46(0.05) 29.44 -0.02 14.58 14.57 -0.940.045 12.88(0.03) 12.89 0.00 14.06 14.06 0.140.050 4.98(0.02) 4.99 0.01 13.86 13.87 1.110.055 1.77(0.01) 1.79 0.02 13.81 13.83 2.64

2y × 2y 0.025 266.54(0.14) 266.30 -0.25 19.44 19.09 -35.380.030 184.39(0.13) 184.16 -0.22 17.16 17.03 -13.290.035 112.03(0.11) 111.87 -0.16 15.56 15.51 -5.300.040 57.64(0.09) 57.55 -0.09 14.55 14.52 -2.350.045 25.15(0.06) 25.11 -0.04 14.02 14.01 -1.120.050 9.68(0.04) 9.68 0.00 13.82 13.82 0.110.055 3.43(0.02) 3.45 0.02 13.78 13.79 1.55

2y × 4y 0.025 512.76(0.26) 512.12 -0.65 19.42 18.94 -48.460.030 354.47(0.25) 353.86 -0.61 17.08 16.88 -19.38

Continued on next page

60




0.035 214.91(0.22) 214.42 -0.49 15.46 15.38 -8.300.040 110.05(0.17) 109.74 -0.31 14.44 14.40 -4.120.045 47.65(0.11) 47.49 -0.17 13.92 13.89 -2.470.050 18.17(0.07) 18.12 -0.04 13.72 13.71 -1.010.055 6.37(0.04) 6.39 0.01 13.68 13.68 0.63

4y × 1y 0.025 129.09(0.09) 128.99 -0.10 18.68 18.57 -10.700.030 93.27(0.08) 93.19 -0.08 16.81 16.76 -5.360.035 62.36(0.07) 62.30 -0.06 15.46 15.44 -2.650.040 38.30(0.06) 38.27 -0.03 14.53 14.52 -1.220.045 21.75(0.05) 21.74 -0.01 13.95 13.94 -0.200.050 11.63(0.03) 11.64 0.01 13.60 13.61 0.610.055 5.98(0.02) 6.00 0.02 13.42 13.43 1.55

4y × 2y 0.025 253.2(0.17) 252.9 -0.28 18.67 18.52 -15.300.030 182.9(0.16) 182.6 -0.25 16.79 16.71 -8.230.035 122.2(0.14) 122 -0.20 15.43 15.39 -4.520.040 74.94(0.11) 74.81 -0.13 14.50 14.48 -2.550.045 42.49(0.09) 42.42 -0.07 13.91 13.90 -1.410.050 22.69(0.07) 22.67 -0.02 13.57 13.57 -0.520.055 11.65(0.05) 11.66 0.01 13.39 13.40 0.34

7y × 1y 0.025 120.3(0.09) 120 -0.33 18.46 18.24 -21.810.030 90.99(0.09) 90.69 -0.30 16.79 16.65 -14.010.035 65.7(0.08) 65.45 -0.25 15.56 15.47 -9.300.040 45.26(0.07) 45.07 -0.20 14.66 14.60 -6.430.045 29.89(0.06) 29.73 -0.16 14.02 13.97 -5.100.050 19.05(0.05) 18.93 -0.12 13.56 13.52 -4.210.055 11.83(0.04) 11.75 -0.08 13.24 13.21 -3.50

61


5.3 Time-dependent LMM-DDSV model and parameter averaging method

In this section, we consider the LMM-DDSV model with time-dependent parameters.Thetime-dependent model is approximated by a time-homogeneous model with constant and“averaged” parameters. The parameter averaging method is used to determined theaveraged parameters, and for the constant parameter model we can apply the analyticalpricing formulas.

5.3.1 Caplets

0.02 0.04 0.060

20

40

60

Strike

pric

e(bp

s)

Tn=7,ave ||λ||=0.13885


0.02 0.04 0.06−0.2

−0.15

−0.1

−0.05

0

Strike

Ana

lytic

−M

C (

bps)

L0=0.04,ave b=0.42096,θ=0.4,ave η=1.9298

0.02 0.04 0.0610

12

14

16

18

strike

Impl

ied

Vol

(%)

MCAnalytic

0.02 0.04 0.06−30

−20

−10

0

strike

Ana

lytic

−M

C (

bps)

Figure 5.5: Caplet pricing based on the parameter averaging method

Figure 5.5 presents the performance of the parameter averaging method for a capletstarting in 7-year and maturing in 7.5-year. The maximum difference in implied volatilityis −22.83 bps. We observe the smallest difference, between the MC implied volatilityand the analytical volatility, around the ATM level. This can be explained by the factthat the parameter averaging method is designed to give the most accurate results forATM-caplets.

62


5.3.2 Swaptions

Table 5.2 presents swaption prices with different maturities and tenors, obtained fromMonte Carlo simulations with time-dependent parameters and the approximation for-mula given by Theorem 4.3.2. The constant parameters for the approximation formulaare calculated by using the parameter averaging method. For the 1-year maturity ITM-swaptions with a 7-year tenor, we see the largest difference as we have observed in thetime-homogeneous case.



1y × 1y 0.025 140.82(0.06) 140.72 -0.11 25.04 24.49 -55.020.030 95.89(0.06) 95.79 -0.10 20.86 20.66 -19.850.035 54.28(0.05) 54.25 -0.03 17.31 17.29 -2.460.040 22.74(0.04) 22.82 0.08 15.29 15.34 5.440.045 8.69(0.03) 8.74 0.05 16.16 16.20 4.000.050 3.74(0.02) 3.73 -0.01 17.83 17.82 -1.290.055 1.74(0.02) 1.71 -0.02 19.30 19.25 -4.93

1y × 2y 0.025 275.87(0.11) 275.67 -0.20 24.20 23.60 -60.390.030 187.29(0.11) 187.09 -0.20 20.04 19.82 -22.090.035 104.85(0.10) 104.75 0.10 16.51 16.46 -4.890.040 42.15(0.08) 42.23 0.08 14.45 14.47 2.680.045 15.20(0.05) 15.23 0.02 15.38 15.39 0.920.050 6.23(0.04) 6.17 -0.06 17.06 17.01 -4.440.055 2.75(0.03) 2.69 -0.06 18.51 18.43 -8.06

1y × 7y 0.025 875.10(0.29) 874.23 -0.87 22.05 20.72 -133.190.030 589.29(0.28) 588.45 -0.84 17.55 17.14 -41.410.035 318.43(0.25) 317.70 -0.73 13.98 13.84 -13.380.040 109.16(0.19) 108.58 -0.57 11.76 11.70 -6.160.045 31.03(0.12) 30.43 -0.60 12.88 12.78 -9.300.050 10.30(0.07) 9.80 -0.50 14.57 14.41 -15.700.055 3.69(0.04) 3.39 -0.31 15.93 15.72 -21.27

2y × 1y 0.025 137.00(0.08) 136.88 -0.12 21.95 21.71 -23.690.030 95.12(0.07) 95.00 -0.12 18.37 18.25 -12.710.035 56.88(0.07) 56.86 -0.02 15.41 15.40 -1.010.040 27.92(0.06) 28.06 0.15 13.82 13.89 7.220.045 13.46(0.05) 13.51 0.05 14.38 14.41 2.910.050 7.34(0.04) 7.28 -0.06 15.71 15.67 -4.150.055 4.36(0.03) 4.26 -0.10 17.01 16.91 -10.17

Continued on next page

63




2y × 2y 0.025 268.01(0.14) 267.79 -0.22 21.24 20.99 -24.700.030 185.32(0.13) 185.07 -0.24 17.70 17.56 -13.700.035 109.48(0.12) 109.40 -0.08 14.72 14.70 -2.680.040 51.95(0.10) 52.15 0.20 13.11 13.16 5.000.045 24.04(0.08) 24.07 0.03 13.71 13.72 0.950.050 12.69(0.06) 12.55 -0.14 15.06 15.01 -5.650.055 7.32(0.05) 7.12 -0.20 16.35 16.24 -11.51

2y × 4y 0.025 513.88(0.24) 513.42 -0.46 20.19 19.88 -30.440.030 353.15(0.23) 352.66 -0.48 16.66 16.50 -15.560.035 204.66(0.21) 204.40 -0.26 13.68 13.63 -4.620.040 91.61(0.18) 91.77 0.16 12.01 12.03 2.100.045 39.42(0.14) 39.31 -0.11 12.67 12.66 -1.710.050 19.57(0.10) 19.31 -0.36 14.06 13.98 -7.880.055 10.77(0.08) 10.37 -0.40 15.33 15.20 -13.38

4y × 1y 0.025 129.22(0.09) 128.90 -0.31 18.81 18.47 -33.720.030 91.91(0.08) 91.65 -0.26 15.92 15.75 -17.520.035 58.56(0.08) 58.47 -0.09 13.74 13.70 -3.990.040 33.25(0.07) 33.36 0.11 12.61 12.65 4.190.045 18.75(0.06) 18.79 0.04 12.76 12.77 1.420.050 11.63(0.03) 11.64 0.01 13.60 13.61 0.610.055 7.52(0.04) 7.34 -0.18 14.38 14.28 -10.94

4y × 2y 0.025 252.47(0.16) 251.81 -0.65 18.29 17.91 -38.870.030 178.72(0.15) 178.13 -0.60 15.40 15.19 -20.960.035 112.55(0.14) 112.22 -0.33 13.19 13.12 -7.840.040 62.35(0.12) 62.36 0.01 12.05 12.05 0.130.045 34.16(0.10) 20.07 -0.11 12.22 12.20 -2.200.050 20.38(0.09) 20.07 -0.31 13.00 12.97 -7.790.055 13.12(0.07) 12.71 -0.41 13.88 13.71 -13.36

7y × 1y 0.025 117.76(0.09) 117.14 -0.62 16.66 16.20 -46.360.030 86.08(0.08) 85.58 -0.50 14.41 14.15 -25.560.035 58.36(0.08) 58.05 -0.31 12.82 12.70 -11.890.040 37.01(0.07) 36.86 -0.15 11.97 11.92 -4.890.045 23.11(0.06) 22.98 -0.13 11.80 11.76 -4.290.050 14.91(0.05) 14.73 -0.19 12.06 11.99 -7.040.055 10.08(0.04) 9.83 -0.25 12.49 12.38 -10.99

64


5.4 Moment explosion for stochastic volatility models

As discussed in Section 4.2, we can map the LMM-DDSV model on the Heston model.However, it is well-known that the Heston model suffers from the possibility of a so-calledmoment explosion [6]. More precisely, for a process X(t) following the Heston dynamics,the moment E(Xw(T )), w ≥ 2 can become infinite within a finite time T . Since theLMM-DDSV model can be mapped onto the Heston model, the LMM-DDSV model cansuffer from the moment explosion as well.

More formally, we consider the process Ln(t), given by Equation (4.6), with ||λn(t)|| =λn. Fix k = (bλn)2

2 w(w − 1), such that k > 0 and define

c = 2k/η2 > 0, a = −2θ/η2, D = a2 − 4c,

then E(Lwn (T )) will be finite for T < T ∗ and infinite for T ≥ T ∗, where the critical timeT ∗ is given by:

• For D ≤ 0, a < 0: T ∗ =∞

• For D ≥ 0, a > 0: T ∗ = γ−1η−2 log(a/2+γa/2−γ

), with γ := 1

2

√D

• For D < 0: T ∗ = 2β−1η−2(π1a<0 + arctan(2β/a)), with β := 12

√−D

For more information we refer to [6]. Given the parameters, we can calculate the criticaltime T ∗ by using the above equations.

Recall from Section 2.4.3, the pricing formula for in-arrears swaps with reset datesT0, T1, · · · , TN−1:

V (0) =N−1∑n=0

τnP (0, Tn)

(Ln(0) + τnEn+1(L2

n(Tn))

1 + τnLn(0)−K

), (5.1)

which implies that the price of in-arrears swaps is also related to the second moment ofthe forward rates. The moment explosion implies that En+1(L2

n(Tn)) can become infiniteif Tn > T ∗. In contrary, the pricing formula for swaps depends only on the first moment,so that swaps do not suffer from moment explosion. To assess the moment explosion,we price payer swaps and in-arrears payer swaps. Therefore, we simulate forward ratesunder the LMM-DDSV model with parameters defined in Table 5.3 (with semi-annuallytenor structure up to 5 years, i.e., TN = 5). An in-arrears swap with strike K resets ondates T1, T2, · · · , T8, T9.

For the parameters of case A in Table 5.3, the critical time is T ∗ = 39.62 which is largerthan TN , and for the parameters of case B, the critical time is T ∗ = 3.65. Therefore,we expect that the in-arrears swap price will not diverge and remains stable for caseA. For case B, the second moment may become infinite so that the Monte Carlo price

65


case A case B

b 1.0 1.0

λn 0.2 0.4

θ 0.4 0.4

η 1.5 2.0

K 0.02 0.02

T ∗ 39.62 3.65

Table 5.3: Simulation parameters and the critical time T ∗

can become distorted and unstable. To determine whether the price from Monte Carlosimulations are reasonable stable, we also price swaps and in-arrears swaps with the ana-lytical formulas. The analytical swap price is obtained by using Equation (2.10) and theanalytical in-arrears swap price is obtained by using Equation (2.17) with the impliedvolatility σn. The results are presented in Table 5.4.

case A case B

Critical time 39.62 3.65

Swaps Price(bps)MC(std.err) 800.85(0.4) 803.95(0.85)

Analytical price 800.22 800.22

In-Arrears Swap Price(bps)MC(std.err) 805.44(0.47) 1011.42(40.62)

Analytical price 803.75 862.75

Table 5.4: Payer swaps and in-arrears swap prices

From Table 5.4 we observe that the Monte Carlo prices and the analytical prices differthe most from each other for the in-arrears swap under settings of case B. For the othercases, the MC and analytical prices are generally close to each other. The differencebetween the MC prices and the analytical prices for both swaps and in-arrears swapsare small for case A. The difference between swap price and in-arrears swap price is alsosmall, as this difference comes from the convexity adjustment. For case B, we observelarger differences between the analytical in-arrears swap price and the MC price. Thus,this example shows that the model suffers from moment explosion, which is clearly vis-ible for an in-arrears swap.

To understand this better, we calculate the Net Present Value (NPV) for the sam-ple paths with diverging forward rates, as shown in Table 5.5. The NPV is given by thesum of discounted cashflows:

NPV =

9∑n=1

CashF lownNumerairen

.

66


More precisely, for swaps the cash-flow at Tn+1 is dependent on the forward rate fixingat Tn, and the (numeraire) bank account at Tn+1 is dependent on the forward rate fixingat Tn. Hence, we have for swaps that the discounted cash-flows equals:

Discounted CF =CFTn+1(F (Tn))

BTn+1(F (Tn)). (5.2)

On the other hand, for in-arrears swaps the cash-flow at Tn+1 is dependent on the forwardrate fixing at Tn+1, and the bank account at Tn+1 is dependent on the forward rate fixingat Tn, so that for in-arrears swap, the discounted cash-flows equals:

Discounted CF =CFTn+1(F (Tn+1))

BTn+1(F (Tn)). (5.3)

The major difference between Equation (5.2) and Equation (5.3) is that when a forwardrate explodes at a certain time, then for the calculation of the discounted cash-flow forswaps, the effect offsets each other in the numerator and denominator of Equation (5.2).For in-arrears swaps, if the forward rate explodes at time Tn+1, then the effect is notoffset by the denominator of Equation (5.3). As a consequence, the NPV of a swap issmaller than the NPV of an in-arrears swap.

Swap In-arrears Swap

n Ln(Tn) B(Tn) Cash Flow Discounted Cash Flow Discounted

0 0.040 1.000 - -

1 0.043 1.020 0.0100 0.00980 0.0114 0.0112

2 0.088 1.042 0.0114 0.01095 0.0338 0.0325

3 0.72 1.088 0.0338 0.0311 0.35 0.321

4 9.64 1.48 0.35 0.236 4.809 3.253

5 1267.21 8.60 4.809 0.559 633.6 73.64

6 72165.5 5459.90 633.6 0.116 36083.74 6.609

7 8100358 1.96E+08 36083.74 0.0002 4050179 0.0206

8 4.43E+08 7.98E+14 4050179 5.08E-09 2.22E+08 2.78E-07

9 9.85E+08 1.77E+23 2.22E+08 1.25E-15 4.93E+08 2.78E-15

10 1.3E+09 8.71E+31 4.93E+08 5.65E-24 6.49E+08 7.45E-24

NPV 0.96 83.89

Table 5.5: Net Present Value (NPV) for swaps and in-arrears swaps

The above table illustrates that the stochastic volatility model suffers from a mo-ment explosion, which is clearly visible with an in-arrears swap. Note that the momentexplosion is not clearly visible for an ordinary swap.

67

Chapter 6

Conclusion

In this thesis we investigated the Libor Market Model (LMM) with Displaced Diffusionand Stochastic Volatility (LMM-DDSV). The reason to investigate the LMM-DDSVmodel is to better fit the market implied volatility curve. A drawback of the standardBGM model [9] is that the BGM model implies a flat volatility curve, which is not con-sistent with the market implied volatility curve, see Figures 3.1a and 3.1b.

In Sections 4.2 and 4.3 we have derived pricing formulas for caplets and swaptions underthe LMM-DDSV model, which is closely related to the Heston model [17] for StochasticVolatility (see Section 4.1). We have seen that there are analytical pricing formulasavailable for European-style of options for the model with constant (time-homogeneous)parameters. We implemented the LMM-DDSV model and extended the model withtime-dependent parameters in Section 4.4. We approximated the time-dependent modelby a time-homogeneous model and by using a parameter averaging method.

In Chapter 5 we have seen that the parameter averaging theorems yield an accurateapproximation of the caplet prices, see Figure 5.5. In addition, we have seen that theapproximations made to derive closed-form pricing formulas for swaptions are accurate,see Table 5.2. The numerical and simulation results also show that the skewness andcurvature of the implied volatility curve can be controlled by the model parameters (seeFigures 5.2, 5.3 and 5.5, and Tables 5.1 and 5.2). The numerical approximation resultsin terms of implied volatilities can become relatively large in some parameter cases, ifthe prices are nearly insensitive to changes in volatility (i.e., when the vega sensitivity∂V∂σ is small). In these cases, a small approximation error in terms of prices may resultinto a relative larger approximation error in terms of implied volatilities.

A drawback of stochastic volatility models is that they can suffer from the so-calledmoment explosion, meaning that the higher moments of a stochastic process can be-come infinite within a finite time. In Section 5.4 we have shown an example where amoment explosion can occur. It is possible to determine whether the moment explosioncan occur or not for a given parameter setting, as the characterization of the critical

68

CHAPTER 6. CONCLUSION

time is given by the results in [6]. This phenomenon is shown by using in-arrears swaps,since the pricing formula for an in-arrears swap depends on the second moment of theforward rates. Comparision between swaps and in-arrear swaps reveals that the NetPresent Value (NPV) for an in-arrears swap can become much larger than the NPV ofa swap.

69

CHAPTER 6. CONCLUSION

Future research direction

We recommend the following directions for further research.

• In this thesis we have investigated the LMM-DDSV model for pricing of interestrate derivatives. A logical next step is to investigate an efficient calibration of theLibor Market Model with stochastic volatility to actual market data. An approachcould be based on a weighted Monte Carlo technique (see Chen, Grzelak andOosterlee [13]). Under this approach, the Monte Carlo weights are not uniformlychosen, but are chosen to replicate the market instruments. However, there iscurrently no consensus how the LMM-DDSV model has to be calibrated in a stableand efficient way. The choice of the constant model parameters is non-trivial forthe LMM-DDSV model. We propose to do future research on the optimal choiceof the model parameters in order to improve the robustness of the calibration.

• We have seen that the LMM-DDSV model is not robust, as there is a potentialissue of moment explosion under this model. It may be interesting to do moreresearch on solving this problem or to use models which avoid this explosion issue.

• In this thesis we have only considered the DDSV model, which is a flexible modelfor generating skewness and curvature in the implied volatilities. There are otherpotential models which are able to generate skewness and curvature in the impliedvolatilities, while still remain practically manageable. For instance, other formsof Stochastic Volatility models with a different local volatility formulation may beworthwhile to investigate, instead of the displaced diffusion formulation. Anotherway is to use LMM with different correlation parameterisations. The main calibra-tion instrument are usually the swaptions, but these instruments do not dependsignificantly on the forward rate correlations [24]. In case we want to price exoticproducts, then considering other correlation structures can become more impor-tant as the prices of exotic products typically depend on the full dynamics of themodel.

70

Appendix A

Theorems and Proofs

A.1 Theorems

In this section, we present some well-known and fundamental theorems which are usedin this thesis. We omit the proofs of these theorems, these can be found in Andersenand Piterbarg [4] and in Karatzas and Shreve [25].

Theorem A.1.1. (Radon-Nikodym Theorem) Let P and P be equivalent probabilitymeasures on a common measure space (Ω,F). There exists a unique (a.s.) non-negativerandom variable R with EP (R) = 1, such that

P (A) = EP (R1A), for allA ∈ F .

The random variable R is called the Radon-Nikodym derivative of P w.r.t. P and isdenoted by dP /dP .

Theorem A.1.2. (Change of Numeraire) Consider two numeraires N(t) and M(t),inducing equivalent martingale measures QN and QM , respectively. If the market iscomplete, then the density process of Radon-Nikodym derivatives relating the two mea-sures is uniquely given by

ς(t) = EQN

t

(dQM

dQN

)=M(t)/M(0)

N(t)/N(0).

For the following Theorem A.1.3 (Girsanov’s Theorem), we consider two measures Pand P (θ) related by a density ςθ(t) = EPt (dP (θ)/dP ), where ςθ(t) is an exponentialmartingale given by the Ito process

dςθ(t)/ςθ(t) = −θ(t)>dW (t),

71

APPENDIX A. THEOREMS AND PROOFS

and where W (t) is a d-dimensional P -Brownian motion and θ(t) is a d-dimensionalprocess. By an application of Ito’s lemma, we can write

ςθ(t) = exp

(−∫ t

0θ(s)>dW (s)− 1

2

∫ t

0θ(s)>θ(s) ds

)= E

(−∫ t

0θ(s)>dW (s)

),

where E(·) is the Doleans exponential. If θ(t) satisfies the Novikov condition, i.e.,

EP[exp

(1

2

∫ t

0θ(s)>θ(s) ds

)]<∞, for all t ∈ [0,∞),

or, if the process is defined only on finite time interval [0, T ],

EP[exp

(1

2

∫ T

0θ(s)>θ(s) ds

)]<∞

then it defines a proper martingale on [0,∞), or on [0, T ].

Theorem A.1.3. (Girsanov’s Theorem) Assume that ςθ(t) defined above is a martingaleon [0, T ]. Then for all t ∈ [0, T ],

W θ(t) = W (t) +

∫ t

0θ(s) ds,

defines a Brownian motion under the measure P (θ) on [0, T ].

A.2 Proof of Theorem 4.1.2

Proof. We include the proof of Theorem 4.1.2 given by Andersen and Piterbarg [4] forthe reader. First note that:

EP(X(T )−K)+ = EP(elogX(T ) − elogK

)+

= EP(elogX(T ) − elogK min(elogX(T )−logK , 1)

)= EPX(T )−KEP min(elogX(T )−logK , 1).

Following Carr and Madan [12], we have

EP(min(elogX(T )−logK , 1)) = e−α logK

∫ ∞−∞

[min(e−(logK−x), 1)eα(logK−x)

][eαxp(x)] dx,

72


where α is a dampening constant which lies in (0, 1) and p(t) is the density of logX(T ).For notational convenience, we define

f1(x) = min(e−x, 1)eαx,

f2(x) = eαxp(x).

If we let F be the Fourier transform and F−1 its inverse, then we have that

EP(min(elogX(T )−logK , 1)) = e−α logK(f1 ∗ f2)(logK)

= e−α logK(F−1(F(f1 ∗ f2)))(logK)

= e−α logK(F−1(Ff1(w)Ff2(w))(logK),

where ∗ is the convolution operator. Next, we calculate Ff1 and Ff2 :

Ff1(w) =

∫ ∞−∞

eiwx min(e−x, 1)eαx dx =

∫ 0

−∞e(α+iw)x dx+

∫ ∞0

e(α+iw−1)x dx

=1

α+ iw− 1

α− 1 + iw=

1

(α+ iw)(1− α− iw),

and

Ff2(w) =

∫ ∞−∞

eiwxeαxp(x) dx =

∫ ∞−∞

e(α+iw)xp(x) dx = E(e(α+iw) logX(T )

)= ΨX(α+ iw, T ).

Hence, we have that

EP(min(elogX(T )−logK , 1)) = e−α logKF−1(Ff1(w)Ff2(w))(logK)

=1

2π

∫ ∞−∞

e−(α+iw) logK ΨX(α+ iw, T )

(α+ iw)(1− α− iw)dw

(A.1)

Combining the fact that X(T ) is martingale under measure P and the result from Equa-tion (A.1), we have

EP(X(T )−K)+ = X(0)− K

2π

∫ ∞−∞

e−(α+iw) logK ΨX(α+ iw, T )

(α+ iw)(1− α− iw)dw. (A.2)

We proceed as follows to derive a type of control variate method. Consider the caseη(t) = 0. Then, the dynamics of z(t) are given by

dz(t) = θ(z0 − z(t))dt

which implies that z(t) = z(0) for all t and the dynamics of X(t) simplify to

dX(t) =√z(0)λ(t)X(t) dW (t)

73


It yields that X(t) is log-normally distributed. Hence, we have that

EP(X(T )−K)+ = X(0)N (d1)−KN (d2) := Black(0, X, T,K, σ) (A.3)

where

σ2 =z0

∫ T0 λ2(t) dt

T

d1 =log(X(0)/K) + σ2T

2

σ√T

.

d2 = d1 − σ√T

We can also use Equation (A.2) to solve the expectation which is given by

EP(X(T )−K)+ = X(0)− K

2π

∫ ∞−∞

e−(α+iw) logK Ψ0X(α+ iw, T )

(α+ iw)(1− α− iw)dw, (A.4)

where

Ψ0X(u, T ) = EP

(eu logX(T )

)= EP

(eu(√

z(0)∫ T0 λ(t) dW (t)− 1

2z(0)

∫ T0 λ2(t) dt

))= e

12u2z(0)

∫ T0 λ2(t) dte−

12z(0)

∫ T0 λ2(t) dt = e

12u(u−1)z(0)

∫ T0 λ2(t) dt.

From Equation (A.3) and (A.4), we get

Black(0, X, T,K, σ) = X(0)− K

2π

∫ ∞−∞

e−(α+iw) logK Ψ0X(α+ iw, T )

(α+ iw)(1− α− iw)dw,

which implies that

∆ := Black(0, X, T,K, σ)−X(0) +K

2π

∫ ∞−∞

e−(α+iw) logKΨ0X(α+ iw, T )

(α+ iw)(1− α− iw)dw (A.5)

equals to 0. Adding Equation (A.5) to Equation (A.2) yields the following

EP(X(T )−K)+ = EP(X(T )−K)+ + ∆

= X(0)− K

2π

∫ ∞−∞

e−(α+iw) logKΨX(α+ iw, T )


+Black(0, X, T,K, σ)−X(0) +K

2π

∫ ∞−∞

e−(α+iw) logKΨ0X(α+ iw, T )


= Black(0, X, T,K, σ)− K

2π

∫ ∞−∞

e−(α+iw) logKq(α+ iw)

(α+ iw)(1− α− iw)dw,

74


where q(u) := ΨX(u, T )−Ψ0X(u, T ).

If we define

H(w) = e−(α+iw) logK q(α+ iw)

(α+ iw)(1− α− iw),

then we can easily check that H(−w) = H(w) from the properties of complex conjugates.Such a function satisfies ∫ ∞

−∞H(w) dw = 2

∫ ∞0

Re(H(w)) dw,


EP(X(T )−K)+ = Black(0, X, T,K, σ)− K

π

∫ ∞0

Re

(e−(α+iw) logKq(α+ iw)

(α+ iw)(1− α− iw)

)dw.

A.3 Proof of Proposition 4.3.1

Proof. Since the swap rate S(t) is a martingale under the swap measure Qj,k−j , the driftterm of the process for S(t) must be zero under this measure. From the definition, S(t)is a function of Lj(t), Lj+1(t), · · · , Lk−1(t) and an application of Ito’s lemma shows that

dS(t) =k−1∑n=j

√z(t)(bLn(t) + (1− b)Ln(0))

∂S(t)

∂Ln(t)λn(t)>dW j,k−j(t)

=√z(t)(bS(t) + (1− b)S(0))

k−1∑n=j

bLn(t) + (1− b)Ln(0)

bS(t) + (1− b)S(0)

∂S(t)

∂Ln(t)λn(t)>dW j,k−j(t)

=√z(t)(bS(t) + (1− b)S(0))

k−1∑n=j

wn(t)λn(t)>dW j,k−j(t),

where

wn(t) =bLn(t) + (1− b)Ln(0)

bS(t) + (1− b)S(0)

∂S(t)

∂Ln(t).

Now we will derive an expression for ∂S(t)/∂Ln(t). Note that S(t) can be written by

S(t) =

∑k−1i=j τiP (t, Ti+1)Li(t)∑k−1r=j τrP (t, Tr+1)

=

k−1∑i=j

τiP (t, Ti+1)∑k−1r=j τrP (t, Tr+1)

Ln(t) =

k−1∑i=j

αi(t)Li(t),

where

αi(t) =τiP (t, Ti+1)∑k−1r=j τrP (t, Tr+1)

.

75


Then, we can easily see that:

∂S(t)

∂Ln(t)= αn(t) +

k−1∑i=j

∂αi(t)

∂Ln(t)Li(t). (A.6)

Note that we have

P (t, Ti+1) =P (t, Ti)

1 + τiLi(t)

=P (t, Ti−1)

(1 + τiLi(t))(1 + τi−1Li−1(t))

=P (t, Tj)∏i

l=j 1 + τlLl(t).

Then, the partial derivative w.r.t. Ln(t) is given by

∂P (t, Ti+1)

∂Ln(t)=

−τn

1+τnLn(t)P (t,Tj)∏i

l=j(1+τlLl(t)), i ≥ n

0 , otherwise

=−τn

1 + τnLn(t)

P (t, Tj)∏il=j(1 + τlLl(t))

H(i− n)

=−τn

1 + τnLn(t)P (t, Ti+1)H(i− n),

where H(·) is the Heaviside function: H(x) = 1 for 0 ≤ x and H(x) = 0 for x < 0. Byusing the above result, for n > j, we can show that:

∂αi(t)

∂Ln(t)=

∂

∂Ln(t)

(τiP (t, Ti+1)∑k−1r=j τrP (t, Tr+1)

)

=τi

∂P (t,Ti+1)∂Ln(t)

∑k−1r=j τrP (t, Tr+1)− P (t, Ti+1) ∂

∂Ln(t)

(∑k−1r=j τrP (t, Tr+1)

)(∑k−1

r=j τrP (t, Tr+1))2

=τi

∂P (t,Ti+1)∂Ln(t)

∑k−1r=j τrP (t, Tr+1)− P (t, Ti+1)

∑k−1r=j τr

∂P (t,Tr+1)∂Ln(t)

(∑k−1


=τi

−τn1+τnLn(t)P (t, Ti+1)H(i− n)∑k−1

r=j τrP (t, Tr+1)−τiP (t, Ti+1)

∑k−1r=j τr

−τn1+τnLn(t)P (t, Tr+1)H(r − n)(∑k−1


=−τn

1 + τnLn(t)

(τiP (t, Ti+1)∑k−1r=j τrP (t, Tr+1)

H(i− n)

− τiP (t, Ti+1)∑k−1r=j τrP (t, Tr+1)

k−1∑r=j

τrP (t, Tr+1)∑k−1r=j τrP (t, Tr+1)

H(r − n)

76


=−τn

1 + τnLn(t)

αi(t)H(i− n)− αi(t)k−1∑r=j

αr(t)H(r − n)

=

−τn1 + τnLn(t)

αi(t)

(H(i− n)−

k−1∑r=n

αr(t)

)

=τn

1 + τnLn(t)αi(t)

−H(i− n) +

k−1∑r=j

αr(t)−n−1∑r=j

αr(t)

,

which simplifies to

∂αi(t)

∂Ln(t)=

τn1 + τnLn(t)

αi(t)

−H(i− n) + 1−n−1∑r=j

αr(t)

. (A.7)

Hence, substituting the result from Equation (A.7) in Equation (A.6) shows:

∂S(t)

∂Ln(t)= αn(t) +

k−1∑i=j

∂αi(t)

∂Ln(t)Li(t)

= αn(t) +

k−1∑i=j

τn1 + τnLn(t)

αi(t)

−H(i− n) + 1−n−1∑r=j

αr(t)

Li(t)

= αn(t) +τn

1 + τnLn(t)

k−1∑i=j

αi(t)

−H(i− n) + 1−n−1∑r=j

αr(t)

Li(t)

= αn(t) +τn

1 + τnLn(t)

k−1∑i=j

αi(t)(−H(i− n) + 1)Li(t)−n−1∑r=j

αr(t)k−1∑i=j

αi(t)Li(t)

= αn(t) +

τn1 + τnLn(t)

n−1∑i=j

αi(t)Li(t)−n−1∑r=j

αr(t)S(t)

= αn(t) +

τn1 + τnLn(t)

n−1∑i=j

αi(t)(Li(t)− S(t)).

77

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Libor market model with stochastic volatility · 2020-06-14 · Interest rates are also a vital tool for monetary policy and within the nancial envi-ronment. However, managing interest