Stochastic Finance - A Numeraire Approachartax.karlin.mff.cuni.cz/.../10_Zahradnik-BinModels.pdf · 2012-01-07 · Introduction to Valuation via Numeraires Stochastic Finance - A

Motivation for Numeraire ApproachIntroduction to Valuation via Numeraires

Stochastic Finance - A Numeraire Approach

Petr Zahradnık

Stochasticke modelovanı v ekonomii a financıch

28th November and 5th December 2011

Petr Zahradnık Numeraire Valuation


1 Motivation for Numeraire Approach




2 Introduction to Valuation via Numeraires







Motivation:

What are the ideas behind derivatives valuation?

Modern Martingale (risk–neutral) approach to valuation mayseem to appear from nowhere. The more unmotivated staysthe Numeraire approach, which generalizes valuation so thatthe original motivation may stay blurred forever.

However, the Numeraire approach turns out to give veryelegant results which seem to appear easily.

Outline:

1 Go through the most standard discrete–time models, showwhy we should accept that prices are some specialexpectations and that prices should be modeled as martingalesfor derivative pricing purposes.

2 Introduce the concept of Numeraire, revisit the Binomialmodel and prepare ground for lectures in continuous time.



Definition

Price XY (t) at time t is a number representing how many units ofan asset Y are required to obtain one unit of an asset X . Thereference asset Y is called a numeraire. Price is always a pairwiserelationship.

Most often, banknotes, say $, are chosen as a numeraire. An assetX , say one Coca–Cola stock, has a price X$ · $. However, not onlyeasily tradeable assets are chosen as numeraires– max0≤s≤t XY (s)can again be a numeraire.

Assets do not have numerical value on their own. We may writeX (t) = Z (t) in the sense of assets (= is rather to be read as anequivalence) when XZ (t) = 1 in the sense of numbers.



Once we have a price of an asset XY (t), we have even an inverseprice YX (t) :

X (t) = XY (t) · Y (t) = XY (t)YX (t) · X (t) ⇒ YX (t) = XY (t)−1

Think of exchange rates as a good motivation. One dollar notedoes note change in time, once we express its price in terms of oneeuro note, this number may change continuously.

Another simple observation:

X (t) = XY (t) · Y (t)

X (t) = XZ (t) · Z (t) = XZ (t)ZY (t) · Y (t) ⇒ XY (t) = XZ (t)ZY (t)

which we call a “change of numeraire formula”



Definition

Let ∆i (t), a predictable process, represent a number of X i (t) unitsheld at time t. A portfolio and a portfolio price are give by:

P(t) =

n∑

i=1

∆i (t)X i (t), PY (t) =

n∑

i=1

∆i (t)X iY (t).

Definition

Let P(0) = 0 and follow a self-financing strategy (at any time onecan exchange only equivalent assets).If there exists an arbitrary T > 0, such that P(T ) ≥ 0 a.s. andP(P(T ) > 0) > 0 (i.e. with positive probability the portfolio isequivalent to an asset which is not worthless), we say that anarbitrage exists in a market.



If an arbitrage is not possible by holding a certain asset, we call theasset a no–arbitrage asset.

Observation

Let V be a contract to deliver X at time T . ObviouslyV (T ) = X (T ) For a no–arbitrage asset, it holds thatV (t) = X (t)∀t ∈ [0,T ].

Observation

The contract to deliver an asset X is allways a no–arbitrage asset,no matter whether X is a no–arbitrage asset or not.

Further, unless stated otherwise, a portfolio consists ofno–arbitrage assets.



Definition

A Contingent Claim is any random variable C that is defined on astochastic basis and E |C | < ∞

Definition

A Derivative (Security) of an asset S is a contingent claim suchthat it is σ∞(S)− measurable.

Definition

A (Europen Vanilla) Call Option on S with strike K and maturityT is a derivative such that at time TC$(T , S(T )) = (S$(T )− K )+



Self–financing portfolio in discrete time: Price transition happensjust before the portfolio is rebalanced so the following must hold:

PY (t + 1)− PY (t) =N∑

i=0

∆i (t)[

X iY (t + 1)− X i

Y (t)]

N∑

i=0

[

∆i (t + 1)−∆i (t)]

X iY (t + 1) = 0

Self–financing portfolion in continuous time: We postulate:

dPY (t) =N∑

i=0

∆i (t)dX iY (t)

By Ito formula (note the similarity to the discrete case), it mustalso hold that

N∑

i=0

[

d[∆i ,X iY ](t) + d∆i (t)X i

Y (t)]

= 0.



Examples of Self–financing portfolios

P(t) =[

1− t

T

]

X +

[

1

T

∫

t

0XY (s)ds

]

Y

is a self–financing portfolio.

P(t) = Φ(d+)X + [−KΦ(d−)]Y ,

where

d± =1

σ√T − t

log

(

XY (t)

K

)

± 1

2σ√T − t,

is a self–financing portfolio too. (It is a portfolio replicating aEuropen vanilla call option)



The following four lectures concentrate on the valuation ofderivatives – contracts such as a vanilla option.

Imagine we live in a simple world with one bond and one stock.There are only two time steps 0 and T . The bond evolves asB$(T ) = (1 + r)B$(0) and the stock evolves randomly – on a oneperiod binomial lattice:

S$(T ) = uS$(0) with probability p;

S$(T ) = dS$(0) with probability q = 1− p.

How would we price a vanilla option, which pays (S$(T )− K )+?

40 years ago, we would have computed an expectation. Black andScholes have shown such an approach is wrong!

First note, that to exclude arbitrage, it must hold thatd < 1 + r < u. (1 + r < d < u: Sell a Bond, get B$(0) banknotes,buy a fraction of stock and realize a risk free profit.)



Black and Scholes (with Merton adding mathematical rigour)formalized the idea of replication. This idea in discrete one–periodmodel (by Cox, Ross and Rubinstein) looks as follows:Without loss of generality assume B(0) = 1. We would like aportfolio consisting of S and B be such that it replicates theoption pay–off, i.e. it is true that

V$(∆,T ) = ∆S(0)S$(T ) + ∆B(0)B$(T ) = C$(T , S(T ))

i.e.

∆S(0)uS$(0) + ∆B(0)B$(T ) = C$(T , uS(0))

∆S(0)dS$(0) + ∆B(0)B$(T ) = C$(T , dS(0))

If such a portfolio exists, than by no–arbitrage arguments thedollar price of the option at time 0 must equal the dollar price ofthe porfolio at time 0– ∆S(0)S$(0) + ∆B(0)



Watch out! We found a price without a word about the probabilityof the Stock moving up. Actually we didn’t need the wordprobability at all. We needed replication only. Our two equationsof two unknowns (∆S and ∆B) imply

∆S(0)S$(0)(u − d) = C$(T , uS$(0))− C$(T , dS$(0))

∆B(0)B$(T )(d − u) = C$(T , uS$(0))d − C$(T , dS$(0))u

which in turn implies

∆S(0) =C$(T , uS$(0))− C$(T , dS$(0))

S$(0)(u − d);

∆B(0) =C$(T , uS$(0))d − C$(T , dS$(0))u

B$(T )(d − u)



However, to have the theory elegant, there should be a way tocompute prices as expectations! For this we introduce an auxiliarymeasure p∗, which makes the bond–discounted stock dollar priceprocess a martingale. The discounting procedure makes a goodeconomic sense and in the general setting must be done to avoidmispricing due to banknotes being an arbitrage asset. Also, we willsee that it means the stock bond–price process is then a martingale(via change of numeraire). By the change of numeraire we alsosee, that the martingale measure really depends on the specificnumeraire.

S$(T ) =1

1 + r(p∗uS$(0) + (1− p∗)dS$(0))

hence p∗ = 1+r−d

u−d, 1− p∗ = u−(1+r)

u−d. Note that this is a

probability measure only because of our assumptions on u, d , r . Ifthese hadn’t hold, we would have not been able to find aprobability measure! (no arbitrage ⇒ ∃ martingale measure) istrue in general, see the First Fundamental Theorem.



Now, from the replication principle, we remember, that

V$(0) =∆S(0)S$(0) + ∆B(0)

=C$(T , uS$(0))− C$(T , dS$(0))

S$(0)(u − d)S$(0)

+C$(T , uS$(0))d − C$(T , dS$(0))u

(1 + r)(d − u)

=1

1 + r

(

1 + r − d

u − dC$(T , uS$(0)) +

u − (1 + r)

u − dC$(T , dS$(0))

)

.

which in the language of our “artificial” martingale measure readsas

V$(0) =1

1 + r(p∗C (T , uS$(0)) + (1− p∗)C$(T , dS$(0)))

= E∗

(

C$(T , S$(T ))

1 + r

)

.

Price of the option can be expressed as an expectation withrespect to a measure given by the numeraire. (?∃?)







Pricing

Let us now move to pricing via numeraires nad martingales ingeneral. Hopefully, now it is clear why it makes sense to do it.



Pricing

Let us now move to pricing via numeraires nad martingales ingeneral. Hopefully, now it is clear why it makes sense to do it.When modeling a financial market, we do not want arbitrage toexist in the model–simply because in reality arbitrage due toeconomic reasons (almost) doesn’t exist. If it appears (a contractis mispriced), it is immediately exploited and the prices are movedso that an arbitrage is not possible anymore.

Fortunately, there is a mathematical rule, which says whether ourmodel would allow arbitrage or not. It is called First FundametalTheorem of Asset Pricing.

It says, that whatever our probabilistic model of financial markets(specifically assets in them) may look like, for it to be reasonableand not allow arbitrage means the same as that there must exist aprobability measure such that it makes all processes of asset priceswith respect to a chosen numeraire martingales!



First Fundamental Theorem of Asset Pricing

Theorem

∃ PY a probability measure s.t. X i

Y(t) are P

Y−martingales for Xan arbitrary no–arbitrage asset and Y a no–arb. asset which isnever worthless ⇐⇒ ∄ arbitrage in the market.

Proof.

Prove the converse of one implication only: If there were anarbitrage, PY (t) wouldn’t be a P

Y−martingale. PY (T ) = Ψ,where Ψ is a non–negative r.v. s.t. PY (Ψ > 0) > 0

This implication is enough for practical purposes. When we modelprice evolutions, we simply check that we are able to make themmartingales. (We may for example only adjust parameters for sucha condition to hold).



Interpretation of the martingale probability measure PY (t): Let Vbe an Arrow–Debreu security, i.e. a contract that pays one unit ofnumeraire when a certain state is reached:

V (T ) = IA(·)Y (T ), i.e. VY (T ) = IA(·)

VY (t) is a PY−martingale, therefore

VY (0) = EY [VY (T )] = P

Y (A).Clearly, the initial market price of the Arrow–Debreu security lies in[0, 1] and the martingale probability indicates how “costly” is anevent A rather than how likely. The price corresponding to theArrow–Debreu security for a single scenario in discrete state spaceis known as an Arrow–Debreu state price. In continuous time, wegeneralize this concept to

PY (A) =

∫

A

dPY

and to the so–called state price density.Petr Zahradnık Numeraire Valuation


Pricing basis

Pricing of basic no–arbitrage assets is easy. The only way to deliverthem is to hold them, hence a contract delivering on unit of assetsX at time T must be worth X (t) at any prior time. (Otherwisethere would be an arbitrage) Imagine a Money Market Account ora Bond being a numeraire and note that banknotes are arbitrageassets!Further we try to exploit the idea of prices of no–arbitrage assetsfor pricing more complicated contracts. We try to price contractsof the form f (X (T ),V (T )) for two no–arbitrage assets X and Y .

Once again–we saw that an arbitrage is not possible when there isa martingale measure. We also saw, that in such a case, a price ofsuch a contract can be computed as an expectation w.r.t. themartingale measure. But what if there are more martingalemeasures?



One–period Trinomial model

The price XY (T ) is assumed to take three possible values:uXY (0), XY (0) or dXY (0), corresponding to scenarios. Withsuitably chosen parameters (which?) the following measure

PY (uXY (0)) =

1− d

u − dξ,PY (XY (0)) = 1−ξ, P

Y (uXY (0)) =u − 1

u − dξ

makes the price process XY (t) a martingale. Therefore, there areactually uncountably many martingale measures! Consider anArrow–Debreu security, which pays one unit of Y ifXY (T ) = uXY (0). If we relied on the expectation w.r.t. themartingale measure, we get that the initial price is equal toPY (uXY (0)). This is not feasible, we don’t want uncountably

many prices to be the correct ones!



Trinomial model continued

Observation

Binomial model is a special case of ξ = 1. The martingale measureis thus unique in the binomial model.

When we get back to our original “economic” replication approachof a derivative VY (X ,Y ), how does it apply? It leads generally tothree linear equations with to uknowns:

VY (T , uX ) = ∆X (0)uXY (0) + ∆Y (0)

VY (T ,X ) = ∆X (0)XY (0) + ∆Y (0)

VY (T , dX ) = ∆X (0)dXY (0) + ∆Y (0).

This means we are not able to replicate the derivative.

Definition

A market model, where every derivative can be replicated, is calledcomplete. It is called incomplete otherwise.



Trinomial model continued

Assume however, that an investor believes a trinomial market is agood model. What do they do to find a price for a derivative? Onepossibility is to introduce another underlying asset and completethe model. For example, assume the market trades anArrow–Debreu security Z , that pays one unit of Y ifXY (1) = uXY (0) happens. Remeber, the quote of ZY (0) tells uswhich martingale measure is the right one. The market is thencomplete if we consider a portfolio

P(0) = ∆X (0)X +∆Y (0)Y +∆Z (0)Z .

Such a portfolio yields three equations of three unknowns.



Second Fundamental Theorem of Asset Pricing

There may be more martingale measures in the model. As said,this unfortunately means, that despite the fact that there is noarbitrage, there might be more “correct” prices of a security. Wewould like to avoid such a situation. We know, that we can“manually” check, whether a market is complete simply bychecking the replication equations for each individual derivative.Mathematically speaking, it seems like we cannot say upfront,whether our model is complete or not. Again, there is a way out ofthis quagmire:

Theorem

A market is complete if and only if the martingale measure PY is

unique.

Proof is quite technical. In the continuous case, it is actually quitedifficult, on must redefine arbitrage and plunge into functionalanalysis with w ∗ − topologies etc.



To be continued...



Fundamental Theorems of Asset Pricing

Theorem (First Fundamental Theorem)

∃ PY a probability measure s.t. X i

Y(t) are P

Y−martingales for Xan arbitrary no–arbitrage asset and Y a no–arb. asset which isnever worthless ⇐⇒ ∄ arbitrage in the market.

There may be more martingale measures in the model. As said,this unfortunately means, that despite the fact that there is noarbitrage, there might be more “correct” prices of a security. Wewould like to avoid such a situation. We know, that we can“manually” check, whether a market is complete simply bychecking the replication equations for each individual derivative.



Fundamental Theorems of Asset Pricing

Mathematically speaking, it seems like we cannot say upfront,whether our model is complete or not. Again, there is a way out ofthis quagmire:

Theorem (Second Fundamental Theorem)

A market is complete if and only if the martingale measure PY is

unique.



Change of measure

In an ideal world where a martingale measure is unique, we have bythe change of numeraire formula:

V = EY [VY (T )] · Y (0) = E

X [VX (T )] · X (0)

Remember the Radon–Nikodym derivative: Martingale measuresare equivalent, hence:

EX [VX (T )] · X (0) = E

Y [VX (T )Z (T )] · X (0)

Since this is true for an arbitrary payoff, we have a financialinterpretation:

VX (T )Z (T ) · X (0) ⇒ Z (T ) =VY (T )

VX (T )YX (0) =

XY (T )

XY (0)



Binomial model revisited

Ideal world with two time steps 0, T , two no–arbitrage assets Xand Y and two possible outcomes for XY (T )– uXY (0) or uXY (0).

We get pY = 1−d

u−d. On the contrary, YX (0) turns into

YX (0)u

orYX (0)

d, which yields pX = u 1−d

u−d. Of course, this result is easily

deduced from the “financial intepretation” of Radon–Nikodymderivative.

Binomial model is easily modified to more periods.



Dollar prices

Computing dollar prices: VBT(0) = E

BT [

V TB(T )

]

, therefore,assuming BT

$ (T ) = 1 (this is more common and actually the onlycorrect in full generality)

V$(0) = VBT (0)BT

$ (0)

= EBT

[V$(T )$BT (T )]BT

$ (0) = EBT

[V$(T )]BT

$ (0)

This formula is of fundamental importance. If we find acorresponding martingale measure for prices evaluation w.r.t. abond maturing at time T, we can immediately price a contractwithout a need to find an interest rate, for example! We get theBond dollar value directly from the market. This is not the casewhen we take a money market as a numeraire. Than we need toknow the interest rate and it must be deterministic to be able tocompute anything.



Exotic Assets

Even self–financing portfolios composed of no–arbtitrage assets are(with some restrictions on ∆s,) no–arbitrage assets and thereforecome with their own martingale probability measure. Let usillustrate it on an asset:

A(T ) =1

2[XY (0) + XY (T )]

Clearly, ∆X (0) = 12 is the self–financing portfolio replicating the

asset.

PA = Z (T )PY =AY (T )

AY (0)PY=

1

2

(

1 +XY (T )

XY (0)PY

)

This is a measure important in pricing all Asian options. Such anaveraging portfolio is made a numeraire and comupations turn outto be much simpler.



Binomial model with an arbitrage asset

To be able to use the fundamental theorems, we must hedge onlythrough no–arbitrage assets, otherwise of–course we have arbitragewhatever we might try to do. When the underlying assets are stockand dollar, we must hedge trought stock and bond, for example.

With American Options, this is generally not feasible, as they maybe excercised at any time, hence we cannot choose bondmaturities. Hence hedging in a bond provides us with a proxy only!

Consider S$ and B$ and their common evolutionS$(T ) = xS$(T − 1), where x = u, d , andB$(T ) = (1 + r)B$(T − 1). We can express these evolutions interms of a no arbitrage assets, for example the bond (but stockworks too!) and get the familiar formulas.



American Options

American options can be excercised at any time. At any time,there is an intrinsic value of the option, the value of immediateexcercise, and continuation value, the value of, well, continuation.

V$(T − 1) = max

(

f $(S$(T )),1

1 + rEBT

T [V$(T + 1)]

)

Informally, we “delete” some of the lower nodes of the tree.American options are much more elegant (mathematicallyspeaking) in continuous time, where the theory of optimal stoppingis much more interesting. (It isn’t only a backward induction).

This is it from the introduction. Let us see the continuous case.



Thank you for attention!

Bibliography:

Jan Vecer, Stochastic Finance: A Numeraire Approach

CRC Press, 2011


Documents

Stochastic Finance - A Numeraire Approachartax.karlin.mff.cuni.cz/.../10_Zahradnik-BinModels.pdf · 2012-01-07 · Introduction to Valuation via Numeraires Stochastic Finance - A