ingale Arbitrage Modeling
fixed-income business.In the popular SABR (Sigma Alpha Beta Rho) stochastic volatilitymodel (Hagan, Kumar, Lesniewski, and Woodward, 2002), which hasbecome the industry standard for interest rate derivatives, a, (Sigma) isassumed to follow the following lognormal process:
do,= ato,dW,u
1
Corr, ( dW,,d y u = p ’
(2.99)
where yuis another (one-dimensional) PN -Brownian Motion that has aconstant instantaneous correlation p (Rho) with W,, a, (Alpha), a deterministic function of time, is the volatility of volatility (or Vol of Vol).The P (Beta) is from the CEV power of the following CEV specification of the volatility(2.100)In the SABR model, analytical approximations can be derived for theequivalent lognormal or Black volatilities.The SABR model can also be combined with the following shiftedlognormal model or a displaced model,(2.101)where 0 I P I 1 is a constant (or a time-dependent non-stochastic parameter).The above model is like a “blend” of a normal model and a lognormal model. With p = 1 , it reduces to a lognormal model, and withp = 0 , it reduces to a normal model.2.5.10 An Example of Martingale Modeling in Real Market
As an example, U S . Treasury yields and LIBOR rates in the swap market are commonly used interest rates. An interesting question as weasked before is: what is the martingale relationship of a LIBOR (zerocoupon) bond with a Treasury (zero-coupon) bond as the numeraire andvice versa? Similarly, how do we, in general, reconcile the martingale
120
Martingale Arbitrage Pricing in Real Market
Chapter 2
relationships with a LIBOR (zero-coupon) bond as the numeraire andwith a Treasury (zero-coupon) bond as the numeraire? A more practicalquestion is: what are the appropriate numeraires for derivatives modelingand pricing?Let’s first look at whether or not the following martingale relationship should hold, i.e.,
where PLIBoR ( t ,2“) , PTreusUy(t, T) time t price of a zero-coupon LIBORbond and a zero-coupon Treasury bond, respectively, both promising apayoff of 1 at the maturity T, and the expectation is taken under the for( t , T ) as the nuward probability measure PT with respect to PTreusurymeraire.We know that‘Treusuiy
( T ,T >=
(2.103)
However, if we simple-mindedly assume that(2.104)
pLIBoR (T7T ) = 1
then, Eq. (2.102) yields‘LIBOR
(t,T , = ‘Treusuiy (t ,T ,
( I IT ,
*
(2.105)
Of course, we know that the above equation is not true. One reasonfor that is that the LIBOR bond is not exactly default-free and Eq.(2.104) is only valid if the LIBOR bond has not defaulted on or prior totime T. Other reasons include that the LIBOR bond is not as liquid as theTreasury bond. Thus, the LIBOR bond should have lower price or PV(or higher yield) than that of the Treasury bond. In other words,‘LIBOR
(t ,)‘ < ‘Treasury
(‘9
T,
(O
‘‘
T,*
(2.106)
A further question is that are there any other subtleties?Let’s for a moment assume hypothetically that the LIBOR bond isindeed default-free. In this case, do we expect that the LIBOR bond hasthe same price or PV (or yield) as that of the Treasury bond?As a matter of fact, in this case, the LIBOR bond should still have
Sec. 2.5
Marfingale Arbitrage Modefing
121
lower price or PV (or higher yield) than that of the Treasury bond. Thisis due to the fact that the income from U.S. Treasury debt instruments(e.g., Treasury bills, notes, and bonds) is exempt from state and localtaxes, though subject to the federal tax. For this reason, the U S . Treasury interest rate should actually be lower than the true “risk-free interestrate” that is default-free, but with no tax benefit.Another related example is to price a derivatives instrument (such asan option) on a Treasury bond. Typically, the payoffs of the Treasurybond need to be discounted by Treasury (zero-coupon) bonds, whereasthe payoffs of the derivatives instrument need to be discounted byLIBOR (zero-coupon) bonds. Similar situation occurs when pricing aderivatives instrument (such as an option) on a convertible bond.More generally, different classes of assets or instruments needed tobe modeled as martingales with different numeraires. Intuitively speaking, different classes of assets or instruments need to be funded with different numeraires.More specifically, this is the case where the price of an underlyingasset U ( t ) and the price of a derivatives instrument V ( t ) follow themartingale relationships with different numeraires ( N u and Nv) andunder different probabiIity measures. In other words,(2.107)
and(2.108)
For instance, the underlying assets of Treasury, agency, municipal,mortgage, etc. all have different corresponding numeraires. The same istrue even for on-the-run and off-the-run Treasury underlying assets.This is the same as the situation whereby different classes of assetsor instruments have different funding or financing rates or rep0 rates forliquid assets, as we discussed before.One way to achieve consistent martingale modeling is to introduce afunding spread multiplier M F (s,t ) such that
122
Martingale Arbitrage Pricing in Real Market
Chapter 2
(2.109)
In a way, this is similar to the modeling of FX derivatives.Thus, the multiplier for the change of probability measure in Eq.(2.75) needs to be modified accordingly. We shall discuss more on thisin the next chapter including its impact on the put-call parity.
2.6
Problems
Problem 2.1. A derivatives instrument is written on a stock paying nodividend with the current price of $100. The derivatives instrument has aperpetual lifetime, unless the stock price reaches a barrier of $200, inwhich case the derivatives instrument gets knocked out (terminated) withan immediate rebate payment of $50. Assume that the interest rate iszero and find the present value of the derivatives instrument.(Hint: use an SFTS to replicate this derivatives instrument. What if thestock price can jump? What if the stock issuer can bankrupt?)Problem 2.2. Same as Problem 2.1, except we make it more general,i.e., the current price of the stock is S, the barrier is H, and the rebate isR. Find the present value of the derivatives instrument and discuss thefollowing scenarios: a) What if the stock price can jump? b) What if thestock issuer can bankrupt?Problem 2.3. Same as Problem 2.1, except the derivatives instrument iswritten on the temperature (with current temperature of 7 0 9 . The derivatives instrument has a perpetual lifetime, unless the temperaturereaches a barrier of l O O F , in which case the derivatives instrument getsknocked out (terminated) with an immediate rebate payment of $50. Canan SFTS be used to replicate this derivatives instrument?
Chapter 3
The BlackScholesFramework andExtensions
Fisher Black, Myron Scholes, and Robert Merton pioneered the application of the risk neutral valuation to derivatives pricing. As a result,Myron Scholes and Robert Merton won the Nobel Prize in Economics in1997.66The Black-Scholes framework is a special case of the martingalemodeling presented in the previous chapter. In this chapter, we presentmore details on the martingale modeling and how it can reduce to theBlack-Scholes framework. We shall also present brief discussions onvarious extensions to the Black-Scholes framework, such as state dependent volatility models (e.g., CEV, shifted lognormal or displaced, andlocal volatility models). We shall also discus the revised put-call parity.
3.1
More on Martingale Models
In this section, we shall present more details on the martingale models66 Fisher Black passed away in 1995, two years before the Nobel Prize was awarded.
Otherwise, he would have been a co-recipient of the prize. Please see Derman (2004)and Mehrling (2005) for their reflections on Fisher Black.The Nobel Prize citation and other information can be found at
http://www.nobel.se/economics/laureates/1997/press.html,http://www.nobel.se/economics/laureates/1997/presentation-speech.html,andhttp://www.nobel.se/economics/laureates/1997/index.html. For the seminal papers oftheir work, please refer to Black and Scholes (1973)and Merton (1973).
124
The Black-Scholes Framework and Extensions
Chapter 3
that we have discussed in the previous chapter and present examples forpricing simple European style derivatives. We divide these martingalemodes into the following common categories: single underlying asset orstate variable and single numeraire, single underlying asset or state variable and multiple numeraires, and multiple underlying assets or statevariables and multiple numeraires.As we discussed earlier, the procedures for martingale modeling startwith the identification of martingale relationships for the derivatives instruments and the underlying (or hedge or replication) instruments, thenare followed by the application of martingale representation theorem and,if necessary, Girsanov’s theorem of change of probability measure.As we shall see that the Black-Scholes framework is a special case ofthe martingale models.3.1.7
Single State Variable and Single Numeraire
This is the case where the price of an underlying asset U ( t ) and theprice of a derivatives instrument V ( t ) follow the martingale relationships with the same numeraire. In other words,
and
where, similar to before, the expectation is taken under the equivalentmartingale measure PN with respect to numeraire N.The classical Black-Scholes model can be derived from this setting.Again, as usual, we first assume that both of these assets do not haveany intermediate cashflows prior to time T. For a European derivativesinstrument, it has a predefined cashflow at time T (corresponding to theexpiry date or the expiry settlement date), i.e.,
where g (.) is a deterministic function.
Sec. 3.1
More on Martingale Models
125
We define the state variable F ( t ) as the numeraire-normalized underlying asset price, i.e.,
or
U(t)=N(t)F(t).
(3.5)
The intuition of the state variable is as follows. If U ( t ) is the priceof the floating leg of a forward starting swap and N ( t ) is the price of thecorresponding annuity, then F ( t ) is the corresponding forward swaprate. This also includes the forward LIBOR as a special case (with U ( t )being the price of the floating leg of a forward rate agreement (FRA)). IfU ( t ) is the price of a non-dividend-paying stock and N ( t ) is the priceof a LIBOR zero-coupon bond (or rep0 zero-coupon bond), then F ( t ) isthe corresponding forward price of the stock. This case covers, e.g.,simple stock options (on a non-dividend paying stock) and simplecaps/floors and swaptions.We further assume thatV ( T >= g ( U ( T ) )
(3.6)
= g ( N ( T ) F ( T ) )7= N W f
(W)
where f (.) is another deterministic function.Thus,
v(t)=N(t)E:[f (F(T))]
(OItIT).
(3.7)
With this setting, if we only need the PV (or V ( 0 ) )and sensitivitieson the PV, the dynamics of the numeraire N ( T ) drops out of the equation (except for the determination of the probability measure) and weonly need to model the state variable F ( t ).In some simple cases, Eq. (3.7) has closed-form solutions or ap-
126
The Black-Scholes Framework and Extensions
Chapter 3
proximations, once we know the mean (or forward value) and the variance (or volatility) of the underlying variable, as well as some model parameters. In general, it can be solved easily with numerical integrations,including using Monte Carlo simulations, quasi-Monte Carlo simulations, replications, and (implicit or explicit) finite difference.This setting covers many cases in derivatives pricing, particularly,for derivatives in the swap market (with single currency and single statevariable or index), such as simple LIBOR based capdfloors and swaptions.A plain vanilla European put or call option with expiry T has the following payoff at time T
(3.8)
where K is the strike, and for a call option w = l and for a put optionw = -1 . More specifically,
Recall from the Martingale Representation Theorem, if F ( f) is continuous with finite variance, then F ( t ) can be represented by a (onedimensional) IF',, -Brownian Motion W,as
Sec. 3.1
More on Martingale Models
127
dF ( t )= o,"dy .
(3.11)
Similar to what we discussed before, for a CEV model,(3.12)and for a shifted lognormal or a displaced model,0:
=op(t)(PF(t)+F(O)(l-P))
Y
(3.13)
PI1 is awhere op( t ), the model volatility, is deterministic and 0 Iconstant.The above models are like a "blend" of a normal model and a lognormal model. With ,8 = 1, they reduce to a lognormal model, and with
P = 0 , they reduce to a normal model. oP( t ) is approximately the corresponding lognormal volatility.We will first discuss the shifted lognormal model. In the shifted lognormal model, when P > 0 , we can define a new variable
F ( t) = F ( t) + F ( 0)- (1 - P ) .P
(3.14)
(3.15)
(3.16)
(3.17)
128
The Black-Scholes Framework and Extensions
F ( t ) + F (0)-
Chapter 3
P
(tlT)Define(3.19)and
(t <T ).
(3.20)
F ( T )= ( t )exp( -iiTi ( T - t )+ Op (w, - ))
(3.21)
Ep
(w, q ) P f op(x) dW,-
E
then,
(t I T )Therefore,
[
F ( T ) = F(t)+F(0)")x exp
P
(-35;( T - t ) + ~p (W, - W,))
(3.22)
- F ( (1O -) P7)(t5T)Further define
I? = K+F(O)-,(1-
P
(3.23)
Sec. 3.1
129
More on Martingale Models
then, Eq. (3.9) becomes
V, ( T ) = N ( T)max( w( 3(T) - g ))(3.24)
and,
fE ( F ( T ) )= max (0( p(T) - K))(3.25)
Thus, Eq. (3.7) can be rewritten as1
r
r
1
(3.26)
(0 I t I T )
By applying Girsanov's theorem of change of probability measure orEq. (2.85), we obtain a key equation
r
1
(3.27)
(0 I t I T)*clmxI*-^
_I__f-
l"x*-i_--*l-
Problem 3.1. Prove the above equation.
-
--*^"""xx.~p
^-*---
Thus, through simple algebra, Eq. (3.26) can be easily rewritten as
130
The Black-Scholes Framework and Extensions
Chapter 3
(3.28)(0 I t 5 T )
where
=
w, -q ,and dl and d2are defined asJr-t
d2 =
;
In( F (t ) / q - c; ( T - t )
FpJT-t
(t<T).
(3.29)
dl = d 2 + C P GIt is easy to see that -wxtT is an N(0,l) variable or a standardnormal variable with 0 mean and 1 standard deviation under the probability measure PN . Thus, Eq. (3.28) can be rewritten as
where Nc ( x ) is the cumulative normal distribution, or more specifically,the probability for an N ( 0,l) variable to be less than x .Here we obtained the price of plain vanilla European options withthe probabilistic approach. We also can obtain the same price by thePDE approach.The original Black-Scholes and Black’s formulas for plain vanillacalls and puts based on a lognormal model are special cases of those ofthe shifted-lognormal model with P = 1 . Particularly,
or
Sec. 3.1
More on Martingale Models
131
where oB is the Black volatility, which is typically implied from themarket observable option prices. Equivalently,
Similarly, for European cash-or-nothing (CON)and asset-or-nothing(AON) digital or binary options, the payoff at the expiry Tare given by(3.34)
Under the shifted lognormal model, the time t values of these options are
(3.35)
(3.36)
(3.37)
132
The Black-Scholes Framework and Extension3
F ( T ) = F ( t )+
f cro ( s ) F (O)dW3
Chapter 3
(3.38)
( t IT )
Define
(t<T )
Cro = F ( 0 )
(3.39)9
and(3.40)thus
F(T)=F(t)+Eo(FT-K)
(tI T )
(3.41)
Therefore,
JT-t
3
N ( t )(To
(3.42)
whered=
F(t)-K
oo6
(3.43)
We can easily find the relationship between the normal volatility 5,and the Black or lognormal volatility oB such that they produce thesame vanilla European option prices. In the at-the-money (ATM) case,the relationship is approximately given by00 4t)oB[1-&oB2(T-t)].
(3.44)
In deriving the above equation, we have used the following Taylorexpansion,
N ~ ( ~ ) = N ~ ( o ) + N ~ ~ ( o ) +~ + N~ ~N ~~ ""(+...(Oo )) ~~(3.45)~~
Sec. 3.1
More on Martingale Models
133
and
N,(O)=$
N,' ( X) =
exp( +x2)
N," (X ) =N," ( X ) =
6-x exp (-+ x2)
.
(3 -46)
Jz-(l-x2)exp(-+x2)
6
Separately, we can obtain approximations for the implied Blackvolatility O, ( F , K , P ) from a CEV model, e g ,(3.47)
This is an example of the volatility skew (but not the volatility smile).This is also a special case of the SABR stochastic volatility model(Hagan, Kumar, Lesniewski, and Woodward, 2002) where the modelvolatility op( 1 ) in Eq. (3.12) is stochastic.3.1.2
Single State Variable and Multiple Numeraires
This setting covers many cases in derivatives pricing, such as for FX derivatives pricing (including regular FX derivatives and Quanto or guaranteed exchange rate (GER) derivatives). It also covers derivatives pricingin the swap market (with single currency and single index), such as various more complicated caps/floors and swaptions (including those within-arrears reset and with CMS legs), as well as derivatives on nonLIBOR indices, such as Treasury, agency, municipal, and mortgage indices. It can also be extended to the cross-currency derivatives and otherinstruments, such as stock options on a stock paying uncertain dividends,which involves multiple state variables and multiple numeraires.A typical scenario for this setting is that the martingale relationshipsof the state variable and the derivatives involve different numeraires. Inmany simple cases, this setting reduces to the Quanto adjustment or the
134
The Black-Scholes Framework and Extensions
Chapter 3
convexity adjustment to the forward value of the state variable.Let’s start with the treatment of simple FX derivatives whose payoffsare based on fiture foreign exchange rate.Let SFx ( t ) denote the time t spot FX exchange rate in terms of thevalue or price of one unit of foreign currency in terms of domestic currency. For example, if the domestic currency U.S. dollar and the foreigncurrency is yen, then S , ( t ) is the U.S. dollar price of one yen, though,in practice, l/SFx(t) or yen price of a U.S.dollar is often what isquoted.Furthermore, let P D ( t , T ) and P F ( t , T ) denote the time t price of adomestic and foreign (default-fiee) zero-coupon bond paying one unit ofdomestic and foreign currency at time T, respectively.Thus,S , ( t ) P F ( f , T ) is a traded asset in the domestic economy or marketwithout intermediate cashflows prior to time T. Consequently, we havethe following martingale relationship,
(0 IsItIT)
where ET,D[.] is the expectation under PTPD, the probability measurewith respect to P D ( I ,T ) as the numeraire.Define the forward FX exchange rate as(3.49)From Eq. (3.48), we obtain the following martingale relationship,
FFX ( s , T ) =E,T’D[F,, ( t , T ) ]
(0 I SI t 5 T ).
(3.50)
This is referred to as the interest rate parity.Let V D( t ) denote the time t value of a derivatives instrument payingout in domestic currency without intermediate cashflows prior to time T.As usual, we have the following martingale relationship,
Sec. 3.1
More on Martingale Models
135
A plain vanilla European FX put or call option with expiry T has thefollowing payoff at time T,
where K is the strike, N," is the principal denominated in foreign currency, and for a foreign currency call and domestic currency put optionw = 1 and for a foreign currency put and domestic currency call optionw=-1.For example, if the domestic currency is U.S. dollar and the foreigncurrency is yen, then w = 1 corresponds to a yen call dollar put optionand w = -1 corresponds to a yen put dollar call option.Since S, ( T )= FFX( T , T ) , then
Similarly, for European cash-or-nothing (CON) and asset-or-nothing(AON) digital or binary options, the payoff at the expiry Tare given by
So far, it is the same as the case of single state variable and singlenumeraire, and we can borrow many equations for that case.However, for more complicated instruments, such as callable PowerReverse Dual Currency (PRDC) notes or swaps, or callable instrumentsin general, we normally need to model the FX exchange rate, the domestic interest rate, and the foreign interest rate as individual (but correlated)risk factors, which is typically achieved by using a 3-factor model.It is also worth mentioning that while many FX derivatives require
The Black-Scholes Framework and Extensions
136
Chapter 3
physical settlement or delivery of the currencies involved, there are nondeliverable FX derivatives, such as NDF (non-deliverable forwards) andNDO (non-deliverable options) whereby the final payoff is settled in onecurrency (the payoff or domestic currency) without the physical deliveryof the foreign currency. The main purpose of the non-deliverable FXderivatives is to provide or enhance the liquidity of the FX derivativeswhen the underlying foreign currency is not liquid enough. In manycases the market for non-deliverable FX derivatives is much bigger thanthat of deliverable FX derivatives, especially when the underlying foreign currency does not have enough supply or is not allowed to freelytrade.There is also an interesting paradox (or Siegel’s paradox) involvingthe FX exchange rate FFX ( t ,T ) in terms of the domestic currency priceof one unit of foreign currency and the FX exchange rate l/FFX( t , T ) interms of the foreign currency price of one unit of domestic currency.Now, let’s take a look at the Quanto or the Guaranteed ExchangeRate (GER) FX derivatives. The motivation of this type of derivatives isto allow investors to take positions in foreign assets or state variables,such as foreign stocks, indices, and interest rates, without significant exposures to the foreign exchange rate. This is achieved by converting future payoffs of foreign assets or state variables denominated in the foreign currency to the domestic currency by a pre-specified (or guaranteed)exchange rate, not the future spot exchange rate SFX( t) .
Sgy
Let V D( t ) denote the time t value of a Quanto derivatives instrument paying out in domestic currency without intermediate cashflowsprior to expiry T. As usual, we have the following martingale relationship,
withV D ( T ) = f ( S F XGER
F (T)),
(3.56)
where U F( t ) is the time t value of the underlying asset or state variablepaying out in foreign currency without intermediate cashflows prior to
Sec. 3.1
More on Martingafe Models
137
expiry T. In practice, Sg!R is usually absorbed in the principal amountand, thus, converting the principal denominated in the foreign currencyinto the principal denominated in the domestic currency.Thus, the time t value of the Quanto derivatives instrument is givenby
The hedge or underlying instrument for the above Quanto derivativesinstrument is S , ( t ) U F( t ) (which is a traded asset in the domesticmarket) or U F( t ) (which is a traded asset in the foreign market). Thus,we have the following martingale relationships,
(3.58)(0 IsI tIT)
andUF(S)
PF ( s , T )
=ET,F
[; (4)]
(0 Is I t I T),
(3.59)
where ETTF[.] is the expectation under P T , F ,the probability measurewith respect to P F ( t ,T ) as the numeraire.By inspecting Eq. (3.58) and (3.59), we can infer the correspondingRadon-Nikodym derivative for the change of probability measure as
(0 I t I T )
where ET,D[.] is the expectation under PT,D, the probability measurewith respect to P D ( t , T ) as the numeraire. Thus, for any stochasticprocess satisfying enough technical conditions, we have
The Black-Scholes Framework and Extensions
138
Chapter 3
To gain intuitions about the above equations, we can let
X ( t ) = U F(4PF ( t , T )
(0 5 t 5 T ) .
(3.62)
Similar to before, the forward price of U F( t ) in the foreign marketis given by
(4
F F ( t , T )= U FPF(t,T)
(0 5 t IT ),
(3.63)
(3.64)or, equivalently,
F F ( s , T ) = E ~ ~ F [ F F ( t , T ) ]( O S s I t I T ) ,
(3.65)
which is obtained through a change of probability measure based on Eq.(3.60). In addition, Eq. (3.56) becomesV D ( T ) = f ( SGERF xF F ( T Y ~ ) ) .(3.66)Thus, the time t value of the Quanto derivatives instrument can be rewritten as
vD (s)= PD (SYT)ET'"Cf(SFXGER F F ( T J ) ) ](3.67)
(0 I s 5 t 5 T )Eq. (3.65) and Eq. (3.67) show that the martingale relationships forthe underlying state variable and the price of the Quanto derivatives instrument are under different probability measures. Therefore, in order tovalue the Quanto derivatives instrument, we need to derive the SDE for
Sec. 3.1
139
More on Martingale Models
F ( t , T ) under the probability measure of PT,D.We can start by evaluating the following forward value,F F ( ~ , T ) ~ E , T , D [ F F ( t , T )(]O I s I t I T ) .
(3.68)
One way to compute this is to use the covariance or convexity adjustment. In other words, sinceC , , , T J ( F ~ ~ ( ~ , T()t, ,F~~) )E,TJ’[F~.=(~,T)F~(~,T)]
[
, (3.69)
- ETiD[FFx( t ,T ) ]ET2D F F ( t , T ) ](0 I s 5 t I T )
thus,
[
E,T,D F F ( t ,T ) ]= F F (S, T )
FFX ( s , T )
(0 I S 5 t I T )
which may require approximations.If FFx( t ,T ) is lognormal, then we can obtain the exact solutions byusing Girsanov’s theorem of change of probability measure. Based onthe martingale relationships that we discussed so far, let’s assume that
dFFx( t , T ) = aFFx(t)FFX(t,T)dW,T,D
(0 I t IT ) ,
(3.71)
andd F F ( t , T )=C?(t,FF)dW,TJ-
(0 I t I T ) ,
(3.72)
whereis a IPT,D -Brownian motion and TTPFis a PT,F-Brownianmotion. Thus,
140
The Black-Scholes Framework and Extensions
Chapter 3
1
1
3
(u)’ du + l o F F x (U ) LIW,T’~S
(3.73)
and
Applying ItG’s lemma to the above equation, it is easy to verify that
and(0 Is I t I T ) .
(3.76)
Girsanov’s theorem of change of probability measure indicates that aBrownian motion in one measure (such as yT,Fin PT,F ) is a Brownianmotion in another measure plus a drift term. In other words,
dW,T3F= p ( t ) d t + d@jT’D
(0 I t I T ) ,
(3.77)
where @jT,D is a PT,D-Brownian motion.
A simple procedure of determining p ( t ) is as follow^.^' From Eq.(3.61) and the fact that KTyFis a PT,F-martingale, we have
(3.78)(0 I s I t I T )
Thus,67 Here we are essentially trying to derive Girsanov’s theorem of change of probabilitymeasure. This may not be needed for reader who is very familiar with this theorem.
Sec. 3.1
141
More on Martingale Models
(3.79)
(0 I s I t 5 T )Applying ItB’s lemma to the above equation and plugging in Eq.(3.75) and Eq. (3.77), we obtain
=-y‘ItT ,F aFFX( t ) d y T , D+$dW,T’F’Is
‘IS