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Chapter 12Change of Numéraire and ForwardMeasures

In this chapter we introduce the notion of numéraire. This allows us to con-sider pricing under random discount rates using forward measures, with thepricing of exchange options (Margrabe formula) and foreign exchange op-tions (Garman-Kohlagen formula) as main applications. A short introduc-tion to the computation of self-financing hedging strategies under change ofnuméraire is also given in Section 12.5. The change of numéraire techniqueand associated forward measures will also be applied to the pricing of bondsand interest rate derivatives such as bond options in Chapter 14.

Contents12.1 Notion of Numéraire . . . . . . . . . . . . . . . . . . . . . . . . . . . 42312.2 Change of Numéraire. . . . . . . . . . . . . . . . . . . . . . . . . . . 42612.3 Foreign Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43512.4 Pricing of Exchange Options . . . . . . . . . . . . . . . . . . . . 44212.5 Hedging by Change of Numéraire . . . . . . . . . . . . . . . 444Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448

12.1 Notion of Numéraire

A numéraire is any strictly positive (Ft)t∈R+ -adapted stochastic process(Nt)t∈R+ that can be taken as a unit of reference when pricing an assetor a claim.

In general, the price St of an asset, when quoted in terms of the numéraireNt, is given by

St := StNt, t ∈ R+.

Deterministic numéraires transformations are easy to handle as a change ofnuméraire by a deterministic factor is a formal algebraic transformation thatdoes not involve any risk. This can be the case for example when a currencyis pegged to another currency, e.g. the exchange rate 6.55957 from Euro to

423

N. Privault

French Franc has been fixed on January 1st, 1999.

On the other hand, a random numéraire may involve risk and allow forarbitrage opportunities.

Examples of numéraire processes (Nt)t∈R+ include:

- Money market account.

Given (rt)t∈R+ a possibly random, time-dependent and (Ft)t∈R+ -adaptedrisk-free interest rate process, let∗

Nt := exp(w t

0rsds

).

In this case,St = St

Nt= e−

r t0 rsdsSt, t ∈ R+,

represents the discounted price of the asset at time 0.

- Currenty exchange rates.

In this case, Nt := Rt denotes the SGD/EUR (SGDEUR=X) exchangerate between a domestic currency (SGD) and a foreign currency (EUR),i.e. one unit of local currency (SGD) corresponds to Rt units in foreigncurrency (EUR). Let

St := StRt, t ∈ R+,

denote the price of a foreign (EUR) asset quoted in units of the lo-cal currency (SGD). For example, if Rt = 0.59 and St = e 1, thenSt = St/Rt = St/0.59 ' S$1.7, and 1/Rt is the foreign EUR/SGD ex-change rate.

- Forward numéraire.

The price P (t, T ) of a bond paying P (T, T ) = $1 at maturity T can betaken as numéraire. In this case we have

Nt := P (t, T ) = IE∗[

e−r Ttrsds

∣∣Ft] , 0 6 t 6 T.

Recall that∗ “Anyone who believes exponential growth can go on forever in a finite world is eithera madman or an economist”, Kenneth E. Boulding, in: Energy Reorganization Act of1973: Hearings, Ninety-third Congress, First Session, on H.R. 11510, page 248, UnitedStates Congress, U.S. Government Printing Office, 1973.

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This version: May 26, 2018http://www.ntu.edu.sg/home/nprivault/indext.html

"

http://www.ntu.edu.sg/home/nprivault/indext.html

http://en.wikipedia.org/wiki/Kenneth_E._Boulding

Change of Numéraire and Forward Measures

My foreign currency account St grew by 5%this year.

Q: Did I achieve a positive return?

A:

(a) Scenario A.

My foreign currency account St grew by 5%this year.

The foreign exchange rate dropped by 10%.

Q: Did I achieve a positive return?

A:

(b) Scenario B.

Fig. 12.1: Why change of numéraire?

t 7−→ e−r t

0 rsdsP (t, T ) = IE∗[

e−r T

0 rsds∣∣Ft] , 0 6 t 6 T,

is an Ft- martingale.

- Annuity numéraires.

Processes of the form

Nt :=n∑k=1

(Tk − Tk−1)P (t, Tk), 0 6 t 6 T1,

where P (t, T1), P (t, T2), . . . , P (t, Tn) are bond prices with maturities T1 <T2 < · · · < Tn arranged according to a tenor structure.

- Combinations of the above: for example a foreign money market ac-count e

r t0 r

fsdsRt, expressed in local (or domestic) units of currency, where

(rft)t∈R+ represents a short term interest rate on the foreign market.

When the numéraire is a random process, the pricing of a claim whose valuehas been transformed under change of numéraire, e.g. under a change of cur-rency, has to take into account the risks existing on the foreign market.

In particular, in order to perform a fair pricing, one has to determine aprobability measure (for example on the foreign market), under which thetransformed (or forward, or deflated) process St = St/Nt will be a martin-gale.

For example in case Nt := er t

0 rsds is the money market account, the risk-neutral measure P∗ is a measure under which the discounted price process

St = StNt

= e−r t

0 rsdsSt, t ∈ R+,

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N. Privault

is a martingale.

In the next section we will see that this property can be extended to anykind of numéraire.

12.2 Change of Numéraire

In this section we review the pricing of options by a change of measure asso-ciated to a numéraire Nt, cf. e.g. [GKR95] and references therein.

Most of the results of this chapter rely on the following assumption, whichexpresses absence of arbitrage. In the foreign exchange setting where Nt =Rt, this condition states that the price of one unit of foreign currenty is amartingale when quoted and discounted in the domestic currency.

Assumption (A) Under the risk-neutral measure P∗, the discountednuméraire

t 7−→Mt := e−r t

0 rsdsNt

is an Ft-martingale.

((A))

Definition 12.1. Given (Nt)t∈[0,T ] a numéraire process, the associated for-ward measure P is defined by

dPdP∗

:= MT

M0= e−

r T0 rsds

NTN0

. (12.1)

Recall that from Section 6.3 the above Relation (12.1) rewrites as

dP = MT

M0dP∗ = e−

r T0 rsds

NTN0

dP∗,

which is equivalent to stating thatwΩX(ω)dP(ω) =

wΩ

e−r T

0 rsdsNTN0

XdP∗

for any (bounded) random variable S or, under a different notation,

IE[X] = IE∗[

e−r T

0 rsdsNTN0

X

],

for all integrable FT -measurable random variables X.

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More generally, by (12.1) and the fact that the process

t 7−→Mt := e−r t

0 rsdsNt

is an Ft-martingale under P∗ under Assumption (A), we find that

IE∗[dPdP∗

∣∣∣ Ft] = IE∗[NTN0

e−r T

0 rsds∣∣∣ Ft] = Nt

N0e−

r t0 rsds = Mt

M0, (12.2)

0 6 t 6 T . In Proposition 12.3 we will show, as a consequence of nextLemma 12.2 below, that for any integrable random claim C we have

IE∗[C e−

r TtrsdsNT

∣∣Ft] = NtIE[C | Ft], 0 6 t 6 T.

Note that (12.2), which is Ft-measurable, should not be confused with (12.3),which is FT -measurable. In the next Lemma 12.2 we compute the probabilitydensity dP|Ft/dP∗|Ft of P|Ft with respect to P∗|Ft .

Lemma 12.2. We have

dP|FtdP∗|Ft

= MT

Mt= e−

r Ttrsds

NTNt

, 0 6 t 6 T. (12.3)

Proof. The proof of (12.3) relies on the abstract version of the Bayes formula.We start by noting that for all integrable Ft-measurable random variable G,by (12.2) and the tower property (18.40) we have

IE[GX

]= IE∗

[GX e−

r T0 rsds

NTN0

]= IE∗

[GNtN0

e−r t

0 rsds IE∗[X e−

r Ttrsds

NTNt

∣∣∣ Ft]]= IE∗

[G IE∗

[dPdP∗

∣∣∣ Ft] IE∗ [X e−r Ttrsds

NTNt

∣∣∣ Ft]]

= IE∗[GdPdP∗

IE∗[X e−

r Ttrsds

NTNt

∣∣∣ Ft]]

= IE[G IE∗

[X e−

r Ttrsds

NTNt

∣∣∣ Ft]] ,for all integrable random variable X, which shows that

IE[X | Ft

]= IE∗

[X e−

r Ttrsds

NTNt

∣∣∣ Ft] ,i.e. (12.3) holds.

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N. Privault

We note that in case the numéraire Nt = er t

0 rsds is equal to the moneymarket account we simply have P = P∗.

Pricing using Change of Numéraire

The change of numéraire technique is specially useful for pricing under ran-dom interest rates, in which case an expectation of the form

IE∗[

e−r TtrsdsC

∣∣∣ Ft]becomes a path integral, see e.g. [Das04] for a recent account of path integralmethods in quantitative finance. The next proposition is the basic result ofthis section, it provides a way to price an option with arbitrary payoff C

under a random discount factor e−r Ttrsds by use of the forward measure. It

will be applied in Chapter 14 to the pricing of bond options and caplets, cf.Propositions 14.1, 14.3 and 14.4 below.Proposition 12.3. An option with integrable claim payoff C ∈ L1(P∗,FT )is priced at time t as

IE∗[

e−r TtrsdsC

∣∣Ft] = NtIE[C

NT

∣∣∣ Ft] , 0 6 t 6 T, (12.4)

provided that C/NT ∈ L1(P,FT ).Proof. By Relation (12.3) in Lemma 12.2 we have

IE∗[

e−r TtrsdsC

∣∣∣ Ft] = IE∗[dP|FtdP∗|Ft

NtNT

C∣∣∣ Ft]

= Nt IE∗[dP|FtdP∗|Ft

C

NT

∣∣∣ Ft]

= NtIE[C

NT

∣∣∣ Ft] , 0 6 t 6 T.

Equivalently we can write

NtIE[C

NT

∣∣∣ Ft] = NtIE∗[C

NT

dP|FtdP∗|Ft

∣∣∣ Ft]= IE∗

[e−

r TtrsdsC

∣∣Ft] , 0 6 t 6 T.

Each application of the formula (12.4) will require toa) identify a suitable numéraire (Nt)t∈R+ , and to428


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b) make sure that the ratio C/NT takes a sufficiently simple form,

in order to allow for the computation of the expectation in the right-handside of (12.4).

Next, we consider further examples of numéraires and associated examplesof option prices.

Examples:

a) Money market account.

Take Nt := er t

0 rsds, where (rt)t∈R+ is a possibly random and time-dependent risk-free interest rate. In this case, Assumption (A) is clearlysatisfied, we have P = P∗, and (12.4) simply reads

IE∗[

e−r TtrsdsC

∣∣Ft] = er t

0 rsds IE∗[

e−r T

0 rsdsC∣∣∣ Ft] , 0 6 t 6 T,

which yields no particular information.

b) Forward numéraire.

Here, Nt := P (t, T ) is the price P (t, T ) of a bond maturing at time T , 0 6t 6 T , and the discounted bond price process

(e−

r t0 rsdsP (t, T )

)t∈[0,T ]

is an Ft-martingale under P∗, i.e. Assumption (A) is satisfied and Nt =P (t, T ) can be taken as numéraire. In this case, (12.4) shows that a randomclaim C can be priced as

IE∗[

e−r TtrsdsC

∣∣∣ Ft] = P (t, T )IE[C∣∣Ft] , 0 6 t 6 T, (12.5)

since P (T, T ) = 1, where the forward measure P satisfies

dPdP∗

= e−r T

0 rsdsP (T, T )P (0, T ) = e−

r T0 rsds

P (0, T ) (12.6)

by (12.1).

c) Annuity numéraires.

We takeNt :=

n∑k=1

(Tk − Tk−1)P (t, Tk)

where P (t, T1), . . . , P (t, Tn) are bond prices with maturities T1 < T2 <· · · < Tn. Here, (12.4) shows that a swaption on the cash flow P (T, Tn)−

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N. Privault

P (T, T1)− κNT can be priced as

IE∗[

e−r Ttrsds(P (T, Tn)− P (T, T1)− κNT )+ ∣∣Ft]

= NtIE[(

P (T, Tn)− P (T, T1)NT

− κ)+ ∣∣∣ Ft] ,

0 6 t 6 T , where (P (T, Tn) − P (T, T1))/NT becomes a swap rate, cf.(13.51) in Proposition 13.11 and Section 14.5.

In the sequel, given (Xt)t∈R+ an asset price process, we define the process offorward (or deflated) prices

Xt := Xt

Nt, 0 6 t 6 T, (12.7)

which represents the values at times t of Xt, expressed in units of thenuméraire Nt. It will be useful to determine the dynamics of

(Xt

)t∈R+

underthe forward measure P.Proposition 12.4. Let (Xt)t∈R+ denote a continuous (Ft)t∈R+-adapted assetprice process such that

t 7−→ e−r t

0 rsdsXt, t ∈ R+,

is a martingale under P∗. Then, under change of numéraire,

the process(Xt

)t∈[0,T ] = (Xt/Nt)t∈[0,T ] of forward prices is an Ft-

martingale under P, provided that it is integrable under P.

Proof. We need to show that

IE[Xt

Nt

∣∣∣ Fs] = Xs

Ns, 0 6 s 6 t, (12.8)

and we achieve this using a standard characterization of conditional expec-tation. Namely, for all bounded Fs-measurable random variables G we notethat under Assumption (A) we have

IE[GXt

Nt

]= IE∗

[GXt

Nt

dPdP∗

]

= IE∗[IE∗[GXt

Nt

dPdP∗

∣∣∣ Ft]]

= IE∗[GXt

NtIE∗[dPdP∗

∣∣∣ Ft]]430


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= IE∗[G e−

r t0 rudu

Xt

N0

]= IE∗

[G e−

r s0 rudu

Xs

N0

]= IE

[GXs

NsIE∗[dPdP∗

∣∣∣ Fs]]

= IE[IE∗[GXs

Ns

dPdP∗

∣∣∣ Fs]]

= IE[GXs

Ns

dPdP∗

]

= IE[GXs

Ns

], 0 6 s 6 t,

becauset 7−→ e−

r t0 rsdsXt

is an Ft-martingale. Finally, the identity

IE[GXt

]= IE

[GXt

Nt

]= IE

[GXs

Ns

]= IE

[GXs

], 0 6 s 6 t,

for all bounded Fs-measurable G, implies (12.8).

Next we will rephrase Proposition 12.4 in Proposition 12.6 using the Girsanovtheorem, which is briefly recalled below.

Girsanov theorem

Recall that, letting

Φt := IE∗[dPdP∗

∣∣∣ Ft] , t ∈ [0, T ], (12.9)

and given (Wt)t∈R a standard Brownian motion under P∗, the Girsanov the-orem∗ shows that the process

(Wt

)t∈R+

defined by

dWt := dWt −1ΦtdΦt · dWt, t ∈ R+, (12.10)

is a standard Brownian motion under P. In case the martingale (Φt)t∈[0,T ]takes the form∗ See e.g. Theorem III-35 page 132 of [Pro04].

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N. Privault

Φt = exp(−

w t

0ψsdWs −

12

w t

0|ψs|2ds

), t ∈ R+,

hencedΦt = −ψtΦtdWt, t ∈ R+,

and by the Itô multiplication Table 4.1, Relation (12.10) reads

dWt = dWt −1ΦtdΦt · dWt

= dWt −1Φt

(−ψtΦtdWt) · dWt

= dWt + ψtdt, t ∈ R+,

and shows that the shifted process(Wt

)t∈R+

=(Wt +

r t0 ψsds

)t∈R+

is a

standard Brownian motion under P, which is consistent with the GirsanovTheorem 6.2. The next result is another application of the Girsanov theorem.

Proposition 12.5. The process(Wt

)t∈R+

defined by

dWt := dWt −1NtdNt · dWt, t ∈ R+, (12.11)

is a standard Brownian motion under P.

Proof. Relation (12.2) shows that Φt defined in (12.9) satisfies

Φt = IE∗[dPdP∗

∣∣∣ Ft]

= IE∗[NTN0

e−r T

0 rsds∣∣∣ Ft]

= NtN0

e−r t

0 rsds, 0 6 t 6 T,

hence

dΦt = d(Φt e−

r t0 rsds

)= −Φtrtdt+ e−

r t0 rsdsdΦt

= −Φtrtdt+ ΦtNtdNt,

which, by (12.10), yields

dWt = dWt −1ΦtdΦt · dWt

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= dWt −1Φt

(−Φtrtdt+ Φt

NtdNt

)· dWt

= dWt −1NtdNt · dWt,

which is (12.11), from Relation (12.10) and the Itô multiplication Table 4.1.

The next proposition confirms the statement of Proposition 12.4, and inaddition it determines the precise dynamics of

(Xt

)t∈R+

under P. See Exer-cise 12.1 for another calculation based on geometric Brownian motion, andExercise 12.7 for an extension to correlated Brownian motions. As a conse-quence, we have the next proposition.

Proposition 12.6. Assume that (Xt)t∈R+ and (Nt)t∈R+ satisfy the stochas-tic differential equations

dXt = rtXtdt+ σXt XtdWt, and dNt = rtNtdt+ σNt NtdWt, (12.12)

where (σXt )t∈R+ and (σNt )t∈R+ are (Ft)t∈R+-adapted volatility processes.Then we have

dXt = (σXt − σNt )XtdWt. (12.13)

Proof. First we note that by (12.11) and (12.12),

dWt = dWt −1NtdNt · dWt = dWt − σNt dt, t ∈ R+,

is a standard Brownian motion under P. Next, by Itô’s calculus and the Itômultiplication Table 4.1 and (12.12) we have

d

(1Nt

)= − 1

N2t

dNt + 1N3t

(dNt)2

= − 1N2t

(rtNtdt+ σNt NtdWt) + |σNt |2

Ntdt

= − 1N2t

(rtNtdt+ σNt Nt(dWt + σNt dt)) + |σNt |2

Ntdt

= − 1Nt

(rtdt+ σNt dWt), (12.14)

hence

dXt = d

(Xt

Nt

)= dXt

Nt+Xtd

(1Nt

)+ dXt · d

(1Nt

)

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N. Privault

= 1Nt

(rtXtdt+ σXt XtdWt)−Xt

Nt

(rtdt+ σNt dWt − |σNt |2dt

)− 1Nt

(rtXtdt+ σXt XtdWt) ·(rtdt+ σNt dWt − |σNt |2dt

)= 1Nt

(rtXtdt+ σXt XtdWt)−Xt

Nt(rtdt+ σNt dWt)

+Xt

Nt|σNt |2dt−

Xt

NtσXt σ

Nt dt

= Xt

NtσXt dWt −

Xt

NtσNt dWt −

Xt

NtσXt σ

Nt dt+Xt

|σNt |2

Ntdt

= Xt

Nt

(σXt dWt − σNt dWt − σXt σNt dt+ |σNt |2dt

)= Xt(σXt − σNt )dWt − Xt(σXt − σNt )σNt dt= Xt(σXt − σNt )dWt,

since dWt = dWt − σNt dt, t ∈ R+.

We end this section with some comments on inverse changes of measure.

Inverse Changes of Measure

In the next proposition we compute conditional inverse density dP∗/dP.

Proposition 12.7. We have

IE[dP∗

dP

∣∣∣ Ft] = N0

Ntexp

(w t

0rsds

)0 6 t 6 T, (12.15)

and the process

t 7−→ N0

Ntexp

(w t

0rsds

), 0 6 t 6 T,

is an Ft-martingale under P.

Proof. For all bounded and Ft-measurable random variables F we have,

IE[FdP∗

dP

]= IE∗ [F ]

= IE∗[FNtNt

]= IE∗

[FNTNt

exp(−

w T

trsds

)]= IE

[FN0

Ntexp

(w t

0rsds

)].

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By (12.14) we also have

d

(1Nt

exp(w t

0rsds

))= − 1

Ntexp

(w t

0rsds

)σNt dWt,

which recovers the second part of Proposition 12.7, i.e. the martingale prop-erty of

t 7−→ 1Nt

exp(w t

0rsds

)under P.

12.3 Foreign Exchange

Currency exchange is a typical application of change of numéraire that illus-trate the principle of absence of arbitrage.

Let Rt denote the foreign exchange rate, i.e. Rt is the (possibly fractional)quantity of local currency that correspond to one unit of foreign currency.

Consider an investor that intends to exploit an “overseas investment op-portunity” by

a) at time 0, changing one unit of local currency into 1/R0 units of foreigncurrency,

b) investing 1/R0 on the foreign market at the rate rf , which will yield theamount etrf

/R0 at time t,c) changing back etrf

/R0 into a quantity etrfRt/R0 = Nt/R0 of his local

currency.

In other words, the foreign money market account etrf is valued etrfRt on

the local (or domestic) market, and its discounted value on the local marketis

e−tr+trfRt, t ∈ R+.

The outcome of this investment will be obtained by a martingale comparisonof etrf

Rt/R0 to the amount ert that could have been obtained by investingon the local market.

Taking

Nt := etrfRt, t ∈ R+, (12.16)

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N. Privault

as numéraire, absence of arbitrage is expressed by Assumption (A), whichstates that the discounted numéraire process

t 7−→ e−rtNt = e−t(r−rf)Rt

is an Ft-martingale under P∗.

Next, we find a characterization of this arbitrage condition under the pa-rameters of the model, and for this we will model foreign exchange rates Rtaccording to a geometric Brownian motion (12.17).∗

Proposition 12.8. Assume that the foreign exchange rate Rt satisfies astochastic differential equation of the form

dRt = µRtdt+ σRtdWt, (12.17)

where (Wt)t∈R+ is a standard Brownian motion under P∗. Under the absenceof arbitrage Assumption (A) for the numéraire (12.16), we have

µ = r − rf , (12.18)

hence the exchange rate process satisfies

dRt = (r − rf)Rtdt+ σRtdWt. (12.19)

under P∗.

Proof. The equation (12.17) has solution

Rt = R0 eµt+σWt−σ2t/2, t ∈ R+,

hence the discounted value of the foreign money market account etrf on thelocal market is

e−tr+trfRt = R0 e(rf−r+µ)t+σWt−σ2t/2, t ∈ R+.

Under the absence of arbitrage Assumption (A), the process e−(r−rf)tRt =e−trNt should be an Ft-martingale under P∗, and this holds provided thatrf − r + µ = 0, which yields (12.18) and (12.19).

As a consequence of Proposition 12.8, under absence of arbitrage a localinvestor who buys a unit of foreign currency in the hope of a higher returnrf >> r will have to face a lower (or even more negative) drift

µ = r − rf << 0∗ Major currencies have started floating against each other since 1973, following the endof the system of fixed exchanged rates agreed upon at the Bretton Woods Conference,July 1-22, 1944.

436


"



in his exchange rate Rt. The drift µ = r−rf is also called the cost of carryingthe foreign currency.

The local money market account Xt := ert is valued ert/Rt on the foreignmarket, and its discounted value on the foreign market is

t 7−→ e(r−rf)t

Rt= Xt

Nt(12.20)

= 1R0

e(r−rf)t−µt−σWt+σ2t/2

= 1R0

e(r−rf)t−µt−σWt−σ2t/2,

where

dWt = dWt −1NtdNt · dWt

= dWt −1RtdRt · dWt

= dWt − σdt, t ∈ R+,

is a standard Brownian motion under P by (12.11). Under absence of arbitragee−(r−rf)tRt is an Ft-martingale under P∗ and (12.20) is an Ft-martingaleunder P by Proposition 12.4, which recovers (12.18).

Proposition 12.9. Under the absence of arbitrage condition (12.18), theinverse exchange rate 1/Rt satisfies

d

(1Rt

)= rf − r

Rtdt− σ

RtdWt, (12.21)

under P , where (Rt)t∈R+ is given by (12.19).

Proof. By (12.18), the exchange rate 1/Rt is written by Itô’s calculus as

d

(1Rt

)= − 1

R2t

(µRtdt+ σRtdWt) + 1R3t

σ2R2tdt

= −(µ− σ2) 1Rtdt− σ

RtdWt

= − µ

Rtdt− σ

RtdWt

= (rf − r) 1Rtdt− σ

RtdWt,

where (Wt)t∈R+ is a standard Brownian motion under P.

" 437



N. Privault

Consequently, under absence of arbitrage, a foreign investor who buys a unitof the local currency in the hope of a higher return r >> rf will have to facea lower (or even more negative) drift −µ = rf − r in his exchange rate 1/Rtas written in (12.21) under P.

Foreign exchange options

We now price a foreign exchange option with payoff (RT − κ)+ under P∗ onthe exchange rate RT by the Black-Scholes formula as in the next proposition,also known as the Garman-Kohlagen [GK83] formula.

Proposition 12.10. (Garman-Kohlagen formula). Consider an exchangerate process (Rt)t∈R+ given by (12.19). The price of the foreign exchangecall option on RT with maturity T and strike price κ is given by

e−(T−t)r IE∗[(RT−κ)+ | Rt] = e−(T−t)rfRtΦ+(t, Rt)−κ e−(T−t)rΦ−(t, Rt),

(12.22)

0 6 t 6 T , where

Φ+(t, x) = Φ

(log(x/κ) + (T − t)(r − rf + σ2/2)

σ√T − t

),

andΦ−(t, x) = Φ

(log(x/κ) + (T − t)(r − rf − σ2/2)

σ√T − t

).

Proof. As a consequence of (12.19) we find the numéraire dynamics

dNt = d( etrfRt)

= rf etrfRtdt+ etr

fdRt

= r etrfRtdt+ σ etr

fRtdWt

= rNtdt+ σNtdWt.

Hence a standard application of the Black-Scholes formula yields

e−(T−t)r IE∗[(RT − κ)+ | Ft] = e−(T−t)r IE∗[( e−T rfNT − κ)+ | Ft]

= e−(T−t)r e−T rfIE∗[(NT − κ eT r

f)+ | Ft]= e−T r

f

(NtΦ

(log(Nt e−T rf

/κ) + (r + σ2/2)(T − t)σ√T − t

)

−κ eT rf−(T−t)rΦ

(log(Nt e−Trf

/κ) + (r − σ2/2)(T − t)σ√T − t

))438


"



= e−T rf(NtΦ

(log(Rt/κ) + (T − t)(r − rf + σ2/2)

σ√T − t

)−κ eT r

f−(T−t)rΦ

(log(Rt/κ) + (T − t)(r − rf − σ2/2)

σ√T − t

))= e−(T−t)rf

RtΦ+(t, Rt)− κ e−(T−t)rΦ−(t, Rt).

Similarly, from (12.21) rewritten as

d

(ertRt

)= rf ert

Rtdt− σ ert

RtdWt,

a foreign exchange call option with payoff (1/RT −κ)+ can be priced under Pin a Black-Scholes model by taking ert/Rt as underlying price, rf as risk-freeinterest rate, and −σ as volatility parameter. In this framework the Black-Scholes formula (5.18) yields

e−(T−t)rfIE[(

1RT− κ)+ ∣∣∣ Rt] (12.23)

= e−(T−t)rfe−rT IE

[( erTRT− κ erT

)+ ∣∣Rt] (12.24)

= e−(T−t)r

RtΦ+

(t,

1Rt

)− κ e−(T−t)rf

Φ−

(t,

1Rt

),

which is the symmetric of (12.22) by exchanging Rt with 1/Rt and r withrf , where

Φ+(t, x) = Φ

(log(x/κ) + (T − t)(rf − r + σ2/2)

σ√T − t

),

andΦ−(t, x) = Φ

(log(x/κ) + (T − t)(rf − r − σ2/2)

σ√T − t

).

Call/put duality for foreign exchange options

Let Nt = etrfRt, where Rt is an exchange rate with respect to a foreign

currency and rf is the foreign market interest rate. From Proposition 12.3and (12.4) we have

IE[

1eT rfRT

(1κ−RT

)+ ∣∣∣ Rt] = 1Nt

e−(T−t)r IE∗[(

1κ−RT

)+ ∣∣∣ Rt] ," 439



N. Privault

and this yields the call/put duality

e−(T−t)rfIE[(

1RT− κ)+ ∣∣∣ Rt] = e−(T−t)rf

IE[κ

RT

(1κ−RT

)+ ∣∣∣ Rt]

= κ etrfIE[

1eT rfRT

(1κ−RT

)+ ∣∣∣ Rt]

= κ

Ntetr

f−(T−t)r IE∗[(

1κ−RT

)+ ∣∣∣ Rt]

= κ

Rte−(T−t)r IE∗

[(1κ−RT

)+ ∣∣∣ Rt] , (12.25)

between a call option with strike price κ and a (possibly fractional) quantityκ/Rt of put option(s) with strike price 1/κ.

In the Black-Scholes case the duality (12.25) can be directly checked byverifying that (12.23) coincides with

κ

Rte−(T−t)r IE∗

[(1κ−RT

)+ ∣∣∣ Rt]

= κ

Rte−(T−t)r e−T r

fIE∗( eT rf

κ− eT r

fRT

)+ ∣∣∣ Rt

= κ

Rte−(T−t)r e−T r

fIE∗( eT rf

κ−NT

)+ ∣∣∣ Rt

= κ

Rt

(e−(T−t)r

κΦp− (t, Rt)− e−(T−t)rf

RtΦp+ (t, Rt)

)= e−(T−t)r

RtΦp− (t, Rt)− κ e−(T−t)rf

Φp+ (t, Rt)

= e−(T−t)r

RtΦ+

(t,

1Rt

)− κ e−(T−t)rf

Φ−

(t,

1Rt

),

whereΦp−(t, x) := Φ

(− log(xκ) + (T − t)(r − rf − σ2/2)

σ√T − t

),

andΦp

+(t, x) := Φ

(− log(xκ) + (T − t)(r − rf + σ2/2)

σ√T − t

).

440


"



Local market Foreign market

Martingales t 7−→ e−rtNt = e−t(r−rf)Rt t 7−→ Xt

Nt= e(r−rf)t

Rt

Options e−(T−t)r IE∗[(

1κ−RT

)+ ∣∣∣ Rt] e−(T−t)rfIE[(

1RT− κ)+ ∣∣∣ Rt]

Table 12.1: Local vs foreign markets.

The foreign exchange call and put options on the local and foreign marketsare linked by the relation

κ e−(T−t)r IE∗[(

1κ−RT

)+ ∣∣∣ Rt] = Rt e−(T−t)rfIE[(

1RT− κ)+ ∣∣∣ Rt] .

Letting κ′ = 1/κ, the put option priced

e−(T−t)r IE∗[

(κ′ −RT )+ ∣∣Rt] = e−(T−t)rκ′Φ+

(t,

1Rt

)− e−(T−t)rf

RtΦ−

(t,

1Rt

)= e−(T−t)rκ′Φp

− (t, Rt)− e−(T−t)rfRtΦ

p+ (t, Rt)

on the local market correspond to a buy back guarantee in currency exchange.In the case of an option at the money with κ′ = Rt with r = rf ' 0 we find

IE∗[

(Rt −RT )+ ∣∣Rt] = Rt

(Φ

(σ√T − t2

)− Φ

(−σ√T − t2

))= Rt

(2Φ(σ√T − t2

)− 1).

For example, if T − t = 30 days, σ = 10%, and Rt denotes the EUR/USD(EURUSD=X) exchange rate between a domestic currency (EUR) and aforeign currency (USD), i.e. one unit of local currency (EUR) corresponds toRt = 1.23 units of foreign currency (USD) we find

IE∗[

(Rt −RT )+ ∣∣Rt] = 1.23(

2Φ(

0.05×√

31/365)− 1)

= 1.23(2× 0.505813− 1)= 0.01429998

" 441



N. Privault

per USD, or 0.011626 per EUR. Based on an actual option price of e 4.5,this would result into an average amount of 4.5/0.011626 'e 387 exhangedat counters.

12.4 Pricing of Exchange Options

Based on Proposition 12.4, we model the process Xt of forward prices as acontinuous martingale under P, written as

dXt = σtdWt, t ∈ R+, (12.26)

where(Wt

)t∈R+

is a standard Brownian motion under P and(σt)t∈R+

is an(Ft)t∈R+ -adapted process. More precisely, we assume that

(Xt

)t∈R+

has thedynamics

dXt = σt(Xt

)dWt, (12.27)

where the function x 7−→ σt(x) is uniformly Lipschitz in t ∈ R+. The Markovproperty of the diffusion process

(Xt

)t∈R+

, cf. Theorem V-6-32 of [Pro04],shows that the conditional expectation IE

[g(XT

) ∣∣Ft] can be written usinga (measurable) function C(t, x) of t and Xt, as

IE[g(XT

) ∣∣Ft] = C(t, Xt), 0 6 t 6 T.

Consequently, a vanilla option with claim payoff C := NT g(XT

)can be priced

as

IE∗[

e−r TtrsdsNT g

(XT

) ∣∣∣ Ft] = NtIE[g(XT

) ∣∣Ft]= NtC(t, Xt), 0 6 t 6 T. (12.28)

In the next Proposition 12.11 we state the Margrabe [Mar78] formula for thepricing of exchange options by the zero interest rate Black-Scholes formula.It will be applied in particular in Proposition 14.3 below for the pricingof bond options. Here, (Nt)t∈R+ denotes any numéraire process satisfyingAssumption (A).

Proposition 12.11. (Margrabe formula). Assume that σt(Xt

)= σ(t)Xt, i.e.

the martingale (Xt)t∈[0,T ] is a (driftless) geometric Brownian motion underP with deterministic volatility (σ(t))t∈[0,T ]. Then we have

IE∗[

e−r Ttrsds (XT − κNT )+ ∣∣Ft] = XtΦ

0+(t, Xt)− κNtΦ0

−(t, Xt),

(12.29)

442


"



t ∈ [0, T ], where

Φ0+(t, x) = Φ

(log(x/κ)v(t, T ) + v(t, T )

2

), Φ0

−(t, x) = Φ

(log(x/κ)v(t, T ) −

v(t, T )2

),

(12.30)and v2(t, T ) =

w T

tσ2(s)ds.

Proof. Taking g(x) = (x− κ)+ in (12.28), the call option with payoff

(XT − κNT )+ = NT(XT − κ

)+,

and floating strike price κNT is priced by (12.28) as

IE∗[

e−r Ttrsds(XT − κNT )+ ∣∣Ft] = IE∗

[e−

r TtrsdsNT

(XT − κ

)+ ∣∣Ft]= NtIE

[(XT − κ

)+ ∣∣Ft]= NtC(t, Xt),

where the function C(t, Xt) is given by the Black-Scholes formula

C(t, x) = xΦ0+(t, x)− κΦ0

−(t, x),

with zero interest rate, since (Xt)t∈[0,T ] is a driftless geometric Brownianmotion which is an Ft-martingale under P, and XT is a lognormal randomvariable with variance coefficient v2(t, T ) =

w T

tσ2(s)ds. Hence we have

IE∗[

e−r Ttrsds (XT − κNT )+ ∣∣Ft] = NtC(t, Xt)

= NtXtΦ0+(t, Xt)− κNtΦ0

−(t, Xt),

t ∈ R+.

In particular, from Proposition 12.6 and (12.13), we can take σ(t) = σXt −σNtwhen (σXt )t∈R+ and (σNt )t∈R+ are deterministic.

Examples:

a) When the short rate process (r(t))t∈[0,T ] is a deterministic function andNt = e−

r Ttr(s)ds, 0 6 t 6 T , we have P = P∗ and Proposition 12.11 yields

Merton’s [Mer73] “zero interest rate” version of the Black-Scholes formula

e−r Ttr(s)ds IE∗

[(XT − κ)+ ∣∣Ft]

= XtΦ0+

(t, e

r Ttr(s)dsXt

)− κ e−

r Ttr(s)dsΦ0

−

(t, e

r Ttr(s)dsXt

),

" 443



N. Privault

where Φ0+ and Φ0 are defined in (12.30) and (Xt)t∈R+ satisfies the equation

dXt

Xt= r(t)dt+ σ(t)dWt, i.e. dXt

Xt

= σ(t)dWt, 0 6 t 6 T.

b) In the case of pricing under a forward numéraire, i.e. when (Nt)t∈[0,T ] =(P (t, T ))t∈[0,T ], we get

IE∗[

e−r Ttrsds (XT − κ)+ ∣∣Ft] = XtΦ+(t, Xt)− κP (t, T )Φ−(t, Xt),

t ∈ R+, since P (T, T ) = 1. In particular, when Xt = P (t, S) the aboveformula allows us to price a bond call option on P (T, S) as

IE∗[

e−r Ttrsds (P (T, S)− κ)+ ∣∣Ft] = P (t, S)Φ+(t, Xt)−κP (t, T )Φ−(t, Xt),

0 6 t 6 T , provided that the martingale Xt = P (t, S)/P (t, T ) under P isgiven by a geometric Brownian motion, cf. Section 14.2.

12.5 Hedging by Change of Numéraire

In this section we reconsider and extend the Black-Scholes self-financing hedg-ing strategies found in (6.31)-(6.32) and Proposition 6.11 of Chapter 6. Forthis, we use the stochastic integral representation of the forward claim payoffsand change of numéraire in order to compute self-financing portfolio strate-gies. Our hedging portfolios will be built on the assets (Xt, Nt), not on Xt

and the money market account Bt = er t

0 rsds, extending the classical hedg-ing portfolios that are available in from the Black-Scholes formula, using atechnique from [Jam96], cf. also [PT12].

Consider a claim with random payoff C, typically an interest rate deriva-tive, cf. Chapter 14. Assume that the forward claim payoff C/NT ∈ L2(Ω)has the stochastic integral representation

C := C

NT= IE

[C

NT

]+

w T

0φtdXt, (12.31)

where(Xt

)t∈[0,T ] is given by (12.26) and

(φt)t∈[0,T ] is a square-integrable

adapted process under P, from which it follows that the forward claim price

Vt := VtNt

= 1Nt

IE∗[

e−r TtrsdsC

∣∣∣ Ft] = IE[C

NT

∣∣∣ Ft] , 0 6 t 6 T,

is an Ft-martingale under P, that can be decomposed as

444


"



Vt = IE[C | Ft

]= IE

[C

NT

]+

w t

0φsdXs, 0 6 t 6 T. (12.32)

The next proposition extends the argument of [Jam96] to the general frame-work of pricing using change of numéraire. Note that this result differs fromthe standard formula that uses the money market account Bt = e

r t0 rsds for

hedging instead of Nt, cf. e.g. [GKR95] pages 453-454. The notion of self-financing portfolio is similar to that of Definition 5.1.Proposition 12.12. Letting ηt := Vt − Xtφt, 0 6 t 6 T , the portfolioallocation

(φt, ηt

)t∈[0,T ] with value

Vt = φtXt + ηtNt, 0 6 t 6 T,

is self-financing in the sense that

dVt = φtdXt + ηtdNt,

and it hedges the claim C, i.e.

Vt = φtXt + ηtNt = IE∗[

e−r TtrsdsC

∣∣∣ Ft] , 0 6 t 6 T. (12.33)

Proof. In order to check that the portfolio allocation(φt, ηt

)t∈[0,T ] hedges

the claim C it suffices to check that (12.33) holds since by (12.4) the price Vtat time t ∈ [0, T ] of the hedging portfolio satisfies

Vt = NtVt = IE∗[

e−r TtrsdsC

∣∣∣ Ft] , 0 6 t 6 T.

Next, we show that the portfolio allocation(φt, ηt

)t∈[0,T ] is self-financing. By

numéraire invariance, cf. e.g. page 184 of [Pro01], we have, using the relationdVt = φtdXt from (12.32),

dVt = d(NtVt)= VtdNt +NtdVt + dNt · dVt= VtdNt +NtφtdXt + φtdNt · dXt

= φtXtdNt +NtφtdXt + φtdNt · dXt +(Vt − φtXt

)dNt

= φtd(NtXt) + ηtdNt

= φtdXt + ηtdNt.

We now consider an application to the forward Delta hedging of Europeantype options with payoff C = g

(XT

)where g : R −→ R and

(Xt

)t∈R+

hasthe Markov property as in (12.27), where σ : R+ × R. Assuming that thefunction C(t, x) defined by" 445



N. Privault

Vt := IE[g(XT

) ∣∣∣ Ft] = C(t, Xt)

is C2 on R+, we have the following corollary of Proposition 12.12, whichextends the Black-Scholes Delta hedging technique to the general change ofnuméraire setup.

Corollary 12.13. Letting ηt = C(t, Xt) − Xt∂C

∂x(t, Xt), 0 6 t 6 T , the

portfolio allocation(∂C∂x

(t, Xt), ηt)t∈[0,T ]

with value

Vt = ηtNt +Xt∂C

∂x(t, Xt), t ∈ R+,

is self-financing and hedges the claim C = NT g(XT

).

Proof. This result follows directly from Proposition 12.12 by noting thatby Itô’s formula, and the martingale property of Vt under P the stochasticintegral representation (12.32) is given by

φt = ∂C

∂x(t, Xt), 0 6 t 6 T.

In the case of an exchange option with payoff function

C = (XT − κNT )+ = NT(XT − κ

)+on the geometric Brownian motion

(Xt

)t∈[0,T ] under P with

σt(Xt

)= σ(t)Xt, (12.34)

where(σ(t)

)t∈[0,T ] is a deterministic function, we have the following corollary

on the hedging of exchange options based on the Margrabe formula (12.29).

Corollary 12.14. The decomposition

IE∗[

e−r Ttrsds (XT − κNT )+

∣∣∣ Ft] = XtΦ0+(t, Xt)− κNtΦ0

−(t, Xt)

yields a self-financing portfolio allocation (Φ0+(t, Xt),−κΦ0

−(t, Xt))t∈[0,T ] inthe assets (Xt, Nt), that hedges the claim C = (XT − κNT )+.

Proof. We apply Corollary 12.13 and the relation

∂C

∂x(t, x) = Φ0

+(t, x), x ∈ R,

446


"



for the function C(t, x) = xΦ0+(t, x)−κΦ0

−(t, x), cf. Relation (5.21) in Propo-sition 5.14.

Note that the Delta hedging method requires the computation of the func-tion C(t, x) and that of the associated finite differences, and may not applyto path-dependent claims.

Examples:

a) When the short rate process (r(t))t∈[0,T ] is a deterministic function andNt = e

r Ttr(s)ds, Corollary 12.14 yields the usual Black-Scholes hedging

strategy(Φ+(t, Xt),−κ e

r T0 r(s)dsΦ−(t,Xt)

)t∈[0,T ]

=(Φ0

+(t, er Ttr(s)dsXt),−κ e

r T0 r(s)dsΦ0

−(t, er Ttr(s)dsXt)

)t∈[0,T ]

,

in the assets (Xt, er t

0 r(s)ds), that hedges the claim C = (XT − κ)+, with

Φ+(t, x) := Φ

log(x/κ) +(r Ttr(s)ds+ (T − t)σ2/2

)σ√T − t

,

and

Φ−(t, x) := Φ

log(x/κ) +(r Ttr(s)ds− (T − t)σ2/2

)σ√T − t

.

b) In case Nt = P (t, T ) and Xt = P (t, S), 0 6 t 6 T < S, Corollary 12.14shows that when

(Xt

)t∈[0,T ] is modeled as the geometric Brownian motion

(12.34) under P, the bond call option with payoff (P (T, S) − κ)+ can behedged as

IE∗[

e−r Ttrsds (P (T, S)− κ)+ ∣∣Ft] = P (t, S)Φ+(t, Xt)−κP (t, T )Φ−(t, Xt)

by the self-financing portfolio allocation

(Φ+(t, Xt),−κΦ−(t, Xt))t∈[0,T ]

in the assets (P (t, S), P (t, T )), i.e. one needs to hold the quantity Φ+(t, Xt)of the bond maturing at time S, and to short a quantity κΦ−(t, Xt) of thebond maturing at time T .

" 447



N. Privault

Exercises

Exercise 12.1 Let (Bt)t∈R+ be a standard Brownian motion started at 0under the risk-neutral measure P∗. Consider a numéraire (Nt)t∈R+ given by

Nt := N0 eηBt−η2t/2, t ∈ R+,

and a risky asset (Xt)t∈R+ given by

Xt := X0 eσBt−σ2t/2, t ∈ R+.

Let P denote the forward measure relative to the numéraire (Nt)t∈R+ , underwhich the process Xt := Xt/Nt of forward prices is known to be a martingale.a) Using the Itô formula, compute

dXt = d(Xt/Nt) = (X0/N0)d(

e(σ−η)Bt−(σ2−η2)t/2).

b) Explain why the exchange option price IE[(XT − λNT )+] at time 0 hasthe Black-Scholes form

e−rT IE[(XT − λNT )+] (12.35)

= X0Φ

(log(X0/λ

)σ√T

+ σ√T

2

)− λN0Φ

(log(X0/λ

)σ√T

− σ√T

2

).

Hints:

(i) Use the change of numéraire identity

e−rT IE[(XT − λNT )+] = N0IE[(XT − λ

)+].

(ii) The forward price Xt is a martingale under the forward measure Prelative to the numéraire (Nt)t∈R+ .

c) Give the value of σ in terms of σ and η.

Exercise 12.2 Consider two zero-coupon bond prices of the form P (t, T ) =F (t, rt) and P (t, S) = G(t, rt), where (rt)t∈R+ is a short term interest rateprocess. Taking Nt := P (t, T ) as a numéraire defining the forward measure P,compute the dynamics of (P (t, S))t∈[0,T ] under P using a standard Brownianmotion

(Wt

)t∈[0,T ] under P.

Exercise 12.3 Forward contract. Using a change of numéraire argument forthe numéraire Nt := P (t, T ), t ∈ [0, T ], compute the price at time t ∈ [0, T ] of448


"



a forward (or future) contract with payoff P (T, S)−K in a bond market withshort term interest rate (rt)t∈R+ . How would you hedge this forward contract?

Exercise 12.4 Bond options. Consider two bonds with maturities T and S,with prices P (t, T ) and P (t, S) given by

dP (t, T )P (t, T ) = rtdt+ ζTt dWt,

anddP (t, S)P (t, S) = rtdt+ ζSt dWt,

where (ζT (s))s∈[0,T ] and (ζS(s))s∈[0,S] are deterministic functions.

a) Show, using Itô’s formula, that

d

(P (t, S)P (t, T )

)= P (t, S)P (t, T ) (ζS(t)− ζT (t))dWt,

where(Wt

)t∈R+

is a standard Brownian motion under P.b) Show that

P (T, S) = P (t, S)P (t, T ) exp

(w T

t(ζS(s)− ζT (s))dWs −

12

w T

t|ζS(s)− ζT (s)|2ds

).

Let P denote the forward measure associated to the numéraire

Nt := P (t, T ), 0 6 t 6 T.

c) Show that for all S, T > 0 the price at time t

IE[

e−r Ttrsds(P (T, S)− κ)+ ∣∣Ft]

of a bond call option on P (T, S) with payoff (P (T, S)− κ)+ is equal to

IE∗[

e−r Ttrsds(P (T, S)− κ)+ ∣∣Ft] (12.36)

= P (t, S)Φ(v

2 + 1v

log P (t, S)κP (t, T )

)− κP (t, T )Φ

(−v2 + 1

vlog P (t, S)

κP (t, T )

),

wherev2 =

w T

t|ζS(s)− ζT (s)|2ds.

d) Compute the self-financing hedging strategy that hedges the bond optionusing a portfolio based on the assets P (t, T ) and P (t, S).

" 449



N. Privault

Exercise 12.5 Consider two risky assets S1 and S2 modeled by the geometricBrownian motions

S1(t) = eσ1Wt+µt and S2(t) = eσ2Wt+µt, t ∈ R+, (12.37)

where (Wt)t∈R+ is a standard Brownian motion under P.

a) Find a condition on r, µ and σ2 so that the discounted price processe−rtS2(t) is a martingale under P.

b) Assume that r − µ = σ22/2, and let

Xt = e(σ22−σ

21)t/2S1(t), t ∈ R+.

Show that the discounted process e−rtXt is a martingale under P.c) Taking Nt = S2(t) as numéraire, show that the forward process X(t) =

Xt/Nt is a martingale under the forward measure P defined by

dPdP

= e−rT NTN0

.

Recall thatWt := Wt − σ2t

is a standard Brownian motion under P.d) Using the relation

e−rT IE[(S1(T )− S2(T ))+] = N0IE[(S1(T )− S2(T ))+/NT ],

compute the pricee−rT IE[(S1(T )− S2(T ))+]

of the exchange option on the assets S1 and S2.

Exercise 12.6 Compute the price e−(T−t)r IE∗[1RT>κ

∣∣ Rt] at time t ∈[0, T ] of a cash-or-nothing “binary” foreign exchange call option with maturityT and strike price κ on the foreign exchange rate process (Rt)t∈R+ given by

dRt = (r − rf)Rtdt+ σRtdWt.

Exercise 12.7 Extension of Proposition 12.6 to correlated Brownian mo-tions. Assume that (St)t∈R+ and (Nt)t∈R+ satisfy the stochastic differentialequations

dSt = rtStdt+ σSt StdWSt , and dNt = ηtNtdt+ σNt NtdW

Nt ,

where (WSt )t∈R+ and (WN

t )t∈R+ have the correlation

450


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dWSt · dWN

t = ρdt,

where ρ ∈ [−1, 1].a) Show that (WN

t )t∈R+ can be written as

WNt = ρWS

t +√

1− ρ2Wt, t ∈ R+,

where (Wt)t∈R+ is a standard Brownian motion under P∗, independent of(WS

t )t∈R+ .b) Letting Xt = St/Nt, show that dXt can be written as

dXt = (rt − ηt + (σNt )2 − ρσNt σSt )Xtdt+ σtXtdWXt ,

where (WXt )t∈R+ is a standard Brownian motion under P∗ and σt is to be

computed.

Exercise 12.8 Quanto options (Exercise 9.5 in [Shr04]). Consider an assetpriced St at time t, with

dSt = rStdt+ σSStdWSt ,

and an exchange rate (Rt)t∈R+ given by

dRt = (r − rf)Rtdt+ σRRtdWRt ,

from (12.18) in Proposition 12.8, where (WRt )t∈R+ is written as

WRt = ρWS

t +√

1− ρ2Wt, t ∈ R+,

where (Wt)t∈R+ is a standard Brownian motion under P∗, independent of(WS

t )t∈R+ , i.e. we havedWR

t · dWSt = ρdt,

where ρ is a correlation coefficient.a) Let

a = r − rf + ρσRσS − (σR)2

and Xt = eatSt/Rt, t ∈ R+, and show by Exercise 12.7 that dXt can bewritten as

dXt = rXtdt+ σXtdWXt ,

where (WXt )t∈R+ is a standard Brownian motion under P∗ and σ is to be

computed.b) Compute the price

e−(T−t)r IE∗[(

STRT− κ)+ ∣∣∣ Ft]

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N. Privault

of the quanto option at time t ∈ [0, T ].

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