changeofnuméraireandforward measures · t∈r+ is a possibly random and time-...

Chapter 16Change of Numéraire and ForwardMeasures

Change of numéraire is a powerful technique for the pricing of options underrandom discount factors by the use of forward measures. It has applicationsto the pricing of interest rate derivatives and other types of options, includingexchange options (Margrabe formula) and foreign exchange options (Garman-Kohlagen formula). The computation of self-financing hedging strategies bychange of numéraire is treated in Section 16.5, and the change of numérairetechnique will be applied to the pricing of interest rate derivatives such asbond options and swaptions in Chapter 19.

16.1 Notion of Numéraire . . . . . . . . . . . . . . . . . . . . . . . . . . . 51116.2 Change of Numéraire. . . . . . . . . . . . . . . . . . . . . . . . . . . 51416.3 Foreign Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52316.4 Pricing Exchange Options . . . . . . . . . . . . . . . . . . . . . . 53116.5 Hedging by Change of Numéraire . . . . . . . . . . . . . . . 534Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538

16.1 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

Ŝt :=StNt

, t ∈ R+.

Deterministic numéraires transformations are easy to handle, as change ofnuméraire by a constant factor is a formal algebraic transformation thatdoes not involve any risk. This can be the case for example when a currency

" 511

This version: September 17, 2020https://www.ntu.edu.sg/home/nprivault/indext.html

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

N. Privault

is pegged to another currency,∗ for example the exchange rate of the Frenchfranc to the Euro was locked at €1 = FRF 6.55957 on 31 December 1998.

On the other hand, a random numéraire may involve new risks, and canallow for arbitrage 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,Ŝt =

StNt

= e−r t

0 rsdsSt, t ∈ R+,

represents the discounted price of the asset at time 0.

- Currency exchange rates

In this case,Nt := Rt denotes e.g. the EUR/SGD (EURSGD=X) exchangerate from a foreign currency (e.g. EUR) to a domestic currency (e.g. SGD),i.e. one unit of foreign currency (EUR) corresponds to Rt units of localcurrency (SGD). Let

Ŝt :=StRt

, t ∈ R+,

denote the price of a local (SG) asset quoted in units of the foreign currency(EUR). For example, if Rt = 1.63 and St = S$1, then

Ŝt =StRt

=1

1.63 × $1 ' €0.61,

and 1/Rt is the domestic SGD/EUR exchange rate. A question of interestis whether a local asset $St, discounted according to a foreign risk-freerate rf and priced in foreign currency as

e−rf t StRt

= e−rf tŜt,

∗ 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.† “Anyone who believes exponential growth can go on forever in a finite world is eithera madman or an economist”, K.E. Boulding (1973), page 248.

512 "


https://en.wikipedia.org/wiki/Kenneth_E._Bouldinghttps://www.ntu.edu.sg/home/nprivault/indext.html

Change of Numéraire and Forward Measures

can be a martingale on the foreign market.

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. 16.1: Why change of numéraire?

- 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

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 numéraire.

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.

" 513



N. Privault

In particular, in order to perform a fair pricing, one has to determine a proba-bility measure under which the transformed (or forward, or deflated) processŜt = St/Nt will be a martingale, see Section 16.3 for details.

For example in case Nt := er t

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

Ŝt =StNt

= e−r t

0 rsdsSt, t ∈ R+,

is a martingale. In the next section, we will see that this property can beextended to any type of numéraire.

16.2 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. Geman et al. (1995) 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 currency is amartingale when quoted and discounted in the domestic currency.

Assumption (A) The discounted numéraire

t 7−→Mt := e−r t

0 rsdsNt

is an Ft-martingale under the risk-neutral probability measure P∗.

((A))Definition 16.1. Given (Nt)t∈[0,T ] a numéraire process, the associated for-ward measure P̂ is defined by

dP̂dP∗ :=

MTM0

= e−r T

0 rsdsNTN0

. (16.1)

Recall that from Section 7.3 the above Relation (16.1) rewrites as

dP̂ = MTM0

dP∗ = e−r T

0 rsdsNTN0

dP∗,

which is equivalent to stating that

ÎE[F ] =w

ΩF (ω)dP̂(ω)

=w

ΩF (ω) e−

r T0 rsds

NTN0

dP∗(ω)

514 "




= IE∗[F e−

r T0 rsds

NTN0

],

for all integrable FT -measurable random variables F . More generally, by(16.1) 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∗[

dP̂dP∗

∣∣∣∣Ft]= IE∗

[NTN0

e−r T

0 rsds

∣∣∣∣Ft] = NtN0 e−r t

0 rsds =MtM0

, (16.2)

0 6 t 6 T . In Proposition 16.5 we will show, as a consequence of nextLemma 16.2 below, that for any integrable random claim payoff C we have

IE∗[C e−

r TtrsdsNT

∣∣Ft] = NtÎE[C | Ft], 0 6 t 6 T .Note that (16.2), which is Ft-measurable, should not be confused with (16.3),which is FT -measurable. In the next Lemma 16.2 we compute the probabilitydensity dP̂|Ft/dP

∗|Ft of P̂|Ft with respect to P

∗|Ft .

Lemma 16.2. We have

dP̂|FtdP∗|Ft

=MTMt

= e−r TtrsdsNT

Nt, 0 6 t 6 T . (16.3)

Proof. The proof of (16.3) relies on an abstract version of the Bayes formula.For all integrable Ft-measurable random variable G, by (16.2), the towerproperty (23.38) of conditional expectations and the characterization (23.49)in Proposition 23.19, we have

ÎE[GX̂

]= IE∗

[GX̂ e−

r T0 rsds

NTN0

]= IE∗

[GNtN0

e−r t

0 rsds IE∗[X̂ e−

r TtrsdsNT

Nt

∣∣∣Ft]]= IE∗

[G IE∗

[dP̂dP∗

∣∣∣Ft] IE∗ [X̂ e− r Tt rsdsNTNt

∣∣∣Ft]]

= IE∗[G

dP̂dP∗ IE

∗[X̂ e−

r TtrsdsNT

Nt

∣∣∣Ft]]

= ÎE[G IE∗

[X̂ e−

r TtrsdsNT

Nt

∣∣∣Ft]] ,for all integrable random variable X̂, which shows that" 515



N. Privault

ÎE[X̂ | Ft

]= IE∗

[X̂ e−

r TtrsdsNT

Nt

∣∣∣Ft] ,i.e. (16.3) holds. �

We note that in case the numéraire Nt = er t

0 rsds is equal to the moneymarket account we simply have P̂ = P∗.Definition 16.3. Given (Xt)t∈R+ an asset price process, we define the pro-cess of forward (or deflated) prices

X̂t :=XtNt

, 0 6 t 6 T . (16.4)

The process(X̂t)t∈R+

epresents the values at times t of Xt, expressed inunits of the numéraire Nt. In the sequel, it will be useful to determine thedynamics of

(X̂t)t∈R+

under the forward measure P̂. The next proposition

shows in particular that the process(

er t

0 rsds/Nt)t∈R+

is an Ft-martingale

under P̂.Proposition 16.4. Let (Xt)t∈R+ denote a continuous (Ft)t∈R+-adapted as-set price process such that

t 7−→ e−r t

0 rsdsXt, t ∈ R+,

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

the deflated process(X̂t)t∈[0,T ] = (Xt/Nt)t∈[0,T ] of forward prices is an

Ft-martingale under P̂,

provided that(X̂t)t∈[0,T ] is integrable under P̂.

Proof. We show that

ÎE[XtNt

∣∣∣∣Fs] = XsNs , 0 6 s 6 t, (16.5)using the characterization (23.49) of conditional expectation in Proposi-tion 23.19. Namely, for all bounded Fs-measurable random variables G wenote that using (16.2) under Assumption (A) we have

ÎE[GXtNt

]= IE∗

[GXtNt

dP̂dP∗

]

= IE∗[IE∗[GXtNt

dP̂dP∗

∣∣∣∣Ft]]

516 "




= IE∗[GXtNt

IE∗[

dP̂dP∗

∣∣∣∣Ft]]

= IE∗[G e−

r t0 rudu

XtN0

]= IE∗

[G e−

r s0 rudu

XsN0

]= IE∗

[GXsNs

IE∗[

dP̂dP∗

∣∣∣∣Fs]]

= IE∗[IE∗[GXsNs

dP̂dP∗

∣∣∣∣Fs]]

= IE∗[GXsNs

dP̂dP∗

]

= ÎE[GXsNs

], 0 6 s 6 t,

becauset 7−→ e−

r t0 rsdsXt

is an Ft-martingale under P∗. Finally, the identity

ÎE[GX̂t

]= ÎE

[GXtNt

]= ÎE

[GXsNs

]= ÎE

[GX̂s

], 0 6 s 6 t,

for all bounded Fs-measurable G, implies (16.5). �

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. Dash (2004) for a recent account of pathintegral methods in quantitative finance. The next proposition is the basicresult of this section, it provides a way to price an option with arbitrarypayoff C under a random discount factor e−

r Ttrsds by use of the forward

measure. It will be applied in Chapter 19 to the pricing of bond options andcaplets, cf. Propositions 19.1, 19.3 and 19.5 below.

Proposition 16.5. An option with integrable claim payoff C ∈ L1(Ω, P∗,FT )is priced at time t as

" 517



N. Privault

IE∗[

e−r TtrsdsC

∣∣∣Ft] = NtÎE [ CNT

∣∣∣Ft] , 0 6 t 6 T , (16.6)provided that C/NT ∈ L1

(Ω, P̂,FT

).

Proof. By Relation (16.3) in Lemma 16.2 we have

IE∗[

e−r TtrsdsC

∣∣∣Ft] = IE∗ [dP̂|FtdP∗|Ft NtNT C∣∣∣∣Ft]

= Nt IE∗[

dP̂|FtdP∗|Ft

C

NT

∣∣∣∣Ft]

= Ntw

Ω

dP̂|FtdP∗|Ft

C

NTdP∗|Ft

= Ntw

Ω

C

NTdP̂|Ft

= NtÎE[C

NT

∣∣∣Ft] , 0 6 t 6 T .Equivalently, we can write

NtÎE[C

NT

∣∣∣Ft] = NtIE∗ [ CNT

dP̂|FtdP∗|Ft

∣∣∣∣Ft]

= IE∗[

e−r TtrsdsC

∣∣Ft] , 0 6 t 6 T .�

Each application of the formula (16.6) will require to:

a) pick a suitable numéraire (Nt)t∈R+ ,b) make sure that the ratio C/NT takes a sufficiently simple form,c) use the Girsanov theorem in order to determine the dynamics of asset

prices under the new probability measure P̂,

so as to compute the expectation under P̂ on the right-hand side of (16.6).

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 clearly

518 "




satisfied, we have P̂ = P∗ and dP∗/dP̂, and (16.6) simply reads

IE∗[

e−r TtrsdsC

∣∣Ft] = er t0 rsds IE∗ [ e− r T0 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, (16.6) shows that a randomclaim payoff C can be priced as

IE∗[

e−r TtrsdsC

∣∣∣Ft] = P (t,T )ÎE[C | Ft], 0 6 t 6 T , (16.7)since P (T ,T ) = 1, where the forward measure P̂ satisfies

dP̂dP∗ = e

−r T

0 rsdsP (T ,T )P (0,T ) =

e−r T

0 rsds

P (0,T ) (16.8)

by (16.1).

c) Annuity numéraires.

We take

Nt :=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, (16.6) shows that a swaption on the cash flow P (T ,Tn)−P (T ,T1)− κNT can be priced as

IE∗[

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

∣∣Ft]= NtÎE

[(P (T ,Tn)− P (T ,T1)

NT− κ)+ ∣∣∣Ft] ,

0 6 t 6 T < T1, where (P (T ,Tn)− P (T ,T1))/NT becomes a swap rate,cf. (18.17) in Proposition 18.8 and Section 19.5.

" 519



N. Privault

Girsanov theorem

Recall that, letting

Φt := IE∗[

dP̂dP∗

∣∣∣∣Ft], t ∈ [0,T ], (16.9)and given (Wt)t∈R a standard Brownian motion under P∗, the GirsanovTheorem∗ shows that the process

(Ŵt)t∈R+

defined by

dŴt := dWt −1

ΦtdΦt • dWt, t ∈ R+, (16.10)

is a standard Brownian motion under P̂ defined by the Girsanov density

ΦT =dP̂dP∗ .

In case the martingale (Φt)t∈[0,T ] takes the form

Φt = exp(−w t

0ψsdWs −

12w t

0|ψs|2ds

), t ∈ R+,

i.e.dΦt = −ψtΦtdWt, t ∈ R+,

the Itô multiplication Table 4.1 shows that Relation (16.10) reads

dŴt = dWt −1

ΦtdΦt • dWt

= dWt −1

Φt(−ψtΦtdWt) • dWt

= dWt + ψtdt, t ∈ R+,

and shows that the shifted process(Ŵt)t∈R+

=(Wt +

r t0 ψsds

)t∈R+

is astandard Brownian motion under P̂, which is consistent with the GirsanovTheorem 7.3. The next result is another application of the Girsanov Theorem.

Proposition 16.6. The process(Ŵt)t∈R+

defined by

dŴt := dWt −1NtdNt • dWt, t ∈ R+, (16.11)

is a standard Brownian motion under P̂.

Proof. Relation (16.2) shows that Φt defined in (16.9) satisfies∗ See e.g. Theorem III-35 page 132 of Protter (2004).

520 "




Φt = IE∗[

dP̂dP∗

∣∣∣∣Ft]= IE∗

[NTN0

e−r T

0 rsds∣∣∣Ft]

=NtN0

e−r t

0 rsds, 0 6 t 6 T ,

hence

dΦt = d(NtN0

e−r t

0 rsds)

= −Φtrtdt+ e−r t

0 rsdsd

(NtN0

)= −Φtrtdt+

ΦtNtdNt,

which, by (16.10), yields

dŴt = dWt −1

ΦtdΦt • dWt

= dWt −1

Φt

(−Φtrtdt+

ΦtNtdNt

)• dWt

= dWt −1NtdNt • dWt,

which is (16.11), from Relation (16.10) and the Itô multiplication Table 4.1.�

The next Proposition 16.7 is consistent with the statement of Proposi-tion 16.4, and in addition it specifies the dynamics of

(X̂t)t∈R+

under P̂using the Girsanov Theorem 16.6. See Exercise 16.1 for another calculationbased on geometric Brownian motion, and Exercise 16.9 for an extension tocorrelated Brownian motions. As a consequence, we have the next proposi-tion.

Proposition 16.7. Assume that (Xt)t∈R+ and (Nt)t∈R+ satisfy the stochas-tic differential equations

dXt = rtXtdt+ σXt XtdWt, and dNt = rtNtdt+ σNt NtdWt, (16.12)

where (σXt )t∈R+ and (σNt )t∈R+ are (Ft)t∈R+-adapted volatility processes and(Wt)t∈R+ is a standard Brownian motion under P∗. Then we have

dX̂t = (σXt − σNt )X̂tdŴt, (16.13)

hence (X̂t)t∈[0,T ] is given by the driftless geometric Brownian motion

" 521



N. Privault

X̂t = X̂0 exp(w t

0(σXs − σNs )dŴs −

12w t

0(σXs − σNs )2ds

), 0 6 t 6 T .

Proof. First we note that by (16.11) and (16.12),

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

is a standard Brownian motion under P̂. Next, by Itô’s calculus and the Itômultiplication Table 4.1 and (16.12) we have

d

(1Nt

)= − 1

N2tdNt +

1N3t

dNt • dNt

= − 1N2t

(rtNtdt+ σNt NtdWt) +

|σNt |2

Ntdt

= − 1N2t

(rtNtdt+ σNt Nt(dŴt + σ

Nt dt)) +

|σNt |2

Ntdt

= − 1Nt

(rtdt+ σNt dŴt), (16.14)

hence

dX̂t = d

(XtNt

)=dXtNt

+Xtd

(1Nt

)+ dXt • d

(1Nt

)=

1Nt

(rtXtdt+ σXt XtdWt)−

XtNt

(rtdt+ σ

Nt dWt − |σNt |2dt

)− 1Nt

(rtXtdt+ σXt XtdWt) ·

(rtdt+ σ

Nt dWt − |σNt |2dt

)=

1Nt

(rtXtdt+ σXt XtdWt)−

XtNt

(rtdt+ σNt dWt)

+XtNt|σNt |2dt−

XtNtσXt σ

Nt dt

=XtNtσXt dWt −

XtNtσNt dWt −

XtNtσXt σ

Nt dt+Xt

|σNt |2

Ntdt

=XtNt

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

)= X̂t(σ

Xt − σNt )dWt − X̂t(σXt − σNt )σNt dt

= X̂t(σXt − σNt )dŴt,

since dŴt = dWt − σNt dt, t ∈ R+. �

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

522 "




Inverse changes of measure

In the next proposition we compute the conditional inverse Radon-Nikodymdensity dP∗/dP̂.

Proposition 16.8. We have

ÎE[

dP∗

dP̂

∣∣∣Ft] = N0Nt

exp(w t

0rsds

)0 6 t 6 T , (16.15)

and the process

t 7−→ N0Nt

exp(w t

0rsds

), 0 6 t 6 T ,

is an Ft-martingale under P̂.

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

ÎE[F

dP∗

dP̂

]= IE∗ [F ]

= IE∗[FNtNt

]= IE∗

[FNTNt

exp(−w Ttrsds

)]= ÎE

[FN0Nt

exp(w t

0rsds

)].

�

By (16.14) we also have

d

(1Nt

exp(w t

0rsds

))= − 1

Ntexp

(w t0rsds

)σNt dŴt,

which recovers the second part of Proposition 16.8, i.e. the martingale prop-erty of

t 7−→ 1Nt

exp(w t

0rsds

)under P̂.

16.3 Foreign Exchange

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

" 523



N. Privault

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 local cur-

rency.

Fig. 16.2: Overseas investment opportunity.∗

In other words, the foreign money market account etrf is valued etrfRt onthe 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 etrfRt/R0 to the amount ert that could have been obtained by investingon the local market.

Taking

Nt := etrfRt, t > 0, (16.16)

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 using the modelparameters r, rf ,µ, by modeling the foreign exchange rates Rt according toa geometric Brownian motion (16.17).∗ For illustration purposes only. Not an advertisement.

524 "




Proposition 16.9. Assume that the foreign exchange rate Rt satisfies astochastic differential equation of the form

dRt = µRtdt+ σRtdWt, (16.17)

where (Wt)t∈R+ is a standard Brownian motion under P∗. Under the absenceof arbitrage Assumption (A) for the numéraire (16.16), we have

µ = r− rf , (16.18)

hence the exchange rate process satisfies

dRt = (r− rf )Rtdt+ σRtdWt. (16.19)

under P∗.

Proof. The equation (16.17) 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−trNt = e−tr+trfRt = R0 e(r

f−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 (16.18) and (16.19). �

As a consequence of Proposition 16.9, 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

N. Privault

where

dŴt = dWt −1NtdNt • dWt

= dWt −1RtdRt • dWt

= dWt − σdt, t ∈ R+,

is a standard Brownian motion under P̂ by (16.11). Under absence of arbi-trage, the process e−(r−rf )tRt is an Ft-martingale under P∗ and (16.20) isan Ft-martingale under P̂ by Proposition 16.4, which recovers (16.18).Proposition 16.10. Under the absence of arbitrage condition (16.18), theinverse exchange rate 1/Rt satisfies

d

(1Rt

)=rf − rRt

dt− σRtdŴt, (16.21)

under P̂ , where (Rt)t∈R+ is given by (16.19).Proof. By (16.18), the exchange rate 1/Rt is written using Itô’s calculus as

d

(1Rt

)= − 1

R2t(µRtdt+ σRtdWt) +

1R3t

σ2R2t dt

= −µ− σ2

Rtdt− σ

RtdWt

= − µRtdt− σ

RtdŴt

=rf − rRt

dt− σRtdŴt,

where (Ŵt)t∈R+ is a standard Brownian motion under P̂. �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 (16.21) under P̂.

Foreign exchange options

We now price a foreign exchange call option with payoff (RT − κ)+ underP∗ on the exchange rate RT by the Black-Scholes formula as in the nextproposition, also known as the Garman and Kohlhagen (1983) formula. Theforeign exchange call option is designed for a local buyer of foreign currency.Proposition 16.11. (Garman and Kohlhagen (1983) formula for call op-tions). Consider the exchange rate process (Rt)t∈R+ given by (16.19). The526 "




price of the foreign exchange call option on RT with maturity T and strikeprice κ > 0 is given in local currency units as

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

(16.22)

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 (16.19), we find the numéraire dynamics

dNt = d( etrfRt)

= rf etrfRtdt+ etrfdRt

= r etrfRtdt+ σ etrfRtdWt

= 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 rf IE∗[(NT − κ eT r

f )+ | Ft]= e−T rf

(NtΦ

(log(Nt e−T r

f/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

))

= e−T rf(NtΦ

(log(Rt/κ) + (T − t)(r− rf + σ2/2)

σ√T − t

)−κ eT rf−(T−t)rΦ

(log(Rt/κ) + (T − t)(r− rf − σ2/2)

σ√T − t

))= e−(T−t)rfRtΦ+(t,Rt)− κ e−(T−t)rΦ−(t,Rt).

�

Similarly, from (16.21) rewritten as

" 527



N. Privault

d

(ertRt

)= rf

ertRt

dt− σ ert

RtdŴt,

a foreign exchange put option with payoff (1/κ−1/RT )+ can be priced underP̂ in a Black-Scholes model by taking ert/Rt as underlying asset price, rf asrisk-free interest rate, and −σ as volatility parameter. The foreign exchangeput option is designed for the foreign seller of local currency, see for examplethe ∗ which is a typical example of a foreign exchangeput option.

Proposition 16.12. (Garman and Kohlhagen (1983) formula for put op-tions). Consider the exchange rate process (Rt)t∈R+ given by (16.19). Theprice of the foreign exchange call option on RT with maturity T and strikeprice 1/κ > 0 is given in foreign currency units as

e−(T−t)rf ÎE[(

1κ− 1RT

)+ ∣∣∣∣Rt] (16.23)=

e−(T−t)rf

κΦ−

(t, 1Rt

)− e

−(T−t)r

RtΦ+

(t, 1Rt

),

0 6 t 6 T , where

Φ+(t,x) := Φ(− log(κx) + (T − t)(r

f − r+ σ2/2)σ√T − t

),

andΦ−(t,x) := Φ

(− log(κx) + (T − t)(r

f − r− σ2/2)σ√T − t

).

Proof. The Black-Scholes formula (6.10) yields


1κ− 1RT

)+ ∣∣∣∣Rt] = e−(T−t)rf e−rT ÎE[( erTκ − erTRT)+ ∣∣∣∣Rt]

=1κ

e−(T−t)rf Φ−(t, 1Rt

)− e

−(T−t)r

RtΦ+

(t, 1Rt

),

which is the symmetric of (16.22) by exchanging Rt with 1/Rt, and r withrf . �

∗ Right-click to open or save the attachment.

528 "



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.Proposition 16.13. The foreign exchange call and put options on the localand foreign markets are linked by the call/put duality relation

e−(T−t)r IE∗[(RT − κ)+

∣∣Rt] = κRt e−(T−t)rf ÎE[( 1κ− 1RT

)+ ∣∣∣Rt] ,(16.24)

between a put option with strike price 1/κ and a (possibly fractional) quantity1/(κRt) of call option(s) with strike price κ.Proof. By application of change of numéraire from Proposition 16.5 and(16.6) we have

ÎE[

1eT rfRT

(RT − κ)+∣∣∣Rt] = 1

Nte−(T−t)r IE∗

[(RT − κ)+

∣∣ Rt],hence


1κ− 1RT

)+ ∣∣∣Rt] = e−(T−t)rf ÎE [ 1κRT

(RT − κ)+∣∣∣Rt]

=1κ

etrf ÎE[

1eT rfRT

(RT − κ)+∣∣∣Rt]

=1κNt

etrf−(T−t)r IE∗[(RT − κ)+

∣∣Rt]=

1κRt

e−(T−t)r IE∗[(RT − κ)+

∣∣Rt].�

In the Black-Scholes case, the duality (16.24) can be directly checked byverifying that (16.23) coincides with

1κRt

e−(T−t)r IE∗[(RT − κ)+ | Rt]

=1κRt

e−(T−t)r e−T rf IE∗[(

eT rfRT − κ eT rf )+ ∣∣Rt]

=1κRt

e−(T−t)r e−T rf IE∗[(NT − κ eT r

f )+ ∣∣Rt]=

1κRt

(e−(T−t)rfRtΦc+(t,Rt)− κ e−(T−t)rΦc−(t,Rt)

)=

1κ

e−(T−t)rf Φc+ (t,Rt)−e−(T−t)rRt

Φc− (t,Rt)

" 529



N. Privault

=1κ


)− e

−(T−t)r

RtΦ+

(t, 1Rt

),

whereΦc+(t,x) := Φ

(log(x/κ) + (T − t)(r− rf + σ2/2)

σ√T − t

),

andΦc−(t,x) := Φ

(log(x/κ) + (T − t)(r− rf − σ2/2)

σ√T − t

).

Local market Foreign market

Measures P∗ P̂

Discount factors t 7−→ e−rt t 7−→ e−rft

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

XtNt

=e(r−r

f )t

Rt

Options e−(T−t)r IE∗[(RT − κ)+

∣∣Rt] e−(T−t)rf ÎE[( 1κ− 1RT

)+ ∣∣∣Rt]

Uses Local purchase of foreign curreny Foreign selling of local curreny

Table 16.1: Local vs foreign exchange options.

Example - the buy back guarantee

The put option priced


1κ− 1RT

)+ ∣∣∣Rt]=

1κ


)− e

−(T−t)r

RtΦ+

(t, 1Rt

)on the foreign market corresponds to a ∗ in currencyexchange. In the case of an option “at the money” with κ = Rt and r = rf '0, we find∗ Right-click to open or save the attachment.

530 "



ÎE[(

1Rt− 1RT

)+ ∣∣∣Rt] = 1Rt×(

Φ(σ√T − t2

)−Φ

(−σ√T − t2

))=

1Rt×(

2Φ(σ√T − t2

)− 1)

.

For example, let Rt denote the USD/EUR (USDEUR=X) exchange rate froma foreign currency (USD) to a local currency (EUR), i.e. one unit of theforeign currency (USD) corresponds to Rt = 1/1.23 units of local currency(EUR). Taking T − t = 30 days and σ = 10%, we find that the foreigncurrency put option allowing the foreign sale of one EURO back into USDsis priced at the money in USD as

ÎE[(

1Rt− 1RT

)+ ∣∣∣Rt] = 1.23(2Φ(0.05×√31/365)− 1)= 1.23(2× 0.505813− 1)= $0.01429998

per USD, or €$0.011626 per exchanged unit of EURO. Based on a displayedoption price of €4.5, this would translate into an amount of 4.5/0.011626 '€387 exchanged on average by customers subscribing to the buy back guar-antee.

16.4 Pricing Exchange Options

Based on Proposition 16.4, we model the process X̂t of forward prices as acontinuous martingale under P̂, written as

dX̂t = σ̂tdŴt, t ∈ R+, (16.25)

where(Ŵt)t∈R+

is a standard Brownian motion under P̂ and (σ̂t)t∈R+ isan (Ft)t∈R+ -adapted stochastic volatility process. More precisely, we assumethat

(X̂t)t∈R+

has the dynamics

dX̂t = σ̂t(X̂t)dŴt, (16.26)

where x 7→ σ̂t(x) is a local volatility function which is Lipschitz in x, uni-formly in t ∈ R+. The Markov property of the diffusion process

(X̂t)t∈R+

,cf. Theorem V-6-32 of Protter (2004), shows that when ĝ is a deterministicpayoff function, the conditional expectation ÎE

[ĝ(X̂T) ∣∣ Ft] can be written

using a (measurable) function Ĉ(t,x) of t and X̂t, as

ÎE[ĝ(X̂T) ∣∣Ft] = Ĉ(t, X̂t), 0 6 t 6 T .

" 531



N. Privault

Consequently, a vanilla option with claim payoff C := NT ĝ(X̂T)can be

priced from Proposition 16.5 as

IE∗[

e−r TtrsdsNT ĝ

(X̂T) ∣∣∣Ft] = NtÎE[ĝ(X̂T ) ∣∣Ft]

= NtĈ(t, X̂t), 0 6 t 6 T . (16.27)

In the next Proposition 16.14 we state the Margrabe (1978) formula for thepricing of exchange options by the zero interest rate Black-Scholes formula.It will be applied in particular in Proposition 19.3 below for the pricingof bond options. Here, (Nt)t∈R+ denotes any numéraire process satisfyingAssumption (A).

Proposition 16.14. (Margrabe (1978) formula). Assume that σ̂t(X̂t)=

σ̂(t)X̂t, i.e. the martingale (X̂t)t∈[0,T ] is a (driftless) geometric Brownianmotion under P̂ with deterministic volatility (σ̂(t))t∈[0,T ]. Then we have

IE∗[

e−r Ttrsds (XT − κNT )+

∣∣Ft] = XtΦ0+(t, X̂t)− κNtΦ0−(t, X̂t),(16.28)

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

),

(16.29)and v2(t,T ) =

w Ttσ̂2(s)ds.

Proof. Taking g(x) := (x− κ)+ in (16.27), the call option with payoff

(XT − κNT )+ = NT(X̂T − κ

)+= NT

(X̂t exp

(w Ttσ̂(t)dŴt −

12w Tt|σ̂(t)|2dt

)− κ)+

,

and floating strike price κNT is priced by (16.27) as

IE∗[

e−r Ttrsds(XT − κNT )+

∣∣Ft] = IE∗ [ e− r Tt rsdsNT (X̂T − κ)+ ∣∣Ft]= NtÎE

[(X̂T − κ

)+ ∣∣Ft]= NtĈ(t, X̂t),

where the function Ĉ(t, X̂t) is given by the Black-Scholes formula

532 "




Ĉ(t,x) = xΦ0+(t,x)− κΦ0−(t,x),

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

w Ttσ̂2(s)ds. Hence we have

IE∗[


∣∣Ft] = NtĈ(t, X̂t)= NtX̂tΦ0+(t, X̂t)− κNtΦ0−(t, X̂t),

t ∈ R+. �

In particular, from Proposition 16.7 and (16.13), we can take σ̂(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 of timeand Nt = e

r Ttr(s)ds, 0 6 t 6 T , we have P̂ = P∗ and Proposition 16.14

yields the Merton (1973) “zero interest rate” version of the Black-Scholesformula

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

),

where Φ0+ and Φ0 are defined in (16.29) and (Xt)t∈R+ satisfies the equa-tion

dXtXt

= r(t)dt+ σ̂(t)dWt, i.e.dX̂t

X̂t= σ̂(t)dWt, 0 6 t 6 T .

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

IE∗[

e−r Ttrsds (XT − κ)+

∣∣Ft] = XtΦ0+(t, X̂t)− κP (t,T )Φ0−(t, X̂t),0 6 t 6 T , since NT = P (T ,T ) = 1. In particular, when Xt = P (t,S)the above formula 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)Φ0+(t, X̂t)−κP (t,T )Φ0−(t, X̂t),0 6 t 6 T , provided that the martingale X̂t = P (t,S)/P (t,T ) under P̂is given by a geometric Brownian motion, cf. Section 19.2.

" 533



N. Privault

16.5 Hedging by Change of Numéraire

In this section we reconsider and extend the Black-Scholes self-financing hedg-ing strategies found in (7.36)-(7.37) and Proposition 7.14 of Chapter 7. Forthis, we use the stochastic integral representation of the forward claim pay-offs and change of numéraire in order to compute self-financing portfoliostrategies. Our hedging portfolios will be built on the assets (Xt,Nt), noton Xt and the money market account Bt = e

r t0 rsds, extending the classi-

cal hedging portfolios that are available in from the Black-Scholes formula,using a technique from Jamshidian (1996), cf. also Privault and Teng (2012).

Consider a claim with random payoff C, typically an interest rate deriva-tive, cf. Chapter 19. Assume that the forward claim payoff C/NT ∈ L2(Ω)has the stochastic integral representation

Ĉ :=C

NT= ÎE

[C

NT

]+

w T0φ̂tdX̂t, (16.30)

where(X̂t)t∈[0,T ] is given by (16.25) and

(φ̂t)t∈[0,T ] is a square-integrable

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

V̂t :=VtNt

=1Nt

IE∗[

e−r TtrsdsC

∣∣∣Ft] = ÎE [ CNT

∣∣∣Ft] , 0 6 t 6 T ,is an Ft-martingale under P̂, that can be decomposed as

V̂t = ÎE[Ĉ | Ft

]= ÎE

[C

NT

]+

w t0φ̂sdX̂s, 0 6 t 6 T . (16.31)

The next Proposition 16.15 extends the argument of Jamshidian (1996) tothe general framework of pricing using change of numéraire. Note that thisresult differs from the standard formula that uses the money market accountBt = e

r t0 rsds for hedging instead ofNt, cf. e.g.Geman et al. (1995) pages 453-

454. The notion of self-financing portfolio is similar to that of Definition 5.9.

Proposition 16.15. Letting η̂t := V̂t − X̂tφ̂t, with φ̂t defined in (16.31),0 6 t 6 T , the portfolio allocation

(φ̂t, η̂t

)t∈[0,T ] with value

Vt = φ̂tXt + η̂tNt, 0 6 t 6 T ,

is self-financing in the sense that

dVt = φ̂tdXt + η̂tdNt,

and it hedges the claim payoff C, i.e.

534 "




Vt = φ̂tXt + η̂tNt = IE∗[

e−r TtrsdsC

∣∣∣Ft] , 0 6 t 6 T . (16.32)Proof. In order to check that the portfolio allocation

(φ̂t, η̂t

)t∈[0,T ] hedges

the claim payoff C it suffices to check that (16.32) holds since by (16.6) theprice Vt at time t ∈ [0,T ] of the hedging portfolio satisfies

Vt = NtV̂t = IE∗[

e−r TtrsdsC

∣∣∣Ft] , 0 6 t 6 T .Next, we show that the portfolio allocation

(φ̂t, η̂t

)t∈[0,T ] is self-financing. By

numéraire invariance, cf. e.g. page 184 of Protter (2001), we have, using therelation dV̂t = φ̂tdX̂t from (16.31),

dVt = d(NtV̂t)

= V̂tdNt +NtdV̂t + dNt • dV̂t

= V̂tdNt +Ntφ̂tdX̂t + φ̂tdNt • dX̂t

= φ̂tX̂tdNt +Ntφ̂tdX̂t + φ̂tdNt • dX̂t +(V̂t − φ̂tX̂t

)dNt

= φ̂td(NtX̂t) + η̂tdNt

= φ̂tdXt + η̂tdNt.

�

We now consider an application to the forward Delta hedging of European-type options with payoff C = NT ĝ

(X̂T)where ĝ : R −→ R and

(X̂t)t∈R+

has the Markov property as in (16.26), where σ̂ : R+ ×R is a deterministicfunction. Assuming that the function Ĉ(t,x) defined by

V̂t := ÎE[ĝ(X̂T) ∣∣Ft] = Ĉ(t, X̂t)

is C2 on R+, we have the following corollary of Proposition 16.15, whichextends the Black-Scholes Delta hedging technique to the general change ofnuméraire setup.

Corollary 16.16. Letting η̂t = Ĉ(t, X̂t) − X̂t∂Ĉ

∂x(t, X̂t), 0 6 t 6 T , the

portfolio allocation(∂Ĉ∂x

(t, X̂t), η̂t)t∈[0,T ]

with value

Vt = η̂tNt +Xt∂Ĉ

∂x(t, X̂t), t ∈ R+,

is self-financing and hedges the claim payoff C = NT ĝ(X̂T).

Proof. This result follows directly from Proposition 16.15 by noting thatby Itô’s formula, and the martingale property of V̂t under P̂ the stochastic" 535



N. Privault

integral representation (16.31) is given by

V̂T = Ĉ

= ĝ(X̂T)

= Ĉ(0, X̂0) +w T

0∂Ĉ

∂t(t, X̂t)dt+

w T0∂Ĉ

∂x(t, X̂t)dX̂t

+12w T

0∂2Ĉ

∂x2(t, X̂t)|σ̂t|2dt

= Ĉ(0, X̂0) +w T

0∂Ĉ

∂x(t, X̂t)dX̂t

= ÎE[C

NT

]+

w T0φ̂tdX̂t, 0 6 t 6 T ,

henceφ̂t =

∂Ĉ

∂x(t, X̂t), 0 6 t 6 T .

�

In the case of an exchange option with payoff function

C = (XT − κNT )+ = NT(X̂T − κ

)+on the geometric Brownian motion

(X̂t)t∈[0,T ] under P̂ with

σ̂t(X̂t)= σ̂(t)X̂t, (16.33)

where(σ̂(t)

)t∈[0,T ] is a deterministic volatility function of time, we have the

following corollary on the hedging of exchange options based on the Margrabe(1978) formula (16.28).

Corollary 16.17. The decomposition

IE∗[


∣∣∣Ft] = XtΦ0+(t, X̂t)− κNtΦ0−(t, X̂t)yields a self-financing portfolio allocation (Φ0+(t, X̂t),−κΦ0−(t, X̂t))t∈[0,T ] inthe assets (Xt,Nt), that hedges the claim payoff C = (XT − κNT )+.

Proof. We apply Corollary 16.16 and the relation

∂Ĉ

∂x(t,x) = Φ0+(t,x), x ∈ R,

for the function Ĉ(t,x) = xΦ0+(t,x) − κΦ0−(t,x), cf. Relation (6.15) inProposition 6.4. �

536 "




Note that the Delta hedging method requires the computation of the functionĈ(t,x) and that of the associated finite differences, and may not apply topath-dependent claims.

Examples:

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

r Ttr(s)ds, Corollary 16.17 yields the usual Black-Scholes

hedging strategy(Φ+(t, X̂t),−κ e

r T0 r(s)dsΦ−(t,Xt)

)t∈[0,T ]

=(

Φ0+(t, er Ttr(s)dsX̂t),−κ e

r T0 r(s)dsΦ0−(t, e

r Ttr(s)dsXt)

)t∈[0,T ]

,

in the assets(Xt, e

r t0 r(s)ds

), that hedges the claim payoff C = (XT − κ)+,

with

Φ+(t,x) := Φ

log(x/κ) +(r Tt r(s)ds+ (T − t)σ

2/2)

σ√T − t

,and

Φ−(t,x) := Φ

log(x/κ) +(r Tt r(s)ds− (T − t)σ

2/2)

σ√T − t

.b) In case Nt = P (t,T ) and Xt = P (t,S), 0 6 t 6 T < S, Corollary 16.17

shows that when(X̂t)t∈[0,T ] is modeled as the geometric Brownian motion

(16.33) under P̂, 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, X̂t)−κP (t,T )Φ−(t, X̂t)by the self-financing portfolio allocation

(Φ+(t, X̂t),−κΦ−(t, X̂t))t∈[0,T ]

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

" 537



N. Privault

Exercises

Exercise 16.1 Let (Bt)t∈R+ be a standard Brownian motion started at 0 un-der the risk-neutral probability 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+,

in a market with risk-free interest rate r = 0. Let P̂ denote the forwardmeasure relative to the numéraire (Nt)t∈R+ , under which the process X̂t :=Xt/Nt of forward prices is known to be a martingale.a) Using the Itô formula, compute

dX̂t = d

(XtNt

)=X0N0

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

IE∗[(XT − λNT )+] (16.34)

= X0Φ

(log(X̂0/λ

)σ̂√T

+σ̂√T

2

)− λN0Φ

(log(X̂0/λ

)σ̂√T

− σ̂√T

2

).

Hints:

(i) Use the change of numéraire identity

IE∗[(XT − λNT )+] = N0ÎE[(X̂T − λ

)+].(ii) The forward price X̂t is a martingale under the forward measure P̂

relative to the numéraire (Nt)t∈R+ .

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

Exercise 16.2 Let(B

(1)t

)t∈R+

and(B

(2)t

)t∈R+

be correlated standard Brow-nian motions started at 0 under the risk-neutral probability measure P∗, withcorrelation Corr

(B

(1)s ,B(2)t

)= ρmin(s, t), i.e. dB(1)t • dB

(2)t = ρdt. Consider

two asset prices(S(1)t

)t∈R+

and(S(2)t

)t∈R+

given by the geometric Brownianmotions538 "




S(1)t := S

(1)0 e

rt+σB(1)t −σ

2t/2, and S(2)t := S(2)0 e

rt+ηB(2)t −η

2t/2, t ∈ R+.

Let P̂2 denote the forward measure with numéraire (Nt)t∈R+ :=(S(2)t

)t∈R+

and Radon-Nikodym density

dP̂2dP∗

= e−rTS(2)T

S(2)0

= eηB(2)T−η2T/2.

a) Using the Girsanov Theorem 16.6, determine the shifts(B̂

(1)t

)t∈R+

and(B̂

(2)t

)t∈R+

of(B

(1)t

)t∈R+

and(B

(2)t

)t∈R+

which are standard Brownianmotions under P̂2.

b) Using the Itô formula, compute

dŜ(1)t = d

(S(1)t

S(2)t

)

=S(1)0

S(2)0

d(

eσB(1)t −ηB

(2)t −(σ

2−η2)t/2),and write the answer in terms of the martingales dB̂(1)t and dB̂

(2)t .

c) Using change of numéraire, explain why the exchange option price

e−rT IE∗[(S(1)T − λS

(2)T

)+]at time 0 has the Black-Scholes form

e−rT IE∗[(S(1)T − λS

(2)T

)+]= S

(1)0 Φ

(log(Ŝ(1)0 /λ

)σ̂√T

+σ̂√T

2

)

−λS(2)0 Φ

(log(Ŝ(1)0 /λ

)σ̂√T

− σ̂√T

2

),

where the value of σ̂ can be expressed in terms of σ and η.

Exercise 16.3 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 measureP̂, compute the dynamics of (P (t,S))t∈[0,T ] under P̂ using a standard Brow-nian motion

(Ŵt)t∈[0,T ] under P̂.

" 539



N. Privault

Exercise 16.4 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 ]of a forward (or future) contract with payoff P (T ,S)−K in a bond marketwith short-term interest rate (rt)t∈R+ . How would you hedge this forwardcontract?

Exercise 16.5 (Question 2.7 page 17 of Downes et al. (2008)). Consider aprice process (St)t∈R+ given by dSt = rStdt+ σStdBt under the risk-neutralprobability measure P∗, where r ∈ R and σ > 0, and the option with payoff

(ST (ST −K))+ = Max(ST (ST −K), 0)

at maturity T .

a) Show that the option payoff can be rewritten as

(ST (ST −K))+ = NT (ST −K)+

for a suitable choice of numéraire process (Nt)t∈[0,T ].b) Rewrite the option price e−(T−t)r IE∗

[(ST (ST −K))+ | Ft

]using a for-

ward measure P̂ and a change of numéraire argument.c) Find the dynamics of (St)t∈R+ under the forward measure P̂.d) Price the option with payoff

(ST (ST −K))+ = Max(ST (ST −K), 0)

at time t ∈ [0,T ] using the Black-Scholes formula.

Exercise 16.6 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+ ζ

Tt dWt,

anddP (t,S)P (t,S) = rtdt+ ζ

St dWt,

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

a) Show, using Itô’s formula, that

d

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

)=P (t,S)P (t,T ) (ζ

S(t)− ζT (t))dŴt,

where(Ŵt)t∈R+

is a standard Brownian motion under P̂.

540 "




b) Show that

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

(w Tt(ζS(s)− ζT (s))dŴs −

12w Tt|ζS(s)− ζT (s)|2ds

).

Let P̂ 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] (16.35)= P (t,S)Φ

(v

2 +1v

log P (t,S)κP (t,T )

)− κP (t,T )Φ

(−v2 +

1v

log P (t,S)κP (t,T )

),

wherev2 =

w Tt|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).

Exercise 16.7 Consider two risky assets S1 and S2 modeled by the geometricBrownian motions

S1(t) = eσ1Wt+µt and S2(t) = eσ2Wt+µt, t ∈ R+, (16.36)

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 process

e−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 X̂(t) =

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

dP̂dP = e

−rT NTN0

.

Recall that" 541



N. Privault

Ŵt := Wt − σ2t

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

e−rT IE∗[(S1(T )− S2(T ))+] = N0ÎE[(S1(T )− S2(T ))+/NT ],

compute the price

e−rT IE∗[(S1(T )− S2(T ))+]

of the exchange option on the assets S1 and S2.

Exercise 16.8 Compute the price e−(T−t)r IE∗[1{RT>κ}

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

dRt = (r− rf )Rtdt+ σRtdWt.

Hint: We have the relation

P∗(x eX+µ > κ

)= Φ

(µ− log(κ/x)√

Var[X ]

)

for X ' N (0,Var[X ]) a centered Gaussian random variable.

Exercise 16.9 Extension of Proposition 16.7 to correlated Brownian mo-tions. Assume that (St)t∈R+ and (Nt)t∈R+ satisfy the stochastic differentialequations

dSt = rtStdt+ σSt StdW

St , and dNt = ηtNtdt+ σNt NtdWNt ,

where (WSt )t∈R+ and (WNt )t∈R+ have the correlation

dWSt • dWNt = ρdt,

where ρ ∈ [−1, 1].a) Show that (WNt )t∈R+ can be written as

WNt = ρWSt +

√1− ρ2Wt, t ∈ R+,

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

b) Letting Xt = St/Nt, show that dXt can be written as

dXt = (rt − ηt + (σNt )2 − ρσNt σSt )Xtdt+ σ̂tXtdWXt ,542 "




where (WXt )t∈R+ is a standard Brownian motion under P∗ and σ̂t is tobe computed.

Exercise 16.10 Quanto options (Exercise 9.5 in Shreve (2004)). Consider anasset priced St at time t, with

dSt = rStdt+ σSStdW

St ,

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

dRt = (r− rf )Rtdt+ σRRtdWRt ,

from (16.18) in Proposition 16.9, where (WRt )t∈R+ is written as

WRt = ρWSt +

√1− ρ2Wt, t ∈ R+,

where (Wt)t∈R+ is a standard Brownian motion under P∗, independent of(WSt )t∈R+ , i.e. we have

dWRt • dWSt = ρdt,

where ρ is a correlation coefficient.

a) Leta = r− rf + ρσRσS − (σR)2

and Xt = eatSt/Rt, t ∈ R+, and show by Exercise 16.9 that dXt can bewritten as

dXt = rXtdt+ σ̂XtdWXt ,

where (WXt )t∈R+ is a standard Brownian motion under P∗ and σ̂ is to bedetermined.

b) Compute the price

e−(T−t)r IE∗[(

STRT− κ)+ ∣∣∣Ft]

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

" 543



pbs@ARFix@533: pbs@ARFix@534: pbs@ARFix@535: pbs@ARFix@536: pbs@ARFix@537: pbs@ARFix@538: pbs@ARFix@539: pbs@ARFix@540: pbs@ARFix@541: pbs@ARFix@542: pbs@ARFix@543: pbs@ARFix@544: pbs@ARFix@545: pbs@ARFix@546: pbs@ARFix@547: pbs@ARFix@548: pbs@ARFix@549: pbs@ARFix@550: pbs@ARFix@551: pbs@ARFix@552: pbs@ARFix@553: pbs@ARFix@554: pbs@ARFix@555: pbs@ARFix@556: pbs@ARFix@557: pbs@ARFix@558: pbs@ARFix@559: pbs@ARFix@560: pbs@ARFix@561: pbs@ARFix@562: pbs@ARFix@563: pbs@ARFix@564: pbs@ARFix@565:

changeofnuméraireandforward measures · t∈r+ is a possibly random and time-...

Documents