Change of Numéraire and Forward Measures [PDF]

Chapter 12

Change of Numéraire and Forward Measures

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

Contents 12.1 Notion of Numéraire . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Change of Numéraire . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Foreign Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Pricing of Exchange Options . . . . . . . . . . . . . . . . . . . . 12.5 Hedging by Change of Numéraire . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

423 426 435 442 444 448

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 asset or a claim. In general, the price St of an asset, when quoted in terms of the numéraire Nt , is given by St Sˆt := , t ∈ R+ . Nt Deterministic numéraires transformations are easy to handle as a change of numéraire by a deterministic factor is a formal algebraic transformation that does not involve any risk. This can be the case for example when a currency is 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 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+ -adapted risk-free interest rate process, let∗ w  t Nt := exp rs ds . 0

In this case,

rt St Sˆt = = e − 0 rs ds St , Nt

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) exchange rate between a domestic currency (SGD) and a foreign currency (EUR), i.e. one unit of local currency (SGD) corresponds to Rt units in foreign currency (EUR). Let St Sˆt := , t ∈ R+ , Rt denote the price of a foreign (EUR) asset quoted in units of the local currency (SGD). For example, if Rt = 0.59 and St = e 1, then Sˆt = St /Rt = St /0.59 ' S$1.7, and 1/Rt is the foreign EUR/SGD exchange rate. - Forward numéraire. The price P (t, T ) of a bond paying P (T, T ) = $1 at maturity T can be taken as numéraire. In this case we have h rT i Nt := P (t, T ) = IE∗ e − t rs ds Ft , 0 6 t 6 T. Recall that “Anyone who believes exponential growth can go on forever in a finite world is either a madman or an economist”, Kenneth E. Boulding, in: Energy Reorganization Act of 1973: Hearings, Ninety-third Congress, First Session, on H.R. 11510, page 248, United States Congress, U.S. Government Printing Office, 1973. ∗

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Change of Numéraire and Forward Measures

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

My foreign currency account St grew by 5% this year. The foreign exchange rate dropped by 10%.

Q: Did I achieve a positive return?

Q: Did I achieve a positive return?

A:

A:

(a) Scenario A.

(b) Scenario B.

Fig. 12.1: Why change of numéraire?

t 7−→ e −

rt 0

rs ds

h rT i P (t, T ) = IE∗ e − 0 rs ds Ft ,

0 6 t 6 T,

is an Ft - martingale. - Annuity numéraires. Processes of the form Nt :=

n X

(Tk − Tk−1 )P (t, Tk ),

0 6 t 6 T1 ,

k=1

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 acrt f count e 0 rs ds Rt , expressed in local (or domestic) units of currency, where (rtf )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 value has been transformed under change of numéraire, e.g. under a change of currency, 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 a probability measure (for example on the foreign market), under which the transformed (or forward, or deflated) process Sˆt = St /Nt will be a martingale. rt

For example in case Nt := e 0 rs ds is the money market account, the riskneutral measure P∗ is a measure under which the discounted price process rt St = e − 0 rs ds St , Sˆt = Nt

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t ∈ R+ , 425

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N. Privault is a martingale. In the next section we will see that this property can be extended to any kind of numéraire.

12.2 Change of Numéraire In this section we review the pricing of options by a change of measure associated to a numéraire Nt , cf. e.g. [GKR95] and references therein. Most of the results of this chapter rely on the following assumption, which expresses absence of arbitrage. In the foreign exchange setting where Nt = Rt , this condition states that the price of one unit of foreign currenty is a martingale when quoted and discounted in the domestic currency. Assumption (A) Under the risk-neutral measure P∗ , the discounted numéraire rt t 7−→ Mt := e − 0 rs ds Nt is an Ft -martingale.

((A)) Definition 12.1. Given (Nt )t∈[0,T ] a numéraire process, the associated forˆ is defined by ward measure P rT ˆ dP MT NT := = e − 0 rs ds . dP∗ M0 N0

(12.1)

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

rT NT ∗ MT ∗ dP = e − 0 rs ds dP , M0 N0

which is equivalent to stating that w

Ω

ˆ X(ω)dP(ω) =

w

Ω

e−

rT 0

rs ds NT

N0

XdP∗

for any (bounded) random variable S or, under a different notation,  r  T NT ˆ IE[X] = IE∗ e − 0 rs ds X , N0 for all integrable FT -measurable random variables X. 426 This version: May 26, 2018 http://www.ntu.edu.sg/home/nprivault/indext.html

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Change of Numéraire and Forward Measures More generally, by (12.1) and the fact that the process t 7−→ Mt := e −

rt 0

rs ds

Nt

is an Ft -martingale under P∗ under Assumption (A), we find that # "   ˆ NT − r T rs ds Nt − r t rs ds dP Mt ∗ IE e 0 e 0 = , Ft = IE∗ Ft = ∗ dP N0 N0 M0

(12.2)

0 6 t 6 T . In Proposition 12.3 we will show, as a consequence of next Lemma 12.2 below, that for any integrable random claim C we have h rT i ˆ IE∗ C e − t rs ds NT Ft = Nt IE[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 probability ˆ |F /dP∗ of P ˆ |F with respect to P∗ . density dP |Ft t t |Ft Lemma 12.2. We have ˆ |F rT dP MT NT t = = e − t rs ds , dP∗|Ft Mt Nt

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   r   ˆ GX ˆ = IE∗ GX ˆ e − 0T rs ds NT IE N0    Nt − r t rs ds ∗ ˆ − r T rs ds NT ∗ 0 e IE X e t = IE G Ft N0 Nt " " # #   rT ˆ d P ∗ ∗ ∗ ˆ − t rs ds NT = IE G IE Ft IE X e Ft dP∗ Nt " #   r ˆ dP ˆ e − tT rs ds NT Ft = IE∗ G ∗ IE∗ X dP Nt    rT N T ∗ ˆ G IE X ˆ e − t rs ds = IE Ft , Nt ˆ which shows that for all integrable random variable X,   r   ˆ X ˆ | Ft = IE∗ X ˆ e − tT rs ds NT Ft , IE Nt i.e. (12.3) holds. "

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N. Privault We note that in case the numéraire Nt = e ˆ = P∗ . market account we simply have P

rt 0

rs ds

is equal to the money

Pricing using Change of Numéraire The change of numéraire technique is specially useful for pricing under random interest rates, in which case an expectation of the form i h rT IE∗ e − t rs ds C Ft becomes a path integral, see e.g. [Das04] for a recent account of path integral methods in quantitative finance. The next proposition is the basic result of this section, it provides a way to price an option with arbitrary payoff C rT under a random discount factor e − t rs ds 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   h rT i ˆ C Ft , IE∗ e − t rs ds C Ft = Nt IE 0 6 t 6 T, (12.4) NT
ˆ FT . provided that C/NT ∈ L1 P, Proof. By Relation (12.3) in Lemma 12.2 we have # " i h rT ˆ |F Nt dP t IE∗ e − t rs ds C Ft = IE∗ C F t dP∗|Ft NT " # ˆ ∗ dP|Ft C = Nt IE Ft dP∗|Ft NT   ˆ C Ft , 0 6 t 6 T. = Nt IE NT Equivalently we can write ˆ Nt IE



" #  ˆ |F C C dP t ∗ Ft = Nt IE Ft NT NT dP∗|Ft h rT i = IE∗ e − t rs ds C Ft ,

0 6 t 6 T. 

Each application of the formula (12.4) will require to a) identify a suitable numéraire (Nt )t∈R+ , and to 428 This version: May 26, 2018 http://www.ntu.edu.sg/home/nprivault/indext.html

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Change of Numéraire and Forward Measures 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-hand side of (12.4). Next, we consider further examples of numéraires and associated examples of option prices. Examples: a) Money market account. rt

Take Nt := e 0 rs ds , where (rt )t∈R+ is a possibly random and timedependent risk-free interest rate. In this case, Assumption (A) is clearly ˆ = P∗ , and (12.4) simply reads satisfied, we have P i h rT h rT rt i IE∗ e − t rs ds C Ft = e 0 rs ds IE∗ e − 0 rs ds C 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 bondmaturing at timeT , 0 6 t 6 T , and the discounted bond price process

e−

rt 0

rs ds

P (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 random claim C can be priced as i h rT   ˆ C Ft , IE∗ e − t rs ds C Ft = P (t, T )IE 0 6 t 6 T, (12.5) ∗

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

rT ˆ P (T, T ) e − 0 rs ds dP = = e − 0 rs ds dP∗ P (0, T ) P (0, T )

(12.6)

by (12.1). c) Annuity numéraires. We take Nt :=

n X

(Tk − Tk−1 )P (t, Tk )

k=1

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 h rT i IE∗ e − t rs ds (P (T, Tn ) − P (T, T1 ) − κNT )+ Ft " + # P (T, Tn ) − P (T, T1 ) ˆ = Nt IE −κ Ft , NT 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 of forward (or deflated) prices ˆ t := Xt , X Nt

0 6 t 6 T,

(12.7)

which represents the values at times t of Xt , expressed in units of the
ˆt numéraire Nt . It will be useful to determine the dynamics of X under t∈R+ ˆ the forward measure P. Proposition 12.4. Let (Xt )t∈R+ denote a continuous (Ft )t∈R+ -adapted asset price process such that t 7−→ e −

rt 0

rs ds

Xt ,

t ∈ R+ ,

is a martingale under P∗ . Then, under change of numéraire,
ˆt the process X = (Xt /Nt )t∈[0,T ] of forward prices is an Ft t∈[0,T ] ˆ provided that it is integrable under P. ˆ martingale under P, Proof. We need to show that   ˆ Xt Fs = Xs , IE Nt Ns

0 6 s 6 t,

(12.8)

and we achieve this using a standard characterization of conditional expectation. Namely, for all bounded Fs -measurable random variables G we note that under Assumption (A) we have " #   ˆ ˆ G Xt = IE∗ G Xt dP IE Nt Nt dP∗ ## " " ˆ Xt dP ∗ ∗ = IE IE G Ft Nt dP∗ " " ## ˆ Xt ∗ dP = IE∗ G IE F t Nt dP∗ 430 This version: May 26, 2018 http://www.ntu.edu.sg/home/nprivault/indext.html

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Change of Numéraire and Forward Measures   rt Xt = IE∗ G e − 0 ru du N0   r ∗ − 0s ru du Xs = IE G e N0 ## " " ˆ Xs ∗ dP F = IE G IE s Ns dP∗ " " ## ˆ Xs dP ∗ = IE IE G Fs Ns dP∗ " # ˆ Xs dP = IE G Ns dP∗   ˆ G Xs , = IE 0 6 s 6 t, Ns because

t 7−→ e −

rt 0

rs ds

Xt

is an Ft -martingale. Finally, the identity         ˆ GXˆt = IE ˆ G Xt = IE ˆ G Xs = IE ˆ GXˆs , IE Nt Ns for all bounded Fs -measurable G, implies (12.8).

0 6 s 6 t, 

Next we will rephrase Proposition 12.4 in Proposition 12.6 using the Girsanov theorem, which is briefly recalled below. Girsanov theorem Recall that, letting " ∗

Φt := IE

# ˆ dP Ft , dP∗

t ∈ [0, T ],

(12.9)

and given (Wt )t∈R a standard Brownian motion under P∗ , the Girsanov the
ˆt orem∗ shows that the process W defined by t∈R +

ˆ t := dWt − 1 dΦt · dWt , dW Φt

t ∈ R+ ,

(12.10)

ˆ In case the martingale (Φt )t∈[0,T ] is a standard Brownian motion under P. takes the form ∗

See e.g. Theorem III-35 page 132 of [Pro04].

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N. Privault  w  t 1wt Φt = exp − ψs dWs − |ψs |2 ds , 0 2 0 hence

dΦt = −ψt Φt dWt ,

t ∈ R+ ,

t ∈ R+ ,

and by the Itô multiplication Table 4.1, Relation (12.10) reads 1 dΦt · dWt Φt 1 (−ψt Φt dWt ) · dWt = dWt − Φt = dWt + ψt dt, t ∈ R+ ,  
rt ˆt and shows that the shifted process W = Wt + 0 ψs ds t∈R+ ˆ t = dWt − dW

t∈R+

is a

ˆ which is consistent with the Girsanov standard Brownian motion under P, Theorem 6.2. The next result is another application of the Girsanov theorem.
ˆt Proposition 12.5. The process W defined by t∈R+ ˆ t := dWt − dW

1 dNt · dWt , Nt

t ∈ R+ ,

(12.11)

ˆ is a standard Brownian motion under P. Proof. Relation (12.2) shows that Φt defined in (12.9) satisfies " # ˆ dP ∗ Φt = IE Ft dP∗   NT − r T rs ds = IE∗ e 0 Ft N0 Nt − r t rs ds e 0 , 0 6 t 6 T, = N0 hence   rt dΦt = d Φt e − 0 rs ds rt

= −Φt rt dt + e − 0 rs ds dΦt Φt dNt , = −Φt rt dt + Nt which, by (12.10), yields ˆ t = dWt − dW

1 dΦt · dWt Φt

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Change of Numéraire and Forward Measures   1 Φt −Φt rt dt + dNt · dWt Φt Nt 1 = dWt − dNt · dWt , Nt

= 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 in ˆ See Exerˆt addition it determines the precise dynamics of X under P. t∈R+ cise 12.1 for another calculation based on geometric Brownian motion, and Exercise 12.7 for an extension to correlated Brownian motions. As a consequence, we have the next proposition. Proposition 12.6. Assume that (Xt )t∈R+ and (Nt )t∈R+ satisfy the stochastic differential equations dXt = rt Xt dt + σtX Xt dWt ,

and

dNt = rt Nt dt + σtN Nt dWt ,

(12.12)

where (σtX )t∈R+ and (σtN )t∈R+ are (Ft )t∈R+ -adapted volatility processes. Then we have ˆ t = (σtX − σtN )X ˆ t dW ˆ t. dX (12.13) Proof. First we note that by (12.11) and (12.12), ˆ t = dWt − dW

1 dNt · dWt = dWt − σtN dt, Nt

t ∈ R+ ,

ˆ Next, by Itô’s calculus and the Itô is a standard Brownian motion under P. multiplication Table 4.1 and (12.12) we have   1 1 1 d = − 2 dNt + 3 (dNt )2 Nt Nt Nt 1 |σ N |2 (rt Nt dt + σtN Nt dWt ) + t dt Nt2 Nt N 2 1 ˆ t + σ N dt)) + |σt | dt = − 2 (rt Nt dt + σtN Nt (dW t Nt Nt 1 ˆ t ), = − (rt dt + σtN dW (12.14) Nt =−

hence ˆt = d dX = "



Xt Nt



dXt + Xt d Nt



1 Nt



+ dXt · d



1 Nt



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N. Privault =

=

= = = =


1 Xt (rt Xt dt + σtX Xt dWt ) − rt dt + σtN dWt − |σtN |2 dt Nt Nt
1 − (rt Xt dt + σtX Xt dWt ) · rt dt + σtN dWt − |σtN |2 dt Nt 1 Xt (rt Xt dt + σtX Xt dWt ) − (rt dt + σtN dWt ) Nt Nt Xt X N Xt σ σ dt + |σtN |2 dt − Nt Nt t t Xt X Xt N Xt X N |σ N |2 σ dWt − σ dWt − σ σ dt + Xt t dt Nt t Nt t Nt t t Nt
Xt X N X N N 2 σ dWt − σt dWt − σt σt dt + |σt | dt Nt t ˆ Xt (σtX − σtN )dWt − Xˆt (σtX − σtN )σtN dt ˆ t, Xˆt (σtX − σtN )dW

ˆ t = dWt − σ N dt, t ∈ R+ . since dW t



We end this section with some comments on inverse changes of measure. Inverse Changes of Measure ˆ In the next proposition we compute conditional inverse density dP∗ /dP. Proposition 12.7. We have w   ∗  t ˆ dP Ft = N0 exp rs ds IE ˆ 0 Nt dP

0 6 t 6 T,

(12.15)

and the process t 7−→

 w t N0 exp rs ds , 0 Nt

0 6 t 6 T,

ˆ is an Ft -martingale under P. Proof. For all bounded and Ft -measurable random variables F we have,   ∗ ˆ F dP = IE∗ [F ] IE ˆ dP   Nt = IE∗ F Nt   w  T NT ∗ = IE F rs ds exp − t Nt  w  t N 0 ˆ F rs ds . = IE exp 0 Nt 434 This version: May 26, 2018 http://www.ntu.edu.sg/home/nprivault/indext.html

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Change of Numéraire and Forward Measures  By (12.14) we also have  w  w  t t 1 1 ˆ t, d exp rs ds =− exp rs ds σtN dW 0 0 Nt Nt which recovers the second part of Proposition 12.7, i.e. the martingale property of w  t 1 exp rs ds t 7−→ 0 Nt ˆ under P.

12.3 Foreign Exchange Currency exchange is a typical application of change of numéraire that illustrate 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 opportunity” by a) at time 0, changing one unit of local currency into 1/R0 units of foreign currency, b) investing 1/R0 on the foreign market at the rate rf , which will yield the f amount e tr /R0 at time t, f f c) changing back e tr /R0 into a quantity e tr Rt /R0 = Nt /R0 of his local currency. f

f

In other words, the foreign money market account e tr is valued e tr Rt on the local (or domestic) market, and its discounted value on the local market is f e −tr+tr Rt , t ∈ R+ . The outcome of this investment will be obtained by a martingale comparison f of e tr Rt /R0 to the amount e rt that could have been obtained by investing on the local market. Taking f

Nt := e tr Rt ,

"

t ∈ R+ ,

(12.16)

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N. Privault as numéraire, absence of arbitrage is expressed by Assumption (A), which states that the discounted numéraire process f

t 7−→ e −rt Nt = e −t(r−r ) Rt is an Ft -martingale under P∗ . Next, we find a characterization of this arbitrage condition under the parameters of the model, and for this we will model foreign exchange rates Rt according to a geometric Brownian motion (12.17).∗ Proposition 12.8. Assume that the foreign exchange rate Rt satisfies a stochastic differential equation of the form dRt = µRt dt + σRt dWt ,

(12.17)

where (Wt )t∈R+ is a standard Brownian motion under P∗ . Under the absence of arbitrage Assumption (A) for the numéraire (12.16), we have µ = r − rf ,

(12.18)

hence the exchange rate process satisfies dRt = (r − rf )Rt dt + σRt dWt .

(12.19)

∗

under P . Proof. The equation (12.17) has solution Rt = R0 e µt+σWt −σ

2

t/2

,

t ∈ R+ , f

hence the discounted value of the foreign money market account e tr on the local market is f

e −tr+tr Rt = R0 e (r

f

−r+µ)t+σWt −σ 2 t/2

,

t ∈ R+ . f

Under the absence of arbitrage Assumption (A), the process e −(r−r )t Rt = e −tr Nt should be an Ft -martingale under P∗ , and this holds provided that rf − r + µ = 0, which yields (12.18) and (12.19).  As a consequence of Proposition 12.8, under absence of arbitrage a local investor who buys a unit of foreign currency in the hope of a higher return rf >> r will have to face a lower (or even more negative) drift µ = r − rf > rf will have to face a lower (or even more negative) drift −µ = rf − r in his exchange rate 1/Rt ˆ as written in (12.21) under P. Foreign exchange options We now price a foreign exchange option with payoff (RT − κ)+ under P∗ on the 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 exchange rate process (Rt )t∈R+ given by (12.19). The price of the foreign exchange call option on RT with maturity T and strike price κ is given by f

e −(T −t)r IE∗ [(RT −κ)+ | Rt ] = e −(T −t) r Rt Φ+ (t, Rt )−κ e −(T −t)r Φ− (t, Rt ), (12.22) 0 6 t 6 T , where Φ+ (t, x) = Φ



Φ− (t, x) = Φ



and

log(x/κ) + (T − t)(r − rf + σ 2 /2) √ σ T −t



log(x/κ) + (T − t)(r − rf − σ 2 /2) √ σ T −t



,

.

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

dNt = d( e tr Rt ) f

f

= rf e tr Rt dt + e tr dRt f

f

= r e tr Rt dt + σ e tr Rt dWt = rNt dt + σNt dWt . Hence a standard application of the Black-Scholes formula yields f

e −(T −t)r IE∗ [(RT − κ)+ | Ft ] = e −(T −t)r IE∗ [( e −T r NT − κ)+ | Ft ]   f f
+ = e −(T −t)r e −T r IE∗ NT − κ e T r | Ft ! f f log(Nt e −T r /κ) + (r + σ 2 /2)(T − t) √ = e −T r Nt Φ σ T −t f

−κ e T r

f

−(T −t)r

Φ

log(Nt e −T r /κ) + (r − σ 2 /2)(T − t) √ σ T −t

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!!

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Change of Numéraire and Forward Measures  log(Rt /κ) + (T − t)(r − rf + σ 2 /2) √ σ T −t   log(Rt /κ) + (T − t)(r − rf − σ 2 /2) T r f −(T −t)r √ −κ e Φ σ T −t

= e −T r

f





Nt Φ

f

= e −(T −t)r Rt Φ+ (t, Rt ) − κ e −(T −t)r Φ− (t, Rt ).  Similarly, from (12.21) rewritten as  rt  e e rt e rt ˆ d = rf dt − σ d Wt , Rt Rt Rt ˆ a foreign exchange call option with payoff (1/RT − κ)+ can be priced under P in a Black-Scholes model by taking e rt /Rt as underlying price, rf as risk-free interest rate, and −σ as volatility parameter. In this framework the BlackScholes formula (5.18) yields " + # f 1 ˆ e −(T −t)r IE −κ (12.23) Rt RT
  rT f ˆ e − κ e rT + Rt = e −(T −t)r e −rT IE RT     f e −(T −t)r 1 1 = Φ+ t, − κ e −(T −t)r Φ− t, , Rt Rt Rt

(12.24)

which is the symmetric of (12.22) by exchanging Rt with 1/Rt and r with rf , where   log(x/κ) + (T − t)(rf − r + σ 2 /2) √ , Φ+ (t, x) = Φ σ T −t and Φ− (t, x) = Φ



log(x/κ) + (T − t)(rf − r − σ 2 /2) √ σ T −t

 .

Call/put duality for foreign exchange options f

Let Nt = e tr Rt , where Rt is an exchange rate with respect to a foreign currency and rf is the foreign market interest rate. From Proposition 12.3 and (12.4) we have " "  + # + # 1 1 1 −(T −t)r ∗ 1 ˆ IE − RT e IE − RT Rt = Rt , Nt κ e T r f RT κ "

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

N. Privault and this yields the call/put duality " " + #  + # κ 1 1 −(T −t)r f ˆ −(T −t)r f ˆ IE e IE −κ − RT Rt = e Rt RT RT κ " #  + f 1 1 ˆ = κ e tr IE − RT Rt f T r e RT κ " + # κ trf −(T −t)r ∗ 1 − RT = e IE Rt Nt κ " + # 1 κ −(T −t)r ∗ − RT e IE (12.25) = Rt , Rt κ between a call option with strike price κ and a (possibly fractional) quantity κ/Rt of put option(s) with strike price 1/κ. In the Black-Scholes case the duality (12.25) can be directly checked by verifying that (12.23) coincides with " + # 1 κ −(T −t)r ∗ e IE − RT Rt Rt κ   !+ f κ −(T −t)r −T rf ∗  e T r  T rf e e IE − e RT = Rt Rt κ   !+ f κ −(T −t)r −T rf ∗  e T r  = e e IE − NT Rt Rt κ   −(T −t)r f e κ Φp− (t, Rt ) − e −(T −t)r Rt Φp+ (t, Rt ) = Rt κ f e −(T −t)r p Φ− (t, Rt ) − κ e −(T −t)r Φp+ (t, Rt ) Rt     f e −(T −t)r 1 1 = Φ+ t, − κ e −(T −t)r Φ− t, , Rt Rt Rt

=

where

and

  log(xκ) + (T − t)(r − rf − σ 2 /2) √ Φp− (t, x) := Φ − , σ T −t   log(xκ) + (T − t)(r − rf + σ 2 /2) √ Φp+ (t, x) := Φ − . σ T −t

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

"

Change of Numéraire and Forward Measures Local market f

Martingales

Options

Foreign market

t 7−→ e −rt Nt = e −t(r−r ) Rt

e

−(T −t)r

∗

IE

"

1 − RT κ

t 7−→

Xt e (r−r = Nt Rt

f

)t

" + # + # 1 −(T −t)r f ˆ −κ IE Rt e Rt RT

Table 12.1: Local vs foreign markets.

The foreign exchange call and put options on the local and foreign markets are linked by the relation " " + # + # f 1 1 ˆ − RT κ e −(T −t)r IE∗ −κ Rt = Rt e −(T −t)r IE Rt . κ RT Letting κ0 = 1/κ, the put option priced       f 1 1 + e −(T −t)r IE∗ (κ0 − RT ) Rt = e −(T −t)r κ0 Φ+ t, − e −(T −t)r Rt Φ− t, Rt Rt f

= e −(T −t)r κ0 Φp− (t, Rt ) − e −(T −t)r Rt Φp+ (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 κ0 = Rt with r = rf ' 0 we find √   √      σ T −t σ T −t + IE∗ (Rt − RT ) Rt = Rt Φ −Φ − 2 2   √   σ T −t = Rt 2Φ −1 . 2 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 a foreign currency (USD), i.e. one unit of local currency (EUR) corresponds to Rt = 1.23 units of foreign currency (USD) we find     p   + IE∗ (Rt − RT ) Rt = 1.23 2Φ 0.05 × 31/365 − 1 = 1.23(2 × 0.505813 − 1) = 0.01429998 "

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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 exhanged at counters.

12.4 Pricing of Exchange Options ˆ t of forward prices as a Based on Proposition 12.4, we model the process X ˆ written as continuous martingale under P, ˆt = σ ˆ t, dX ˆt dW

(12.26)

t ∈ R+ ,

ˆ and σ is a standard Brownian motion under P ˆt is an t∈R+
t∈R+ ˆt (Ft )t∈R+ -adapted process. More precisely, we assume that X has the t∈R+ dynamics
ˆ t dW ˆ t, ˆt = σ (12.27) dX ˆt X ˆt where W







where the function x 7−→ σ ˆt (x) is uniformly Lipschitz in t ∈ R+ . The Markov
ˆt property of the diffusion process X , cf. Theorem V-6-32 of [Pro04], t∈R+
  ˆ gˆ X ˆ T Ft can be written using shows that the conditional expectation IE ˆ ˆ a (measurable) function C(t, x) of t and Xt , as 
 ˆ gˆ X ˆ X ˆ t ), ˆ T Ft = C(t, IE

0 6 t 6 T.


ˆ T can be priced Consequently, a vanilla option with claim payoff C := NT gˆ X as h rT
i 
 ˆ gˆ X ˆ T Ft = Nt IE ˆ T Ft IE∗ e − t rs ds NT gˆ X ˆ X ˆ t ), = Nt C(t,

0 6 t 6 T. (12.28)

In the next Proposition 12.11 we state the Margrabe [Mar78] formula for the pricing of exchange options by the zero interest rate Black-Scholes formula. It will be applied in particular in Proposition 14.3 below for the pricing of bond options. Here, (Nt )t∈R+ denotes any numéraire process satisfying Assumption (A).
ˆt = σ ˆ t , i.e. Proposition 12.11. (Margrabe formula). Assume that σ ˆt X ˆ (t)X ˆ t )t∈[0,T ] is a (driftless) geometric Brownian motion under the martingale (X ˆ with deterministic volatility (ˆ P σ (t))t∈[0,T ] . Then we have

h rT i + ˆ t ) − κNt Φ0− (t, X ˆ t ), IE∗ e − t rs ds (XT − κNT ) Ft = Xt Φ0+ (t, X (12.29) 442 This version: May 26, 2018 http://www.ntu.edu.sg/home/nprivault/indext.html

"

Change of Numéraire and Forward Measures t ∈ [0, T ], where   log(x/κ) v(t, T ) Φ0+ (t, x) = Φ + , v(t, T ) 2 and v 2 (t, T ) =

wT t

Φ0− (t, x) = Φ



 log(x/κ) v(t, T ) − , v(t, T ) 2 (12.30)

σ ˆ 2 (s)ds.

Proof. Taking g(x) = (x − κ)+ in (12.28), the call option with payoff
ˆT − κ +, (XT − κNT )+ = NT X and floating strike price κNT is priced by (12.28) as h rT h rT i
i ˆ T − κ + Ft IE∗ e − t rs ds (XT − κNT )+ Ft = IE∗ e − t rs ds NT X h
i ˆ X ˆ T − κ + Ft = Nt IE ˆ X ˆ t ), = Nt C(t, ˆ X ˆ t ) is given by the Black-Scholes formula where the function C(t, ˆ x) = xΦ0+ (t, x) − κΦ0− (t, x), C(t, ˆ t )t∈[0,T ] is a driftless geometric Brownian with zero interest rate, since (X ˆ and X ˆ T is a lognormal random motion which is an Ft -martingale under P, wT 2 2 variable with variance coefficient v (t, T ) = σ ˆ (s)ds. Hence we have t

h

IE∗ e −

rT t

i rs ds ˆ X ˆt) (XT − κNT ) Ft = Nt C(t, +

ˆ t Φ0+ (t, X ˆ t ) − κNt Φ0− (t, X ˆ t ), = Nt X t ∈ R+ .



In particular, from Proposition 12.6 and (12.13), we can take σ ˆ (t) = σtX − σtN when (σtX )t∈R+ and (σtN )t∈R+ are deterministic. Examples: a) When the short rate process (r(t))t∈[0,T ] is a deterministic function and rT ˆ = P∗ and Proposition 12.11 yields Nt = e − t r(s)ds , 0 6 t 6 T , we have P Merton’s [Mer73] “zero interest rate” version of the Black-Scholes formula h i rT + e − t r(s)ds IE∗ (XT − κ) Ft  rT   rT  rT = Xt Φ0+ t, e t r(s)ds Xt − κ e − t r(s)ds Φ0− t, e t r(s)ds Xt , "

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

N. Privault where Φ0+ and Φ0 are defined in (12.30) and (Xt )t∈R+ satisfies the equation dXt = r(t)dt + σ ˆ (t)dWt , Xt

i.e.

ˆt dX =σ ˆ (t)dWt , ˆt X

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 h rT i + ˆ t ) − κP (t, T )Φ− (t, X ˆ t ), IE∗ e − t rs ds (XT − κ) Ft = Xt Φ+ (t, X t ∈ R+ , since 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 h rT i + ˆ t )−κP (t, T )Φ− (t, X ˆ t ), IE∗ e − t rs ds (P (T, S) − κ) Ft = P (t, S)Φ+ (t, X ˆ is ˆ t = P (t, S)/P (t, T ) under P 0 6 t 6 T , provided that the martingale X given 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 hedging strategies found in (6.31)-(6.32) and Proposition 6.11 of Chapter 6. For this, we use the stochastic integral representation of the forward claim payoffs and change of numéraire in order to compute self-financing portfolio strategies. Our hedging portfolios will be built on the assets (Xt , Nt ), not on Xt rt and the money market account Bt = e 0 rs ds , extending the classical hedging portfolios that are available in from the Black-Scholes formula, using a technique from [Jam96], cf. also [PT12]. Consider a claim with random payoff C, typically an interest rate derivative, cf. Chapter 14. Assume that the forward claim payoff C/NT ∈ L2 (Ω) has the stochastic integral representation   w T C ˆ C + ˆt, = IE φˆt dX (12.31) Cˆ := 0 NT NT
is given by (12.26) and φˆt t∈[0,T ] is a square-integrable ˆ from which it follows that the forward claim price adapted process under P,   Vt 1 ∗ h − r T rs ds i ˆ C Ft , 0 6 t 6 T, Vˆt := = IE e t C Ft = IE Nt Nt NT ˆt where X




t∈[0,T ]

ˆ that can be decomposed as is an Ft -martingale under P, 444 This version: May 26, 2018 http://www.ntu.edu.sg/home/nprivault/indext.html

"

Change of Numéraire and Forward Measures   ˆ Cˆ | Ft = IE ˆ Vˆt = IE



C NT



+

wt 0

ˆs, φˆs dX

0 6 t 6 T.

(12.32)

The next proposition extends the argument of [Jam96] to the general framework of pricing using change of numéraire. Note that this result differs from rt the standard formula that uses the money market account Bt = e 0 rs ds for hedging instead of Nt , cf. e.g. [GKR95] pages 453-454. The notion of selffinancing portfolio is similar to that of Definition 5.1. ˆ t φˆt , 0 6 t 6 T , the portfolio Proposition 12.12. Letting ηˆt := Vˆt − X
allocation φˆt , ηˆt t∈[0,T ] with value Vt = φˆt Xt + ηˆt Nt ,

0 6 t 6 T,

is self-financing in the sense that dVt = φˆt dXt + ηˆt dNt , and it hedges the claim C, i.e. i h rT Vt = φˆt Xt + ηˆt Nt = IE∗ e − t rs ds C Ft ,

0 6 t 6 T. (12.33)
ˆ Proof. In order to check that the portfolio allocation φt , ηˆt t∈[0,T ] hedges the claim C it suffices to check that (12.33) holds since by (12.4) the price Vt at time t ∈ [0, T ] of the hedging portfolio satisfies i h rT Vt = Nt Vˆt = IE∗ e − t rs ds C Ft , 0 6 t 6 T.
Next, we show that the portfolio allocation φˆt , ηˆt t∈[0,T ] is self-financing. By numéraire invariance, cf. e.g. page 184 of [Pro01], we have, using the relation ˆ t from (12.32), dVˆt = φˆt dX dVt = d(Nt Vˆt ) = Vˆt dNt + Nt dVˆt + dNt · dVˆt ˆ t + φˆt dNt · dX ˆt = Vˆt dNt + Nt φˆt dX


ˆ t dNt + Nt φˆt dX ˆ t + φˆt dNt · dX ˆ t + Vˆt − φˆt X ˆ t dNt = φˆt X ˆ t ) + ηˆt dNt = φˆt d(Nt X

= φˆt dXt + ηˆt dNt .  We now consider an application to the
forward Delta hedging of
European ˆ T where gˆ : R −→ R and X ˆt type options with payoff C = gˆ X has t∈R+ the Markov property as in (12.27), where σ ˆ : R+ × R. Assuming that the ˆ x) defined by function C(t, 445 "

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N. Privault h
i ˆ g X ˆ X ˆt) ˆ T Ft = C(t, Vˆt := IE is C 2 on R+ , we have the following corollary of Proposition 12.12, which extends the Black-Scholes Delta hedging technique to the general change of numéraire setup. ˆ ˆ X ˆt) − X ˆ t ∂ C (t, X ˆ t ), 0 6 t 6 T , the Corollary 12.13. Letting ηˆt = C(t, ∂x  ∂ Cˆ  ˆ t ), ηˆt (t, X with value portfolio allocation ∂x t∈[0,T ] Vt = ηˆt Nt + Xt

∂ Cˆ ˆ t ), (t, X ∂x

t ∈ R+ ,


ˆT . is self-financing and hedges the claim C = NT gˆ X Proof. This result follows directly from Proposition 12.12 by noting that ˆ the stochastic by Itô’s formula, and the martingale property of Vˆt under P integral representation (12.32) is given by ∂ Cˆ ˆ t ), φˆt = (t, X ∂x

0 6 t 6 T. 

In the case of an exchange option with payoff function
+ ˆT − κ + C = (XT − κNT ) = NT X ˆt on the geometric Brownian motion X


t∈[0,T ]

ˆ with under P


ˆt = σ ˆt, σ ˆt X ˆ (t)X

(12.34)
where σ ˆ (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 i h rT + ˆ t ) − κNt Φ0 (t, X ˆt) IE∗ e − t rs ds (XT − κNT ) Ft = Xt Φ0+ (t, X − ˆ t ), −κΦ0 (t, X ˆ t ))t∈[0,T ] in yields a self-financing portfolio allocation (Φ0+ (t, X − + the assets (Xt , Nt ), that hedges the claim C = (XT − κNT ) . Proof. We apply Corollary 12.13 and the relation ∂ Cˆ (t, x) = Φ0+ (t, x), ∂x 446 This version: May 26, 2018 http://www.ntu.edu.sg/home/nprivault/indext.html

x ∈ R, "

Change of Numéraire and Forward Measures ˆ x) = xΦ0+ (t, x) − κΦ0− (t, x), cf. Relation (5.21) in Propofor the function C(t, sition 5.14.  Note that the Delta hedging method requires the computation of the funcˆ x) and that of the associated finite differences, and may not apply tion C(t, to path-dependent claims. Examples: a) When the short rate process (r(t))t∈[0,T ] is a deterministic function and rT Nt = e t r(s)ds , Corollary 12.14 yields the usual Black-Scholes hedging strategy   r ˆ t ), −κ e 0T r(s)ds Φ− (t, Xt ) Φ+ (t, X t∈[0,T ]   rT rT rT 0 r(s)ds ˆ r(s)ds 0 t 0 = Φ+ (t, e Φ− (t, e t r(s)ds Xt ) , Xt ), −κ e t∈[0,T ]

in the assets (Xt , e

rt

+

), that hedges the claim C = (XT − κ) , with r   T log(x/κ) + t r(s)ds + (T − t)σ 2 /2 , √ Φ+ (t, x) := Φ  σ T −t 0

r(s)ds

and  Φ− (t, x) := Φ 

log(x/κ) +

r

 r(s)ds − (T − t)σ 2 /2 . √ σ T −t

T t

b) In case Nt = P (t, T )
and Xt = P (t, S), 0 6 t 6 T < S, Corollary 12.14 ˆt shows that when X is modeled as the geometric Brownian motion t∈[0,T ] ˆ the bond call option with payoff (P (T, S) − κ)+ can be (12.34) under P, hedged as h rT i + ˆ t )−κP (t, T )Φ− (t, X ˆt) IE∗ e − t rs ds (P (T, S) − κ) Ft = P (t, S)Φ+ (t, X by the self-financing portfolio allocation ˆ t ), −κΦ− (t, X ˆ t ))t∈[0,T ] (Φ+ (t, X ˆt) in the assets (P (t, S), P (t, T )), i.e. one needs to hold the quantity Φ+ (t, X ˆ t ) of the of the bond maturing at time S, and to short a quantity κΦ− (t, X bond maturing at time T .

"

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N. Privault

Exercises Exercise 12.1 Let (Bt )t∈R+ be a standard Brownian motion started at 0 under the risk-neutral measure P∗ . Consider a numéraire (Nt )t∈R+ given by Nt := N0 e ηBt −η

2

t/2

,

t ∈ R+ ,

and a risky asset (Xt )t∈R+ given by Xt := X0 e σBt −σ

2

t/2

,

t ∈ R+ .

ˆ denote the forward measure relative to the numéraire (Nt )t∈R , under Let P + ˆ t := Xt /Nt of forward prices is known to be a martingale. which the process X a) Using the Itô formula, compute   ˆ t = d(Xt /Nt ) = (X0 /N0 )d e (σ−η)Bt −(σ2 −η2 )t/2 . dX b) Explain why the exchange option price IE[(XT − λNT )+ ] at time 0 has the Black-Scholes form e −rT IE[(XT − λNT )+ ]
√ ! ˆ 0 /λ log X σ ˆ T √ + − λN0 Φ = X0 Φ 2 σ ˆ T

(12.35)
√ ! ˆ 0 /λ log X σ ˆ T √ − . 2 σ ˆ T

Hints: (i) Use the change of numéraire identity 
 ˆ X ˆT − λ + . e −rT IE[(XT − λNT )+ ] = N0 IE ˆ ˆ t is a martingale under the forward measure P (ii) The forward price X relative to the numéraire (Nt )t∈R+ . c) Give the value of σ ˆ 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 rate ˆ process. Taking Nt := P (t, T ) as a numéraire defining the forward measure P, ˆ using a standard Brownian compute the dynamics of (P (t, S))t∈[0,T ] under P
ˆ ˆt motion W under P. t∈[0,T ] Exercise 12.3 Forward contract. Using a change of numéraire argument for the numéraire Nt := P (t, T ), t ∈ [0, T ], compute the price at time t ∈ [0, T ] of 448 " This version: May 26, 2018 http://www.ntu.edu.sg/home/nprivault/indext.html

Change of Numéraire and Forward Measures a forward (or future) contract with payoff P (T, S)−K in a bond market with short 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 ) = rt dt + ζtT dWt , P (t, T ) and

dP (t, S) = rt dt + ζtS dWt , P (t, S)

where (ζ T (s))s∈[0,T ] and (ζ S (s))s∈[0,S] are deterministic functions. a) Show, using Itô’s formula, that   P (t, S) P (t, S) S ˆ t, d (ζ (t) − ζ T (t))dW = P (t, T ) P (t, T ) ˆt where W b) Show that P (T, S) =


t∈R+

ˆ is a standard Brownian motion under P.

w  wT T P (t, S) ˆs − 1 exp (ζ S (s) − ζ T (s))dW |ζ S (s) − ζ T (s)|2 ds . t P (t, T ) 2 t

ˆ denote the forward measure associated to the numéraire Let P Nt := P (t, T ),

0 6 t 6 T.

c) Show that for all S, T > 0 the price at time t h rT i IE e − t rs ds (P (T, S) − κ)+ Ft of a bond call option on P (T, S) with payoff (P (T, S) − κ)+ is equal to h rT i IE∗ e − t rs ds (P (T, S) − κ)+ Ft (12.36)     1 P (t, S) v 1 P (t, S) v + log − κP (t, T )Φ − + log , = P (t, S)Φ 2 v κP (t, T ) 2 v κP (t, T ) where v2 =

wT t

|ζ S (s) − ζ T (s)|2 ds.

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

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

N. Privault Exercise 12.5 Consider two risky assets S1 and S2 modeled by the geometric Brownian motions S1 (t) = e σ1 Wt +µt

and

S2 (t) = e σ2 Wt +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 process e −rt S2 (t) is a martingale under P. b) Assume that r − µ = σ22 /2, and let 2

2

Xt = e (σ2 −σ1 )t/2 S1 (t),

t ∈ R+ .

Show that the discounted process e −rt Xt is a martingale under P. ˆ c) Taking Nt = S2 (t) as numéraire, show that the forward process X(t) = ˆ defined by Xt /Nt is a martingale under the forward measure P ˆ dP NT = e −rT . dP N0 Recall that

ˆ t := Wt − σ2 t W

ˆ is a standard Brownian motion under P. d) Using the relation ˆ 1 (T ) − S2 (T ))+ /NT ], e −rT IE[(S1 (T ) − S2 (T ))+ ] = N0 IE[(S compute the price

e −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∗ 1{RT >κ} Rt at time t ∈ [0, T ] of a cash-or-nothing “binary” foreign exchange call option with maturity T and strike price κ on the foreign exchange rate process (Rt )t∈R+ given by dRt = (r − rf )Rt dt + σRt dWt . Exercise 12.7 Extension of Proposition 12.6 to correlated Brownian motions. Assume that (St )t∈R+ and (Nt )t∈R+ satisfy the stochastic differential equations dSt = rt St dt + σtS St dWtS ,

and dNt = ηt Nt dt + σtN Nt dWtN ,

where (WtS )t∈R+ and (WtN )t∈R+ have the correlation 450 This version: May 26, 2018 http://www.ntu.edu.sg/home/nprivault/indext.html

"

Change of Numéraire and Forward Measures dWtS · dWtN = ρdt, where ρ ∈ [−1, 1]. a) Show that (WtN )t∈R+ can be written as WtN = ρWtS +

p

1 − ρ2 W t ,

t ∈ R+ ,

where (Wt )t∈R+ is a standard Brownian motion under P∗ , independent of (WtS )t∈R+ . b) Letting Xt = St /Nt , show that dXt can be written as dXt = (rt − ηt + (σtN )2 − ρσtN σtS )Xt dt + σ ˆt Xt dWtX , ˆt is to be where (WtX )t∈R+ is a standard Brownian motion under P∗ and σ computed. Exercise 12.8 Quanto options (Exercise 9.5 in [Shr04]). Consider an asset priced St at time t, with dSt = rSt dt + σ S St dWtS , and an exchange rate (Rt )t∈R+ given by dRt = (r − rf )Rt dt + σ R Rt dWtR , from (12.18) in Proposition 12.8, where (WtR )t∈R+ is written as WtR = ρWtS +

p

1 − ρ2 W t ,

t ∈ R+ ,

where (Wt )t∈R+ is a standard Brownian motion under P∗ , independent of (WtS )t∈R+ , i.e. we have dWtR · dWtS = ρdt, where ρ is a correlation coefficient. a) Let

a = r − rf + ρσ R σ S − (σ R )2

and Xt = e at St /Rt , t ∈ R+ , and show by Exercise 12.7 that dXt can be written as dXt = rXt dt + σ ˆ Xt dWtX , where (WtX )t∈R+ is a standard Brownian motion under P∗ and σ ˆ is to be computed. b) Compute the price " + # ST −κ e −(T −t)r IE∗ Ft RT "

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

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

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

"