changeofnuméraireandforward measures · t∈r+ is a possibly random and time-...
Post on 25-Jul-2020
6 Views
Preview:
TRANSCRIPT
-
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 "
This version: September 17, 2020https://www.ntu.edu.sg/home/nprivault/indext.html
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
This version: September 17, 2020https://www.ntu.edu.sg/home/nprivault/indext.html
https://www.ntu.edu.sg/home/nprivault/indext.html
-
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 "
This version: September 17, 2020https://www.ntu.edu.sg/home/nprivault/indext.html
https://www.ntu.edu.sg/home/nprivault/indext.html
-
Change of Numéraire and Forward Measures
= 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
This version: September 17, 2020https://www.ntu.edu.sg/home/nprivault/indext.html
https://www.ntu.edu.sg/home/nprivault/indext.html
-
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 "
This version: September 17, 2020https://www.ntu.edu.sg/home/nprivault/indext.html
https://www.ntu.edu.sg/home/nprivault/indext.html
-
Change of Numéraire and Forward Measures
= 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
This version: September 17, 2020https://www.ntu.edu.sg/home/nprivault/indext.html
https://www.ntu.edu.sg/home/nprivault/indext.html
-
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 "
This version: September 17, 2020https://www.ntu.edu.sg/home/nprivault/indext.html
https://www.ntu.edu.sg/home/nprivault/indext.html
-
Change of Numéraire and Forward Measures
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
This version: September 17, 2020https://www.ntu.edu.sg/home/nprivault/indext.html
https://www.ntu.edu.sg/home/nprivault/indext.html
-
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 "
This version: September 17, 2020https://www.ntu.edu.sg/home/nprivault/indext.html
https://www.ntu.edu.sg/home/nprivault/indext.html
-
Change of Numéraire and Forward Measures
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
This version: September 17, 2020https://www.ntu.edu.sg/home/nprivault/indext.html
https://www.ntu.edu.sg/home/nprivault/indext.html
-
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 "
This version: September 17, 2020https://www.ntu.edu.sg/home/nprivault/indext.html
https://www.ntu.edu.sg/home/nprivault/indext.html
-
Change of Numéraire and Forward Measures
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
This version: September 17, 2020https://www.ntu.edu.sg/home/nprivault/indext.html
https://www.ntu.edu.sg/home/nprivault/indext.html
-
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 "
This version: September 17, 2020https://www.ntu.edu.sg/home/nprivault/indext.html
https://www.ntu.edu.sg/home/nprivault/indext.html
-
Change of Numéraire and Forward Measures
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 "
This version: September 17, 2020https://www.ntu.edu.sg/home/nprivault/indext.html
https://www.ntu.edu.sg/home/nprivault/indext.html
-
Change of Numéraire and Forward Measures
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
This version: September 17, 2020https://www.ntu.edu.sg/home/nprivault/indext.html
https://www.ntu.edu.sg/home/nprivault/indext.html
-
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
e−(T−t)rf ÎE[(
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 "
This version: September 17, 2020https://www.ntu.edu.sg/home/nprivault/indext.html
-
https://www.ntu.edu.sg/home/nprivault/indext.html
-
Change of Numéraire and Forward Measures
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
e−(T−t)rf ÎE[(
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
This version: September 17, 2020https://www.ntu.edu.sg/home/nprivault/indext.html
https://www.ntu.edu.sg/home/nprivault/indext.html
-
N. Privault
=1κ
e−(T−t)rf Φ−(t, 1Rt
)− 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
e−(T−t)rf ÎE[(
1κ− 1RT
)+ ∣∣∣Rt]=
1κ
e−(T−t)rf Φ−(t, 1Rt
)− 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 "
This version: September 17, 2020https://www.ntu.edu.sg/home/nprivault/indext.html
-
https://www.ntu.edu.sg/home/nprivault/indext.html
-
Change of Numéraire and Forward Measures
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
This version: September 17, 2020https://www.ntu.edu.sg/home/nprivault/indext.html
https://www.ntu.edu.sg/home/nprivault/indext.html
-
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 "
This version: September 17, 2020https://www.ntu.edu.sg/home/nprivault/indext.html
https://www.ntu.edu.sg/home/nprivault/indext.html
-
Change of Numéraire and Forward Measures
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∗[
e−r Ttrsds (XT − κNT )+
∣∣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
This version: September 17, 2020https://www.ntu.edu.sg/home/nprivault/indext.html
https://www.ntu.edu.sg/home/nprivault/indext.html
-
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 "
This version: September 17, 2020https://www.ntu.edu.sg/home/nprivault/indext.html
https://www.ntu.edu.sg/home/nprivault/indext.html
-
Change of Numéraire and Forward Measures
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
This version: September 17, 2020https://www.ntu.edu.sg/home/nprivault/indext.html
https://www.ntu.edu.sg/home/nprivault/indext.html
-
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∗[
e−r Ttrsds (XT − κNT )+
∣∣∣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 "
This version: September 17, 2020https://www.ntu.edu.sg/home/nprivault/indext.html
https://www.ntu.edu.sg/home/nprivault/indext.html
-
Change of Numéraire and Forward Measures
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
This version: September 17, 2020https://www.ntu.edu.sg/home/nprivault/indext.html
https://www.ntu.edu.sg/home/nprivault/indext.html
-
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 "
This version: September 17, 2020https://www.ntu.edu.sg/home/nprivault/indext.html
https://www.ntu.edu.sg/home/nprivault/indext.html
-
Change of Numéraire and Forward Measures
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
This version: September 17, 2020https://www.ntu.edu.sg/home/nprivault/indext.html
https://www.ntu.edu.sg/home/nprivault/indext.html
-
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 "
This version: September 17, 2020https://www.ntu.edu.sg/home/nprivault/indext.html
https://www.ntu.edu.sg/home/nprivault/indext.html
-
Change of Numéraire and Forward Measures
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
This version: September 17, 2020https://www.ntu.edu.sg/home/nprivault/indext.html
https://www.ntu.edu.sg/home/nprivault/indext.html
-
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 "
This version: September 17, 2020https://www.ntu.edu.sg/home/nprivault/indext.html
https://www.ntu.edu.sg/home/nprivault/indext.html
-
Change of Numéraire and Forward Measures
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
This version: September 17, 2020https://www.ntu.edu.sg/home/nprivault/indext.html
https://www.ntu.edu.sg/home/nprivault/indext.html
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:
top related