Chapter IV Application to Finance 1. The Simple Black Scholes Market
4. Pricing of Random Payoffs at Fixed Future Dates
LetB be a market containing zero coupon bonds of all maturitiesT ∈(0, T∗] and ξ a local deflator for B such thatξBj is a martingale, for each zero coupon bond Bj inB.
4.a European options. Fix a timeT ∈(0, T∗] and letB|T denote the market based on the filtered probabilityspace (Ω,(Ft)t∈[0,T], P) resulting from B byrestricting the securityprocesses in B to the interval [0, T]. Thus all trading stops at time T in the marketB|T.
Lemma. Let τ : Ω→[0, T] be an optional time andφ a trading strategy satisfying Vt(φ) =t
τφ(s)ãdB(s),P-as. on the set[τ < t], for all t∈[0, T]. Then the trading strategy χ= 1]]τ,T]]φ is self-financing.
Proof. Note thatVt(χ) = 1[τ <t]Vt(φ) especiallyV0(χ) = 0. Thus we must show that Vt(χ) =t
0χ(s)ãdB(s),P-as., for allt∈[0, T]. Consider suchtand note thatχ= 0 and sot
0χ(s)ãdB(s) = 0 =Vt(χ),P-as. on the set [t≤τ] (III.2.d.4). Likewise on the set [τ < t] we have
Vt(χ) =Vt(φ) = t
τ
φ(s)ãdB(s) = t
0
1]]τ,T]](s)φ(s)ãdB(s) = t
0
χ(s)ãdB(s).
4.a.0. Let ψ1,ψ2 be self-financing trading strategies with VT(ψ1) =VT(ψ2). Then (a) Vt(ψ1) =Vt(ψ2), for allt∈[0, T], or
(b) there is arbitrage in the marketB|T.
Proof. Assume that (b) is not true, set ψ =ψ1−ψ2 and let > 0. To establish (a) it will suffice to show that the optional timeτ = inf{t∈[0, T]|Vt(ψ)> } ∧T satisfies P(τ < T) = 0. Let ρ be the trading strategywhich buys one share of the zero coupon bond B(t, T) maturing at time T and holds this to timeT. Set Z =Vτ(φ)/B(τ, T) and consider the trading strategy
χ= 1]]τ,T]]Zρ−1]]τ,T]]ψ= 1]]τ,T]]φ,
where φ= 1]]τ,T]]Zρ−ψ. Following the strategyχ we wait until timeτ. If τ =T we do nothing. If τ < T we short the portfolio ψ (long ψ2, shortψ1) and invest the proceeds into the zero coupon bond B(t, T). The self-financing propertyofψ implies that
Vt(ψ) =V0(ψ) + t
0
ψ(s)ãdB(s), ∀t∈[0, T], and so Vτ(ψ) =V0(ψ) +
τ 0
ψ(s)ãdB(s).
Subtraction yieldst
τψ(s)ãdB(s) =Vt(ψ)−Vτ(ψ),P-as. on the set [τ < t]. Likewise the Fτ-measurabilityof Z and III.2.d.0.(i) implythat t
τ1]]τ,T]](s)Zρ(s)ãdB(s) =
252 4.a European options.
t
0Z1]]τ,T]](s)dB(s, T) =Z(B(t, T)−B(t∧τ, T)). It follows that t
τ
φ(s)ãdB(s) =Z(B(t, T)−B(τ, T))−
Vt(ψ)−Vτ(ψ)
=ZB(t, T)−Vt(ψ) =Vt(φ),
P-as. on the set [τ < t]. The preceding Lemma now shows that the strategy χis self-financing. Since V0(χ) = 0 andVT(χ) =Z > 0, on the set [τ < T], we must have P(τ < T) = 0.
Remark. From example 1.c.3 we know that such arbitrage χ can coexist with the local deflatorξforB. Thus we cannot conclude alternative (a) in 4.a.0. Additional assumptions onφ,θ are necessary.
4.a.1 Law of One Price. Let φ, θ be trading strategies. If ξ(t)
Vt(φ)−Vt(θ) is a P-martingale, then VT(φ) =VT(θ)implies that Vt(φ) =Vt(θ), for allt∈[0, T].
Proof. SetDt=Vt(φ)−Vt(θ) and assume thatDT = 0. The martingale property ofξ(t)Dtthen implies thatDt= 0, that is,Vt(φ) =Vt(θ), for allt∈[0, T].
Remark. If φ, θ are self-financing, then the processesξ(t)Vt(φ) andξ(t)Vt(θ) and hence the differenceξ(t)
Vt(φ)−Vt(θ)
areP-local martingales (3.c.0). If in addition the maximal function supt∈[0,T]ξ(t)Vt(φ)−Vt(θ)is integrable, then ξ(t)
Vt(φ)− Vt(θ)
is a P-martingale (I.8.a.4).
Options and replication. AEuropean optionexercisable at timeT is a nonnegative, FT-measurable random variableh. The quantityh(ω) is the payoff received by the option holder at time T if the market is in state ω. The nonnegativityof h is the mathematical expression of the fact that holders of options traded in existing financial markets do not exercise an option unless it is in the money, that is, unless this results in a positive payoff. TheFT-measurabilitycorresponds to the fact that the uncertaintyregarding the payoff h(ω), if exercised, is resolved bytimeT.
A trading strategy θ in B|T is said to replicate the option h, if it is self- financing, ξ(t)Vt(θ) is a martingale and VT(θ) = h. Necessarilythen ξ(t)Vt(θ) = EP
ξ(T)VT(θ)|Ft
, that is
Vt(θ) =ξ(t)−1EP
ξ(T)h|Ft
, t∈[0, T]. (0)
The optionhis said to bereplicableif and onlyif there exists a trading strategyθ inB|T replicatingh. Recall that for a self-financing trading strategyθ, the process ξ(t)Vt(θ) is automaticallya P-local martingale; however to obtain formula (0) we need the stronger martingale condition.
Assume now thathis replicated bythe trading strategyθ. To determine what the priceπt(h) of the claimhat timet≤T should be, let us assume that we decide to tradehin our market according to some price processπ(t). To avoid arbitrage we must haveπ(T) =h=VT(θ) and soπ(t) =Vt(θ) =ξ(t)−1EP
ξ(T)h|Ft
, for all
t ∈[0, T] (4.a.0 with ψ1 =θ and ψ2 being the buyand hold strategyinvesting in h). This leads us to define thearbitrage priceofhas
πt(h) =ξ(t)−1EP
ξ(T)h| Ft
, t∈[0, T]. (1)
For later use we define πt(h) as in (1), even if his not replicable. However then πt(h) can no longer be interpreted as the arbitrage price process of the claim h.
Using 3.d.0.(d) withA(t) =B(t, T) (thenA(T) = 1), we see that πt(h) =B(t, T)EPT
h|Ft
, t∈[0, T], (2)
wherePT denotes the forward martingale measure at dateT. IfBcontains a riskless bondB0 such thatξB0is a martingale (and hence the spot martingale measureP0
is defined), a similar application of 3.d.0.(d) withA(t) =B0(t) yields πt(h) =B0(t)EP0
h/B0(T)| Ft
, t∈[0, T]. (3)
Formula (2) points to the importance of the forward martingale measure PT. The switch between forward martingale measures at different dates S < T is made as follows:
4.a.2. Let 0≤t≤S < T and assume thathisFS-measurable. Then
(a) B(t, S)EPS[B(S, T)h| Ft] =B(t, T)EPT [h| Ft], if h≥0 orh∈L1(PT).
(b) B(t, S)EPS[h|Ft] =B(t, T)EPT[h/B(S, T)| Ft], if h≥0 orh∈L1(PS).
Remark. Looking at formula (a) considerhas a random payoff occurring at time T. The right hand side evaluates this payoff at timetaccording to (2). The left hand side first discounts this payoff back to time S and then evaluates the discounted payoff at time t according to (2). A similar interpretation is possible for formula (b). Consider has a payoff occurring at timeS.
Proof. (b) Applythe symmetric numeraire change formula 3.d.1.(d) to the market B|S and the numerairesA(t) =B(t, S) andC(t) =B(t, T). Noting thatA(S) = 1 the symmetric numeraire change formula 3.d.1.(d)
A(t)EPA[h/A(S)|Ft] =C(t)EPC[h/C(S)|Ft] yieldsB(t, S)EPS[h|Ft] =B(t, T)EPT[h/B(S, T)|Ft].
(a) SimplyreplacehwithB(S, T)hin (b).
4.a.3. Let θ be a self-financing trading strategy such thatVT(θ) =hand A∈ S+ a numeraire such thatξA is a martingale. Thenθreplicateshif and only ifVtA(θ) = Vt(θ)/A(t) is aPA-martingale.
Proof. According to 3.d.0.(b),VtA(θ) is aPA-martingale if and onlyifξ(t)Vt(θ) is a P-martingale.
254 4.c Option to exchange assets.
4.b Forward contracts and forward prices. Af orward contract for deliveryof an assetZinB at dateT with strike priceKobliges the holder of the contract to buy this asset at timeT forK dollars therebyinducing a single cash flowh=ZT −K at time T. The f orward price at time t for deliveryof the asset Z at time T is defined to be that strike price K, which makes the value of the forward contract equal to zero at time t. The payoffh=ZT−K at timeT can be implemented at time zero via the following self-financing trading strategy θ:
At time t buyone unit of the asset Z, sell K units of the zero coupon bond maturing at timeT and hold until timeT.
The Law of One Price suggests that the arbitrage priceπt(h) of this forward contract at time t≤T be defined as πt(h) =Vt(θ) = Zt−KB(t, T). Setting this equal to zero and solving for K, we obtain the forward priceFZ(t, T) at time tfor delivery of the asset Z at timeT as
FZ(t, T) = Zt
B(t, T). (0)
IfZt/B(t, T) is in fact aPT-martingale (rather than onlya local martingale), then (0) can be rewritten as
FZ(t, T) =EPT
ZT|Ft
. (1)
4.c Option to exchange assets. Fix K > 0 and let h be the European option to receive one unit of the asset S1 in exchange forK units of the assetS2 at timeT. The payoffhof this option is given byh= (S1(T)−KS2(T))+. IfS2(t) =B(t, T) is the zero coupon bond maturing at timeT, thenh= (S1(T)−K)+ is the European call on S1 with strike price K exercisable at timeT, that is, the payoff at time T of the right to buyone share of S1forK dollars at timeT.
Let us introduce the exercise set A = {ω ∈ Ω | S1(T)(ω) > KS2(T)(ω)}, that is the set of all states in which our option is exercised at time T. Then h= (S1(T)−KS2(T)) 1A and, assuming that this claim is replicable, its arbitrage price processπt(h) can be written as
πt(h) =ξ(t)−1EP[ξ(T)S1(T)1A| Ft]−Kξ(t)−1EP[ξ(T)S2(T)1A| Ft]. (0) Let us now use the assets S1, S2 themselves as numeraires. Assume that the processes ξS1, ξS2 are martingales and hence the probabilities PS1, PS2 defined.
Using 3.d.0.(d) with h = Sj(T)1A, we can write ξ(t)−1EP[ξ(T)Sj(T)1A| Ft] = Sj(t)EPSj[1A|Ft],j= 1,2, and thus (0) becomes
πt(h) =S1(t)EPS1[1A|Ft]−KS2(t)EPS2[1A|Ft]. (1) To get a more specific formula we need to make further assumptions on our market model. Assume that we are in the setting of the general Black-Scholes model of 3.g with a deflator ξsatisfying
dξ(t)
ξ(t) =−r(t)dt−φ(t)ãdW(t),
whereWtis a Brownian motion with respect to the market probabilityPgenerating the (augmented) filtration (Ft). The assetsSj are assumed to follow the dynamics
dSj(t)
Sj(t) =àj(t)dt+σj(t)ãdW(t), j = 1,2. (2)
Then d(ξSj)(t)
(ξSj)(t) =
σj(t)−φ(t)
ãdW(t), j= 1,2 (3) (see 3.g.eq.(4)). Set Zt=log
S1(t)/S2(t)
and write the exercise setA as A=
ZT > log(K)
. (4)
To evaluate the conditional expectations in (1) we must find the distribution ofZT under the probabilitiesPSj, j= 1,2. From (3) it follows that
d log ξSj
(t) =−1
2αj(t)2dt+αj(t)ãdW(t), (5) where αj(t) =σj(t)−φ(t). Thus
dZ(t) =d log(ξS1)(t)−d log(ξS2)(t)
= 1 2
α2(t)2− α1(t)2
dt+ (α1(t)−α2(t))ãdWt. (6) To determine the conditional expectationEPS
1[1A|Ft] we use III.4.c.0 to determine the dynamics of Z under the measure PS1. Bydefinition of the measure PS1 we have
Mt:= d(PS1|Ft)
d(P|Ft) = (ξS1)(t)
(ξS1)(0), t∈[0, T], and consequently, from (3), dMt
Mt
=α1(t)ãdWt. According to III.4.c.0 it follows that WtS1 = Wt−t
0α1(s)ds is a PS1-Brownian motion on
Ω,F,(Ft), PS1
. Obviously dWt =α1(t)dt+dWtS1. Substituting this into (6) we find that
d Z(t) = 1 2
α22− α12
+ (α1−α2)ãα1
dt+ (α1−α2)ãdWtS1
= 1
2α1−α22dt+ (α1−α2)ãdWtS1,
(7)
with respect to the measure PS1 (i.e., for a PS1-Brownian motion WtS1). We now make the following assumption: The process α1(t)−α2(t) = σ1(t)−σ2(t) is nonstochastic. Then III.6.c.3 can be applied to the dynamics (7) to compute
256 4.c Option to exchange assets.
the conditional expectationEPS
1
1A|Ft
. Because of the special nature of the drift term in (7) the quantitiesm(t, T), Σ(t, T) from III.6.c.3 become
Σ2(t, T) =T
t (α1−α2)(s)2ds and m(t, T) = 12Σ2(t, T) and it follows that
EPS1
1A|Ft
=N(d1) where d1= log
S1(t)/KS2(t)
+12Σ2(t, T)
Σ(t, T) .
A similar computation starting from the equation d Z(t) =−1
2(α1−α2)2dt+ (α1−α2)ãdWtS2 under the measure PS2 yieldsEPS
2[1A|Ft] =N(d2), where d2=log
S1(t)/KS2(t)
−12Σ2(t, T)
Σ(t, T) =d1−Σ(t, T).
Thus we have the option price
πt(h) =S1(t)N(d1)−K S2(t)N(d2),
where d1,d2 are given as above. We can summarize these findings as follows:
4.c.0 Margrabe’s Formula. Assume that the asset prices Sj(t), j = 1,2, follow the dynamics (2) above and that the process σ1−σ2 is deterministic and satisfies Σ2(t, T) = T
t (σ1 −σ2)(s)2ds > 0, for all t ∈ [0, T). Then the option h = S1(T)−KS2(T)+
to exchange assets has arbitrage price process πt(h) =S1(t)N(d1)−KS2(t)N(d2), t∈[0, T),
where N is the standard normal cumulative distribution function and d1,d2 are as above.
Remark. From (6) it follows that the quantityΣ2(t, T) can also be written as Σ2(t, T) =ZTt =
log S1/S2
T t
and is thus a measure of the aggregate (percentage) volatilityof S1/S2 on the interval [t, T] under the market probability P, that is, realized in the market. If S2(t) is the zero coupon bond B(t, T), then h =
S1(T)−K+
is the European call onS1with strike priceK. In this case the quotientS1(t)/S2(t) is thef orward price FS1(t, T) of S1 deliverable at time T and consequentlythe crucial quantity (σ1−σ2)(t)the volatilityof the forward price ofS1and not ofS1 itself.
The reader will note that the term structure of interest rates does not enter our formula unless one of the assets S1, S2 is a zero coupon bond. The reason is that the claim hcan be replicated bya trading strategyθ investing in the assets S1,S2only. Indeed our formula for the priceπt(h) indicates as possible weights
θ1=N(d1) and θ2=−KN(d2)
forS1 andS2respectively. Let us verifythat this strategyis self-financing. Bythe numeraire invariance of the self-financing condition it will suffice to show that θis self-financing in the marketξB, that is, thatd
θã(ξB)
=θãd(ξB). Now in general d
θã(ξB)
=θãd(ξB) +ξBãdθ+dθ, ξB (8) and so we must show that ξBãdθ+dθ, ξB= 0. (9) Since
θã(ξB)
(t) =ξ(t)πt(h) =EP
ξ(T)h| Ft
is a martingale,d
θã(ξB) and θãd(ξB) are known to be the stochastic differentials of local martingales. Thus (8) shows that the bounded variation terms on the left of (9) automaticallycancel. It will thus suffice to show that the local martingale part of the stochastic differential Bãdθ=S1dθ1+S2dθ2vanishes. We mayassume thatK= 1. Then
θ1=N(d1) = 1
√2π
d1(t,S1,S2)
−∞
e−u2/2du, θ2=−N(d2) =− 1
√2π
d2(t,S1,S2)
−∞
e−u2/2du, where
d1(t, s1, s2) =log(s1)−log(s2) +12Σ2(t, T)
Σ(t, T) , and
d2(t, s1, s2) =log(s1)−log(s2)−12Σ2(t, T)
Σ(t, T) , fors1, s2∈R.
WritingX ∼Y ifX−Y is a bounded variation process and using similar notation for stochastic differentials, Ito’s formula yields
dθ1∼∂N(d1)
∂s1
(t, S1, S2)dS1+∂N(d1)
∂s2
(t, S1, S2)dS2, and dθ2∼ −∂N(d2)
∂s1 (t, S1, S2)dS1−∂N(d2)
∂s2 (t, S1, S2)dS2.
(10)
Here ∂N(d1)
∂s1
= (2π)−12e−d21/2∂d1
∂s1
= (2π)−12e−d21/2 1 s1Σ(t, T).
Likewise ∂N(d1)
∂s2 =−(2π)−12e−d21/2 1 s2Σ(t, T).
258 4.c Option to exchange assets.
Similarly
∂N(d2)
∂s1 = (2π)−12e−d22/2 1
s1Σ(t, T), ∂N(d2)
∂s2 =−(2π)−12e−d22/2 1 s2Σ(t, T). Thus, from (10),
(2π)12Σ(t, T)S1dθ1∼e−d21/2dS1−e−d21/2S1
S2
dS2, and (2π)12Σ(t, T)S2dθ2∼ −e−d22/2S2
S1
dS1+e−d22/2dS2, which, upon addition, yields
(2π)12Σ(t, T)
S1dθ1+S2dθ2
∼
e−d21/2−e−d21/2S2
S1
dS1+
e−d22/2−e−d22/2S1
S2
dS2∼
e−d21/2
1−e12(d21−d22)S2 S1
dS1+e−d22/2
1−e12(d22−d21)S1 S2
dS2.
(11)
Noting now that 1
2(d21−d22) = (d1−d2)d1+d2
2 = Σ(t, T)d1+d2
2 =log S1/S2
,
we see that the coefficients ofdS1,dS2on the right of (11) vanish and consequently Bãdθ=S1dθ1+S2dθ2∼0, as desired. Thus θis self-financing. The self-financing propertyof θ is no accident. See 4.e.0 below. Byits verydefinition this strategy replicates the arbitrage price ofhon the entire interval [0, T). The reader will have noticed that the weights θ1, θ2 are not defined at time t =T. Thus Vt(θ) is not defined fort=T. Howeverξ(t)Vt(θ) =EP
ξ(T)h| Ft
is a continuous martingale and the martingale convergence theorem I.7.c.0 shows that ξ(t)Vt(θ)→ξ(T)hand henceVt(θ)→halmost surely, ast↑T. Thus defining the weightsθ1(T),θ2(T) in anyfashion such that VT(θ) =hwill extend the self-financing propertyof θ from the interval [0, T) to all of [0, T].
In the case of a European call onS1(S2(t) =B(t, T)) the replicating strategy invests inS1and the zero coupon bondB(t, T) expiring at timeT. Indeed, if interest rates are stochastic the call cannotbe replicated investing inS1 and the risk-free bond except under rather restrictive assumptions and even then the corresponding portfolio weights have to be chosen differently(see 4.f.3 below).
4.d Valuation of non-path-dependent European options in Gaussian models.
Consider a European option h of the form h = f(S1(T), S2(T), . . . , Sk(T)) exer- cisable at timeT, whereS1(t), S2(t), . . . , Sk(t) are anyassets (possiblyzero coupon bonds). Weassumethat the claimhisattainableand that we are in the setting of the general Black Scholes market of section 3.g. Let Fj(t) =Sj(t)/B(t, T) denote the forward price of the assetSj deliverable at timeT and recall that PT denotes the forward martingale measure at time T. The price process πt(h) ofh can then be computed as
πt(h) =B(t, T)EPT[h|Ft], t∈[0, T]. (0) SinceSj(T) =Fj(T) the optionhcan also be written as
h=f
F1(T), F2(T), . . . , Fk(T)
. (1)
The forward pricesFj(t) arePT-local martingales and hence (in the context of 3.g) follow a driftless dynamics
dFj(t)
Fj(t) =γj(t)ãdWtT, equivalently d(logFj(t)) =−1
2γj(t)2dt+γj(t)ãdWtT
(2)
under the forward martingale measurePT, that is, for some Brownian motionWtT on (Ω,F,(Ft)t∈[0,T], PT). Integration yields
Fj(t) =Fj(0)exp t
0
γj(s)ãdWsT −1 2
t 0
γj(s)2ds
, t∈[0, T]. (3) The use of forward prices and the forward martingale measure eliminates interest rates from explicit consideration. All the necessaryinformation about interest rates is contained in the numeraire asset A(t) = B(t, T). To make (3) useful for the computation of the conditional expectation (0) we make the following
(G)Gaussian assumption: The volatilityprocesses γj are nonstochastic.
The forward price Fj(t) is then a log-Gaussian process with respect to the for- ward martingale measurePT (III.6.d.4). Likewise assumption (G) implies that the deflated processesξSj are log Gaussian processes with respect to the market prob- abilityP (3.g.eq.(4) and III.6.d.4).
To compute the conditional expectation (0) withh as in (1) it will be conve- nient to write the vector
F1(T), F2(T), . . . , Fk(T)
as a function of some vector measurable with respect toFtand another vector independent ofFt. Using (3) for t=t, T we see that
Fj(T) =Fj(t)exp T
t
γj(s)ãdWsT −1 2
T t
γj(s)2ds
=Fj(t)exp
ζj(t, T)−1 2Cjj
, where
(4)
260 4.d Valuation of non-path-dependent European options in Gaussian models.
ζj(t, T) =T
t γj(s)ãdWsT and Cij =T
t γi(s)ãγj(s)ds=log(Fi), log(Fj)Tt. Fixt∈[0, T]. Combining (0), (1) and (4) we obtain
πt(h) =B(t, T)EPT
f
F1(t)eζ1(t,T)−12C11, . . . , Fk(t)eζk(t,T)−12Ckk
| Ft
. (5) Note now that the vector (F1(t), . . . , Fk(t)) is Ft-measurable, while the vector (ζ1(t, T), . . . , ζk(t, T)) is independent of Ft with distribution N(0, C) (III.6.a.2, III.6.c.2). Thus the conditional expectation (5) is computed byintegrating out the vector (ζ1(t, T), . . . , ζk(t, T)) according to its distribution while leaving the vector (F1(t), . . . , Fk(t)) unaffected (I.2.b.11); in short
πt(h) =B(t, T)
Rk
f
F1(t)ex1−12C11, . . . , Fk(t)exk−12Ckk
N(0, C)(dx). (6) Let us now reduce this integral to an integral with respect to the standard multi- normal distribution N(0, I)(dx) =nk(x)dx= (2π)−k2e−12x2dx.
To do this we represent the covariance matrixCin the formC=AA, for some k×kmatrixA, that is we write
Cij=T
t γi(s)ãγj(s)ds=θiãθj,
where θj =cj(A) is the jth column of the matrix A. Especiallythen Cii =θi2. Using II.1.a.7 we can now rewrite (6) as
πt(h) =B(t, T)
Rk
f
F1(t)eθ1ãx−12θ12, . . . , Fk(t)eθkãx−12θk2
nk(x)dx. (7) Replacingxwith−xand noting thate−θjãx−12θj2=nk(x+θj)/nk(x), (7) can be rewritten as
πt(h) =B(t, T)
Rk
f
F1(t)nk(x+θ1)
nk(x) , . . . , Fk(t)nk(x+θk) nk(x)
nk(x)dx. (8) 4.d.0 Theorem. Assume that the assets S1(t), . . . , Sk(t) follow the dynamics (2) and that the Gaussian assumption (G) holds. Then the price process πt(h) of an attainable European claim h=f(S1(T), . . . , Sk(T))maturing at timeT is given by equation (8), where Fj(t) =Sj(t)/B(t, T) is the forward price of the asset Sj, the vectors θj=θj(t, T)∈Rk are chosen so that
Cij =log(Fi), log(Fj)Tt =T
t γi(s)ãγj(s)ds=θiãθj
andnk(x) = (2π)−k2e−12x2 is the standard normal density in Rk.
Homogeneous case. In case the function f = f(s1, s2, . . . , sk) is homogeneous of degree one, formula (8) simplifies as follows:
πt(h) =
Rk
f
S1(t)nk(x+θ1), . . . , Sk(t)nk(x+θk)
dx. (9)
The zero coupon bondB(t, T) drops out and anyexplicit dependence on the rate of interest disappears. We have seen this before in the formula for the price of an option to exchange assets. In the Black Scholes call price formula the rate of interest enters onlysince one of the assets is in fact the zero coupon bond with the same maturityas the call option.
We will apply4.d.0 to several options depending on two assets S1,S2 (k=2).
Recall that N(d) =d
−∞n1(t)dt denotes the (one dimensional) cumulative normal distribution function and that the two dimensional standard normal density n2 satisfiesn2(x) =n2(x1, x2) =n1(x1)n1(x2). The following Lemma will be useful:
4.d.1 Lemma. Letrbe a real number,θ, w∈R2 andG={x∈R2|xãw≤r} ⊆R2.
Then
G
n2(x+θ)dx=N
r+θãw w
.
Proof: Let e1 = (1,0) ∈ R2 and A be the (linear) rotation ofR2 which satisfies Aw=we1. ThenAis a unitarymap, that is,A=A−1. Consider the substitution x=A−1u. Using the rotational invariance of Lebesgue measure, the fact thatAis an isometryand that the standard normal densityn2(x) depends onxonlythrough the normx, it follows that
G
n2(x+θ)dx=
AG
n2(A−1u+θ)du=
AG
n2(A−1(u+Aθ))du
=
AG
n2(u+Aθ)du.
Here u∈AG⇐⇒ Au∈G⇐⇒ (Au)ãw≤r⇐⇒ uã(Aw)≤r⇐⇒ u1w ≤r.
Thus AG={u∈R2 |u1 ≤r/w }. Set u= (u1, u2), Aθ= (α1, α2)∈R2. Then n2(u+Aθ) =n1(u1+α1)n1(u2+α2) and the special nature of the domainAGnow implies that
G
n2(x+θ)dx=
AG
n2(u+Aθ)du
=
u1≤r/wn1(u1+α1)du1
R
n1(u2+α2)du2
.
Since the second integral in this product is equal to one it follows that
G
n2(x+θ)dx=
t≤r/w+α1
n1(t)dt=N r
w+α1
. (10)
Finally α1= (Aθ)ãe1=θã(Ae1) =θã(A−1e1) =θã w w and so 4.d.1 follows from (10).
262 4.d Valuation of non-path-dependent European options in Gaussian models.
Consider now an option h = f(S1, S2) which depends on two assets. Here the dimensionk= 2 and the vectorsθ1, θ2∈R2 satisfy
θ12= T
t
γ1(s)2ds=log(F1)Tt, θ22= T
t
γ2(s)2ds=log(F2)Tt,
and θ1ãθ2=
T t
γ1(s)ãγ2(s)ds=log(F1), log(F2)Tt, from which it follows that Σ2(t, T) := θ1−θ22 = T
t γ1(s)−γ2(s)2ds. Set Y(t) =S1(t)/S2(t) =F1(t)/F2(t) andZ(t) =log(Y(t)). From (2)
dZ(t) =−1 2
γ1(t)2− γ2(t)2 +
γ1(t)−γ2(t)
ãdWtT,
and sodZt=γ1(t)−γ2(t)2dt. Thus Σ2(t, T) =T
t γ1(s)−γ2(s)2ds=ZTt
as in Margrabe’s formula 4.c.0.
4.d.2 Example. Option to exchange assets. The option to receive, at timeT, one unit of asset S1 in exchange forKunits of asset S2has payoff
h=f(S1(T), S2(T)) = (S1(T)−KS2(T))+= (S1(T)−KS2(T))1[S1(T)≥KS2(T)]
which is homogeneous of degree one inS1,S2. Let us see if we can derive Margrabe’s formula 4.c.0 from (9) above. Enteringf(s1, s2) = (s1−Ks2)1[s1≥Ks2]into (9) yields
πt(h) =
G
S1(t)n2(x+θ1)−KS2(t)n2(x+θ2) dx
=S1(t)
G
n2(x+θ1)dx−KS2(t)
G
n2(x+θ2)dx, where G =
x∈R2 |S1(t)n2(x+θ1) ≥KS2(t)n2(x+θ2)
. Thus x∈G if and onlyif
n2(x+θ1) n2(x+θ2)=exp
−1 2
x+θ12− x+θ22
≥ KS2(t) S1(t) , equivalently 1
2
x+θ12− x+θ22
≤log
S1(t) KS2(t)
,
that is xã(θ1−θ2)≤log
S1(t) KS2(t)
−1 2
θ12− θ22 . ThusG={x∈R2|xãw≤r}, wherew=θ1−θ2 and
r=log
S1(t) KS2(t)
−1 2
θ12− θ22 .
From 4.d.1
Gn2(x+θ1)dx=N(d1) and
Gn2(x+θ2)dx=N(d2) and so πt(h) =S1(t)N(d1)−KS2(t)N(d2),
where d1= (r+θ1ãw)
w andd2 = (r+θ2ãw)
w. Recalling from (10) that w=θ1−θ2= Σ(t, T) and observing that
r+θ1ãw=log
S1(t) KS2(t)
+1
2w2 and r+θ2ãw=log
S1(t) KS2(t)
−1 2w2, it follows that
d1=log
S1(t)/KS2(t)
+12Σ2(t, T)
Σ(t, T) and d2=d1−Σ(t, T), as in formula 4.c.0 above. Note that we can also write these quantities as
d1,2= log(F1(t)/F2(t))−log(K)±12
log(F1/F2)T
t
log(F1/F2)T t
.
4.d.3 Example. Digital option. Consider the option h= 1[S1(T)≥KS2(T)]. Since the function f(s1, s2) = 1[s1≥Ks2] satisfiesf(αs1, αs2) = f(s1, s2), 4.d.0 simplifies to
πt(h) =B(t, T)
R2
f
S1(t)n2(x+θ1), S2(t)n2(x+θ2)
n2(x)dx
=B(t, T)
G
n2(x)dx=B(t, T)N r
w
, where G,r andware as in 4.d.2. Setρ=r/w. Asw=
log(F1/F2)Tt and r=log
S1(t) KS2(t)
−1 2
θ12− θ22
=log
F1(t)/F2(t)
−log(K) +1 2
log(F2)Tt − log(F1)Tt
,
we obtainπt(h) =B(t, T)N(ρ), where ρ=log
F1(t)/F2(t)
−log(K) +12
log(F2)Tt − log(F1)Tt
log(F1/F2)Tt
. (11) Comparing this with the formulaπt(h) =B(t, T)EPT(h|Ft) shows thatN(ρ) is the (conditional) exercise probabilityN(ρ) = EPT(h|Ft) =PT
S1(T)≥KS2(T)| Ft
under the forward martingale measurePT. In caseS2(t) =B(t, T) the option payoff becomes h= 1[S1(T)≥K],F2(t) = 1 and consequently(11) simplifies to
ρ= log F1(t)
−log(K)−12log(F1)Tt
log(F1)Tt
.
264 4.d Valuation of non-path-dependent European options in Gaussian models.
4.d.4 Example. Power option. Let nowh=S1(T)λS2(T)à1[S1(T)≥KS2(T)], where λ, à∈R. LetG,randw be as in 4.d.2. As the functionf(s1, s2) =sλ1sà21[s1≥Ks2] satisfiesf(αs1, αs2) =αλ+àf(s1, s2), 4.d.0 simplifies to
πt(h) =B(t, T)1−(λ+à)
R2
f
S1(t)n2(x+θ1), S2(t)n2(x+θ2)
n2(x)1−(λ+à)dx
=B(t, T)1−(λ+à)S1(t)λS2(t)à
G
n2(x+θ1)λn2(x+θ2)àn2(x)1−(λ+à)dx
=B(t, T)F1(t)λF2(t)à
G
n2(x+θ1)λn2(x+θ2)àn2(x)1−(λ+à)dx.
SetU =exp
−12
λ(1−λ)θ21+à(1−à)θ22−2λà θ1ãθ2
. Bystraightforward computation nk(x+θ1)λnk(x+θ2)ànk(x)1−(λ+à) = U nk(x+λθ1+àθ2) and so, using 4.d.1,
πt(h) =U B(t, T)F1(t)λF2(t)à
G
nk(x+λθ1+àθ2)dx
=U B(t, T)F1(t)λF2(t)àN w−1
r+ (λθ1+àθ2)ãw .
Sincew=θ1−θ2 andr=log(F1(t)/F2(t))−log(K)−12(θ12− θ22 we have r+(λθ1+àθ2)ãw=r+λθ12+ (à−λ)θ1ãθ2−àθ22
=log(F1(t)/F2(t))−log(K) + (λ−12)θ12+ (à−λ)θ1ãθ2+ (12−à)θ22. Recallingθ12=log(F1)Tt,θ22=log(F2)Tt,θ1ãθ2=log(F1), log(F2)Tt and w=log(F1/F2)Tt, the option price assumes the form
πt(h) =U B(t, T)F1(t)λF2(t)àN(q), where U =exp
−12
λ(1−λ)log(F1)Tt +à(1−à)log(F2)Tt −2λàlog(F1), log(F2)Tt
and q= log
F1(t)/F2(t)
−log(K) + (λ−12)log(F1)Tt + (12−à)log(F2)Tt
+ (à−λ)log(F1), log(F2)Tt
log(F1/F2)Tt.
Remark. Herelog(Fj)Tt is the aggregate percentage volatilityof the forward price Fj that is left from current time t to the time T of expiryof the option. It is incorrect to use the volatilities of the cash prices Sj instead. The two are the same onlyif the zero coupon bondA(t) =B(t, T) is a bounded variation process.
Note thatlog(Fj)Tt =T
t γj(s)2ds,log(F1), log(F2)Tt =T
t γ1(s)ãγ2(s)dsand log(F1/F2)Tt =log(F1)Tt +log(F2)Tt −2log(F1), log(F2)Tt
=T
t γ1(s)−γ2(s)2ds.
The following notation is frequentlyemployed in the literature: set σj =γj (the numerical volatilityof the forward priceFj(t)) andρij =
γi/γi
ã
γj/γj . The dynamics of the forward prices Fj(t) then becomes
dFj(t) =Fj(t)σj(t)dVjT(t)
for PT-Brownian motionsVjT(t) defined by dVjT(t) =γj(t)−1γj(t)ãdWtT which are one dimensional and correlated by dViT, VjTt=ρij(t)dt(III.6.b.0). Then
log(Fj)Tt = T
t
σ2j(s)ds, log(F1), log(F2)Tt = T
t
(σ1σ2ρ12)(s)ds
and log(F1/F2)Tt = T
t
(σ12+σ22−2σ1σ2ρ12)(s)ds.
4.e Delta hedging. A replicating strategyθ for a European optionhis also called a hedge for h with the interpretation that a seller of h will trade in the market according toθto hedge the random payoff hat timeT.
Assume now thatAis a numeraire asset such thatξAis a martingale. Then the local martingale measure PA is defined. Assume that the market B contains only finitelymanysecurities and thatAis a securityofBand writeB= (A, B1, . . . , Bn).
For a process X(t) we writeXA(t) =X(t)/A(t) as usual.
Now let h be a European option exercisable at time T and write πt(h) = ξ(t)−1EP
ξ(T)h|Ft
,t∈[0, T]. Thenξ(t)πt(h) is aP-martingale and henceπAt(h) a PA-martingale (3.d.0.(b)).
4.e.0 Delta Hedging. Assume that the process πtA(h)can be written in the form πAt(h) =F
t, BA(t)
=F
t, B1A(t), . . . , BnA(t)
, t∈[0, T], (0) for some function F =F(t, b) = F(t, b1, . . . , bn) ∈ C1,2
[0, T]×Rn+
. Let θ(t) = K(t), H1(t), . . . , Hn(t)
be the trading strategy investing in B = (A, B1, . . . , Bn) defined by
Hj(t) = ∂F
∂bj
t, BjA(t)
and K(t) =F
t, BA(t)
−n
j=1Hj(t)BjA(t).
Thenθ is a replicating strategy forh.
Remark. HereF =F(t, b)∈C1,2([0, T]×Rn+) is to be interpreted as in III.3.a.1, that is,F is continuous on [0, T]×Rn+, the partial derivative∂f /∂texists on (0, T)×R+n