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Pricing Theory of Path DependentOptions
Hong-Kun XuDepartment of Applied Mathematics
National Sun Yat-sen University
Kaohsiung 80424, Taiwan
E-mail: [email protected]
Abstract
Path-dependent options have payoffs depending on the history of the under-lying asset’s price, as opposed to the vanilla European and common Americanoptions which have payoff depending only on the underlying asset’s price at theexpiration time (or the time at which the option is exercised).
Path-dependent options are highly customized instruments that are usuallycreated by the over-the-counter desks of major derivatives dealers in order tohelp their clients solve very specific types of business problems. These optionswhich typically have unusual features are the fastest-growing segment of theoptions market. There are many different kinds of path-dependent options inexistence today, and new ones are being created all the time.
The pricing theory for path-dependent options is is highly nontrivial, andvery much complicated indeed. In this lecture, I will first briefly introduce theBlack-Scholes-Merton model for the vanilla European and American options,including all basics which are required to understand the Black-Scholes-Mertontheory, in particular, the fundamental concept of risk-neutral pricing formulaand the change of numeraire technique. I will then move on to talk about thepricing of several types of path-dependent options, such as chooser’s option,power and powered options, digital option, barrier option, lookback option,Asian option, and so on.
1
Part I: The Black-Scholes-Merton Model
2
1 Elements of Stochastic Calculus
1.1 Brownian Motion
Definition 1.1. (Brownian motion) A stochastic process Wtt≥0 defined on a prob-ability space (Ω,F ,P) is said to be a Brownian motion if the following propertieshold.
(i) For any 0 = t0 < t1 < t2 < · · · < tn, the increments
Wt1 −Wt0 ,Wt2 −Wt1 , · · · ,Wtn −Wtn−1
are independent and normally distributed with mean zero and variance ti+1− tifor 0 ≤ i ≤ n− 1:
E[Wti+1−Wti ] = 0,
Var(Wti+1−Wti) = ti+1 − ti.
(ii) Wt is continuous in t ≥ 0.
(iii) W0 = 0.
Proposition 1.2. The Brownian motion Wt possesses the properties:
(i) E[Wt] = 0 for all t.
(ii) Var(Wt) = E[W 2t ] = t for all t; Wt ∼ N(0, t), where N(µ, σ2) is the distribution
function of a normal random variable X with mean µ and variance σ2; that is,
FX(x) = PX ≤ x =
∫ x
−∞
1√2πσ2
exp
(−(v − µ)2
2σ2
)dv, x ∈ R.
In particular, N(0, 1) distributed random variable is called standard normallydistributed and its distribution function is denoted by N(x); that is,
N(x) ==
∫ x
−∞
1√2π
e−12v2
dv, x ∈ R.
(iii) E[WsWt] = mins, t for all s, t ≥ 0.
3
1.2 Martingale
Definition 1.3. (Filtration) Let (Ω,F ,P) be a probability space. Suppose, for eacht ≥ 0, Ft is a σ-algebra of subsets of Ω. We say that F = Ft is a filtration if
• Ft is sub-σ-algebra of F ;
• Fs ⊂ Ft for all 0 ≤ s < t.
A probability space with a filtration is said to be a filtered probability space.
The filtration with which shall be involved is generated by a given Brownianmotion Wtt≥0. In other words, for each t ≥ 0, we have Ft = σ(Wu : 0 ≤ u ≤t ∪ N ), where N is the set of P-null sets of Ω.
Definition 1.4. (Martingale) Let Xtt≥0 be a stochastic process on a filtered prob-ability space (Ω,F ,P, Ft). Assume E[|Xt|] < ∞ for each t ≥ 0. We say that
• X is a martingale relative to the filtration Ft if, for each 0 ≤ s ≤ t,
E[Xt|Fs] = Xs.
• X is a supermartingale relative to the filtration Ft if, for each 0 ≤ s ≤ t,
E[Xt|Fs] ≤ Xs.
• X is a submartingale relative to the filtration Ft if, for each 0 ≤ s ≤ t,
E[Xt|Fs] ≥ Xs.
Proposition 1.5. Let Wt be a Brownian motion on the filtered probability space(Ω,P,F ,F), where F is the natural filtration for Wt. Then
(i) Wt is a martingale.
(ii) W 2t − t is a martingale.
(iii)exp(σWt − 1
2σ2t)
is a martingale.
(iv) Wt accumulates quadratic variations linearly; more precisely, we have [W,W ]t =t (a.s.) for t > 0. This fact is rewritten as dWtdWt = dt. Similarly, we havedWtdt = 0 and dtdWt = 0.
In the computation of option pricing, the following independence lemma is oftenutilized.
4
Lemma 1.6. (Independence lemma.) Let (Ω,F ,P) be a probability space and G asub-σ-algebra of F . Suppose the random variables X1, · · · , Xm are G-measurable andthe random variables Xm+1, · · · , Xm+n are independent of G. Let f(x1, · · · , xm+n)be a real-valued function of m + n variables. Define a function g(x1, · · · , xm) of mvariables by
g(x1, · · · , xm) := E[f(x1, · · · , xm, Xm+1, · · · , Xm+n)].
ThenE[f(X1, · · · , Xm, · · · , Xm+n)|G] = g(X1, · · · , Xm).
5
1.3 Ito’s Calculus
Asset price process is modeled by stochastic processes. So we need stochastic calculus.In particular, we need Ito’s formula.
Lemma 1.7. (Ito’s Formula) Let f(t, x) be a real-valued function such that f is C1
in t and C2 in x. Let Xt is an Ito stochastic process of the form
dXt = µ(t,Xt)dt + σ(t,Xt)dWt,
where µ(t, x) and σ(t, x) are given functions of (t, x). Then
df(t, Xt) =∂f
∂t(t,Xt)dt +
∂f
∂x(t,Xt)dXt +
1
2
∂2f
∂x2(t,Xt)dXtdXt
=
(∂f
∂t(t,Xt) + µ(t,Xt)
∂f
∂x(t,Xt) +
1
2σ2(t,Xt)
∂2f
∂x2(t,Xt)
)dt
+σ(t,Xt)∂f
∂x(t,Xt)dWt.
We also need the following product formula (or integration by parts).
Proposition 1.8. Let Xt and Yt be two Ito processes
dXt = Ktdt + HtdWt, dYt = K ′tdt + H ′
tdWt.
Thend(XtYt) = (dXt)Yt + Xt(dYt) + dXtdYt,
where dXtdYt = HtH′tdt.
6
2 The Black-Scholes-Merton Model
2.1 Geometric Brownian Motion
In the Black-Scholes-Merton model, the market consists of two tradable assets. Thefirst asset is riskless and is called a bond or money market account, denoted by B.Let Bt denote its value at time t. We assume that the evolution of Bt follows thedeterministic differential equation
dBt = rBtdt, (2.1)
where r is a positive constant interpreted as the risk-free rate of interest. We alwaysassume B0 = 1 so that the solution of the ODE (2.2) is
Bt = ert. (2.2)
The second asset is a risky one which is referred to as stock. The value per unit(or price) at time t is denoted St. We assume that St follows a geometric Brownianmotion (GBM) driven by the stochastic differential equation (SDE):
dSt = St(µdt + σdWt), (2.3)
where µ and σ are two constants and where Wt is a (standard) Brownian motion.
Proposition 2.1. The solution to the SDE (2.3) is given by
St = S0 exp
(σWt +
(µ− 1
2σ2
)t
). (2.4)
Remark 2.2. The model given by (2.4) is also called lognormal because log St isnormally distributed. (The log is understood to have the natural base.) Indeed, wehave
log St = log S0 +
(µ− 1
2σ2
)t + σWt. (2.5)
Thus log St is normally distributed with mean log S0 +(µ− 1
2σ2
)t and variance σ2t.
Let F = Ftt≥0 be the natural filtration generated by a Brownian motion Wtt≥0;that is, Ft = σ(Wu0≤u≤t ∪N ), with N being the collection of all P-null sets.
7
2.2 European Options
A European option is an agreement which gives the holder the right, but not theobligation, to buy (or sell) one share of the underlying asset for a specified price (calledthe strike price, denoted by K) and at a specified time (known as the expiration time(or expiry, or maturity) denoted by T ). An option to buy is a call and an option tosell is a put.
Let St be the time-t value of the underlying asset (referred to as stock in a generalmeaning). Obviously, at the expiration time T , the payoff of a call is
(ST −K)+ =
ST −K, if ST > K0, if ST ≤ K.
Similarly, the payoff of a put is given by
(K − ST )+ =
0, if ST ≥ KK − ST , if ST < K.
An option has a value at any time t prior to the expiration time T . How to determineit ‘reasonably’ is one of the fundamental issues in quantitative finance, in particular,in mathematical finance. By ‘reasonably’ we mean to price the option in an arbitrage-free way.
8
2.3 Replicating Trading Strategy
Let F = Ftt≥0 be the filtration generated by the Brownian motion Wt0≥0 whichdrives the stock price process St as depicted in GBM.
Definition 2.3. A trading strategy is a process ϕ = ϕt, where ϕt = (αt, βt),0 ≤ t ≤ T , which is adapted to the filtration F = Ft, where αt and βt are realnumbers which denote the units of the bond B and the number of shares of the assetS. The value process at time t of ϕ is defined as
V ϕt = αtBt + βtSt.
We assume that ∫ T
0
|αt|dt < ∞,
∫ T
0
β2t dt < ∞ (2.6)
almost surely, so that the integrals
∫ T
0
αtdBt =
∫ T
0
αtrertdt
and ∫ T
0
βtdSt =
∫ T
0
(βtStµ)dt +
∫ T
0
σβtStdWt
are well-defined.
Definition 2.4. A trading strategy ϕ is said to be self-financing if the followingcondition is satisfied:
dV ϕt = αtdBt + βtdSt.
The integral form is
V ϕt = α0B0 + β0S0 +
∫ t
0
αudBu +
∫ t
0
βudSu.
Recall that the discounted price and discounted value process:
St = e−rtSt, V ϕt = e−rtV ϕ
t .
Proposition 2.5. A trading strategy ϕ = (αt, βt) is self-financing if and only if itsatisfies the condition:
dV ϕt = βtdSt. (2.7)
The integral form is
V ϕt = V ϕ
0 +
∫ t
0
βudSu a.s.
9
Proof. By Ito′s Lemma (product rule), we have
dV ϕt = d(e−rt)V ϕ
t + e−rtdV ϕt + d(e−rt)dV ϕ
t
= −re−rtV ϕt dt + e−rtdV ϕ
t
= −rV ϕt dt + e−rtdV ϕ
t
and
dSt = d(e−rt)St + e−rtdSt + d(e−rt)dSt
= −re−rtStdt + e−rtdSt
= −rStdt + e−rtdSt.
Similarly,dSt = rStdt + ertdSt.
Note that Bt = ert. Now we see that ϕ is self-financing if and only if dV ϕt = αtdBt +
βtdSt. This is equivalent to
dV ϕt = −re−rtV ϕ
t dt + e−rtdV ϕt
= −re−rt(αtert + βtSt)dt + e−rt(αtdBt + βtdSt)
= −re−rtβtStdt + e−rtβtdSt
= βt(−rStdt + e−rtdSt)
= βtdSt.
Recall that a contingent claim is a random payment made out at a future timeT . To our purpose, a contingent claim is an FT -measurable and nonnegative randomvariable X. For example, X = (ST − K)+ for a European call option, and X =(K − ST )+ for a European put.
Definition 2.6. We say that a trading strategy ϕ replicates (or hedges) a contingentclaim X if X = V ϕ
T . We say that a market model is complete if every contingentclaim can be replicated by a self-financing trading strategy. In other words, if X isa contingent claim, then there exists a self-financing trading strategy ϕ such thatX = V ϕ
T .
We will see that the Black-Scholes-Merton model is complete.
10
2.4 The Black-Scholes-Merton Partial Differential Equation
Let X be a contingent claim and let V be its value process with time-t value givenby Vt = v(t, S), where v(t, x) is a (deterministic) function of (t, x) (referred to as thevalue function of X), and where S is the price at time t of the underlying stock St
which follows the GBM. Use Ito’s formula to get
dVt =
(v′t + µStv
′x +
1
2σ2S2
t v′′xx
)dt + σStv
′xdWt (2.8)
where all partial derivatives are evaluated at (t, St). The differential of the discounted
value process Vt := e−rtVt is
dVt = e−rt
(−rv + v′t + µStv
′x +
1
2σ2S2
t v′′xx
)dt
+σe−rtStvxdWt. (2.9)
Suppose now ϕ = (α, β) is a self-financing trading strategy and V ϕt is its discounted
value process. Then we have
dV ϕt = βtdSt = βte
−rt(−rStdt + dSt)
= βte−rtSt((µ− r)dt + σdWt) (2.10)
In order for the portfolio ϕ to replicate the claim X, we must have e−rtVt = e−rtV ϕt .
In particular, dVt = dV ϕt . Therefore, comparing the dWt terms in (2.9) and (2.10)
yields thatβt = vx(t, St)
which then implies from the dt terms that the value function v(t, x) satisfies thefollowing partial differential equation:
v′t(t, x) + rxv′x(t, x) +1
2σ2x2v′′xx(t, x) = rv(t, x) (2.11)
for 0 ≤ t ≤ T and x ≥ 0. Equation (2.11) is called the Black-Scholes-Merton partialdifferential equation.
11
2.5 Change of Probability Measures
In option pricing, we must change probability measures from one to another. This isbecause the physical probability measure P plays no role in the no-arbitrage pricingtheory.
Definition 2.7. Let (Ω,P,F) be a probability space. A probability measure Q on(Ω,F) is said to be absolutely continuous with respect P if
A ∈ F , P(A) = 0 ⇒ Q(A) = 0.
Q is said to be equivalent to P if
A ∈ F , P(A) = 0 ⇔ Q(A) = 0.
(Note: We use E to reserve the notation for the expectation under the (physical)probability measure P.)
Theorem 2.8. Let (Ω,P,F) be a probability space and let Q be a probability measureon (Ω,F). Then Q is absolutely continuous with respect P if and only if there existsa nonnegative random variable Z such that
∀A ∈ F , Q(A) =
∫
A
Z(ω)dP(ω). (2.12)
Note: Z is called the density (or the Radon-Nikodyn derivative) of Q with respectto P, written Z = dQ
dP . Also, Q is equivalent to P iff P(Z > 0) = 1; in this case, theRadon-Nikodyn derivative of P with respect to Q is Z−1.
The Randon-Nikodym derivative process Zt is defined as
Zt = E[Z|Ft], 0 ≤ t ≤ T, (2.13)
written asdQdP
∣∣∣∣Ft
= Zt.
Then Z = ZT and Zt is a P-martingale. As a matter of fact, we have for 0 ≤ s < t ≤ T ,
E[Zt|Fs] = E[E[Z|Ft]|Fs]
= E[Z|Fs]
= Zs.
Lemma 2.9. Let Zt be the Randon-Nikodym derivative process of Q with respect toP.
(i) If 0 ≤ t ≤ T and if X is Ft-measurable, then EQ[X] = E[XZt].
(ii) If 0 ≤ s < t ≤ T and if X is Ft-measurable, then EQ[X|Fs] = 1ZsE[XZt|Fs].
12
2.6 The Girsanov Theorem
We have a Brownian motion Wt under the physical probability measure P. However,we need to change to another equivalent probability measure Q, and want to know ifWt is kept to be a Brownian motion under the new measure Q?
Theorem 2.10. Let (Ω,P,F ,FW ) be a filtered probability space, where FW = Ftis the natural filtration for a Brownian motion Wt.
Let θt0≤t≤T be an adapted process satisfying the condition
E[exp
(1
2
∫ T
0
θ2sds
)]< ∞.
Define a process Lt by
Lt := exp
(−
∫ t
0
θsdWs − 1
2
∫ t
0
θ2sds
), 0 ≤ t ≤ T.
Define a new probability measure P(L) with density LT ; that is,
P(L)(A) =
∫
A
LT (ω)dP(ω), A ∈ F .
Then the shifted process given by
W(L)t := Wt +
∫ t
0
θsds, 0 ≤ t ≤ T
is a P(L)-Brownian motion.
In the particular case where θt ≡ θ, a constant, we have that
W θt := Wt + θt
is a Brownian motion under the equivalent probability measure Pθ defined by
Pθ(A) =
∫
A
LT (ω)dP(ω), A ∈ F
where
LT = exp
(−θWT − 1
2θ2T
).
13
2.7 Change of Numeraire
2.7.1 Change of Numeraire
Change of probability measures is often carried out by change of numeraires.A numeraire is the unit of account which is used to denominate other assets.
The most often numeraire is perhaps the currency of a country. However, change ofnumeraire is necessary due to fiance considerations. A model can be complicated orsimple depending on the choice of the numeraire for the method.
In principle, any attainable and positively priced asset N can be taken as anumeraire and denominate all other assets in terms of the chosen numeraire. Weassume that a numeraire does not pay dividends. A numeraire induces an equivalentmartingale measure.
Definition 2.11. A probability measure P∗, equivalent to the physical measure P, issaid to be an equivalent martingale measure (EMM) with respect to a numeraire Nif every underlying asset S normalized by N is a P∗-martingale:
St
Nt
= E∗[
Su
Nu
∣∣∣∣Ft
], 0 ≤ t ≤ u.
Suppose P(N) and P(M) are martingale measures with respect to numeraires Nand M , respectively. Then the relation between the change of EMMs P(N) and P(M)
is given by
dP(M) =MT /M0
NT /N0
dP(N).
Hence,
E(M)[X|Ft] = E(N)
[MT /Mt
NT /Nt
X
∣∣∣∣Ft
]
where X ∈ FT .In terms of numeraires, no-arbitrageness and completeness can be described as
follows.
Theorem 2.12. (i) A market is arbitrage-free if and only if, for each numeraire,there is an equivalent martingale measure.
(ii) An arbitrage-free market is complete if and only of there exists exactly oneequivalent martingale measure corresponding to every numeraire.
It can be proved that in a complete market, changes of equivalent measures areactually through changes of numeraires.
We consider the following two numeraires:
• money market account (MMA) B,
• underlying risky asset S.
14
2.7.2 MMA as numeraire
First let us choose the money market account B as a numeraire; thus we use B tonormalize other assets. In this case we introduce the discounted price of the riskyasset S as follows:
St =St
Bt
= e−rtSt.
Recall under the physical probability measure P, St follows GBM:
dSt = St(µdt + σdWt) (2.14)
where Wt is a P-Brownian motion.Using Ito’s Lemma we get
dSt = σSt(dWt + θdt)
where
θ =µ− r
σ.
LetWt = Wt + θt. (2.15)
Then we getdSt = σStdWt. (2.16)
The solution isSt = S0e
σfWt− 12σ2t. (2.17)
Returning to St we obtain
St = S0eσfWt+(r− 1
2σ2)t. (2.18)
This is the solution of the SDE
dSt = St(rdt + σdWt).
We would ask if Wt is also a Brownian motion under an appropriate probabilitymeasure. The answer is yes, due to the following Girsanov’s theorem.
Corollary 2.13. Wt defined by (2.15) is a P-Brownian motion, where P is definedby
P(A) =
∫
A
ZdP,
withZ = e−θWT− 1
2θ2T .
It follows from (2.17) that
15
Corollary 2.14. The discounted asset price process St is a P-martingale:
E[Su|Ft] = St, 0 ≤ t ≤ u.
Therefore, P is an EMM.
Definition 2.15. The EMM P is called the risk-neutral probability measure, due tothe specific choice of MMA as the numeraire.
16
2.7.3 Underling Risky Asset as numeraire
We now look at the case where we choose the underlying asset S as the numeraireand use it to normalize other assets (i.e., the MMA B):
Bt
St
=ert
St
.
Since, under the physical probability measure P, the risky asset S follows theGBM:
dSt = St(µdt + σdWt) (2.19)
where Wt is a P-Browian motion, upon setting
f(t, x) =ert
x, t ≥ 0, x > 0
and using Ito’s Lemma, we obtain
d
(Bt
St
)= df(t, St)
= ftdt + fxdSt +1
2fxxdStdSt
=
((r + σ2)dt− 1
St
dSt
)f(t, St)
= ((r − µ + σ2)dt− σdWt)f(t, St)
= −σf(t, St)(dWt + θdt).
Here
θ =µ− r − σ2
σ.
By Girsanov’s theorem, we see that
Wt := Wt + θt
is a standard Brownian motion under the equivalent measure P which is defined by
P(A) =
∫
A
ZdP (2.20)
where the density Z is given by
Z = e−θWT− 12θ2T .
Moreover,
d
(Bt
St
)= −σ
(Bt
St
)dWt.
17
This shows that the normalized price process
Bt
St
is a P-martingale, and therefore, P is an EMM.It is also not hard to find that under the EMM P, the asset S follows the GBM:
dSt = St((r + σ2)dt + σdWt). (2.21)
Thus, under P,
St = S0eσWt+(r+ 1
2σ2)t.
18
2.8 The Representation of Brownian Motion
Theorem 2.16. Let Xt0≤t≤T be a square-integrable martingale with respect to thenatural filtration F = F generated by a Brownian motion Wt0≤t≤T . Then thereexists an adapted process Ht0≤t≤T such that
E[∫ T
0
H2t dt
]< ∞
and for all t ∈ [0, T ],
Xt = X0 +
∫ t
0
HsdWs (a.s.).
19
2.9 The Risk-Neutral Pricing Formula
Let Wt0≤t≤T be a Brownian motion on a probability space (Ω,F ,P).
Theorem 2.17. In the Black-Scholes-Merton model, any contingent claim X whichis a nonnegative and FT -measurable random variable in the space L2(Ω,F , P), isreplicable. Moreover, its value at time t is given by the following risk-neutral pricingformula:
Vt = E[e−r(T−t)X|Ft
]. (2.22)
Proof. First we prove that X is replicable.Recall that under the risk-neutral probability measure P, the discounted stock
price St = e−rtSt is a martingale.Let now Vt be the time-t value of the claim and define a P-martingale Mt by
Mt = E[e−rT X|Ft
].
The the natural filtration FW for Wt is also the the natural filtration for Wt. Sowe can use the Representation of Brownian martingale (Theorem 2.16), there existsan adapted process Kt such that
E[∫ T
0
K2t dt
]< ∞
and, for all t ∈ [0, T ],
Mt = M0 +
∫ t
0
KsdWs.
Define a trading strategy ϕ = (αt, βt) by
βt =Kt
σSt
, αt = Mt − βtSt.
Then ϕ is self-financing; indeed, the discounted value process of ϕ satisfies
V ϕt = αt + βtSt = Mt
= M0 +
∫ t
0
KsdWs
= M0 +
∫ t
0
βsσSsdWs
= M0 +
∫ t
0
βsdSs.
By Proposition 2.5, ϕ is self-financing. It is straightforward that ϕ replicates theclaim X since
V ϕT = erT MT = erT E
[e−rT X|FT
]= X.
20
Next we prove the risk-neutral pricing formula (2.22).From the above proof we have seen that the discounted value process V ϕ
t is
actually a P-martingale. It turns out that, for all t ∈ [0, T ],
V ϕt = E[V ϕ
T |Ft].
Consequently,V ϕ
t = E[e−rT X|Ft].
This shows that the time-t value of the claim is independent of the choice of thereplicating strategy ϕ, and by removing the discount, we obtain
Vt = E[e−r(T−t)X|Ft].
Corollary 2.18. If the contingent claim X is of the form X = h(ST ), then
Vt = F (t, St),
where, with τ = T − t,
F (t, x) = e−rτ
∫ ∞
−∞
1√2π
h
(x exp
((r − 1
2σ2)τ + σ
√τy
))e−
12y2
dy. (2.23)
Proof. By the risk-neutral pricing formula (2.22), we have
Vt = E[e−r(T−t)h(ST )|Ft]
= e−r(T−t)E[h
(St exp
(σ(WT − Wt) + (r − 1
2σ2)(T − t)
)) ∣∣∣∣Ft
].
However, since WT − Wt =√
τZ, with τ = T − t and Z ∼ N(0, 1) independent of Ft.By the Independence Lemma, we find that
Vt = F (t, St)
with F (t, x) given by
F (t, x) = e−r(T−t)E[h
(x exp
(σ√
τZ + (r − 1
2σ2)(T − t)
))]. (2.24)
Finally, noticing Z ∼ N(0, 1), we get (2.23) from (2.24).
In particular, if the claim is the payoff of a European call (i.e., X = h(ST ) =(ST −K)+), we get the following famous Black-Scholes formula.
21
Theorem 2.19. The time-t value of a European call option with payoff (ST −K)+
is given by the Black-Scholes formula:
Ct = StN(d1(τ, St))−Ke−rτN(d2(τ, St)), (2.25)
where τ = T − t,
d1(τ, x) =log(x/K) + (r + 1
2σ2)τ
σ√
τ,
d2(τ, x) = d1(τ, x)− σ√
τ =log(x/K) + (r − 1
2σ2)τ
σ√
τ.
Similarly, the time-t value of a European put option with payoff (K − ST )+ is givenby the formula:
Pt = Ke−rτN(−d2(τ, St))− StN(−d1(τ, St)). (2.26)
Proof. Since the derivative is a call, we have h(x) = (x − K)+. It turns out thatCt = F (t, St), where by (2.23),
F (t, x) = e−rτ
∫ ∞
−∞
1√2π
(x exp
((r − 1
2σ2)τ + σ
√τy
)−K
)+
e−12y2
dy. (2.27)
It is easily seen that the integrand is positive if and only if
x > − log(x/K) + (r − 12σ2)τ
σ√
τ= −d2(t, x).
It follows that
F (t, x) = e−rτ
∫ ∞
−d2(t,x)
1√2π
(x exp
((r − 1
2σ2)τ + σ
√τy
)−K
)e−
12y2
dy
= xe−rτ
∫ ∞
−d2(t,x)
1√2π
exp
((r − 1
2σ2)τ + σ
√τy − 1
2y2
)dy
−Ke−rτ
∫ ∞
−d2(t,x)
1√2π
e−12y2
dy
≡ xe−rτI1 −Ke−rτI2. (2.28)
Using the substitution v = −y, we immediately get
I2 =
∫ d2(t,x)
−∞
1√2π
e−12v2
dv = N(d2(t, x)). (2.29)
As for the integral I1, we have
I1 = erτ
∫ ∞
−d2(t,x)
1√2π
exp
(−1
2(y − σ
√τ)2
)dy.
22
Using the substitution v = y − σ√
τ , we get
I1 = erτ
∫ ∞
−d2(t,x)−σ√
τ
1√2π
exp
(−1
2v2
)dv
= erτ
∫ d1(t,x)
−∞
1√2π
e−12v2
dv
= erτN(d1(t, x)) (2.30)
Combining (2.28), (2.29) and (8.17), we get
F (t, x) = xN(d1(τ, x))−Ke−rτN(d2(τ, x)).
Therefore, Ct = F (t, St) is given by the BS formula (2.25).
Proposition 2.20. (Put-call parity)
Pt + St − Ct = e−r(T−t)K.
23
2.9.1 The Fundamental Theorems of Asset Pricing
Theorem 2.21. (The First Fundamental Theorems of Asset Pricing) If a marketmodel has a risk-neutral probability measure, then it does not have arbitrage.
Theorem 2.22. (The Second Fundamental Theorems of Asset Pricing) A marketmodel, with a risk-neutral probability measure, is complete if and only if the risk-neutral probability measure is unique.
The BSM model is complete.
24
2.10 Martingale Valuation
In a complete market (e.g., the BSM model), every contingent claim can be replicatedby a self-financing trading strategy and thus its values can uniquely been determinedvia martingale valuation.
Theorem 2.23. Let P(N) be the martingale measure with respect to an numeraire N .Then for any contingent claim X, we have that the value at time t of this claim is
Vt = EP(N)
[Nt
NT
X
∣∣∣∣Ft
]. (2.31)
Note that the risk-neutral valuation method is a martingale valuation methodcorresponding to the choice of the MMA as the numeraire.
25
2.11 Derivation of Black-Scholes Formula via Change of Numeraire
Introducing the indicator
IA(x) =
1, if x ∈ A,0, if x 6∈ A,
we can rewrite risk-neutral pricing formula (2.22) as
Ct = e−r(T−t)(E[(ST −K)+|Ft]
)
= e−r(T−t)(E[IST >KST |Ft]−KPST > K|Ft
). (2.32)
Under the risk-neutral probability measure P, we have by (2.18),
ST = St exp
(σ(WT − Wt) + (r − 1
2σ2)(T − t)
).
Let
Y = −WT − Wt√T − t
.
Then Y is independent of Ft and
Y ∼ N(0, 1).
Rewrite ST as, with τ = T − t,
ST = St exp
(−σ√
τY + (r − 1
2σ2)τ
).
Solving the inequality
ST = St exp
(−σ√
τY + (r − 1
2σ2)τ
)> K
we get
Y <1
σ√
τ
[log
St
K+
(r − 1
2σ2
)τ
]= d2(τ, St).
It follows that
PST > K|Ft = PY < d2(τ, St) = N(d2(τ, St)). (2.33)
In order to compute the first term in Ct (i.e., the expected value E[IST >KST |Ft]),we use the change of numeraire technique: change the numeraire from the MMA tothe asset S; hence, change the EMMs from P = P(B) to P = P(S) given by
dP = ZdP,
26
where
Z =SN/S0
BN/B0
= eσWT− 12σ2T
since, under P, ST = S0eσfWT +(r− 1
2σ2)T . Moreover, we have
E[IST >KST |Ft] = E[
BT /Bt
ST /St
IST >KST
∣∣∣∣Ft
]
= er(T−t)StE[IST >K
∣∣Ft
]
= er(T−t)StP [ST > K| Ft] .
Since, under P, St satisfies the SDE:
dSt = St((r + σ2)dt + σdWt), (2.34)
whereWt = Wt − σt
is a P-Brownian motion. Note that under P,
ST = Steσ(cWT−cWt)+(r+ 1
2σ2)(T−t)
= Ste−σ√
τY +(r+ 12σ2)τ
where
Y = −WT − Wt
σ√
τ∼ N(0, 1)
is independent of Ft.The argument for computing PST > K|Ft works for computing PST > K|Ft.
However, now we have (under P),
ST > K ⇔ Y <log St
K+ (r + 1
2σ2)τ
σ√
τ= d1(τ, St).
Consequently,E[IST >KST |Ft] = er(T−t)Std1(τ, St). (2.35)
Substituting (2.33) and (2.35) into (2.32), we arrive at the Black-Scholes formula: fora European call option:
Ct = StN(d1(τ, St))− e−r(T−t)KN(d2(τ, St)).
27
2.12 American Options
Definition 2.24. An American option gives the holder the right, but not the obliga-tion, to buy (or sell) a share of stock S at any time up to and including the expirationtime T for the predetermined strike price of K. That is, if the holder chooses to ex-ercise the option at time t ∈ [0, T ], then he or she receives the amount (St −K)+ fora call (or (K − St)
+ for a put) option.
We can extend the above definition of American options to a more general case.Recall that F = Ft is the natural filtration generated by the Brownian motionwhich drives the stock price process St.Definition 2.25. An American derivative security is a nonnegative stochastic processG = Gt0≤t≤T defined on the filtered probability space (Ω,P,F ,F) such that, foreach t ∈ [0, T ], Gt is Ft-measurable (i.e., G is adapted). If the holder chooses toexercise the security at time t, she/he receives the amount Gt. For an American call,we have Gt = (St −K)+ and for an American put, we have Gt = (K − St)
+, for allt ∈ [0, T ].
To price American options, we need the concept of stopping times.
Definition 2.26. A stopping time τ is a random variable from Ω to [0,∞] satisfyingthe property that, for every t ≥ 0,
τ ≤ t := ω ∈ Ω : τ(ω) ≤ t ∈ Ft.
Example 2.27. (First passage time for a continuous process) Let Xt be an adaptedprocess with continuous paths, let m ∈ R, and set
τm = inft ≥ 0 : Xt = m.
That is, τm is the first time that the process Xt reaches the level m. (If there is notime t such that Xt = m, then τm = ∞.) Then τm is a stopping time.
Definition 2.28. Given a stopping time τ . The stopped process of a process Xtis defined as the process
Xt∧τ (ω) = Xt∧τ(ω)(ω).
Here t ∧ τ = mint, τ.Theorem 2.29. (Optional sampling)
• A martingale stopped at a stopping time is a martingale.
• A supermartingale stopped at a stopping time is a supermartingale.
• A submartingale stopped at a stopping time is a submartingale.
28
Definition 2.30. The price at time t of the American put option is defined to be
v(t, x) := maxτ∈Tt,T
E[e−r(τ−t)(K − Sτ )+|St = x]. (2.36)
Here Tt,T is family of stopping times taken values in [t, T ] ∪ ∞.It is known that if the underlying asset S does not pay dividends, the an American
call has the same value as its European counterpart. It is also known that the priceof an American put is characterized as follows.
Theorem 2.31. In the Black-Scholes environment, we have
(i) The discounted value process
e−ruv(u, Su), t ≤ u ≤ T
is a supermartingale.
(ii) The stopped process
e−r(u∧τ∗)v(u ∧ τ∗, Su∧τ∗), t ≤ u ≤ T
is a martingale.
There is no closed-form solution for an American put option, in general. However,in the case of an perpetual put, there is indeed a closed-form solution. A perpetualAmerican put (PAP) has no expiration time (or the expiration time is ∞).
Definition 2.32. The price of the perpetual American put is
v(x) = maxτ∈T
E[e−rτ (K − Sτ )
+]
(2.37)
where x = S0 is the initial stock price and T is the family of stopping times takenvalues in [0, T ] ∪ ∞. (If τ = ∞, e−rτ (K − Sτ )
+ is interpreted as zero.)
Proposition 2.33. The price function v(x) of PAP is given by
v(x) =
K − x, if 0 ≤ x ≤ L∗,
(K − L∗)(
xL∗
)− 2rσ2
, if x ≥ L∗
where L∗ = 2r+σ2
σ2 K.
29
Part II: Pricing Theory for Exotic and PathDependent Options
The European and American options which we have investigated in part I havea common feature that the payoffs depend only on the underlying asset price atthe time that the option is exercised. These options are known as vanilla (or plainvanilla) options. Other options are called exotic. Exotic options are usually tradedover-the-counter (OTC).
Path-dependent options are those options whose payoffs depend on the history ofunderlying asset’s prices, namely, on the paths that the asset reaches the price whenexercised. Exotic options are often path-dependent, but can be path-independent.
Exotic options are highly customized instruments that are usually created by theover-the-counter desks of major derivatives dealers in order to help their clients solvevery specific types of business problems. These options which typically have unusualfeatures are the fastest-growing segment of the options market. There are manydifferent kinds of exotic options in existence today, and new ones are being createdall the time. This article will examine a number of them and how they’re used totraverse distinct financial obstacles.
Digital Options
Digital options (also known as binary or cash-or-nothing options) have a fixedpayoff if the option ends up being in the money at expiration, regardless of the extentto which it is in the money. For example, let’s assume a digital option is writtenthat pays $1,000,000 if the price of ABC, Inc.’s, stock is above $75 per share at theend of one year. It doesn’t matter if the stock is at 75.125 or $220 when the optionexpires, the payoff will still be $1,000,000. The digital option’s value is computedas the payoff multiplied by the probability that the option will be in the money atexpiration, discounted back to today.
Digital options have several business applications. Perhaps a company has anexecutive bonus program that awards its senior management $5,000,000 if its stockrises by fifty percent over the next two years. The company could hedge the cost ofthe compensation program by purchasing a binary option with a $5,000,000 payoff.For this option, the payoff will either be $5,000,000 if the option ends up in the moneyor nothing if it doesn’t.
Power and powered options
These are options for which the payoffs depend on the underlying asset raised tocertain powers.
30
Chooser Options
With a chooser option, the long is provided the opportunity to decide whetherthe option is to be a call or a put at some time after the option is actually purchased.The ’choose date’ can be any date after the option is created, up to and includingits expiration date. Chooser options are used as volatility plays and are purchasedwhen the buyer expects the volatility of the underlying security to increase, but isuncertain as to the direction.
Compound Options
Compound options provide their owners with the right to buy or sell anotheroption. These options create positions with greater leverage than traditional options.There are four basic types of compound options:
• Call on call the right to buy a call.
• Put on call the right to sell a call.
• Call on put the right to buy a put.
• Put on put the right to sell a put.
There is only one common business application that compound options are used forto hedge bids for business projects that may or may not be accepted.
Lookback Options
Lookback options give their owners the right to buy or sell the underlying securityat the most attractive price that it actually trades in the cash market over a specifiedperiod of time. This time period is typically but not always the same time frameas the option’s life. As such, the strike prices of these options can, and do, change.When a look-back option is first created, the strike price is equal to the then-currentmarket value. As the price of the underlying changes, the strike price is reset towhatever the lowest value is at which the underlying security trades (in the case ofa call option) or whatever the highest value is at which the underlying trades (if it’sa put option). Look-backs are also known as reset options, because the strike priceis constantly reset to the most attractive strike reached during the time period. Bytheir very nature, look-backs are always either at the money or in the money.
The primary business application of look-back options is to allow portfolio man-agers to smooth out their results. For example, in years when they have significantlyoutperformed their index, they may buy look-back calls and puts that expire the nextyear. By doing so, they’ll take some of the return from the current year and use it to
31
‘stack the deck’ in their favor for the following year.
Barrier Options
Barrier options are options that are either activated or deactivated when the priceof the underlying security passes through some predefined value (the barrier). Barrieroptions have eight different varieties:
• Up and in call a call option that’s activated if the price of the underlying risesabove a certain price level.
• Up and out call a call option that’s deactivated if the price of the underlyingrises above a certain price level.
• Down and in call a call option that’s activated if the price of the underlyingfalls below a certain price level.
• Down and out call a call option that’s deactivated if the price of the underlyingfalls below a certain price level.
• Up and in put a put option that’s activated if the price of the underlying risesabove a certain price level.
• Up and out put a put option that’s deactivated if the price of the underlyingrises above a certain price level.
• Down and in put a put option that’s activated if the price of the underlyingfalls below a certain price level.
• Down and out put a put option that’s deactivated if the price of the underlyingfalls below a certain price level.
Asian Options
Asian options are very similar to look-backs, with the exception that while look-backs are based on the highest or lowest price over a period of time, Asians are basedon an average price. These options can be divided into two categories: Asian strikeoptions and Asian expiration options. With an Asian strike option, the strike price isnot a set price; instead, it’s the average of the prices at which the underlying securitytrades over a specified period of time. For example, the formula for the strike priceof an Asian strike call option might be calculated as “the average of all of the pricesat which the underlying trades between the option’s inception and its expiration.”
An Asian expiration option’s terminal value also is not a single fixed value, butis instead the average of the prices at which the underlying trades over a specifiedperiod of time. It might be calculated in this manner: ”the average of the closing
32
prices over the final two weeks before the option expires.”
Restrike Options
The strike price of these options changes if the price of the underlying passesthrough a barrier price. They’re typically written so that the strike price of callsis lowered and the strike price of puts is raised. For example, a call option’s initialstrike price of $100 might drop to a restrike price of $65 if the price of the underlyingsecurity falls below $75.
Instalment Options
Instalment options are a simple extension of a plain vanilla contract with theadded touch of being able to pay the premium of the option over a period of time.The holder of an instalment option does not pay the total amount of premium up-front; instead, the holder pays a portion of the premium up-front, and thereafter, inthe way of instalments. One can look at it as a series of compound options as theholder of the option is also able to cancel the instalment option at any of the premiumpayment periods.
Bermuda Options
Bermuda options are a hybrid between American and European options. UnlikeAmerican options (which can be exercised at any time during a specified period) andEuropean options (which can be exercised only at maturity), Bermuda options maybe exercised prior to maturity, but only on certain dates. The most common appli-cation of these options is to hedge the embedded call options found in bonds. Sincecallable bonds can normally only be called on certain days, investors who own themdon’t need to hedge the call risk every single day only those specific call dates.
33
3 Digital Options
Digital options (also known as binary options or cash-or-nothing options) have a fixedpayoff if the option ends up being in the money at expiration, regardless of the extentto which it is in the money. For example, let’s assume a digital option is written thatpays $1,000,000 if the price of ABC, Inc.’s, stock is above $75 per share at the end ofone year. It doesn’t matter if the stock is at $75.125 or $220 when the option expires,the payoff will still be $1,000,000.
Digital options have several business applications. Perhaps a company has anexecutive bonus program that awards its senior management $1,000,000 if its stockrises by fifty percent over the next two years. The company could hedge the cost ofthe compensation program by purchasing a digital option with a $1,000,000 payoff.For this option, the payoff will either be $1,000,000 if the option ends up in the moneyor nothing if it doesn’t.
The payoff of a digital call is given by
CT = IST≥K =
1, if ST ≥ K,0, if ST < K.
Thus, by the risk-neutral pricing formula,
Ct = e−r(T−t)E[CT |Ft] = e−r(T−t)E[IST≥K|Ft].
But, under the risk-neutral probability measure P,
ST = St exp
(σ(WT − Wt) + (r − 1
2σ2)τ
)
= St exp
(−σ√
τY + (r − 1
2σ2)τ
),
where τ = T − t, and
Y = −WT − Wt√τ
∼ N(0, 1)
independent of Ft. Solving the inequality
ST = St exp
(−σ√
τY + (r − 1
2σ2)τ
)> K
we get,
Y <1
σ√
τ
[log
St
K+
(r − 1
2σ2
)τ
]=: d2(τ, St).
It follows that
Ct = e−r(T−t)PST > K|Ft= e−r(T−t)PY < d2(τ, St)= e−r(T−t)N(d2(τ, St)).
34
4 Power and Powered Options
4.1 Power Options
A power option pays the owner the value of the stock raised to some power less astrike price only if the value of the underlying is greater than the strike price; thatis, the payoff of the power option is (we consider the European style call case):
CT = (SaT −K)+
where a > 0 is a constant. The case of a = 1 is exactly the the plain vanilla calloption which has been discussed.
If we choose the money market account Bt = ert as the numeraire, then the time-tvalue of this option is given by
Ct = e−r(T−t)E[(SaT −K)+|Ft]. (4.1)
Under the risk-neutral measure P, with τ = T − t,
ST = Steσ(fWT−fWt)+(r− 1
2σ2)τ ,
SaT = Sa
t eaσ(fWT−fWt)+a(r− 12σ2)τ .
Put
Y = −WT − Wt
σ√
τ∼ N(0, 1)
independent of Ft. Then
SaT = Sa
t e−aσ√
τY +a(r− 12σ2)τ .
Solving the inequalitySa
t e−aσ√
τY +a(r− 12σ2)τ > K
we get
Y <1
σ√
τ
[log
(St
K1/a
)+
(r − 1
2σ2
)τ
]=: d
(a)2 (τ, St).
35
By the Independence Lemma, we get Ct = v(t, St), where
v(t, x) = e−rτ E[(
xae−aσ√
τY +a(r− 12σ2)τ −K
)+]
= e−rτ
∫ d(a)2 (τ,x)
−∞
1√2π
(xae−aσ
√τy+a(r− 1
2σ2)τ −K
)e−
12y2
dy
= e−rτxa
∫ d(a)2 (τ,x)
−∞
1√2π
e−12y2−aσ
√τy+a(r− 1
2σ2)τdy
−e−rτK
∫ d(a)2 (τ,x)
−∞
1√2π
e−12y2
dy
= e−rτxa
∫ d(a)2 (τ,x)
−∞
1√2π
e−12y2−aσ
√τy+a(r− 1
2σ2)τdy
−e−rτKN(d(a)2 (τ, x)).
Since
∫ d(a)2 (τ,x)
−∞
1√2π
e−12y2−aσ
√τy+a(r− 1
2σ2)τdy
= eaτ(r+ 12(a−1)σ2)
∫ d(a)2 (τ,x)
−∞
1√2π
e−12(y+aσ
√τ)2dy
= eaτ(r+ 12(a−1)σ2)
∫ d(a)+aσ
√τ
2 (τ,x)
−∞
1√2π
e−12v2
dv
= eaτ(r+ 12(a−1)σ2)N(d
(a)1 (τ, x))
where
d(a)1 (τ, x) = d
(a)2 (τ, x) + aσ
√τ
=1
σ√
τ
[log
( x
K1/a
)+
(r + (a− 1
2)σ2
)τ
].
Therefore,
v(t, x) = e(a−1)(r+ 12aσ2)xaN(d
(a)1 (τ, x))− e−rτKN(d
(a)2 (τ, x)). (4.2)
It is immediately clear that if a = 1, the (4.2) is reduced to the Black-Scholes formulafor a European call option.
36
4.2 Powered Option
A powered option has the payoff
hT = [(ST −K)+]a = (ST −K)aIST >K (4.3)
where a ≥ 1 is an integer. The payoff hT can be rewritten as
hT =a∑
j=0
(a
j
)Sa−j
T (−K)jIST >K.
The risk-neutral valuation theory implies that the time-t value of the powered optionis given by the formula
ht = e−r(T−t)E[(ST −K)aIST >K|Ft
].
Noticing the computation of E[SaT ISa
T >K|Ft], we get, with τ = T − t,
ht = e−r(T−t)
a∑j=0
(a
j
)(−K)jE
[Sa−j
T IST >K|Ft
].
Under P,
ST = Steσ(fWT−fWt)+(r− 1
2σ2)(T−t)
= Ste−σ√
τY +(r− 12σ2)τ
where
Y = −WT − Wt√τ
∼ N(0, 1)
independent of Ft. Thus, ST > K iff Y < d2(τ, St), where
d2(τ, x) =1
σ√
τ
[log
x
K+ (r − 1
2σ2)τ
].
Since for 0 ≤ j ≤ a,
Sa−jT = Sa−j
t e−σ(a−j)√
τY +(a−j)(r− 12σ2)τ ,
we obtain by applying the Independence Lemma,
E[Sa−j
T IST >K|Ft
]= Sa−j
t gj(t, St),
37
where
gj(t, x) =
∫ d2(τ,x)
−∞e−σ(a−j)
√τy+(a−j)(r− 1
2σ2)τ 1√
2πe−
12y2
dy
= e(a−j)τ [r+ 12σ2(a−j−1)]
∫ d2(τ,x)
∞
1√2π
e−12(y+σ(a−j)
√τ)2dy
= e(a−j)τ [r+ 12σ2(a−j−1)]
∫ d2(τ,x)+σ(a−j)√
τ
∞
1√2π
e−12v2
dv
= e(a−j)τ [r+ 12σ2(a−j−1)]N
(d
(j)1 (τ, x)
)
where
d(j)1 (τ, x) = d2(τ, x) + σ(a− j)
√τ
=1
σ√
τ
[log
x
K+
(r + σ2(a− j − 1
2)
)τ
].
Therefore,
ht = e−rτ
a∑j=0
(a
j
)(−K)jSa−j
t e(a−j)τ [r+ 12σ2(a−j−1)]N
(d
(j)1 (τ, St)
)
=a∑
j=0
(a
j
)(−K)jSa−j
t e(a−j−1)[r+ 12σ2(a−j)]τN
(d
(j)1 (τ, St)
).
38
5 Chooser’s Option
Definition 5.1. Assume the BS model and consider a European call and a Europeanput having the same strike price K and expiration time T . Let 0 < t0 < T . A chooseroption gives the holder the right at time t0 to choose to own either the call or theput. The value at time t0 of the chooser option is maxCt0 , Pt0, where Ct0 and Pt0
are the value of the call and put at time t0, respectively. Alternatively, we have thetime-t0 of the chooser option is
maxCt0 , Pt0 = maxCt0 , e
−r(T−t0)K − St0 + Ct0
= Ct0 + max0, e−r(T−t0)K − St0
= Ct0 +(e−r(T−t0)K − St0
)+.
Let Ct(T, K) and Pt(T, K) denote the time-t values of the European call and putwith strike price K and expiration time T .
Since a rational investor will choose the option which has a higher value, thechooser option has the value maxCt0 , Pt0 at time t0, where Ct0 = Ct0(T, K) andPt0 = Pt0(T, K) are the values of the call and put at time t0, respectively.
Also, it is not hard to see that the payoff at time T is thus given by
X = (ST −K)+ICt0≥Pt0 + (K − ST )+ICt0<Pt0.
The risk-neutral pricing theory gives that the value of the chooser option at timet = 0 is
V0 = e−rT E[X].
Since it is easy to find that
X = (ST −K)+ + (K − ST )ICt0<Pt0,
we get
V0 = e−rT E[(ST −K)+] + e−rT E[(K − ST )ICt0<Pt0]
= C0 + e−rT E[(K − ST )ICt0<Pt0].
We infer that
e−rT E[(K − ST )ICt0<Pt0] = E[e−rT (K − ST )ICt0<Pt0]
= E[E[e−rT (K − ST )ICt0<Pt0]|Ft0 ]
= E[ICt0<Pt0E[e−rT (K − ST )|Ft0 ]]
= E[ICt0<Pt0(e−rT K − E[e−rT ST |Ft0 ])]
= E[ICt0<Pt0(e−rT K − e−rt0St0)]
= e−rt0E[ICt0<Pt0(e−r(T−t0)K − St0)].
39
For each t ∈ [0, T ], using the put-call parity
Pt + St − Ct = Ke−r(T−t),
Pt > Ct iff e−r(T−t)K − St > 0. Hence,
ICt<Pt(e−r(T−t)K − St) = (e−r(T−t)K − St)
+.
It turns out that
e−rT E[(K − ST )ICt0<Pt0] = e−rt0E[(e−r(T−t0)K − St0)+].
Therefore,
V0 = C0 + e−rt0E[(e−r(T−t0)K − St0)+]
= C0 + E[e−rt0(e−r(T−t0)K − St0)+].
This shows that the value of the chooser option at time t = 0 is the sum of thevalue of a call expiring at time T with strike K and the value of a put expiring attime t0 with strike e−r(T−t0)K.
We next derive a Black-Scholes formula. Since
St = S0eσfWt+(r− 1
2σ2)t = S0e
σ√
tZ+(r− 12σ2)t,
where we write Wt =√
tZ with Z ∼ N(0, 1), we have
Pt > Ct iff e−r(T−t)K > S0eσ√
tZ+(r− 12σ2)t.
Solving for Z, we get
Z < − 1
σ√
t
[log
S0
K+
(r − σ2t
2T
)T
]=: dt.
40
It follows that
e−rt0E[(e−r(T−t0)K − St0)+]
= e−rt0
∫ ∞
−∞
(e−r(T−t0)K − S0e
σ√
t0y+(r− 12σ2)t0
)+ 1√2π
e−12y2
dy
= e−rt0
∫ dt0
−∞
(e−r(T−t0)K − S0e
σ√
t0y+(r− 12σ2)t0
) 1√2π
e−12y2
dy
=
∫ dt0
−∞
(e−rT K − S0e
σ√
t0y− 12σ2t0
) 1√2π
e−12y2
dy
= e−rT KN(dt0)− S0
∫ dt0
−∞
1√2π
e−12y2+σ
√t0y− 1
2σ2t0dy
= e−rT KN(dt0)− S0
∫ dt0
−∞
1√2π
e−12(y−σ
√t0)2dy
= e−rT KN(dt0)− S0
∫ dt0−σ√
t0
−∞
1√2π
e−12v2
dv
= e−rT KN(dt0)− S0N(dt0 − σ√
t0).
We therefore eventually arrive at the formula:
V0 = C0 + e−rT KN(dt0)− S0N(dt0 − σ√
t0)
= S0N(d1)− e−rT KN(d2) + e−rT KN(dt0)− S0N(dt0 − σ√
t0),
where
d1 =1
σ√
T
[log
S0
K+
(r +
1
2σ2
)T
],
d2 =1
σ√
T
[log
S0
K+
(r − 1
2σ2
)T
]= d1 − σ
√T .
What is the time-t value, Vt?We have
Vt = e−r(T−t)E[X|Ft], t ∈ [0, T ].
Here X, once again, is the payoff given by
X = (ST −K)+ICt0≥Pt0 + (K − ST )+ICt0<Pt0= (ST −K)+ + (K − ST )ICt0<Pt0.
Putting τ = T − t, we have
Vt = e−rτ E[(ST −K)+|Ft] + e−rτ E[(K − ST )ICt0<Pt0|Ft]
= Ct(T,K) + e−rτ E[(K − ST )ICt0<Pt0|Ft].
41
We distinguish two cases:
Case (i): 0 ≤ t < t0. We then deduce that
E[(K − ST )ICt0<Pt0|Ft] = E[(K − ST )ICt0<Pt0]
= E[E[(K − ST )ICt0<Pt0|Ft0 ]|Ft]
= E[ICt0<Pt0E[(K − ST )|Ft0 ]|Ft]
= E[ICt0<Pt0(K − erT E[e−rT ST |Ft0 ])|Ft]
= E[ICt0<Pt0(K − er(T−t0)St0)|Ft].
The put-call parityPt0 − Ct0 = Ke−r(T−t0) − St0
asserts that Pt0 > Ct0 iff K − er(T−t0)St0 > 0.It thus follows that
E[(K − ST )ICt0<Pt0|Ft] = er(T−t0)E[(e−r(T−t0)K − St0)+|Ft]
and thatVt = Ct(T, K) + e−r(t0−t)E[(e−r(T−t0)K − St0)
+|Ft]. (5.1)
The second term is the time-t value of a European put with expiration time t0 andstrike price e−r(T−t0)K, Pt(t0, e
−r(T−t0)K). Hence,
Vt = Ct(T, K) + Pt(t0, e−r(T−t0)K). (5.2)
Using Black-Scholes formula, we get
Vt = StN(d1(T − t, St, K))− e−r(T−t)KN(d2(T − t, St, K))
+e−r(T−t)KN(−d2(t0 − t, St, e−r(T−t0)K))
−StN(−d1(t0 − t, St, e−r(T−t0)K)).
Here,
d1(τ, x,K) =1
σ√
τ
[log
x
K+ (r +
1
2σ2)τ
],
d2(τ, x, K) =1
σ√
τ
[log
x
K+ (r − 1
2σ2)τ
].
Case (ii): t > t0. Then ICt0<Pt0 is Ft-measurable; thus
Vt = Ct(T, K) + e−rτ E[(K − ST )ICt0<Pt0|Ft]
= Ct(T, K) + ICt0<Pt0e−r(T−t)E[(K − ST )|Ft]
= Ct(T, K) + ICt0<Pt0e−r(T−t)(K − E[ST |Ft])
= Ct(T, K) + ICt0<Pt0e−r(T−t)(K − erT e−rtSt)
= Ct(T, K) + ISt0<e−r(T−t0)K(e−r(T−t)K − St).
42
6 Compound Options
Definition 6.1. A compound option is an option written on another option (i.e., theunderlying asset is an option); namely, a compound option is an option on option.
Thus there are four main types of compound options: (1) a call on a call; (2) acall on a put; (3) a put on a call; and (4) a put on a put. A compound option has twoexpiration times and two strike prices. Consider a call on a call. On the first exercisetime T1, the holder of this call on call compound option has the right to exercise the(first) call option by paying the first strike price of K1 and receive the (second) calloption which gives the holder to buy the underlying asset for the second strike priceK2 at the (second) expiration time T2. The compound option will be exercised onthe first exercise date only if the value of the option on that date is greater than thefirst strike price.
Use Ci(t) = Ci(t, Ti, Ki) to denote the value at time t of a call option expiring atTi with strike price Ki, where i = 1, 2.
Since the second call is based on the underlying asset S which follows the geometricBrownian motion, we have by the Black-Scholes formula
C2(t) = E[e−r(T2−t)(ST2 −K2)+|Ft]
= StN(d1(T2 − t, St, K2))
−K2e−r(T2−t)N(d2(T2 − t, St, K2)) (6.1)
where, for τ, x > 0,
d1(τ, x, K2) =log x
K2+ (r + 1
2σ2)τ
σ√
τ,
d2(τ, x,K2) =log x
K2+ (r − 1
2σ2)τ
σ√
τ= d1(τ, x,K2)− σ
√τ .
By the risk-neutral pricing theory, we have that the time-t value of the compoundoption is given by, for t ∈ [0, T1],
Vt = E[e−r(T1−t)(C2(T1)−K1)+|Ft]. (6.2)
Note that the value of the second call C2(t) can be rewritten as
C2(t) = c2(t, St),
where c2(t, x) is known as the value function of the (second) call C2. We indeed have
c2(t, x) = xN(d1(T2 − t, x, K2))−K2e−r(T2−t)N(d2(T2 − t, x, K2)). (6.3)
It is not hard to find, due to strict increasingness and continuity, that there existsa unique solution x∗ to the equation:
c2(T1, x∗) = K1. (6.4)
43
In another way, x∗ is the stock price such that the (second) option is at the money.Due again to increasingness, we have
C2(T1) = c2(T1, ST1) > K1 iff ST1 > x∗.
Since, under the risk-neutral probability measure P ,
ST1 = St exp
(σ(WT1 − Wt) + (r − 1
2σ2)(T1 − t)
),
setting
Z = −WT1 − Wt√T1 − t
∼ N(0, 1) (independent of Ft),
we can write
ST1 = St exp
(−σ
√T1 − tZ + (r − 1
2σ2)(T1 − t)
). (6.5)
Therefore, the inequality ST1 > x∗ gives that
Z <log St
x∗ + (r − 12σ2)(T1 − t)
σ√
T1 − t= d2(T1 − t, St, x
∗). (6.6)
C2(T1) = c2(T1, ST1)
= ST1N(d1(T2 − T1, ST1 , K2))
−K2e−r(T2−T1)N(d2(T2 − T1, ST1 , K2)).
where, by virtue of (6.5),
d1(T2 − T1, ST1 , K2) =log
ST1
K2+ (r + 1
2σ2)(T2 − T1)
σ√
T2 − T1
=log St
K2+ (r + 1
2σ2)(T2 − T1)
σ√
T2 − T1
+−σ√
T1 − tZ + (r − 12σ2)(T1 − t)
σ√
T2 − T1
= d1(T2 − T1, St, K2)− αtZ + βt,
d2(T2 − T1, ST1 , K2) =log
ST1
K2+ (r − 1
2σ2)(T2 − T1)
σ√
T2 − T1
=log St
K2+ (r − 1
2σ2)(T2 − T1)
σ√
T2 − T1
+−σ√
T1 − tZ + (r − 12σ2)(T1 − t)
σ√
T2 − T1
= d2(T2 − T1, St, K2)− αtZ + βt,
44
where
αt =
√T1 − t√T2 − T1
, βt =T1 − t
σ√
T2 − T1
(r − 1
2σ2
).
We can therefore write
C2(T1) = St exp
(−σ
√T1 − tZ + (r − 1
2σ2)(T1 − t)
)
×N (d1(T2 − T1, St, K2)− αtZ + βt)
−K2e−r(T2−T1)N (d2(T2 − T1, St, K2)− αtZ + βt) . (6.7)
Since Z is independent of Ft, by the Independence Lemma, we get
Vt = e−r(T1−t)E[(C2(T1)−K1)+|Ft] = v(t, St), (6.8)
where
v(t, x) = e−r(T1−t)E[
xe−σ√
T1−tZ+(r− 12σ2)(T1−t)
×N (d1(T2 − T1, x, K2)− αtZ + βt)
−K2e−r(T2−T1)N (d2(T2 − T1, x, K2)− αtZ + βt)−K1
+
× IZ<d2(T1−t,x,x∗)]
= e−r(T1−t)
∫ d2(T1−t,x,x∗)
−∞
1√2π
e−12z2
xe−σ
√T1−tz+(r− 1
2σ2)(T1−t)
×N (d1(T2 − T1, x, K2)− αtz + βt)
−K2e−r(T2−T1)N (d2(T2 − T1, x, K2)− αtz + βt)−K1
dz
= J1 − J2 − J3 (6.9)
where
J1 = e−r(T1−t)x
∫ d2(T1−t,x,x∗)
−∞
1√2π
e−12z2−σ
√T1−tz+(r− 1
2σ2)(T1−t)
×N (d1(T2 − T1, x, K2)− αtz + βt) dz,
J2 = e−r(T2−t)K2
∫ d2(T1−t,x,x∗)
−∞
1√2π
e−12z2
×N (d2(T2 − T1, x, K2)− αtz + βt) dz,
J3 = e−r(T1−t)K1
∫ d2(T1−t,x,x∗)
−∞
1√2π
e−12z2
dz
= e−r(T1−t)K1N (d2(T1 − t, x, x∗)) . (6.10)
45
To evaluate J2, we set
ρt =
√T1 − t√T2 − t
; hence√
1− ρ2t =
√T2 − T1√T2 − t
.
It is then not hard to find that
d2(T2 − T1, x, K2)− αtz + βt
=log St
K2+ (r − 1
2σ2)(T2 − t)− σ
√Tt − tz
σ√
T2 − T1
=σ√
T2 − td2(T2 − t, x, K2)− σ√
Tt − tz
σ√
T2 − T1
=d2(T2 − t, x,K2)− ρtz√
1− ρ2t
.
It follows that
J2 = e−r(T2−t)K2
∫ d2(T1−t,x,x∗)
−∞N
(d2(T2 − t, x, K2)− ρtz√
1− ρ2t
)1√2π
e−12z2
dz
= e−r(T2−t)K2N2(d2(T1 − t, x, x∗), d2(T2 − t, x, K2); ρt), (6.11)
where we use N2(x, y; ρ) to denote the bivariate cumulative normal distribution; thatis,
N2(x, y; ρ) =
∫ x
−∞N
(y − ρz√1− ρ2
)1√2π
e−12z2
dz.
Next compute J1. We have
J1 = xe−r(T1−t)
∫ d2(T1−t,x,x∗)
−∞
1√2π
e−12z2−σ
√T1−tz+(r− 1
2σ2)(T1−t)
×N (d1(T2 − T1, x, K2)− αtz + βt) dz
= x
∫ d2(T1−t,x,x∗)
−∞
1√2π
e−12(z+σ
√T1−t)2
×N (d1(T2 − T1, x, K2)− αtz + βt) dz
= x
∫ d2(T1−t,x,x∗)+σ√
T1−t
−∞
1√2π
e−12v2
×N(d1(T2 − T1, x,K2)− αt(v − σ
√T1 − t) + βt
)dv
= x
∫ d1(T1−t,x,x∗)
−∞
1√2π
e−12v2
×N(d1(T2 − T1, x,K2)− αtv + σαt
√T1 − t) + βt
)dv, (6.12)
46
whered1(T1 − t, x, x∗) = d2(T1 − t, x, x∗) + σ
√T1 − t.
However,
d1(T2 − T1, x, K2) =log x
K2+ (r + 1
2σ2)(T2 − T1)
σ√
T2 − T1
=log x
K2+ (r + 1
2σ2)(T2 − t)− (r + 1
2σ2)(T1 − t)
σ√
T2 − T1
=d1(T2 − t, x,K2)σ
√T2 − t− (r + 1
2σ2)(T1 − t)
σ√
T2 − T1
.
Since √1− ρ2
t =
√T2 − T1√T2 − t
,
we get
d1(T2 − T1, x, K2) =d1(T2 − t, x, K2)√
1− ρ2t
− T1 − t
σ√
T2 − T1
(r +
1
2σ2
).
Moreover, noting that
αt =
√T1 − t√T2 − T1
=ρt√
1− ρ2t
,
we derive that
d1(T2 − T1, x,K2)− αtv + σαt
√T1 − t + βt
=d1(T2 − t, x, K2)√
1− ρ2t
− ρt√1− ρ2
t
v
− T1 − t
σ√
T2 − T1
(r +
1
2σ2
)+
σ(T1 − t)√T2 − T1
+T1 − t
σ√
T2 − T1
(r − 1
2σ2
)
=d1(T2 − t, x, K2)− ρtv√
1− ρ2t
. (6.13)
Substituting (6.13) into (6.12) yields
J1 = x
∫ d1(T1−t,x,x∗)
−∞
1√2π
e−12v2
N
(d1(T2 − t, x,K2)− ρtv√
1− ρ2t
)dv
= xN2 (d1(T1 − t, x, x∗), d1(T2 − t, x, K2); ρt) . (6.14)
Finally, substituting (6.10), (6.11) and (6.14) into (6.9), we get the value functionv(t, x) of the compound option:
v(t, x) = xN2 (d1(T1 − t, x, x∗), d1(T2 − t, x,K2); ρt)
−e−r(T2−t)K2N2(d2(T1 − t, x, x∗), d2(T2 − t, x,K2); ρt)
−e−r(T1−t)K1N (d2(T1 − t, x, x∗)) . (6.15)
47
7 Barrier Options
7.1 Introduction
Barrier options are path-dependent options which are either initiated or exterminatedupon when the underlying asset reaches a certain barrier level; that is, they are eitherknocked in or knocked out.
An option contract that may only be exercised when the underlying asset reachessome barrier price. A barrier option may either be a knock-in or a knock-out. Aknock-in may only be exercised when the underlying asset rises above or falls belowthe barrier price. On the other hand, a knock-out automatically expires when theunderlying asset rises above or falls below the barrier price. It is important to notethat the barrier price is distinct from the exercise price, though, theoretically, theymay be set at the same amount. See also: Exotic option
In finance, a barrier option is a type of financial option where the option to exercisedepends on the underlying crossing or reaching a given barrier level. Barrier optionsare always cheaper than a similar option without barrier. Barrier options were createdto provide the insurance value of an option without charging as much premium. Forexample, if you believe that IBM will go up this year, but are willing to bet that itwon’t go above $100, then you can buy the barrier and pay less premium than thevanilla option.
Barrier options are path-dependent exotics that are similar in some ways to ordi-nary options. There are put and call, as well as European and American varieties.But they become activated or, on the contrary, null and void only if the underlierreaches a predetermined level (barrier). “In” options start their lives worthless andonly become active in the event a predetermined knock-in barrier price is breached.”Out” options start their lives active and become null and void in the event a certainknock-out barrier price is breached. In either case, if the option expires inactive, thenthere may be a cash rebate paid out. This could be nothing, in which case the optionends up worthless, or it could be some fraction of the premium. The four main typesof barrier options are:
• Up-and-out: spot price starts below the barrier level and has to move up forthe option to be knocked out.
• Down-and-out: spot price starts above the barrier level and has to move downfor the option to become null and void.
• Up-and-in: spot price starts below the barrier level and has to move up for theoption to become activated.
• Down-and-in: spot price starts above the barrier level and has to move downfor the option to become activated.
48
For example, a European call option may be written on an underlying with spotprice of $100, and a knockout barrier of $120. This option behaves in every waylike a vanilla European call, except if the spot price ever moves above $120, theoption “knocks out” and the contract is null and void. Note that the option does notreactivate if the spot price falls below $120 again. Once it is out, it’s out for good.In-out parity is the barrier option’s answer to put-call parity. If we combine one “in”option and one “out” barrier option with the same strikes and expirations, we get theprice of a vanilla option: C = Cin+Cout. A simple arbitrage argument-simultaneouslyholding the “in” and the “out” option guarantees that one and only one of the twowill pay off. The argument only works for European options without rebate.
49
7.2 Up-and-Out Call
7.2.1 Definition
Definition 7.1. A up-and-out call is a call option for which a barrier level of price(above the current price of the underlying asset) is set for the underlying asset andwhich becomes worthless as soon as the underlying asset price reaches the barrierlevel.
Recall that the underlying asset S follows the GBM:
dSt = rStdt + σStdWt (7.1)
where Wt is a standard Brownian motion under the risk-neutral probability measureP.
Consider a European call option with strike price K and expiration time T . As-sume the up-and-out barrier level is L. We assume K < L (otherwise if K ≥ L, thenthe option must knock out in order to be in the money (i.e., ST > K ≥ L) and hencecould only pay off zero).
The solution to Eq. (7.1) is given as
St = S0eσfWt+(r− 1
2σ2)t = S0e
σcWt (7.2)
where
Wt := αt + Wt, α =1
σ
(r − 1
2σ2
).
SetMT = max
0≤u≤TWu.
Then we find thatST := max
0≤u≤TSu = S0e
σcMT .
Since the up-and-out call knocks out if and only if ST > L; i.e., S0eσcMT > L. If
S0eσcMT ≤ L, the option pays off
VT := (ST −K)+ = (S0eσcWT −K)+.
Consequently, we find that the payoff of the up-and-out call is given by
(S0eσcWT −K)+IS0eσcMT≤L.
Since(S0e
σcWT −K)+ = (S0eσcWT −K)IS0eσcWT≥K,
50
we get
VT = (S0eσcWT −K)IS0eσcWT≥K, S0eσcMT≤L
= (S0eσcWT −K)IcWT≥a, cMT≤b,
where
a =1
σlog
K
S0
, b =1
σlog
L
S0
.
51
7.2.2 The Black-Scholes-Merton Equation
Theorem 7.2. Let v(t, x) denote the price at time t of the up-and-out call under theassumption that the call has not knocked out prior to time t and St = x. Then v(t, x)satisfies the Black-Scholes partial differential equation
v′t(t, x) + rxv′x(t, x) +1
2σ2x2v′′xx(t, x) = rv(t, x), 0 ≤ t < T, 0 ≤ x ≤ B (7.3)
and the boundary conditions:
v(t, 0) = 0, 0 ≤ t ≤ T, (7.4)
v(t, L) = 0, 0 ≤ t < T, (7.5)
v(T, x) = (x−K)+, 0 ≤ x ≤ L. (7.6)
Proof. Assume 0 < S0 < L. Then the risk-neutral pricing formula gives us that thetime-t value of the up-and-out call is
Vt = E[e−r(T−t)VT |Ft], 0 ≤ t ≤ T. (7.7)
We have that the discounted value process
Vt = e−rtVt = E[e−rT VT |Ft], 0 ≤ t ≤ T (7.8)
is a martingale.Being Markovian, Vt can be written as
Vt = v(t, St), 0 ≤ t ≤ T, (7.9)
where v(t, x) is a function in Theorem 7.2. However, (7.9) can’t hold for all 0 ≤ t ≤ Tand x ≥ 0 since v(t, St) is value of the option that has not knocked out at time t andVt is the value of the option without any assumption.
Observe that as soon as the stock price S reaches the barrier level L, the optionbecomes worthless. So we define the stopping time ρ by
ρ = inft ≥ 0 : St = L.
Thus, St < L for 0 ≤ t < ρ and Sρ = L, and the option knocks out at time ρ so thatv(ρ, Sρ) = 0. Consequently, the relation (7.9) holds for 0 ≤ t ≤ ρ and the discountedprocess
e−rtv(t, St)
is a martingale on the interval [0, ρ]. However, the stopped process
e−r(t∧ρ)Vt∧ρ =
e−rtVt, if 0 ≤ t ≤ ρ,e−rρVρ, if ρ < t ≤ T ,
(7.10)
52
is a martingale on the interval [0, T ].By Ito, we have
d(e−rtv(t, St)) = −re−rtv(t, St)dt + e−rtdv(t, St)
= −re−rtv(t, St)dt + e−rt[v′t(t, St)dt + v′x(t, St)dSt +1
2v′′xx(t, St)dStdSt]
= −re−rtv(t, St)dt + e−rt[v′t(t, St)dt + v′x(t, St)dSt +1
2σ2S2
t v′′xx(t, St)dt]
= e−rt[−rv(t, St) + v′t(t, St) +1
2σ2S2
t v′′xx(t, St)]dt + e−rtv′x(t, St)dSt
= e−rt[−rv(t, St) + v′t(t, St) + rStv′x(t, St) +
1
2σ2S2
t v′′xx(t, St)]dt
+e−rtσStv′x(t, St)dWt
Being a martingale on [0, ρ], the dt term must be vanish. Hence we get
−rv(t, St) + v′t(t, St) + rStv′x(t, St) +
1
2σ2S2
t v′′xx(t, St) = 0. (7.11)
Since (t, St) can take any values in the set (t, x) : 0 ≤ t < T, 0 ≤ x ≤ L, replacingSt by x in (7.11) gives the equation (7.3) for 0 ≤ t ≤ ρ.
53
7.2.3 Price of Up-and-Out Calls
We compute V0, the price at time zero of the up-and-out call option. We have
V0 = E[e−rT VT ] = E[e−rT (S0e
σcWT −K)IcWT≥a, cMT≤b
], (7.12)
where
a =1
σlog
K
S0
, b =1
σlog
L
S0
. (7.13)
Lemma 7.3. Under P, the joint density of the pair (MT , WT ) is
fcMT ,cWT(m,w) =
2(2m− w)
T√
2πTexp
(αw − 1
2α2T − 1
2T(2m− w)2
), w ≤ m, m ≥ 0
(7.14)and zero for other values of (m,w).
Now return to the computation of (7.12). If a ≥ 0, we must integrate overthe region (m,w) : a ≤ w ≤ m ≤ b; if a < 0, we integrate over the region(m,w) : a ≤ w ≤ m, 0 ≤ m ≤ b. In both cases, the region is described as(m,w) : a ≤ w ≤ m, w+ ≤ m ≤ b. Otherwise, the region over which we integratehas zero area, and the time-0 value of the call is zero rather than the integral computedbelow. We also assume S0 > 0 so that a and b are finite.
When 0 < S0 ≤ B, we have
V0 =
∫ ∞
−∞
∫ ∞
−∞e−rT (S0e
σw −K)Iw≥a, m≤bfcMT ,cWT(m,w)dmdw.
Since fcMT ,cWT(m,w) = 0 for (m,w) 6∈ (m,w) : w ≤ m, m ≥ 0, plus the condition
w ≥ a, m ≤ b, we see that the above integral is indeed over the region
(m,w) : a ≤ w ≤ m, w+ ≤ m ≤ b.
Therefore, (integrate with respect to m first) we get
V0 =
∫ b
a
∫ b
w+
e−rT (S0eσw −K)
2(2m− w)
T√
2πTexp
(αw − 1
2α2T − 1
2T(2m− w)2
)dmdw
= −∫ b
a
e−rT (S0eσw −K)
1√2πT
exp
(αw − 1
2α2T − 1
2T(2m− w)2
) ∣∣∣∣m=b
m=w+
dw
=1√2πT
∫ b
a
(S0eσw −K) exp
(−rT + αw − 1
2α2T − 1
2Tw2
)dw
− 1√2πT
∫ b
a
(S0eσw −K) exp
(−rT + αw − 1
2α2T − 1
2T(2b− w)2
)dw
= S0I1 −KI2 − S0I3 + KI4, (7.15)
54
where
I1 =1√2πT
∫ b
a
exp
(σw − rT + αw − 1
2α2T − 1
2Tw2
)dw
I2 =1√2πT
∫ b
a
exp
(−rT + αw − 1
2α2T − 1
2Tw2
)dw
I3 =1√2πT
∫ b
a
exp
(σw − rT + αw − 1
2α2T − 1
2T(2b− w)2
)dw
=1√2πT
∫ b
a
exp
(σw − rT + αw − 1
2α2T − 2
Tb2 +
2
Tbw − 1
2Tw2
)dw
I4 =1√2πT
∫ b
a
exp
(−rT + αw − 1
2α2T − 1
2T(2b− w)2
)dw
=1√2πT
∫ b
a
exp
(−rT + αw − 1
2α2T − 2
Tb2 +
2
Tbw − 1
2Tw2
)dw.
Each of these integrals is of the form
1√2πT
∫ b
a
exp
(β + γw − 1
2Tw2
)dw =
1√2πT
∫ b
a
exp
(− 1
2T(w − γT )2 +
1
2γT + β
)dw
= e12γ2T+β 1√
2π
∫ 1√T
(b−γT )
1√T
(a−γT )
e−12y2
dy,
where we have used the substitution y = w−γ√
T√T
. Using the standard cumulative
distribution function property N(−z) + N(z) = 1 and (7.13), we get
1√2πT
∫ b
a
exp
(β + γw − 1
2Tw2
)dw = e
12γ2T+β
[N
(b− γT√
T
)−N
(a− γT√
T
)]
= e12γ2T+β
[N
(−a + γT√T
)−N
(−b + γT√T
)]
= e12γ2T+β
[N
(1
σ√
T
[log
S0
K+ γσT
])
−N
(1
σ√
T
[log
S0
L+ γσT
]).
Set
δ±(τ, s) =1
σ√
τ
[log s +
(r ± 1
2σ2
)τ
]. (7.16)
Now for the integral I1, we set β = −rT − 12α2T and γ = α + σ so that 1
2γ2T + β0
and γσ = r + 12σ2. Consequently,
I1 = N
(δ+
(T,
S0
K
))−N
(δ+
(T,
S0
L
)). (7.17)
55
For the integral I2, we take β = −rT − 12α2T and γ = α. Then, since 1
2γ2T + β =
−rT and γσ = r − 12σ2, we get
I2 = N
(δ−
(T,
S0
K
))−N
(δ−
(T,
S0
L
)). (7.18)
For the integral I3, we take β = −rT − 12α2T − 2b2
Tand γ = α + σ + 2b
T. Then, since
1
2γ2T + β = log
(S0
L
)− 2rσ2−1
and
γσT =
(r +
1
2σ2
)T + log
(S0
L
)2
,
we get
I3 =
(S0
L
)− 2rσ2−1 [
N
(δ+
(T,
L2
KS0
))−N
(δ+
(T,
L
S0
))]. (7.19)
For the integral I4, we take β = −rT − 12α2T − 2b2
Tand γ = α + 2b
T. Then, since
1
2γ2T + β = −rT + log
(S0
L
)− 2rσ2 +1
and
γσT =
(r − 1
2σ2
)T + log
(S0
L
)2
,
we get
I4 = e−rT
(S0
L
)− 2rσ2 +1 [
N
(δ−
(T,
L2
KS0
))−N
(δ−
(T,
L
S0
))]. (7.20)
Eventually, substituting (7.17)-(7.20) into (7.15), we obtain
V0 = S0
[N
(δ+
(T,
S0
K
))−N
(δ+
(T,
S0
L
))]
−e−rT K
[N
(δ−
(T,
S0
K
))−N
(δ−
(T,
S0
L
))]
−L
(S0
L
)− 2rσ2
[N
(δ+
(T,
L2
KS0
))−N
(δ+
(T,
L
S0
))]
+e−rT K
(S0
L
)− 2rσ2 +1 [
N
(δ−
(T,
L2
KS0
))−N
(δ−
(T,
L
S0
))].(7.21)
56
Now for t ∈ [0, T ), put x = St and assume that the option is still alive so that0 < x ≤ L (since, if x > L, the up-and-out call has already knocked out). To getVt, we replace, in (7.21), S0 and T with St = x and τ := T − t (the time remains tomaturity), respectively. Therefore, Vt = v(t, St), where
v(t, x) = x[N
(δ+
(τ,
x
K
))−N
(δ+
(τ,
x
L
))]
−e−rτK[N
(δ−
(τ,
x
K
))−N
(δ−
(τ,
x
L
))]
−L(x
L
)− 2rσ2
[N
(δ+
(τ,
L2
Kx
))−N
(δ+
(τ,
L
x
))]
+e−rτK(x
L
)− 2rσ2 +1
[N
(δ−
(τ,
L2
Kx
))−N
(δ−
(τ,
L
x
))](7.22)
for 0 ≤ t < T and 0 < x ≤ L. While v(t, x) = 0 if x > L.
57
8 Lookback Options
A lookback option is a path-dependent option whose payoff is based on the maximum(or minimum) stock price realized over some interval of time prior to expiration.
A lookback option that allows investors to “look back” at the underlying pricesoccurring over the life of the option, and then exercise based on the underlying asset’soptimal value.
Investopedia explains Lookback Option: This type of option reduces uncertaintiesassociated with the timing of market entry.
There are two types of lookback options:
1. Fixed - The option’s strike price is fixed at purchase. However, the option isnot exercised at the market price: in the case of a call, the option holder can look backover the life of the option and choose to exercise at the point when the underlyingasset was priced at its highest over the life of the option. In the case of a put, theoption can be exercised at the asset’s lowest price. The option settles at the selectedpast market price and against the fixed strike.
2. Floating - The option’s strike price is fixed at maturity. For a call, the strikeprice is fixed at the lowest price reached during the life of the option, and, for a put,it is fixed at the highest price. The option settles at market and against the floatingstrike.
While lookback options are appealing to investors, they can be expensive and arealso considered to be quite speculative.
58
8.1 Floating Strike Lookback Option
We again use the following notation:
Wt = αt + Wt, α =1
σ
(r − 1
2σ2
),
Mt = max0≤u≤t
Wu,
Yt ≡ St = max0≤u≤t
Su = S0eσcMt ,
Zt = min0≤u≤t
Su.
RecallSt = S0e
σfWt+(r− 12σ2)t = S0e
σcWt .
A floating lookback call option with expiration time T has payoff given by
CT = ST − ZT .
A floating lookback put option has payoff given by
PT = YT − ST .
We will consider the case of a floating lookback put option; the case of a floatinglookback call option being similar.
According to the risk-neutral pricing formula, we have that the value of the optionat time t is
Vt = E[e−r(T−t)(YT − ST )|Ft] = v(t, St, Yt), (8.1)
where v(t, x, y) is a function of (t, x, y) referred to as the value function and being tobe determined.
59
8.2 The Black-Scholes-Merton Equation
Theorem 8.1. The value function v(t, x, y) for a floating strike put option satisfiesthe following Black-Scholes equation:
v′t(t, x, y) + rxv′x(t, x, y) +1
2σ2x2v′′xx(t, x, y) = rv(t, x, y), 0 ≤ t < T, 0 ≤ x ≤ y,
(8.2)and the boundary conditions:
v(t, 0, y) = e−r(T−t)y, 0 ≤ t ≤ T, y ≥ 0, (8.3)
vy(t, y, y) = 0, 0 ≤ t ≤ T, y > 0, (8.4)
v(T, x, y) = y − x, 0 ≤ x ≤ y. (8.5)
Proof. By Ito’s formula (for two-dimension), we get
d(e−rtv(t, St, Yt)) = −re−rtv(t, St, Yt)dt + e−rtdv(t, St, Yt)
= −re−rtv(t, St, Yt)dt + e−rt(v′tdt + v′xdSt + v′ydYt
+1
2v′′xxdStdSt + v′′xydStdYt +
1
2v′′yydYtdYt).
We will use the facts (to be derived later on):
dYtdYt = 0, dStdYt = 0.
We then get, for dSt = St(rdt + σdWt),
d(e−rtv(t, St, Yt)) = −re−rtvdt + e−rt(v′tdt + v′xdSt + v′ydYt +1
2σ2S2
t v′′xxdt)
= e−rt
(−rv + v′t + rStv
′x +
1
2σ2S2
t v′′xx
)dt
+e−rtσStv′xdWt + e−rtv′ydYt.
Here the values of v and its partial derivative are all evaluated at (t, St, Yt).In order to be a martingale, the dt term must be zero and this leads to the PDE
(8.2) when putting x = St and y = Yt. The new feature is that the term v′y(t, St, Yt)dYt
must be also zero, and therefore, we get
d(e−rtv(t, St, Yt)) = e−rtσStv′x(t, St, Yt)dWt.
Lemma 8.2. We have
(a) Yt has zero quadratic variation: dYtdYt = 0.
60
(b) The mixed quadratic variation of Yt and St is zero: dStdYt = 0; namely,
σm :=m∑
j=1
(Ytj − Ytj−1)(Stj − Stj−1
) → 0
as |π| := max1≤j≤m(tj − tj−1) → 0, where π : 0 = t0 < t1 < · · · < tm = T is apartition of [0, T ].
(Note: Item (b) is not trivial since dYt is not a dt term; namely, there is no nonzeroprocess θt such that dYt = dθt.)
Proof. (a) Observe Yt is continuous and increasing. Let now 0 = t0 < t1 < · · · <tm = T be a partition of [0, T ]. We have
m∑j=1
(Ytj − Ytj−1)2 ≤ max
1≤j≤m(Ytj − Ytj−1
)m∑
j=1
(Ytj − Ytj−1)
= max1≤j≤m
(Ytj − Ytj−1)(YT − Y0).
Due to (uniform) continuity, max1≤j≤m(Ytj − Ytj−1) → 0 as max1≤j≤m(tj − tj−1) → 0.
We therefore obtain∑m
j=1(Ytj − Ytj−1)2 → 0 as max1≤j≤m(tj − tj−1) → 0.
(b) We have
σm =∑
1≤j≤mYtj
>Ytj−1
(Ytj − Ytj−1)(Stj − Stj−1
).
61
8.3 Reduction of Dimension
Lemma 8.3. The price function v(t, x, y) satisfies the following positive homogeneity(linear scaling) property
v(t, λx, λy) = λv(t, x, y), λ > 0. (8.6)
Proof. We have
v(t, λSt, λYt) = E[e−r(T−t)(λYT − λST )|Ft]
= λE[e−r(T−t)(YT − ST )|Ft]
= λv(t, St, Yt).
Putting x = St and y = Yt yields (8.6).
In particular, we have
v(t, x, y) = yv
(t,
x
y, 1
)= yu
(t,
x
y
)(8.7)
for 0 ≤ t ≤ T, 0 ≤ x ≤ y, y > 0, where
u(t, z) := v(t, z, 1), 0 ≤ t ≤ T, 0 ≤ z ≤ 1. (8.8)
From (8.7) we obtain
v′t(t, x, y) = yu′t
(t,
x
y
),
v′x(t, x, y) = u′z
(t,
x
y
),
v′′xx(t, x, y) =1
yu′′zz
(t,
x
y
),
v′y(t, x, y) = u
(t,
x
y
)− x
yu′z
(t,
x
y
).
Substituting these partial derivatives into the Black-Scholes equation (8.2), we get
y
[− ru
(t,
x
y
)+ u′t
(t,
x
y
)+ r · x
yu′z
(t,
x
y
)+
1
2σ2
(x
y
)2
u′′zz
(t,
x
y
)]= 0.
Canceling y and making the change of variable z = xy, we obtain that u(t, z) satisfies
the BS equation:
ut(t, z) + rzuz(t, z) +1
2σ2z2uzz(t, z) = rz, 0 ≤ t < T, 0 < z < 1. (8.9)
62
The boundary conditions that u(t, z) satisfies are obtained from the boundary condi-tions (8.3)-(8.5) that v(t, x, y) satisfies. We find that
e−r(T−t)y = v(t, 0, y) = yu(t, 0)
is reduced to the condition:
u(t, 0) = e−r(T−t), 0 ≤ t ≤ T ; (8.10)
0 = vy(t, y, y) = u(t, 1)− uz(t, 1)
impliesu(t, 1) = uz(t, 1), 0 ≤ t < T ; (8.11)
and
y − x = v(T, x, y) = yu
(T,
x
y
)
impliesu(T, z) = 1− z, 0 ≤ z ≤ 1. (8.12)
The PDE (8.9) together with the boundary conditions (8.10)-8.12) uniquely deter-mines u(t, z) and hence v(t, x, y).
63
8.4 Price of Lookback Options
For t ∈ [0, T ], we set τ = T − t. We observe
Yt = S0eσcMt , YT = S0e
σcMT = Yteσ(cMT−cMt).
If maxt≤u≤T Wu > Wt ≡ max0≤u≤t Wu, then
MT − Mt = maxt≤u≤T
Wu − Mt.
On the other hand, if maxt≤u≤T Wu ≤ Wt, then MT ≡ max0≤u≤T Wu = Wt ≡max0≤u≤t Wu; hence
MT − Mt = 0.
Consequently,
MT − Mt =
[max
t≤u≤TWu − Mt
]+
=
[max
t≤u≤T(Wu − Wt)− (Mt − Wt)
]+
.
Since St = S0eσcWt and Yt = S0e
σcMt , we get
σ(MT − Mt) =
[max
t≤u≤T(Wu − Wt)− log
Yt
St
]+
. (8.13)
Therefore, we can rewrite the value Vt as
Vt = E[e−r(T−t)YT |Ft]− E[e−r(T−t)ST |Ft] (8.14)
= e−rτ E
[Yt exp
[max
t≤u≤T(Wu − Wt)− log
Yt
St
]+ ∣∣∣∣Ft
]
−ertE[e−rT ST |Ft] (8.15)
≡ I1 + I2. (8.16)
The second term equalsI2 = −erte−rtSt = St.
For the first term, using the rule of taking what is known, we have
I1 = e−rτYtE
[exp
[max
t≤u≤T(Wu − Wt)− log
Yt
St
]+ ∣∣∣∣Ft
]. (8.17)
Since Wu − Wt is independent of Ft for t ≤ u ≤ T , we can use the IndependenceLemma to rewrite the conditional expectation in (8.17) as
g(St, Yt),
64
where
g(x, y) = E
[exp
[max
t≤u≤T(Wu − Wt)− log
y
x
]+]
. (8.18)
Putting all of these together yields
Vt = e−rτYtg(St, Yt)− St = v(t, St, Yt), (8.19)
wherev(t, x, y) = e−rτyg(x, y)− x. (8.20)
So the key is to compute the function g(x, y) determined by (8.18). Since Wu− Wt ∼Wu−t − W0 = Wu−t in the sense of distribution, we see that
maxt≤u≤T
(Wu − Wt) = max0≤u≤τ
Wu = Mτ
and hence
g(x, y) = E[exp
[σMτ − log
y
x
]+]
= E[exp
[σMτ − log
y
x
]+IσcMτ≥log y
x
]
+E[exp
[σMτ − log
y
x
]+
IσcMτ <log yx
]
= E[exp
[σMτ − log
y
x
]IσcMτ≥log y
x
]
+E[IσcMτ <log y
x
]
=x
yE
[exp
σMτ
IσcMτ≥log y
x
]+ P
[Mτ <
1
σlog
y
x
](8.21)
≡ g1(x, y) + g2(x, y).
Lemma 8.4. The cumulative distribution function of Mt under P is
PMt ≤ m = N
(m− αt√
t
)− e2αmN
(−m− αt√t
), m ≥ 0. (8.22)
The density function, under P, of the random variable Mt is given by
ffMt(m) =
2√2πt
exp
(− 1
2t(m− αt)2
)− 2αe2αmN
(−m− αt√t
), m ≥ 0 (8.23)
and zero for m < 0.
Proof. See Shreve [9], page 297.
65
To compute the second term of (8.21), setting in (8.22)
t := τ, m :=1
σlog
y
x,
we get, as α = 1σ(r − 1
2σ2),
m− αt√t
=1√τ
[1
σlog
y
x− ατ
]
=1
σ√
τ
[log
y
x−
(r − 1
2σ2
)τ
]
= − 1
σ√
τ
[log
x
y+
(r − 1
2σ2
)τ
]
= −δ−
(τ,
x
y
),
−m− αt√t
=1√τ
[− 1
σlog
y
x− ατ
]
=1
σ√
τ
[− log
y
x−
(r − 1
2σ2
)τ
]
= −δ−(τ,
y
x
),
e2αm = exp
(2α
σlog
y
x
)
= exp
[(2r
σ2− 1
)log
y
x
]
=(y
x
) 2rσ2−1
.
It then follows from (8.22) that
g2(x, y) = P[Mτ <
1
σlog
y
x
]= N
(−δ−
(τ,
x
y
))−
(y
x
) 2rσ2−1
N(−δ−
(τ,
y
x
)).
(8.24)To compute the first term of (8.21) (i.e. g1(x, y)), we use the density function fcMτ
66
given as in (8.23). We have
g1(x, y) =x
yE
[exp
σMτ
IσcMτ≥log y
x
]
=x
y
∫ ∞
1σ
log yx
eσmfcMτ(m)dm
=x
y
∫ ∞
1σ
log yx
2√2πτ
eσm− 12τ
(m−ατ)2dm
−x
y
∫ ∞
1σ
log yx
2αe(σ+2α)mN
(−m− ατ√τ
)dm (8.25)
≡ I1 − I2.
I1 =x
y
∫ ∞
1σ
log yx
2√2πτ
eσm− 12τ
(m−ατ)2dm
=x
y
∫ ∞
( 1σ
log yx−ατ)/
√τ
2√2π
eσ(ατ+√
τu)− 12u2
du (u =m− ατ√
τ)
=x
y
∫ ∞
( 1σ
log yx−ατ)/
√τ
2√2π
e−12(u−σ
√τ)2+στ(α+ 1
2σ)du
=2x
yeστ(α+ 1
2σ)
∫ ∞
1√τ (
1σ
log yx−ατ)−σ
√τ
1√2π
e−12v2
dv (v = u− σ√
τ).
However, since α = 1σ(r − 1
2σ2), we have
στ(α +1
2σ) = τ(σα +
1
2σ2) = rτ
and
1√τ
(1
σlog
y
x− ατ
)− σ
√τ =
1
σ√
τ
(log
y
x− στ(α + σ)
)
=1
σ√
τ
(log
y
x− στ(
1
σ(r − 1
2σ2) + σ)
)
=1
σ√
τ
(log
y
x− τ(r +
1
2σ2)
)
= − 1
σ√
τ
(log
x
y+ τ(r +
1
2σ2)
)
= −δ+
(τ,
x
y
).
67
We therefore obtain that
I1 =2x
yerτ
∫ ∞
−δ+(τ, xy )
1√2π
e−12v2
dv
=2x
yerτ
∫ δ+(τ, xy )
−∞
1√2π
e−12v2
dv
=2x
yerτN
(δ+
(τ,
x
y
)). (8.26)
We next compute the integral I2 in (8.25).
I2 =x
y
∫ ∞
1σ
log yx
2αe(σ+2α)mN
(−m− ατ√τ
)dm
=x
y
∫ ∞
1σ
log yx
2αe2rσ
mN
(−m− ατ√τ
)dm
for σ + 2α = 2rσ
. Recall
N(u) =
∫ u
−∞
1√2π
e−12v2
dv, u ∈ R.
We can rewrite
I2 =2αx√2πy
∫ ∞
1σ
log yx
∫ −m−ατ√τ
−∞e
2rσ
m− 12v2
dvdm.
We shall interchange the order of the above iterated integral. The region over whichthe integral is computed is described as
D :=
(m, v) : −∞ < v ≤ −m− ατ√
τ,
1
σlog
y
x≤ m < ∞
.
Since the intersection of the two lines
m =1
σlog
y
x, v =
−m− ατ√τ
is given by
m =1
σlog
y
x,
v = − 1√τ
(1
σlog
y
x+ ατ
)
= − 1
σ√
τ
(log
y
x+ (r − 1
2σ2)τ
)
= −δ−(τ,
y
x
),
68
we can describe the integral domain D is an alternative was as follows: Change thedescription of D to (we can draw a picture to get it easily)
D :=
(m, v) :
1
σlog
y
x≤ m ≤ −√τv − ατ, −∞ < v ≤ −δ−
(τ,
y
x
).
We therefore obtain
I2 =2αx√2πy
∫ −δ−(τ, yx)
−∞
∫ −√τv−ατ
1σ
log yx
e2rσ
m− 12v2
dmdv.
The inner integral is
∫ −√τv−ατ
1σ
log yx
e2rσ
m− 12v2
dm =σ
2re
2rσ
m− 12v2
∣∣∣∣−√τv−ατ
m= 1σ
log yx
=σ
2re−
12v2
[e−
2rσ
(√
τv+ατ) −(y
x
) 2rσ2
].
It follows that
I2 =2αx√2πy
· σ
2r
∫ −δ−(τ, yx)
−∞e−
12v2
[e−
2rσ
(√
τv+ατ) −(y
x
) 2rσ2
]dv
=ασx
ry
∫ −δ−(τ, yx)
−∞
1√2π
e−12v2− 2r
σ(√
τv+ατ)dv (8.27)
−ασx
ry
(y
x
) 2rσ2
∫ −δ−(τ, yx)
−∞
1√2π
e−12v2
dv. (8.28)
Since
−1
2v2 − 2r
σ(√
τv + ατ) = −1
2
(v +
2r√
τ
σ
)2
+2rτ(r − ασ)
σ2
= −1
2
(v +
2r√
τ
σ
)2
+ rτ,
we find the integral in (8.27) is, via the substitution u := v + 2r√
τσ
,
∫ −δ−(τ, yx)
−∞
1√2π
e−12v2− 2r
σ(√
τv+ατ)dv = erτ
∫ −δ−(τ, yx)+ 2r
√τ
σ
−∞
1√2π
e−12u2
du
= erτN
(−δ−
(τ,
y
x
)+
2r√
τ
σ
)
= erτN
(δ+
(τ,
x
y
))
69
since
−δ−(τ,
y
x
)+
2r√
τ
σ= − 1
σ√
τ
[log
y
x+
(r − 1
2σ2
)τ
]+
2r√
τ
σ
=1
σ√
τ
[− log
y
x−
(r − 1
2σ2
)τ + 2rτ
]
=1
σ√
τ
[log
x
y+
(r +
1
2σ2
)τ
]
= δ+
(τ,
x
y
).
Back to (8.27) and (8.28), we get
I2 =ασx
ryerτN
(δ+
(τ,
x
y
))− ασ
r
(y
x
) 2rσ2−1
N(−δ−
(τ,
y
x
)). (8.29)
So we obtain
g1(x, y) = I1 − I2
=2x
yerτN
(δ+
(τ,
x
y
))− ασx
ryerτN
(δ+
(τ,
x
y
))
+ασ
r
(y
x
) 2rσ2−1
N(−δ−
(τ,
y
x
)). (8.30)
Combining (8.24) and (8.30) yields
g(x, y) = g1(x, y) + g2(x, y)
=2x
yerτN
(δ+
(τ,
x
y
))− ασx
ryerτN
(δ+
(τ,
x
y
))
+ασ
r
(y
x
) 2rσ2−1
N(−δ−
(τ,
y
x
))
+N
(−δ−
(τ,
x
y
))−
(y
x
) 2rσ2−1
N(−δ−
(τ,
y
x
)). (8.31)
By (8.20), we get
v(t, x, y) = e−rτyg(x, y)− x
= e−rτy
2x
yerτN
(δ+
(τ,
x
y
))− ασx
ryerτN
(δ+
(τ,
x
y
))
+ασ
r
(y
x
) 2rσ2−1
N(−δ−
(τ,
y
x
))
+N
(−δ−
(τ,
x
y
))−
(y
x
) 2rσ2−1
N(−δ−
(τ,
y
x
)) − x. (8.32)
70
After simplifications in (8.32), noticing ασr
= 1− σ2
2r, we get the following formula
for 0 ≤ t < T, 0 < x ≤ y,
v(t, x, y) =
(1 +
σ2
2r
)xN
(δ+
(τ,
x
y
))+ e−rτyN
(−δ−
(τ,
x
y
))
−σ2
2re−rτ
(x
y
) 2rσ2
xN(−δ−
(τ,
y
x
))− x. (8.33)
71
9 Asian Options
An Asian option is a path-dependent option whose payoff depends on the averageof the underlying asset price over a certain time interval. It can be European orAmerican. The time interval can be the entire interval of the option’s life from theinitiation to the expiration, or beginning from some time later than the initiationuntil the option’s expiration. The average can be arithmetic or geometric.
We assume that the underlying asset (stock) S(t) follows the geometric Brownianmotion:
dS(t) = rS(t)dt + σS(t)dW (t),
where W (t), 0 ≤ t ≤ T , is a Brownian motion under the risk-neutral probabilitymeasure P, and the interest rate r and the volatility σ are both assumed to beconstant.
Introducing the processes Y (t) and I(t) defined by
Y (t) =
∫ t
0
S(u)du, I(t) =
∫ t
0
log S(u)du,
we can classify Asian options as follows
1. Arithmetic average floating strike call option:
V (T ) =
(S(T )− 1
TY (T )
)+
.
2. Arithmetic average fixed strike call option:
V (T ) =
(1
TY (T )−K
)+
,
where K is the strike price.
3. Geometric average floating strike call option:
V (T ) =(S(T )− eI(T )/T
)+.
4. Geometric average fixed strike call option:
V (T ) =(eI(T )/T −K
)+.
The reasons to introduce Asian options are probably two:
72
• It costs less than its European counterparts. For instance, consider the Asianarithmetic average fixed strike call option. The value at time t of this option is
V (t) = E
[e−r(T−t)
(1
T
∫ T
0
S(u)du−K
)+ ∣∣∣∣F(t)
]
where E is the expectation under the risk-neutral probability measure P. It isknown that
V (t) ≤ E [e−r(T−t)(S(T )−K)+|F(t)
].
The right side is the time t value of the European call option with expirationtime T and strike price of K.
• It makes more difficult for anyone to significantly affect the payoff by manipu-lating the underlying asset price.
73
9.1 Valuation
The valuation of Asian options is always complicated. No closed form solution exists,in general. The difficulties come from the path-dependence on the stock price.
Let W (t), 0 ≤ t ≤, be a Brownian motion on the probability space
(Ω,F ,P,F(t))
where the filtration F(t) is generated by W (t) satisfying the usual conditions, andwhere P is the risk-neutral probability measure.
74
9.1.1 European Arithmetic Average Fixed Strike Asian Call
Consider a European fixed strike Asian call option with payoff
V (T ) =
(1
T
∫ T
0
S(u)du−K
)+
.
Let V (t) denote the time t value of the Asian call. That is,
V (t) = E
[e−r(T−t)
(1
T
∫ T
0
S(u)du−K
)+ ∣∣∣∣F(t)
].
In general, there is no closed form pricing formula for Asian options, except somespecial cases. For instance, if
1
T
∫ T
0
S(u)du ≥ K a. s.
then we have
V (t) =1
rT
re−r(T−t)
∫ t
0
S(u)du + S(t)(1− e−r(T−t)
)−Ke−r(T−t).
Since V (T ) depends on the whole path of the stock S, we can’t use the Markovproperty to claim that V (t) is a function of t and S(t) (only depending on the stockprice at time t). A (standard) approach is to introduce the augment process
Y (t) =
∫ t
0
S(u)du.
Thus (S, Y ) satisfies a two-dimensional SDE
dS(t) = S(t)(rdt + σdW (t)),
dY (t) = S(t)dt.
Thus (S(t), Y (t)) is a two-dimensional Markov process. Now the payoff
V (T ) =
(1
TY (T )−K
)+
depends only on T and (S(T ), Y (T )), the final value of the Markov process (S(t), Y (t)).Hence we can use Markov property to conclude that there exists a function v(t, x, y)such that
V (t) = v(t, S(t), Y (t)) = E[e−(T−t)V (T )
∣∣F(t)]
75
and, since e−rtv(t, S(t), Y (t)) is a two-dimensional P-martingale, v(t, x, y) satisfies thepartial differential equation
vt(t, x, y) + rxvx(t, x, y) + xvy(t, x, y) +1
2σ2x2vxx(t, x, y) = rv(t, x, y)
for 0 ≤ t < T, x ≥ 0, y ∈ R, with the boundary conditions
v(t, 0, y) = e−r(T−t)(
yT−K
)+, 0 ≤ t < T, y ∈ R,
limy↓−∞ v(t, x, y) = 0, 0 ≤ t < T, x ≥ 0,
v(T, x, y) =(
yT−K
)+, y ∈ R.
This boundary value problem for the Asian partial differential equation does not havea closed form solution; numerical methods need.
Through appropriate change of variables, the above partial differential equationcan have simpler form. Indeed, setting
u(t, x, y) = xmeqtv
(T − 2t
σ2, x,
2y
σ2
),
wherem =
r
σ2, q = m + m2
we can reduce the above PDE to
x2∂2u
∂x2+ x
∂u
∂y= x2∂u
∂t
and the final and boundary conditions are reduced to
u(0, x, y) = xm(
2yσ2 −K
)+, x ∈ R+, y ∈ R,
limy↓−∞ u(t, x, y) = 0, 0 ≤ t < T, x ≥ 0,
u(t, 0, y) = 0, 0 ≤ t < T, y ∈ R.
76
9.1.2 European Geometric Average Fixed Strike Asian Call
A European geometric average fixed strike Asian call has payoff
V (T ) =(eI(T )/T −K
)+
where
I(t) =
∫ t
0
log S(u)du, 0 < t ≤ T.
The time t value of this option is
V (t) = E[e−r(T−t)V (T )|F(t)
]
= E[e−r(T−t)
(eI(T )/T −K
)+ |F(t)].
Similarly to the arithmetic average situation, being solution to a two-dimensionalSDE, the pair (S(t), I(t)) has the Markov property. Moreover, since the payoff V (T )depends only on the final time T and the final value of (S(t), I(t)), there exists afunction v(t, x, I) such that
V (t) = v(t, S(t), I(t))
and since e−rtv(t, S(t), I(t)) is a martingale, we see that v satisfies the partial differ-ential equation
−rv + vt + rxvx + (log x)vI +1
2σ2x2vxx = 0
with the final condition
v(T, x, I) =(eI/T −K
)+, I, x ≥ 0.
(Here all partial derivatives are evaluated at (t, x, I).)Again appropriate change of variables gives a simpler form of the above PDE.
Indeed, letting
x := x =
√2
σlog x, y =
√2
σI
and
u(t, x, y) = exp
2r − σ2
2√
2σx +
(2r + σ2
2√
2σ
)2
t
v
(T − t, e
σ√2t,
σ√2y
),
we reduce the above PDE to the Kolmogorov equation
∂2u
∂x2+ x
∂u
∂y=
∂u
∂t
77
with the initial condition
u(0, x, y) = exp
2r − σ2
2√
2σx
(exp
σy√2T
−K
)+
.
We can evaluate the value at time 0 of the option as follows. Since the stock Sfollows the GBM:
S(t) = S(0) exp
σW (t) +
(r − 1
2σ2
)t
,
we have
log S(t) = log S(0) + σW (t) +
(r − 1
2σ2
)t.
It follows that
I(T ) =
∫ T
0
log S(u)du = T log S(0) + σ
∫ T
0
W (u)du +1
2(r − 1
2σ2)T 2.
Noticing ∫ T
0
W (u)du = T
√T
3Z, Z ∼ N(0, 1),
we deduce that the value at time 0 of the option is
V (0) = e−rTE
[(exp
I(T )
T
−K
)+]
= e−rTE
[(S(0) exp
σ
T
∫ T
0
W (u)du +1
2(r − 1
2σ2)T
−K
)+]
= e−rTE
[(S(0) exp
σ
√T
3Z +
1
2(r − 1
2σ2)T
−K
)+]
= e−rT 1√2π
∫ ∞
−∞
(S(0) exp
σ
√T
3u +
1
2(r − 1
2σ2)T
−K
)+
e−12u2
du.
The integrand is positive if and only if
u > − log(S(0)/K) + (r − 12σ2)(T/2)
σ√
T/3=: −ρ2.
We obtainV (0) = S(0)e−(r+ 1
6σ2)T
2 N(ρ1)−Ke−rT N(ρ2),
where
ρ1 =log S(0)
K+
(r + 1
6σ2
)T2
σ√
T/3,
78
ρ2 = ρ1 − σ√
T/3 =log S(0)
K+
(r − 1
2σ2
)T2
σ√
T/3
We next compute the time-t value, Vt, given by
Vt = e−r(T−t)E
[(exp
(1
T
∫ T
0
log Sudu
)−K
)+∣∣∣∣∣Ft
]. (9.1)
Since, for u ≥ t,
Su = Steσ(fWu−fWt)+(r− 1
2σ2)(u−t)
we have
log Su = log St + σ(Wu − Wt) + (r − 1
2σ2)(u− t).
1
T
∫ T
0
log Sudu =1
T
∫ t
0
log Sudu +1
T
∫ T
t
log Sudu
=1
TI(t) +
T − t
Tlog St
+1
T
∫ T
t
[σ(Wu − Wt) + (r − 1
2σ2)(u− t)
]du
=1
TI(t) +
T − t
Tlog St +
1
2T(r − 1
2σ2)(T − t)2 +
σ
T
∫ T
t
Wu−tdu
=1
TI(t) +
T − t
Tlog St +
1
2T(r − 1
2σ2)(T − t)2 +
σ
T
∫ T−t
0
Wvdv.
But we have ∫ T−t
0
Wvdv =
√(T − t)3
3Z
where Z ∼ N(0, 1) is independent of Ft. Therefore, we get
1
T
∫ T
0
log Sudu =1
TI(t) +
T − t
Tlog St +
1
2T(r − 1
2σ2)(T − t)2 +
σ
T
√(T − t)3
3Z.
By (9.1), we get
Vt = e−r(T−t)E
[(S
T−tT
t exp
(1
TI(t) +
1
2T(r − 1
2σ2)(T − t)2 +
σ
T
√(T − t)3
3Z
)−K
)+∣∣∣∣∣Ft
]
= v(t, x, y),
where x = St an y = I(t), and
v(t, x, y) = e−r(T−t)E
[(x
T−tT exp
(1
Ty +
1
2T(r − 1
2σ2)(T − t)2 +
σ
T
√(T − t)3
3Z
)−K
)+].
79
We need to determine the solution to the inequality:
xT−t
T exp
(1
Ty +
1
2T(r − 1
2σ2)(T − t)2 +
σ
T
√(T − t)3
3z
)> K.
Namely,
z > − log x1−t/T
K+ 1
Ty + 1
2T(r − 1
2σ2)(T − t)2
σT
√(T−t)3
3
=: −d2(t, x, y).
For brevity, put
α(t, y) =1
Ty +
1
2T(r − 1
2σ2)(T − t)2.
We then get
v(t, x, y) = e−r(T−t)E
[(x
T−tT exp
(α(t, y) +
σ
T
√(T − t)3
3Z
)−K
)+]
= e−r(T−t)
∫ ∞
−∞
(x
T−tT exp
(α(t, y) +
σ
T
√(T − t)3
3z
)−K
)+1√2π
e−12z2
dz
= e−r(T−t)
∫ ∞
−d(t,x,y)
(x
T−tT exp
(α(t, y) +
σ
T
√(T − t)3
3z
)−K
)1√2π
e−12z2
dz
= e−r(T−t)
∫ ∞
−d(t,x,y)
xT−t
T exp
(α(t, y) +
σ
T
√(T − t)3
3z
)1√2π
e−12z2
dz
−e−r(T−t)K
∫ ∞
−d(t,x,y)
1√2π
e−12z2
dz
= J1 − J2.
We have
J2 = e−r(T−t)K
∫ ∞
−d2(t,x,y)
1√2π
e−12z2
dz
= e−r(T−t)K
∫ d2(t,x,y)
−∞
1√2π
e−12z2
dz
= e−r(T−t)KN(d2(t, x, y)).
80
J1 = e−r(T−t)
∫ ∞
−d2(t,x,y)
xT−t
T exp
(α(t, y) +
σ
T
√(T − t)3
3z
)1√2π
e−12z2
dz
= e−r(T−t)xT−t
T
∫ d2(t,x,y)
−∞
1√2π
exp
(α(t, y)− σ
T
√(T − t)3
3z − 1
2z2
)dz
= e−r(T−t)xT−t
T exp
(σ2(T − t)3
6T 2+ α(t, y)
)
×∫ d2(t,x,y)
−∞
1√2π
exp
−1
2
(z +
σ
T
√(T − t)3
3
)2 dz
Making a change of variable by
u = z +σ
T
√(T − t)3
3
and setting
d1(t, x, y) = d2(t, x, y) +σ
T
√(T − t)3
3,
we get
J1 = e−r(T−t)xT−t
T exp
(σ2(T − t)3
6T 2+ α(t, y)
) ∫ d1(t,x,y)
−∞
1√2π
exp
(−1
2u2
)du
= e−r(T−t)xT−t
T exp
(σ2(T − t)3
6T 2+ α(t, y)
)N(d1(t, x, y))
Therefore, we obtain
v(t, x, y) = e−r(T−t)xT−t
T exp
(σ2(T − t)3
6T 2+ α(t, y)
)N(d1(t, x, y))−e−r(T−t)KN(d2(t, x, y))
(9.2)and
Vt = e−r(T−t)ST−t
Tt exp
(σ2(T − t)3
6T 2+ α(t, It)
)N(d1(t, St, It))−e−r(T−t)KN(d2(t, St, It))
(9.3)
Consider an Asian call option with payoff
V (T ) =
(exp
(1
c
∫ T
T−c
log S(u)du
)−K
)+
(1)
81
where 0 < c ≤ T . If c = T , then the geometric average is taken over the entireinterval of the option
V (T ) =
(exp
(1
T
∫ T
0
log S(u)du
)−K
)+
.
Let’s find the closed-form formula with payoff (1) for a general time t ∈ [0, T ].For t ≤ u ≤ T , we have
S(u) = S(t) expσ(W (u)−W (t)) + (r − 1
2σ2)(u− t),
log S(u) = log S(t) + σ(W (u)−W (t)) + (r − 1
2σ2)(u− t),
1
c
∫ T
t
log S(u)du =T − t
clog S(t) +
σ
c
∫ T
t
(W (u)−W (t))du
+1
c(r − 1
2σ2)
∫ T
t
(u− t)du
=T − t
clog S(t) +
σ
c
∫ T
t
(W (u)−W (t))du +(T − t)2
2c(r − 1
2σ2).
Let
X =σ
c
∫ T
t
(W (u)−W (t))du.
Then X is normally distributed with mean zero. Note that X is independent of F(t).We next compute its variance. Noting
( c
σX
)2
=
(∫ T
t
(W (u)−W (t))du
)2
=
∫ T
t
∫ T
t
(W (u)−W (t))(W (s)−W (t))dsdu
=
∫ T
t
∫ T
t
(W (u)W (s)−W (u)W (t)−W (t)W (s)−W 2(t))dsdu
82
we deduce that (noting E[W (u)W (v)] = min(u, v))
Var(X) = E[X2]
=σ2
c2
∫ T
t
∫ T
t
(min(u, s)− t)dsdu
=σ2
c2
∫ T
t
(∫ u
t
(s− t)ds +
∫ T
u
(u− t)ds
)du
=σ2
c2
∫ T
t
(1
2(u− t)2 − (u− t)(u− T )
)du
=σ2
c2
(1
6(T − t)3 − 1
3(T 3 − t3) +
1
2(t + T )(T 2 − t2)− tT (T − t)
)
=σ2
c2· 1
3(T − t)3.
Thus, in the sense of distribution, we have
X =σ
c
√T − t
3(T − t)Z, Z = N(0, 1).
It follows that
exp
(1
c
∫ T
t
log S(u)du
)= S(t)
T−tc exp
(σ
c
√T − t
3(T − t)Z +
(T − t)2
2c(r − 1
2σ2)
).
Hence
V (t) = e−(T−t)E
[(exp
(1
c
∫ T
T−c
log S(u)du
)−K
)+ ∣∣∣∣F(t)
]
= e−(T−t)E
[(exp
(1
c
∫ t
T−c
log S(u)du
)exp
(1
c
∫ T
t
log S(u)du
)−K
)+ ∣∣∣∣F(t)
]
= e−(T−t)E[
S(t)T−t
c exp
(1
c
∫ t
T−c
log S(u)du
)
× exp
(σ
c
√T − t
3(T − t)Z +
(T − t)2
2c(r − 1
2σ2)
)−K
+∣∣∣∣F(t)
]
We need the following lemma.
Lemma.
E[γ(exp(σZ+µ)−k)+] = γ exp(µ+1
2σ2)N
(log(γ/k) + µ + σ2
σ
)−kN
(log(γ/k) + µ
σ
).
83
Here
N(x) =1√2π
∫ x
−∞e−
12y2
dy.
We therefore obtain that (for T − c ≤ t ≤ T )
V (t) = e−r(T−t)
[S(t)
T−tc exp
(1
c
∫ t
T−c
log S(u)du
)
× exp
((T − t)2
2c(r − 1
2σ2) +
1
2
σ2(T − t)3
3c2
)N(d1)−KN(d2)
],
where
d1 =c
σ(T − t)
√3
T − t
(log
S(t)T−t
c
K+
1
c
∫ t
T−c
log S(u)du
)+
(T − t)2
2c(r − 1
2σ2) +
σ2(T − t)3
3c2
,
and
d2 = d1 − σ
c
√(T − t)2
3.
Next we discuss the case where 0 ≤ t ≤ T − c. Again we have for T − c ≤ u ≤ T ,
S(u) = S(t) expσ(W (u)−W (t)) + (r − 1
2σ2)(u− t),
log S(u) = log S(t) + σ(W (u)−W (t)) + (r − 1
2σ2)(u− t),
1
c
∫ T
T−c
log S(u)du = log S(t) +σ
c
∫ T
T−c
(W (u)−W (t))du
+1
c(r − 1
2σ2)
∫ T
T−c
(u− t)du
= log S(t) +σ
c
∫ T
T−c
(W (u)−W (t))du + (r − 1
2σ2)
1
2c
[(T − t)2 − (T − c− t)2
]
= log S(t) +σ
c
∫ T
T−c
(W (u)−W (t))du + (r − 1
2σ2)(T − t− 1
2c).
Let
X =σ
c
∫ T
T−c
(W (u)−W (t))du.
Then X is normally distributed and independent of F(t) since t ≤ T − c.
84
( c
σX
)2
=
(∫ T
T−c
(W (u)−W (t))du
)2
=
∫ T
T−c
∫ T
T−c
(W (u)−W (t))(W (s)−W (t))dsdu
=
∫ T
T−c
∫ T
T−c
(W (u)W (s)−W (u)W (t)−W (t)W (s)−W 2(t))dsdu
we deduce that (noting E[W (u)W (v)] = min(u, v))
Var(X) = E[X2]
=σ2
c2
∫ T
T−c
∫ T
T−c
(min(u, s)− t)dsdu
=σ2
c2
∫ T
T−c
(∫ u
T−c
(s− t)ds +
∫ T
u
(u− t)ds
)du
=σ2
c2
∫ T
T−c
(1
2[(u− t)2 − (T − c− t)2]− (u− t)(u− T )
)du
=σ2
c2
(1
6[(T − t)3 − (T − t− c)3]− c
2(T − t− c)2
−1
3(T 3 − (T − c)3) +
1
2(t + T )(T 2 − (T − c)2)− ctT
)
=σ2
c2
(1
6(T − t)3 − 1
6(T − t− c)2(T − t + 2c) +
1
2c2(T − t)− 1
3c3
)
=: ξ2(t, T ).
Thus,X = ξ(t, T )Z,
where Z ∼ N(0, 1) is independent of F(t). Consequently,
V (t) = e−r(T−t)E
[(exp
(1
c
∫ T
T−c
log S(u)du
)−K
)+ ∣∣∣∣F(t)
]
= e−r(T−t)E
[(S(t) exp
(X + (r − 1
2σ2)(T − t− 1
2c)
)−K
)+]
= e−r(T−t)E
[(S(t) exp
(ξ(t, T )Z + (r − 1
2σ2)(T − t− 1
2c)
)−K
)+]
.
By the Lemma we obtain (for 0 ≤ t ≤ T − c)
V (t) = e−r(T−t)
[S(t) exp
((r − 1
2σ2)(T − t− 1
2c) +
1
2ξ2(t, T )
)N(d1)−KN(d2)
],
85
where
d1 =1
ξ(t, T )
[log
S(t)
K+ (r − 1
2σ2)(T − t− 1
2c) + ξ2(t, T )
]
andd2 = d1 − ξ(t, T ).
86
9.1.3 Change of Numeraire
The difficulty in pricing Asian options arises from the path-dependence of the payoff.We next consider a European style Asian call option with fixed strike payoff
V (T ) =
(1
c
∫ T
T−c
S(u)du−K
)+
where c is constant such that 0 < c ≤ T.
Definition. (Asian Forward Contract) An Asian forward contract has the followingpayoff at the expiration time T
F =1
c
∫ T
T−c
S(u)du−K.
Clearly, the payoff of the Asian option if the positive part F+ of F .
Proposition. The Asian forward contract can be replicated.
Proof. Define λ by
λ(t) =
0, 0 ≤ t ≤ T − c,tc, T − c ≤ t ≤ T,
and q(t) by
q(t) = e−rT
∫ T
t
ersdλ(s).
That is,
q(t) =
1cr
(1− r−rc) , 0 ≤ t ≤ T − c,1cr
[1− e−r(T−t)
], T − c ≤ t ≤ T,
Out self-financing portfolio has an initial value given by
X(0) =1
rc
(1− e−rc
)S(t)− e−rT K,
and follows the SDE
dX(t) = q(t)dS(t) + r(X(t)− q(t)S(t))dt.
It follows that
X(t) =
1rc
(1− e−rc) S(t)− e−rT K, 0 ≤ t ≤ T − c,1rc
(1− e−r(T−t))S(t) + e−r(T−t)[
1c
∫ t
T−cS(u)du−K
], T − c ≤ t ≤ T.
In particular,
X(T ) =1
c
∫ T
T−c
S(u)du−K.
87
Theorem. (Pricing Formula) The time t value of the Asian option with the payoffgiven as above is
V (t) = S(t)EQ[X+(T )
S(T )
∣∣∣∣F(t)
],
where Q is the equivalent measure defined by
dQdP
∣∣∣∣t
=S(t)
S(0)ert=: Z(t).
Proof.
V (t) = E[e−r(T−t)V (T )|F(t)
]
= ertE[e−rT X+(T )|F(t)
]
=S(t)
Z(t)E
[Z(T )
X+(T )
S(T )
∣∣∣∣F(t)
]
= S(t)EQ[X+(T )
S(T )
∣∣∣∣F(t)
].
88
9.2 The Partial Differential Equation for Asian Options
Let
Y (t) =X(t)
S(t).
It is not hard to find, by Ito’s Lemma, that Y (t) is a martingale under Q. Indeed,
dY (t) = σ(q(t)− Y (t))dWQ(t),
where WQ = W (t)− σt is a Brownian motion by Girsanov’s theorem.The Markov property implies the existence of a function g(t, x) such that
g(t, Y (t)) = EQ[Y +(T )|F(t)]
with the boundary condition
g(T, y) = y+, y ∈ R.
By Ito’s Lemma we get
dg(t, Y (t)) = gt(t, Y (t))dt + gy(t, Y (t))dY (t) +1
2gyy(t, Y (t))dY (t)dY (t)
=
[gt(t, Y (t)) +
1
2σ2(q(t)− Y (t))2gyy(t, Y (t))
]dt
+σ(q(t)− Y (t))gy(t, Y (t))dWQ(t).
Since g(t, Y (t)) is martingale, the dt term in the differential dg(t, Y (t)) has to be zero.This implies that g(t, x) satisfies the partial differential equation:
gt(t, x) +1
2σ2(q(t)− x)2gxx(t, x) = 0, 0 ≤ t < T, x ∈ R.
The final and boundary conditions are
g(T, x) = x+, x ∈ R,
limx→−∞
g(t, x) = 0, 0 ≤ t ≤ T,
limx→∞
[g(t, x)− x] = 0, 0 ≤ t ≤ T.
89
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