4. Option pricing models under the Black-Scholes framework
Riskless hedging principle
Writer of a call option – hedges his exposure by holding certain units
of the underlying asset in order to create a riskless portfolio.
In an efficient market with no riskless arbitrage opportunity, a riskless
portfolio must earn rate of return equals the riskless interest rate.
1
Dynamic replication strategy
How to replicate an option dynamically by a portfolio of the riskless
asset in the form of money market account and the risky underlying
asset?
The cost of constructing the replicating portfolio gives the fair price
of an option.
Risk neutrality argument
The two tradeable securities, option and asset, are hedgeable with
each other. Hedgeable securities should have the same market price
of risk.
2
Black-Scholes’ assumptions on the financial market
(i) Trading takes place continuously in time.
(ii) The riskless interest rate r is known and constant over time.
(iii) The asset pays no dividend.
(iv) There are no transaction costs in buying or selling the asset or
the option, and no taxes.
(v) The assets are perfectly divisible.
(vi) There are no penalties to short selling and the full use of pro-
ceeds is permitted.
(vii) There are no arbitrage opportunities.
3
The stochastic process of the asset price S is assumed to follow the
Geometric Brownian motion
dS
S= ρ dt+ σ dZ.
Consider a portfolio which involves short selling of one unit of a
European call option and long holding of ∆ units of the underlying
asset. The value of the portfolio Π is given by
Π = −c+ ∆S,
where c = c(S, t) denotes the call price.
Since both c and Π are random variables, we apply the Ito lemma
to compute their stochastic differentials as follows:
dc =∂c
∂tdt+
∂c
∂SdS +
σ2
2S2 ∂
2c
∂S2dt
4
dΠ = −dc+ ∆ dS
=
(−∂c∂t
− σ2
2S2 ∂
2c
∂S2
)dt+
(∆ − ∂c
∂S
)dS
=
[−∂c∂t
− σ2
2S2 ∂
2c
∂S2+
(∆ − ∂c
∂S
)ρS
]dt+
(∆ − ∂c
∂S
)σS dZ.
Why the differential Sd∆ does not enter into dΠ? By virtue of
the assumption of following a self-financing trading strategy, the
contribution to dΠ due to Sd∆ is offset by the accompanying pur-
chase/sale of units of options.
5
If we choose ∆ =∂c
∂S, then the portfolio becomes a riskless hedge in-
stantaneously
(since
∂c
∂Schanges continuously with time
). By virtue
of “no arbitrage”, the hedged portfolio should earn the riskless in-
terest rate.
By setting dΠ = rΠ dt
dΠ =
(−∂c∂t
− σ2
2S2 ∂
2c
∂S2
)dt = rΠ dt = r
(−c+ S
∂c
∂S
)dt.
Black-Scholes equation:∂c
∂t+σ2
2S2 ∂
2c
∂S2+ rS
∂c
∂S− rc = 0.
Terminal payoff: c(S, T) = max(S −X,0).
The parameter ρ (expected rate of return) does not appear in the
governing equation and the auxiliary condition.
6
• 5 parameters in the option model: S, T,X, r and σ; only σ is
unobservable.
Deficiencies in the model
1. Geometric Brownian motion assumption? Actual asset price
dynamics is much more complicated.
2. Continuous hedging at all times
— trading usually involves transaction costs.
3. Interest rate should be stochastic instead of deterministic.
7
Dynamic replication strategy (Merton’s approach)
QS(t) = number of units of asset
QV (t) = number of units of option
MS(t) = dollar value of QS(t) units of asset
MV (t) = dollar value of QV (t) units of option
M(t) = value of riskless asset invested in money market account
• Construction of a self-financing and dynamically hedged portfo-
lio containing risky asset, option and riskless asset.
8
• Dynamic replication: Composition is allowed to change at all
times in the replication process.
• The self-financing portfolio is set up with zero initial net invest-
ment cost and no additional funds added or withdrawn after-
wards.
The zero net investment condition at time t is
Π(t) = MS(t) +MV (t) +M(t)
= QS(t)S +QV (t)V +M(t) = 0.
Differential of option value V :
dV =∂V
∂tdt+
∂V
∂SdS +
σ2
2S2∂
2V
∂S2dt
=
(∂V
∂t+ ρS
∂V
∂S+σ2
2S2∂
2V
∂S2
)dt+ σS
∂V
∂SdZ.
9
Formally, we write
dV
V= ρV dt+ σV dZ
where
ρV =
∂V∂t + ρS∂V∂S + σ2
2 S2∂2V∂S2
Vand σV =
σS∂V∂SV
.
dΠ(t) = [QS(t) dS +QV (t) dV + rM(t) dt]
+ [SdQS(t) + V dQS(t) + dM(t)]︸ ︷︷ ︸zero due to self-financing trading strategy
10
The instantaneous portfolio return dΠ(t) can be expressed in terms
of MS(t) and MV (t) as follows:
dΠ(t) = QS(t) dS +QV (t) dV + rM(t) dt
= MS(t)dS
S+MV (t)
dV
V+ rM(t) dt
= [(ρ− r)MS(t) + (ρV − r)MV (t)] dt
+ [σMS(t) + σVMV (t)] dZ.
We then choose MS(t) and MV (t) such that the stochastic term
becomes zero.
From the relation:
σMS(t) + σVMV (t) = σSQS(t) +σS∂V∂SV
V QV (t) = 0,
we obtain
QS(t)
QV (t)= −∂V
∂S.
11
Taking the choice of QV (t) = −1, and knowing
0 = Π(t) = −V + ∆S +M(t)
we obtain
V = ∆S +M(t), where ∆ =∂V
∂S.
Since the replicating portfolio is self-financing and replicates the
terminal payoff, by virtue of no-arbitrage argument, the initial cost
of setting up this replicating portfolio of risky asset and riskless asset
must be equal to the value of the option being replicated.
12
The dynamic replicating portfolio is riskless and requires no net
investment, so dΠ(t) = 0.
0 = [(ρ− r)MS(t) + (ρV − r)MV (t)] dt.
PuttingQS(t)
QV (T)= −∂V
∂S, we obtain
(ρ− r)S∂V
∂S= (ρV − r)V.
Replacing ρV by
[∂V
∂t+ ρS
∂V
∂S+σ2
2S2∂
2V
∂S2
]/V , we obtain the Black-
Scholes equation
∂V
∂t+σ2
2S2∂
2V
∂S2+ rS
∂V
∂S− rV = 0.
13
Alternative perspective on risk neutral valuation
From ρV =
∂V∂t + ρS∂V∂S + σ2
2 S2∂2V∂S2
V, we obtain
∂V
∂t+σ2
2S2∂
2V
∂S2+ ρS
∂V
∂S− ρV V = 0.
We need to calibrate the parameters ρ and ρV , or find some other
means to avoid such nuisance.
Combining σV =σS∂V∂SV
and (ρ− r)S∂V
∂S= (ρV − r)V , we obtain
ρV − r
σV︸ ︷︷ ︸λV
=ρ− r
σ︸ ︷︷ ︸λS
⇒ Black-Scholes equation.
14
• The market price of risk is the rate of extra return above r per
unit risk.
• Two hedgeable securities should have the same market price of
risk.
• The Black-Scholes equation can be obtained by setting ρ =
ρV = r (implying zero market price of risk).
• In the world of zero market price of risk, investors are said to be
risk neutral since they do not demand extra returns on holding
risky assets.
• Option valuation can be performed in the risk neutral world by
artificially taking the expected rate of returns of the asset and
option to be r.
15
Arguments of risk neutrality
• We find the price of a derivative relative to that of the underlying
asset ⇒ mathematical relationship between the prices is invariant
to the risk preference.
• Be careful that the actual rate of return of the underlying as-
set would affect the asset price and thus indirectly affects the
absolute derivative price.
• We simply use the convenience of risk neutrality to arrive at the
mathematical relationship but actual risk neutrality behaviors of
the investors are not necessary in the derivation of option prices.
16
“How we came up with the option formula?” — Black (1989)
⋆ It started with tinkering and ended with delayed recognition.
⋆ The expected return on a warrant should depend on the risk of
the warrant in the same way that a common stock’s expected
return depends on its risk.
⋆ I spent many, many days trying to find the solution to that (dif-
ferential) equation. I have a PhD in applied mathematics, but
had never spent much time on differential equations, so I didn’t
know the standard methods used to solve problems like that. I
have an A.B. in physics, but I didn’t recognize the equation as
a version of the heat equation, which has well-known solutions.
17
Continuous time securities model
• Uncertainty in the financial market is modeled by the filtered
probability space (Ω,F , (Ft)0≤t≤T , P), where Ω is a sample space,
F is a σ-algebra on Ω, P is a probability measure on (Ω,F),Ftis the filtration and FT = F.
• There are M + 1 securities whose price processes are modeled
by adapted stochastic processes Sm(t), m = 0,1, · · · ,M .
• We define hm(t) to be the number of units of the mth security
held in the portfolio.
• The trading strategy H(t) is the vector stochastic process (h0(t)
h1(t) · · ·hM(t))T , where H(t) is a (M+1)-dimensional predictable
process since the portfolio composition is determined by the in-
vestor based on the information available before time t.
18
• The value process associated with a trading strategy H(t) is
defined by
V (t) =M∑
m=0
hm(t)Sm(t), 0 ≤ t ≤ T,
and the gain process G(t) is given by
G(t) =M∑
m=0
∫ t
0hm(u) dSm(u), 0 ≤ t ≤ T.
• Similar to that in discrete models, H(t) is self-financing if and
only if
V (t) = V (0) +G(t).
19
• We use S0(t) to denote the money market account process that
grows at the riskless interest rate r(t), that is,
dS0(t) = r(t)S0(t) dt.
• The discounted security price process S∗m(t) is defined as
S∗m(t) = Sm(t)/S0(t), m = 1,2, · · · ,M.
• The discounted value process V ∗(t) is defined by dividing V (t)
by S0(t). The discounted gain process G∗(t) is defined by
G∗(t) = V ∗(t) − V ∗(0).
20
No-arbitrage principle and equivalent martingale measure
• A self-financing trading strategy H represents an arbitrage op-
portunity if and only if (i) G∗(T) ≥ 0 and (ii) EPG∗(T) > 0 where
P is the actual probability measure of the states of occurrence
associated with the securities model.
• A probability measure Q on the space (Ω,F) is said to be an
equivalent martingale measure if it satisfies
(i) Q is equivalent to P , that is, both P and Q have the same
null set;(ii) the discounted security price processes S∗
m(t),m = 1,2, · · · ,Mare martingales under Q, that is,
EQ[S∗m(u)|Ft] = S∗
m(t), for all 0 ≤ t ≤ u ≤ T.
21
Theorem
Let Y be an attainable contingent claim generated by some trading
strategy H and assume that an equivalent martingale measure Q
exists, then for each time t,0 ≤ t ≤ T , the arbitrage price of Y is
given by
V (t;H) = S0(t)EQ
[Y
S0(T)
∣∣∣∣∣Ft].
The validity of the Theorem is readily seen if we consider the dis-
counted value process V ∗(t;H) to be a martingale under Q. This
leads to
V (t;H) = S0(t)V∗(t;H) = S0(t)EQ[V ∗(T ;H)|Ft].
Furthermore, by observing that V ∗(T ;H) = Y/S0(T), so the risk
neutral valuation formula follows.
22
Change of numeraire
• The choice of S0(t) as the numeraire is not unique in order that
the risk neutral valuation formula holds.
• Let N(t) be a numeraire whereby we have the existence of an
equivalent probability measure QN such that all security prices
discounted with respect to N(t) are QN-martingale. In addition,
if a contingent claim Y is attainable under (S0(t), Q), then it is
also attainable under (N(t), QN).
23
The arbitrage price of any security given by the risk neutral valuation
formula under both measures should agree. We then have
S0(t)EQ
[Y
S0(T)
∣∣∣∣∣Ft]= N(t)EQN
[Y
N(T)
∣∣∣∣∣Ft].
To effect the change of measure from QN to Q, we multiplyY
N(T)by the Radon-Nikodym derivative so that
S0(t)EQ
[Y
S0(T)
∣∣∣∣∣Ft]= N(t)EQ
[Y
N(T)
dQNdQ
∣∣∣∣∣Ft].
By comparing like terms, we obtain
dQNdQ
=N(T)
N(t)
/S0(T)
S0(t).
24
Black-Scholes model revisited
The price processes of S(t) and M(t) are governed by
dS(t)
S(t)= ρ dt+ σ dZ
dM(t) = rM(t) dt.
The price process of S∗(t) = S(t)/M(t) becomes
dS∗(t)S∗(t)
= (ρ− r)dt+ σ dZ.
We would like to find the equivalent martingale measure Q such
that the discounted asset price S∗ is Q-martingale. By the Girsanov
Theorem, suppose we choose γ(t) in the Radon-Nikodym derivative
such that
γ(t) =ρ− r
σ,
then Z is a Brownian motion under the probability measure Q and
dZ = dZ +ρ− r
σdt.
25
Under the Q-measure, the process of S∗(t) now becomes
dS∗(t)S∗(t)
= σ dZ,
hence S∗(t) is Q-martingale. The asset price S(t) under the Q-
measure is governed by
dS(t)
S(t)= r dt+ σ dZ.
When the money market account is used as the numeraire, the cor-
responding equivalent martingale measure is called the risk neutral
measure and the drift rate of S under the Q-measure is called the
risk neutral drift rate.
26
The arbitrage price of a derivative is given by
V (S, t) = e−r(T−t)Et,SQ [h(ST )]
where Et,SQ is the expectation under the risk neutral measure Q
conditional on the filtration Ft and St = S. By the Feynman-Kac
representation formula, the governing equation of V (S, t) is given
by
∂V
∂t+σ2
2S2∂
2V
∂S2+ rS
∂V
∂S− rV = 0.
Consider the European call option whose terminal payoff is max(ST−X,0). The call price c(S, t) is given by
c(S, t) = e−r(T−t)EQ[max(ST −X,0)]
= e−r(T−t)EQ[ST1ST≥X] −XEQ[1ST≥X].
27
Exchange rate process under domestic risk neutral measure
• Consider a foreign currency option whose payoff function de-
pends on the exchange rate F , which is defined to be the do-
mestic currency price of one unit of foreign currency.
• Let Md and Mf denote the money market account process in
the domestic market and foreign market, respectively. The pro-
cesses of Md(t),Mf(t) and F(t) are governed by
dMd(t) = rMd(t) dt, dMf(t) = rfMf(t) dt,dF(t)
F(t)= µdt+σ dZF ,
where r and rf denote the riskless domestic and foreign interest
rates, respectively.
28
• We may treat the domestic money market account and the for-
eign money market account in domestic dollars (whose value
is given by FMf) as traded securities in the domestic currency
world.
• With reference to the domestic equivalent martingale measure,
Md is used as the numeraire.
• By Ito’s lemma, the relative price process X(t) = F(t)Mf(t)/Md(t)
is governed by
dX(t)
X(t)= (rf − r+ µ) dt+ σ dZF .
29
• With the choice of γ = (rf − r+µ)/σ in the Girsanov Theorem,
we define
dZd = dZF + γ dt,
where Zd is a Brownian process under Qd.
• Under the domestic equivalent martingale measure Qd, the pro-
cess of X now becomes
dX(t)
X(t)= σ dZd
so that X is Qd-martingale.
• The exchange rate process F under the Qd-measure is given by
dF(t)
F(t)= (r − rf) dt+ σ dZd.
• The risk neutral drift rate of F under Qd is found to be r − rf .
30
Recall that the Black-Scholes equation for a European vanilla call
option takes the form
∂c
∂τ=σ2
2S2 ∂
2c
∂S2+ rS
∂c
∂S− rc, 0 < S <∞, τ > 0, τ = T − t.
Initial condition (payoff at expiry)
c(S,0) = max(S −X,0), X is the strike price.
Using the transformation: y = lnS and c(y, τ) = e−rτw(y, τ), the
Black-Scholes equation is transformed into
∂w
∂τ=σ2
2
∂2w
∂y2+
(r − σ2
2
)∂w
∂y, −∞ < y <∞, τ > 0.
The initial condition for the model now becomes
w(y,0) = max(ey −X,0).
31
Green function approach
The infinite domain Green function is known to be
φ(y, τ) =1
σ√
2πτexp
−
[y+ (r − σ2
2 )τ ]2
2σ2τ
.
Here, φ(y, τ) satisfies the initial condition:
limτ→0+
φ(y, τ) = δ(y),
where δ(y) is the Dirac function representing a unit impulse at the
origin.
The initial condition can be expressed as
w(y,0) =
∫ ∞
−∞w(ξ, 0)δ(y − ξ) dξ,
so that w(y,0) can be considered as the superposition of impulses
with varying magnitude w(ξ, 0) ranging from ξ → −∞ to ξ → ∞.
32
• Since the Black-Scholes equation is linear, the response in po-
sition y and at time to expiry τ due to an impulse of magnitude
w(ξ, 0) in position ξ at τ = 0 is given by w(ξ,0)φ(y − ξ, τ).
• From the principle of superposition for a linear differential equa-
tion, the solution is obtained by summing up the responses due
to these impulses.
c(y, τ) = e−rτw(y, τ)
= e−rτ∫ ∞
−∞w(ξ, 0) φ(y − ξ, τ) dξ
= e−rτ∫ ∞
lnX(eξ −X)
1
σ√
2πτ
exp
−
[y+ (r − σ2
2 )τ − ξ]2
2σ2τ
dξ.
33
Note that
∫ ∞
lnXeξ
1
σ√
2πτexp
−
[y+ (r − σ2
2 )τ − ξ]2
2σ2τ
dξ
= exp(y+ rτ)∫ ∞
lnX
1
σ√
2πτexp
−
[y+
(r+ σ2
2
)τ − ξ
]2
2σ2τ
dξ
= erτSN
ln S
X + (r+ σ2
2 )τ
σ√τ
, y = lnS;
∫ ∞
lnX
1
σ√
2πτexp
−
[y+ (r − σ2
2 )τ − ξ]2
2σ2τ
dξ
= N
y+ (r − σ2
2 )τ − lnX
σ√τ
= N
ln S
X + (r − σ2
2 )τ
σ√τ
, y = lnS.
34
Hence, the price formula of the European call option is found to be
c(S, τ) = SN(d1) −Xe−rτN(d2),
where
d1 =ln SX + (r+ σ2
2 )τ
σ√τ
, d2 = d1 − σ√τ .
The call value lies within the bounds
max(S −Xe−rτ ,0) ≤ c(S, τ) ≤ S, S ≥ 0, τ ≥ 0,
35
36
c(S, τ) = e−rτEQ[(ST −X)1ST≥X]
= e−rτ∫ ∞
0max(ST −X,0)ψ(ST , T ;S, t) dST .
• Under the risk neutral measure,
lnSTS
=
(r − σ2
2
)τ + σZ(τ)
so that lnSTS
is normally distributed with mean
(r − σ2
2
)τ and
variance σ2τ, τ = T − t.
• From the density function of a normal random variable, the
transition density function is given by
ψ(ST , T ;S, t) =1
STσ√
2πτexp
−
[lnSTS −
(r − σ2
2
)τ
]2
2σ2τ
.
37
If we compare the price formula with the expectation representation
we deduce that
N(d2) = EQ[1ST≥X] = Q[ST ≥ X]
SN(d1) = e−rτEQ[ST1ST≥X].
• N(d2) is recognized as the probability under the risk neutral
measure Q that the call expires in-the-money, so Xe−rτN(d2)
represents the present value of the risk neutral expectation of
payment paid by the option holder at expiry.
• SN(d1) is the discounted risk neutral expectation of the terminal
asset price conditional on the call being in-the-money at expiry.
38
Delta - derivative with respect to asset price
c =∂c
∂S= N(d1) + S
1√2π
e−d212∂d1∂S
−Xe−rτ1√2π
e−d222∂d2∂S
= N(d1) +1
σ√
2πτ[e−
d212 − e−(rτ+ln S
X)e−d222 ]
= N(d1) > 0.
Knowing that a European call can be replicated by ∆ units of asset
and riskless asset in the form of money market account, the factor
N(d1) in front of S in the call price formula thus gives the hedge
ratio ∆.
39
• c is an increasing function of S since∂
∂SN(d1) is always posi-
tive. Also, the value of c is bounded between 0 and 1.
• The curve of c against S changes concavity at
Sc = X exp
(−(r+
3σ2
2
)τ
)
so that the curve is concave upward for 0 ≤ S < Sc and concave
downward for Sc < S <∞.
limτ→∞
∂c
∂S= 1 for all values of S,
while
limτ→0+
∂c
∂S=
1 if S > X12 if S = X0 if S < X
.
40
Variation of the delta of the European call value with respect to the
asset price S. The curve changes concavity at S = Xe−(r+3σ2
2
)τ.
41
Variation of the delta of the European call value with respect to
time to expiry τ . The delta value always tends to one from below
when the time to expiry tends to infinity. The delta value tends to
different asymptotic limits as time comes close to expiry, depending
on the moneyness of the option.
42
Continuous dividend yield models
Let q denote the constant continuous dividend yield, that is, the
holder receives dividend of amount equal to qS dt within the interval
dt. The asset price dynamics is assumed to follow the Geometric
Brownian MotiondS
S= ρ dt+ σ dZ.
We form a riskless hedging portfolio by short selling one unit of the
European call and long holding units of the underlying asset. The
differential of the portfolio value Π is given by
43
dΠ = −dc+ dS + qS dt
=
(−∂c∂t
− σ2
2S2 ∂
2c
∂S2+ qS
)dt+
(− ∂c
∂S
)dS.
The last term qS dt is the wealth added to the portfolio due to
the dividend payment received. By choosing =∂c
∂S, we obtain a
riskless hedge for the portfolio. The hedged portfolio should earn
the riskless interest rate.
We then have
dΠ =
(−∂c∂t
− σ2
2S2 ∂
2c
∂S2+ qS
∂c
∂S
)dt = r
(−c+ S
∂c
∂S
)dt,
which leads to
∂c
∂τ=σ2
2S2 ∂
2c
∂S2+(r− q)S
∂c
∂S− rc, τ = T − t, 0 < S < ∞, τ > 0.
44
Martingale pricing approach
Suppose all the dividend yields received are used to purchase addi-
tional units of asset, then the wealth process of holding one unit of
asset initially is given by
St = eqtSt,
where eqt represents the growth factor in the number of units. The
wealth process St follows
dSt
St= (ρ+ q) dt+ σ dZ.
We would like to find the equivalent risk neutral measure Q under
which the discounted wealth process S∗t is Q-martingale. We choose
γ(t) in the Radon-Nikodym derivative to be
γ(t) =ρ+ q − r
σ.
45
Now Z is Brownian process under Q and
dZ = dZ +ρ+ q − r
σdt.
Also, S∗t becomes Q-martingale since
dS∗t
S∗t
= σ dZ.
The asset price St under the equivalent risk neutral measure Q be-
comesdSt
St= (r − q) dt+ σ dZ.
Hence, the risk neutral drift rate of St is r − q.
Analogy with foreign currency options
The continuous yield model is also applicable to options on foreign
currencies where the continuous dividend yield can be considered as
the yield due to the interest earned by the foreign currency at the
foreign interest rate rf .
46
Call and put price formulas
The price of a European call option on a continuous dividend paying
asset can be obtained by changing S to Se−qτ in the price formula.
This rule of transformation is justified since the drift rate of the
dividend yield paying asset under the risk neutral measure is r − q.
Now, the European call price formula with continuous dividend yield
q is found to be
c = Se−qτN(d1) −Xe−rτN(d2),
where
d1 =ln SX + (r − q+ σ2
2 )τ
σ√τ
, d2 = d1 − σ√τ .
47
Similarly, the European put formula with continuous dividend yield
q can be deduced from the Black-Scholes put price formula to be
p = Xe−rτN(−d2) − Se−qτN(−d1).
The new put and call prices satisfy the put-call parity relation
p = c− Se−qτ +Xe−rτ .
Furthermore, the following put-call symmetry relation can also be
deduced from the above call and put price formulas
c(S, τ ;X, r, q) = p(X, τ ;S, q, r),
48
• The put price formula can be obtained from the corresponding
call price formula by interchanging S with X and r with q in the
formula. Recall that a call option entitles its holder the right to
exchange the riskless asset for the risky asset, and vice versa for
a put option. The dividend yield earned from the risky asset is
q while that from the riskless asset is r.
• If we interchange the roles of the riskless asset and risky asset
in a call option, the call becomes a put option, thus giving the
justification for the put-call symmetry relation.
49
Time dependent parameters
Suppose the model parameters become deterministic functions of
time, the Black-Scholes equation has to be modified as follows
∂V
∂τ=σ2(τ)
2S2 ∂
2V
∂S2+[r(τ)−q(τ)] S ∂V
∂S−r(τ)V, 0 < S < ∞, τ > 0,
where V is the price of the derivative security.
When we apply the following transformations: y = lnS and w =
e∫ τ0 r(u) duV , then
∂w
∂τ=σ2(τ)
2
∂2w
∂y2+
[r(τ) − q(τ) − σ2(τ)
2
]∂w
∂y.
Consider the following form of the fundamental solution
f(y, τ) =1√
2πs(τ)exp
(−[y+ e(τ)]2
2s(τ)
),
50
it can be shown that f(y, τ) satisfies the parabolic equation
∂f
∂τ=
1
2s′(τ)
∂2f
∂y2+ e′(τ)
∂f
∂y.
Suppose we let
s(τ) =
∫ τ
0σ2(u) du
e(τ) =∫ τ
0[r(u) − q(u)] du− s(τ)
2,
one can deduce that the fundamental solution is given by
φ(y, τ) =1√
2π∫ τ0 σ
2(u) duexp
−
y+∫ τ0 [r(u) − q(u) − σ2(u)
2 ] du22∫ τ0 σ
2(u) du
.
Given the initial condition w(y,0), the solution can be expressed as
w(y, τ) =
∫ ∞
−∞w(ξ,0) φ(y − ξ, τ) dξ.
51
Note that the time dependency of the coefficients r(τ), q(τ) and
σ2(τ) will not affect the spatial integration with respect to ξ. We
make the following substitutions in the option price formulas
r is replaced by1
τ
∫ τ
0r(u) du
q is replaced by1
τ
∫ τ
0q(u) du
σ2 is replaced by1
τ
∫ τ
0σ2(u) du.
For example, the European call price formula is modified as follows:
c = Se−∫ τ0 q(u) du N(d1) −Xe−
∫ τ0 r(u) duN(d2)
where
d1 =ln SX +
∫ τ0 [r(u) − q(u) + σ2(u)
2 ] du√∫ τ
0 σ2(u) du
, d2 = d1 −√∫ τ
0σ2(u) du.
52
Implied volatilities
The only unobservable parameter in the Black-Scholes formulas is
the volatility value, σ. By inputting an estimated volatility value, we
obtain the option price. Conversely, given the market price of an
option, we can back out the corresponding Black-Scholes implied
volatiltiy .
• Several implied volatility values obtained simultaneously from
different options (varying strikes and maturities) on the same
underlying asset provide the market view about the volatility of
the stochastic movement of the asset price.
• Given the market prices of European call options with different
maturities (all have the strike prices of 105, current asset price
is 106.25 and short-term interest rate over the period is flat at
5.6%).
maturity 1-month 3-month 7-monthValue 3.50 5.76 7.97
Implied volatility 21.2% 30.5% 19.4%
53
Time dependent volatility
The Black-Scholes formulas remain valid for time dependent volatil-
ity except that
√1
T − t
∫ T
tσ(τ)2 dτ is used to replace σ.
How to obtain σ(t) given the implied volatility measured at time t∗
of a European option expiring at time t. Now
σimp(t∗, t) =
√1
t− t∗
∫ t
t∗σ(τ)2 dτ
so that∫ t
t∗σ(τ)2 dτ = σ2
imp(t∗, t)(t− t∗).
Differentiate with respect to t, we obtain
σ(t) =
√
σimp(t∗, t)2 + 2(t− t∗)σimp(t∗, t)
∂σimp(t∗, t)
∂t.
54
Practically, we do not have a continuous differentiable implied volatil-
ity function σimp(t∗, t), but rather implied volatilities are available at
discrete instants ti. Suppose we assume σ(t) to be piecewise con-
stant over (ti−1, ti), then
(ti − t∗)σ2imp(t
∗, ti) − (ti−1 − t∗)σ2imp(t
∗, ti−1)
=
∫ titi−1
σ2(τ) dτ = σ2(t)(ti − ti−1), ti−1 < t < ti,
σ(t) =
√√√√(ti − t∗)σ2imp(t
∗, ti−1) − (ti−1 − t∗)σ2imp(t
∗, ti−1)
ti − ti−1, ti−1 < t < ti.
55
Quanto-prewashing techniques
1. Consider two assets whose dynamics follow the lognormal pro-
cesses
df
f= µf dt+ σf dZf
dg
g= µg dt+ σg dZg.
By the Ito Lemma
d(fg)
fg= (µf + µg + ρσfσg) dt+ σ dZ
where σ2 = σ2f + σ2
g + 2ρσfσg;
d(f/g)
f/g= (µf − µg − ρσfσg + σ2
g ) dt+ σ dZ
where σ2 = σ2f + σ2
g − 2ρσfσg.
56
Proof
d(fg) = f dg+ g df + ρσfσgfg dt︸ ︷︷ ︸arising from dfdg and
observing dZfdZg = ρ dt
d(fg)
fg=
dg
g+df
f+ ρσfσg dt
= (µf + µg+ ρσfσg) dt+ σf dZf + σg dZg. (1)
Observe that the sum of two Brownian processes remains to be
Brownian. Recall the formula
VAR(X + Y ) = VAR(X) + VAR(Y ) + 2COV(X, Y ).
Hence, the sum of σf dZf + σg dZg can be expressed as σ dZ, where
σ2 = σ2f + σ2
g + 2ρσfσg.
57
d
(1
g
)= −dg
g2+
2
g3dg2
2; (2)
d(1g
)
1g
= −µg dt+ σ2g dt− σg dZg. (3)
Replacing g by 1/g in formula (1), we obtain
d(f/g)
f/g= (µf − µg − ρσfσg + σ2
g ) dt+ σf dZf − σg dZg. (4)
Recall the formula: VAR(X−Y ) = VAR(X)+VAR(Y )−2COV(X, Y ).
The sum of σf dZf − σg dZy can be expressed as σ dZ, where
σ2 = σ2f + σ2
g − 2ρσfσg.
58
2. S = foreign asset price in foreign currency
F = exchange rate
= domestic currency price of one unit of foreign currency
S∗ = FS = foreign asset price in domestic currency
q = dividend yield of the asset.
Under the domestic risk neutral measure Qd, the risk neutral
drift rate of S∗ and F are
δdS∗ = rd − q and δdF = rd − rf .
Under the foreign Qf , the risk neutral drift rate for S and 1/F
are
δfS = rf − q and δ
f1/F
= rf − rd.
59
Quanto prewashing is to find δdS, that is, the risk neutral drift rate
of the price of the asset in foreign currency under Qd.
Recall the formula:
δdS∗ = δdFS = δdF + δdS + ρσFσS
where the dynamics of S and F under Qd are
dS
S= δdS dt+ σS dZ
dS
dF
F= δdF dt+ σF dZ
dF ,
where ZdS and ZdF are Qd-Brownian process.
60
We obtain
δdS = δdS∗ − δdF − ρσFσS
= (rd − q) − (rd − rf) − ρσFσS
= (rf − q) − ρσFσS.
Comparing with δfS = rf − q and δdS, there is an extra term −ρσFσS.
The risk neutral drift rate of the asset is changed by the amount
−ρσFσS when the risk neutral measure is changed from the foreign
currency world to the domestic currency world.
61
Siegel’s paradox
Given that the price dynamics of F under Qd is
dF
F= (rd − rf) dt+ σF dZd,
then the process for 1/F is
d(1/F)
1/F= (rf − rd + σ2
F ) dt− σF dZd.
This is seen as a puzzle to many people since the risk neutral drift
rate for 1/F should be rf − rd instead of rf − rd + σ2F .
We observe directly from the above SDE’s that
σF = σ1/F and ρF,1/F = −1.
62
An interesting application of Siegel’s paradox
Suppose the terminal payoff of an exchange rate option is FT1FT>K.Let V d(F, t) denote the value of the option in the domestic currency
world. Define
V f(Ft, t) = V d(Ft, t)/Ft,
so that the terminal payoff of the exchange rate option in foreign
currency world is 1FT>K. Now
V f(F, t) = e−rf(T−t)EQft [1FT>K|Ft = F ].
63
From δd1/F = δ
f1/F
+ σ2F and observing σF = σ1/F , we deduce that
δfF = δdF + σ2
F .
This is easily seen if we interchange the foreign and domestic cur-
rency worlds. We obtain
V d(F, t) = FV f(F, t) = e−rf(T−t)FN(d)
where
d =ln FK +
(δfF − σ2
F2
)τ
σ√τ
=ln FK +
(rd − rf +
σ2F2
)τ
σ√τ
.
64
Foreign exchange options
Under the domestic risk neutral measure Qd, the exchange rate
process follows
dF
F= (rd − rf) dt+ σF dZF .
Suppose the terminal payoff is max(FT − Xd,0), then the price of
the exchange rate call option is
V (F, τ) = Fe−rfτN(d1) −Xde−rdτN(d2),
where
d1 =ln FXd
+
(rd − rf +
σ2F2
)τ
σF√τ
, d2 = d1 − σF√τ .
65
Equity options with exchange rate risk exposure
Quanto options are contingent claims whose payoff is determined
by a financial price or index in one currency but the actual payout
is done in another currency.
1. Foreign equity call struck in foreign currency
c1(ST , F,0) = FT max(ST −Xf ,0).
2. Foreign equity call struck in domestic currency
c2(ST , F,0) = max(FTST −Xd,0).
3. Foreign exchange rate foreign equity call
c3(ST , F,0) = F0 max(ST −Xf ,0).
66
Under the domestic risk neutral measure Qd
dS
S= δdS dt+ σS dZS
dF
F= δdF dt+ σF dZF
S∗ = FS = asset price in domestic currency
dS∗
S∗ = δdS∗ dt+ σS∗ dZS∗.
where ZS, ZF , and ZS∗ are all Qd-Brownian processes.
By Ito’s lemma,
δdS∗ = δdS + δdF + ρSFσSσF
σ2S∗ = σ2
S + σ2F + 2ρSFσSσF .
Under the risk neutral measures,
δdS∗ = rd − q, δdF = rd − rf , δfS = rf − q,
δdS = δdS∗ − δdF − ρSFσSσF = rf − q − ρSFσSσF = δfS − ρSFσSσF .
67
1. Define
c1(S, F, τ)/F = c1(S, τ) so that c1(S,0) = max(S −Xf ,0).
This call option behaves like the usual vanilla call option in the
foreign currency world so that
c1(S,0) = Se−qτN(d(1)1 ) −Xfe
−rfτN(d(1)2 )
where
d(1)1 =
ln SXf
+
(δfS +
σ2S2
)τ
σS√τ
, d(1)2 = d
(1)1 − σS
√τ .
68
2. c2(S, F,0) = max(S∗T −Xd,0)
c2(S, F, τ) = S∗e−qτN(d(2)1 ) −Xde
−rdτN(d(2)2 )
where
d(2)1 =
ln S∗Xd
+
(δdS∗ +
σ2S∗2
)τ
σ∗S√τ
, d(2)2 = d
(2)1 − σ∗S
√τ ,
σ2S∗ = σ2
S + σ2F + 2ρSFσSσF .
69
3. For c3(S,0) = F0 max(ST −Xf ,0), the payoff is denominated in
the domestic currency world, so the risk neutral drift rate is of
the stock price δdS. The call price is
c3(S, τ) = F0e−rdτ [Seδ
dSτN(d
(3)1 ) −XfN(d
(3)2 )]
where
d(3)1 =
ln SXf
+
(δdS +
σ2S2
)τ
σS√τ
, d(3)2 = d
(3)1 − σS
√τ .
• The price formula does not depend on the exchange rate F since
the exchange rate has been chosen to be the fixed value F0.
• The currency exposure is reflected through the dependence on
σF and correlation coefficient ρSF .
70
Digital quanto option relating 3 currency worlds
FS\U = SGD currency price of one unit of USD currency
FH\S = HKD currency price of one unit of SGD currency
• Digital quanto option payoff: pay one HKD if FS\U is above
some strike level K.
• We may interest FS\U as the price process of a tradeable asset
in SGD. The dynamics is governed by
dFS\UFS\U
= (rSGD − rUSD) dt+ σFS\U dZSFS\U
.
71
• Given δSFS\U= rSGD − rUSD, how to find δHFS\U
, which is the risk
neutral drift rate of the SGD asset denominated in Hong Kong
dollar?
• By the quanto-prewashing technique
δHFS\U= δSFS\U
− ρσFS\UσFH\S .
• Digital option value = EQH
t
[1FS\U>K
]= N(d)
where
d =
lnFS\UK +
δHFS\U −
σ2FS\U2
τ
σFS\U√τ
.
72
Exchange options
Exchange asset 2 for asset 1 so that the terminal payoff is
V (S1, S2,0) = max(S1 − S2,0).
Assume
dS1
S1= δS1
dt+ σ1 dZ1, δS1= r − q1,
dS2
S2= δS2
dt+ σ2 dZ2, δS2= r − q2.
Define S =S1
S2, σ2S = σ2
1−2ρ12σ1σ2+σ22. Write
V (S1, S2,0)
S2= max
(S1
S2− 1,0
).
V (S1, S2, τ) = e−rτ[S1e
δS1τN(d1) − S2eδS2τN(d2)
], where
d1 =ln S1S2
+
[(δS1
− δS2
)+
σ2S2
]τ
σS√τ
, d2 = d1 − σS√τ .
73
Hints on the proof of the price formula
Under the risk neutral measure
dS
S= (δS1
− δS2− ρ12σ1σ2 + σ2
2) dt+ σ1 dZ1 − σ2 dZ2.
It is convenient to use S2(t)eq2t as numeraire, where
S2(t) = S2(0)e
(δS2−
σ222
)t+σ2Z2(t)
or
S2(t)eq2t
S2(0)e−rt = e−
σ222 t+σ2Z2(t).
We can take γ = −σ2 in Girsanov Theorem so that
S2(t)eq2t
S2(0)e−rt = e−
12σ
22t+σ2Z2(t) =
dQ∗
dQ.
74
We then have dZ2 = dZ2 − σ2 dt where dZ2 is under Q∗. In a similar
manner, we obtain
dZ1 = dZ1 − ρ12σ2 dt.
Putting everything together,
dS
S= (δS1
− δS2) dt+ (σ1 dZ1 − σ2 dZ2)
and
σ1 dZ1 − σ2 dZ2 = σ dZ
where
σ2 = σ21 + σ2
2 − 2ρ12σ1σ2.
75
Use of the exchange option formula to price quanto options
Consider
C2(S, F,0) = F max(S −X,0) = max(S∗ −XF,0)
C4(S, F,0) = Smax(F −X,0) = max(S∗ −XS,0)
where S∗ = FS. Both can be considered as exchange options.
Though an exchange option appears to be a two-state option, it
can be reduced to an one-state pricing model when the similarity
variable S =S1
S2is defined. Similarly, the two-state quanto options
can be reduced to one-state pricing models.
76
For valuation of C4(S, F, τ), we consider the similarity variableS∗
S=
F . Note that
δdS∗ = δdS + δdF + ρSFσSσF , σ = σF ,
and the corresponding difference in the risk neutral drift rates in Qdis
δdS∗ − δdS = δdF + ρSFσSσF = rd − rf + ρSFσSσF .
77
