278 Frequently Asked Questions In Quantitative Finance
V=e−r(T−t)( 2 π(T−t))−d/^2 (Det)−^1 /^2 (σ 1 ···σd)−^1
∫∞
0
···
∫∞
0
Payoff(S′ 1 ···S′d)
S′ 1 ···S′d
×exp
(
−
1
2
αT−^1 α
)
dS 1 ′···dSd′
where
αi=
1
σi(T−t)^1 /^2
(
ln
(
Si
S′i
)
+
(
r−Di−
σi^2
2
)
(T−t)
)
,
is the correlation matrix and there is a continuous
dividend yield ofDion each asset.
Stochastic volatility
If the risk-neutral volatility is modelled by
dσ=(p−λq)dt+qdX 2 ,
whereλis the market price of volatility risk, with the
stock model still being
dS=μSdt+σSdX 1 ,
with correlation between them ofρ, then the option-
pricing equation is
∂V
∂t
+^12 σ^2 S^2
∂^2 V
∂S^2
+ρσSq
∂^2 V
∂S∂σ
+^12 q^2
∂^2 V
∂σ^2
+rS
∂V
∂S
+(p−λq)
∂V
∂σ
−rV= 0.
This pricing equation can be interpreted as representing
the present value of the expected payoff under risk-neutral
random walks for bothSandσ. So for a call option, for
example, we can price via the expected payoff
V(S,σ,t)=e−r(T−t)EQt[max(ST−K,0)].
