262 Frequently Asked Questions In Quantitative Finance
today’s asset price andt∗today’s date. Note that here
the arguments ofVare the ‘variables’ strike,K,and
expiration,T.
If we differentiate this with respect toKwe get
∂V
∂K
=−e−r(T−t
∗)
∫∞
K
p(S∗,t∗;S,T)dS.
After another differentiation, we arrive at this equation
for the probability density function in terms of the
option prices
p(S∗,t∗;K,T)=er(T−t
∗)∂^2 V
∂K^2
.
We also know that the forward equation for the tran-
sition probability density function, the Fokker–Planck
equation, is
∂p
∂T
=^12
∂^2
∂S^2
(σ^2 S^2 p)−
∂
∂S
(rSp).
Hereσ(S,t) is evaluated att=T.Wealsohave
∂V
∂T
=−rV+e−r(T−t
∗)
∫∞
K
(S−K)
∂p
∂T
dS.
This can be written as
∂V
∂T
=−rV+e−r(T−t
∗)
∫∞
K
(
1
2
∂^2 (σ^2 S^2 p)
∂S^2
−
∂(rSp)
∂S
)
×(S−K)dS.
using the forward equation. Integrating this by parts
twice we get
∂V
∂T
=−rV+^12 e−r(T−t
∗)
σ^2 K^2 p+re−r(T−t
∗)
∫∞
K
Sp dS.
In this expressionσ(S,t)hasS=Kandt=T.After
some simple manipulations we get
∂V
∂T
=^12 σ^2 K^2
∂^2 V
∂K^2
−rK
∂V
∂K
.
