162 Frequently Asked Questions In Quantitative Finance
In the stochastic volatility world we can look at the sec-
ond derivative of option value with respect to volatility,
and if it is positive we would expect to have to pay for
this convexity, that is option values will be relatively
higher wherever this quantity is largest. For a call or
put in the world of constant volatility we have
∂^2 V
∂σ^2
=S
√
T−t
d 1 d 2 e−D(T−t)e−d
2
1 /^2
√
2 πσ
.
This function is plotted in Figure 2-10 forS=100,T−
t=1,σ= 0 .2,r= 0 .05 andD=0. Observe how it is
positive away from the money, and small at the money.
(Of course, this is a bit of a cheat because on one hand
I am talking about random volatility and yet using a
formula that is only correct for constant volatility.)
d^2 V/d vol^2
-20
0
20
40
60
80
100
120
140
75 80 85 90 95 100 105 110 115 120 125
Strike
Figure 2-10:∂^2 V/∂σ^2 versus strike.
