Stochastic Volatility Modeling

Variance swaps 181

A.1 Vega/gamma relationship in the Black-Scholes model

Denote by

V

the vega:

V =

dP

bσ

d

bσ

, where

P

bσ

is the Black-Scholes price with

implied volatility

bσ

. Taking the derivative of the Black-Scholes equation (1.4) with

respect to bσ yields:

dV

dt

+ (r − q)S

dV

dS

+

bσ

2

2

S

2

d

2

V

dS

2

− rV = −bσS

2

d

2

P

bσ

dS

2

(5.63)

At maturity,

V (t = T, S) = 0, ∀S

.

V

is thus only generated by the source term in

(5.63):

V (t, S) = bσ

Z

T

t

E

t,S



e

−r(τ −t)

S

2

τ

d

2

P

bσ

dS

2

(τ, S

τ

)



dτ (5.64)

Setting x = ln S, the Black-Scholes equation reads:

dP

bσ

dt

+



r − q −

bσ

2

2



dP

bσ

dx

+

bσ

2

2

d

2

P

bσ

dx

2

= rP

bσ

(5.65)

Take the derivative of (5.65)

n

times with respect to

x

. Since neither

r

, nor

q

, nor

bσ

depend on x,

d

n

P

bσ