Chapter 2: FAQs 109
wheredX 1 anddX 2 are correlated Brownian motions
with correlationρ(S,σ,t).
Using Girsanov you can get the governing equation in
three steps:
1.Under a pricing measureQ, Girsanov plus the fact
thatSis traded implies that
dX 1 =dX ̃ 1 −
μ−r
σ
dt
and
dX 2 =dX ̃ 2 −λ(S,σ,t)dt,
whereλis the market price of volatility risk
2.Apply Ito’s formula to the discounted option priceˆ
V(S,σ,t)=e−r(T−t)F(S,σ,t), expanding underQ,
using the formulæ fordSanddVobtained from the
Girsanov transformation
3.Since the option is traded, the coefficient of thedt
term in its Itˆo expansion must also be zero; this
yields the relevant equation
Girsanov and the idea of change of measure are par-
ticularly important in the fixed-income world where
practitioners often have to deal with many different
measures at the same time, corresponding to different
maturities. This is the reason for the popularity of the
BGM model and its ilk.
References and Further Reading
Joshi, M 2003The Concepts and Practice of Mathematical
Finance.CUP
Lewis, A 2000Option Valuation under Stochastic Volatility.
Finance Press
Neftci, S 1996An Introduction to the Mathematics of Financial
Derivatives. Academic Press
