Steps of getting governing equation of Girsanov’s Theorem

Question

Define the steps of getting governing equation of Girsanov’s Theorem?

Answer 1

By using Girsanov you can determine the governing equation within three steps:

1. Under a pricing measure Q, Girsanov plus the fact that S is traded implies that

dX₁= dX‾₁ - ((µ - r)/σ) dt

and

dX₂ = dX‾₂ - (λ(S, σ, t))dt,

Here λ is the market price of volatility risk.

2. Apply Itoˆ's formula to the discounted option price

V(S, σ, t) = e^-r(T-t)F(S, σ, t), expanding in Q,usingthe

formule for dS and dV acquired by the Girsanov transformation

3. As the option is traded, the coefficient of the dt term in its Itˆ o expansion should also be zero; this yields the relevant equation

Girsanov and the concept of change of measure are mainly important into the fixed-income world where practitioners frequently have to deal with many different measures at similar time, corresponding to various maturities. It is the purpose for the popularity of the BGM model and its ilk.