April 2005 q5b

[r_v.s]

How do we get the SDE of rt under Q?

 

[Graham Aylott]

You need to use the Cameron-Martin-Girsanov (CMG) Theorem from Chapter 15 Section 1. The CMG Theorem essentially tells us that the relationship between the standard Brownian motion (SBM) under the real-world "P" probabilities, Z(t), and the SBM under the risk-neutral "Q" probabilities, Z~(t), is:

Z~(t) = Z(t) + gamma*t

or equally:

dZ~(t) = dZ(t) + gamma*dt

where gamma is the market price of risk, which in the context of term structure models is:

gamma = [m(t,T) - r(t)] / S(t,T)

- see page 12 of Chapter 17.

So, all you need to do is replace dZ(t) in the SDE for r(t) under P given in the question with:

dZ(t) = dZ~(t) - gamma*dt

to get the corresponding SDE for r(t) under Q.

In this case:

gamma = mu*r(t)/sigma

and you obtain:

dr(t) = sigma*dZ~(t) under Q.