Cameron-Martin-Girsanov Theorem

[kylie jane]

Hi,

Can someone explain what is meant by the following statement page 11 chapter 14.
"Note that the converse of the Cameron-Martin-Girsanov Theorem tells us that we can change the drift but not the volatility of the Brownian motion."

Do they mean: we can change the drift of Brownian motion and get equivalence, but if we change the diffusion or volatility parameter we don't get equivalence??

Thanks!

 

[John Potter]

It means that we can substitute dZt (change in standard Brownian motion in real world)

with dZ~t - MPR dt

(change in standard Brownian motion in risk-neutral world) + Market price of risk times change in time

eg for geo BM (see p46 of Tables)

MPR = (mu - r) / sigma

It's the same for Bond prices:

dB = B { stuff dt + more stuff dZt}

MPR = (stuff - rt) / more stuff

make sure you know this formula:

dZ~t = dZt + MPR dt

 

[Jammy]

It means that we can substitute dZt (change in standard Brownian motion in real world)

with dZ~t - MPR dt

(change in standard Brownian motion in risk-neutral world) + Market price of risk times change in time

eg for geo BM (see p46 of Tables)

MPR = (mu - r) / sigma

It's the same for Bond prices:

dB = B { stuff dt + more stuff dZt}

MPR = (stuff - rt) / more stuff

make sure you know this formula:

dZ~t = dZt + MPR dt

Hi John,
Can you draw out how the corollary implies this?

Thank you

 

[Mark Mitchell]

The statement of the CMG theorem is:

Z~t = Zt + integral from 0 to t of gamma_s ds.

By considering the changes in each term as we move forward over a small time interval, this is directly equivalent to:

dZ~t = dZt + gamma_t dt

This is exactly the formula in John's post, where gamma_t is called MPR.

 

[Jammy]

Hi Mark, thanks for replying.

I was under the impression that the gamma_t in the corollary to the CMG in the core material was just meant to represent any previsible process, not the mkt price of risk per say.

I understand that when we substitute 'dWt= dW~t - gamma_t times dt' in the sde for say a GBM, the drift co-efficient changes while volatility co-efficient remains unchanged.

 

[Mark Mitchell]

Yes - the statement of the CMG theorem just refers to gamma_t, and says this must be a previsible process, but does not link it to the market price of risk. However, on the two occasions we actually see the CMG theorem in action (in Ch15 relating to geometric Brownian motion, and Ch17 relating to short rate models), the gamma_t we need is the market price of risk. This is why John uses this terminology in his original post. I believe that gamma_t is the market price of risk, in general, but this is not proved in CT8.