Just trying to clarify a few things for myself and I'm having great difficulty working out the Girsanov transform required to adapt a Q-risk neutral cash-account-numeraired process to a Q^T-forward-measure-Bond numeraired process (bond expiry T obv.)Can work out the Radon-Nikodym derivative but not the transform. If anybody could help it would be much appreciated. Thank you

the radon-nkiodym derivative pins down the girsanov transformation.what do you mean by the following?QuoteCan work out the Radon-Nikodym derivative but not the transform.

Apologies if it wasn't clear enough. The R-N deriv iswhere Q is risk-neutal and Q^T is forward, B(t) is the cash account. P(t,T) is bond. What mathematical steps do I take to find the Girsanov transform? Thanks for your help

had you asked what the asset value process under Q^T would be given its Q dynamics, you would get the following:under Q^T (by existence, we assume the free of arbitrage), the discounted (the numeriare is the T-bond) value processes of all traded assets are marginagles. then the discounted the dynamics has the the form of: dS = S dW_t, where W_t is a Q^T-Brownian motion, by the martingale representation theorem.

What I am trying to clear up is the steps taken by Jamshidian (An Exact Bond Option Formula) and later the pioneering market model papers about the forward measure approach. 1. The decision is made that the T-expiry bond is a good numeraire for T-expiry contingent claims. Yep.2. Have a risk-neutral, Q-dynamics of the underlying spot rate (Jamshidian essentially uses Vasicek's).3. To price under Q^T you need the change of measure (Radon-Nikodym derivative) that will give you the form of the expectation under Q^T. You also need the Girsanov transform that will adapt the dynamics of any Q-processes to Q^T processes. (The spot rate process needs to be adapted)The Radon-Nikodym derivative is easy to arrive at using change of numeraire results (Bjork, Baxter & Rennie). The details of how this then reveals the Girsanov transform is very opaque to me and this is what I'm not sure about.I know the result (the transform is the volatility of the bond P(t,T)). Just not the steps. Apologies if this seems trivial but I remain no closer to understanding the crucial Radon-Nikodym to Girsanov transform bit.Thanks

ok, i now understand what your question is. i hope you will be satisfied by the following.1) under Q for a genearl no-arbitrage asset value process with dynamics: dS_t = r_t S_t dt + v_t S_t dW_t where r_t is the (stochastic) short rate process, v_t is the (probably stochastic) volatility process and W_t is a standard Q-Bm. we use this S_t process as the numeraire (particualr example will be the T-bond in your case.).2) then by definition the likelihood process is L_t = \frac{S_t}{S_0 B_t}. using ito's formula and the Q dynamics of S_t, we end up with dL_t = v_t L_t dW_t. as you said, we identify that the volatility is the girsanov kernal. we are done.

Thanks, for your help mjyI think I'm there (A fun Sunday!). I think my problem was a) being very reluctant to want to use Ito on L_t it looked messy andb) not always understanding mgales deeply enough.In the end I went for the argument \frac{P(t,T)}{B(t)} is a martingale under Q^T and worked out the SDE it followed. The volatility trips neatly out of one simple application of Ito to the bond and thus the transform.This result can then be verified using your suggestion in 2).For other readers Brigo & Mercurio's analysis of change of measures was useful for me.