Sometimes we have to find expectations under a forward martingale measure, for example the forward Libor is calculated as Libor(0,t,T) = E[Libor(t,T) | F_0]. In the more general case we want to computeE[ f(DF(t,T)) | F_0]under a forward martingale measure, where 0<t<T and f is some function of the forward discount factor DF(t,T). I am trying to calculate this expectation using Monte Carlo sims.The question is: for each simulation path, what do I have to substitute for DF(t,T)? Is it just exp{- int_t^T r(s) ds}, where the rates are simulated in a usual way starting from a given r(0) and using, say, a (calibrated) Vasicek model?Many thanks for your responses.

QuoteOriginally posted by: TensorIs it just exp{- int_t^T r(s) ds}, where the rates are simulated in a usual way starting from a given r(0) and using, say, a (calibrated) Vasicek model?Exactly.Note indeed that some models (Vasicek, HW, CIR, etc) have a closed formula for the (forward) price of a zero bond P(t,T) =E[exp{- int_t^T r(s) ds} | F_t].In these cases you need only to simulate from r(0) to r(t) and use the formula to obtain P(t,T) as a function of r(t)P.

Paolos, thanks! I felt that I am going in the right direction.But where in the simulation process (and in the formula E[ f(DF(t,T)) | F_0]) is the magic expression "forward martingale measure" hidden?Say, I want to compute the same expectation E[ f(DF(t,T)) | F_0] under a local (or any other) measure . How would the simulation procedure change?

Hi Tensor,everytime you change the numeraire and pass from a "risk neutral" to a forward (or any other equivalent) measureyou need also to change the drift of the SDE according to Girsanov's theorem.This means that under the forward measure you cannot simulate the rates using the "risk neutral" dynamics:dr(t)=k[B-r(t)]dt + sigma dW(t)but the forward T dynamics:dr(t)=k[B-r(t)+m(t)]dt + sigma dW(t)where m(t)=-sigma*sigma*{1-e^[-k(T-t)]}/kI've assumed the Vasicek model for sake of simplicity but the same applies to any dynamicsHope it helpsPaolo

Thanks Paolos, I'll try to derive this formula.

Or you can look at the book of Brigo & Mercurio, where this issue is well described.