Ho-Lee in terms of HJM

In page 152 of Baxter and Rennie, I'm unable to understand how the equation g(x,t,T) was arrived at. I would really appreciate it and obliged if someone can take me through the steps.

Thank you so much!

 

Anyone?

As far as I worked out, the difference in my calculations and the actual answer lies in selecting the market price of risk under measure Q. Still not able to arrive at the solution...

 

After countless hours of breaking my head, I think I finally cracked it with this method:

The change of measure from P to Q for dr(t) is computed by:
1) Compute the equation for r(t) from dr(t).
2) Multiply dW(t) to the diffusion term of r(t) and integrate the diffusion term from t to T. This will be deterministic value.
3) Multiply the diffusion term from dr(t) with answer in 2 - let's call this M.
4) Now under risk neutral measure Q,
dr(t) = equation dr(t) under P (with W(t) replaced by W(t)bar) - M*d(t).

 

All the best for your exam.

 

Thanks Edwin! You beat the hell out of CA1 too!

 

Thanks Oxymoron, I can only study like a bastard!

 

Tested this approach with Ho-Lee, Vasicek and Hull and White. I'm able to derive forward rates under Q and thus, prices of bonds with ease. It should be easily extendable to Black-Derman-Toy, Black-Karasinski etc.

 

Hi Oxymoron,

Just to add some theory behind it, to make forward rate a martingale , the numeraire here needed is the bond price maturing at the same duration. We also have to set the MRP equal to the bond price volatility, which under different models is sigma*B(t,T).

Hence the drift term we need to add here is to make Wt under Q from P will be sigma*sigma*B(t,T).

for Vasicek this results into adding sigma2*B(t,T)

 

Perfect.