Short rate model under HJM

randomwalker · September 14th, 2007, 3:25 pm

I am reading up on baxter and rennie, and chapter 5.4's formulates the short rate models under the HJM framework. This function serves as the link between the short rate and forward rate:g(x,t,T) = -log E_Q (exp(-integrate from t to T (r_s) ds) | r_t = x) = integrate from t to T [f(t,u)] duThe Ho and Lee model is given by dr(t) = sigma dW(t) + teta (t) dtnow on the next page, I don't know how it comes from that to thisg(x,t,T) = x(T-t)-1/6 sigma^2 (T-t)^3 + integrate from t to T [ (T - s) teta (s) ds]Can somebody please help me with this? Thanks in advance!

Speedy · September 17th, 2007, 10:29 pm

You could write down the integrated version of the Ho and Lee SDE:for s >= t, r_s = r_t + sigma (W_s - W_t) + Integral{from t to s} theta_u duYou can plug this into your first g(x,t,T) equation. You'll need to change the order of the integrals for the theta terms.For the Brownian motion, you could use thatIntegral{from t to T} (W_s - W_t) ds has the same law asIntegral{from 0 to T-t} (W_u) du