I was trying to replicate the convexity adjustment between FRAs and futures under the Ho-Lee model : convexity adjustment Ho-Lee
As part of doing this the first step was to replicate the SDE for the bond, however, I end up with a different formula to the one Hull calculates in the above pdf.
The Bond SDE formula which Hull uses is,
By using applying Ito on P, what I get is a different SDE
dP(t,T) = r(t)P(t,T)dt - (T-t)σP(t,T)dz where P(t,T) = A(t,T)e^[-r(T-t)] and the Ho-Lee diffusion eq for the short term rate is dr = θ(t)dt + σ dz
by replacing dr with Ho-Lee SDE, this boils down to ()dt+()dz but the ()dt term of the Bond SDE ends up being very different to Hull's formula,
dP = [r * P(t,T) + (dA/dt)e^[-r(T-t)]]dt - (T-t)P(t,T)dr + [0.5(T-t)^2 P(t,T)σ^2 ]dz
What am I missing/how can I bridge the gap I get by applying Ito and Hull's Bond SDE ?
[rP + (dA/dt)e^(-r(T-t)) - (T-t)Pθ + 0.5P (T-t)^2σ^2]dt