I have a problem in implementing bond option price in Hull white model using MC. I am trying to price a bond option where option matures at time S and bond matures at T (S
I am doing the following approach, please tell me if it is wrong:

1.Simulate short rate process r(t) beginning from 0 till S using a modified Euler scheme which compensates drift due to change of numeriare.

2.I plug in the terminal value r(S) into bond pricing formula P(t1,t2,r(S))

where

t1=start time

t2=end time

At this point, I am not sure whether I should use P(0,T-S,r(S)) or P(S,T,r(S)).



Can someone please help or tell me what I am doing wrong? Thanks in advance.

Regards
EDIT : corrected typo changed T->S

Dear asd, you don't need to calculate bond price for each simulation. Make use of affine property of short rate models. i.e, bond price depends on r(T).

For example, let's say:
P0(t,T,r0) : initial bond price of original rate
P1(t,T,rt) : bond price simulated in MC simulation

where P(t,T,r) = A(t,T)*exp(-B(t,T)*r) is formula for bond price

At each simulation A(t,T) and B(t,T) are not changing.Hence you can store values P0(t,T,r0) and B(t,T) before beginning the MC loop and use this formula to get bond price during each simulation when your rt is changing:

P1 = P0*exp(-B(t,T)*(r0-rt))

Example code in VBA : Bond Option pricing in Hull White using Monte Carlo simulaltion

Hope it helps!

(sorry for necroposting)

I have also implemented a pricing function for a ZC-bond put option in Matlab. I used the formulas from Brigo & Mercurio, "Interest Rate Models", chapter 3.3.2. You can access the code here - I wonder how your code differs from mine:
http://www.quantcode.com/modules/newbb/viewtopic.php?topic_id=471&forum=1

Regarding (1), could you show your work? I am interested in seeing your Euler scheme to compare it with my own work.