HiI am looking at Monte Carlo (not tree) on the Hull-White 1-factor (extended Vasicek) model. I've generated paths with say 10 timesteps and now I want to test my code, so I want to price some simple option or bond which has an analytic solution. So the 2 ideas that come to mind are a bond option and a caplet. 1. Bond option. suppose at t=3 I have the right to buy a bond that matures at t=5 for K. There's a formula akin to Black's formula in Hull. But on a Monte Carlo sample path, at exercise time t=3 how do I know how much the bond is worth? I can see the short rate at t=3 which gets me to t=4, but I don't think I can assume that I can see the short rate at t=4 to roll to t=5. Do I use the HW bond price formula at t=3? But then I need the zero curve at t=3, which is what I'm trying to find...2. Caplet. Again there's a nice formula, but to use it I need the volatility of the forward interest rates. According to Hull these decrease because of the mean reversion, but what is the exact formula for the volatility of forward interest rate? I found a formula in Hull on the bond option section for the volatility of the forward bond price, but what about the volatility of the rate?Thanks!-Neal

Hello,it sounds like you have reverted input and output to your model: typically you will have the Black volatilities as input, say from the market. Then you use the 'closed form' solution in the 1 factor model for bond option, caplets or swaptions to calibrate your model parameters. Only then you price your caplet with MC on the calibrated model and try to match the Black prices you've input before. You can dummy out the Black inputs if you don't have real market data, for testing at least. I'm probably reading too much into your brief post, but you seem to be comparing your numerical solution within the model to the exact solution within the model, leaving out any calibration issues. This is a good initial testing step but will only reveal implementation errors but not how well you are matching market instruments (just a reminder, in the case that was not already obvious to you).QuoteOriginally posted by: nealsmithHiI am looking at Monte Carlo (not tree) on the Hull-White 1-factor (extended Vasicek) model. I've generated paths with say 10 timesteps and now I want to test my code, so I want to price some simple option or bond which has an analytic solution. So the 2 ideas that come to mind are a bond option and a caplet. 1. Bond option. suppose at t=3 I have the right to buy a bond that matures at t=5 for K. There's a formula akin to Black's formula in Hull. But on a Monte Carlo sample path, at exercise time t=3 how do I know how much the bond is worth? I can see the short rate at t=3 which gets me to t=4, but I don't think I can assume that I can see the short rate at t=4 to roll to t=5. Do I use the HW bond price formula at t=3? But then I need the zero curve at t=3, which is what I'm trying to find...In MC you go forward in time. At payoff you then know (remember) the reference (long rate) from the earlier fixing/reset date and cay pay-out accordingly. At reset/fixing time you have the short-rate. In the 1f HW model you have a closed form expression for zero coupon bonds B(t_fix, T_mat, r_t_fix), isn't that equivalent to the zero curve seen from t_fix you were looking for? Quote2. Caplet. Again there's a nice formula, but to use it I need the volatility of the forward interest rates. According to Hull these decrease because of the mean reversion, but what is the exact formula for the volatility of forward interest rate? I found a formula in Hull on the bond option section for the volatility of the forward bond price, but what about the volatility of the rate?Thanks!-NealMaybe your post is in the wrong sub forum and therefore attracts no better responses. You might consider googling for a walk-through guide of pricing under 1F HW, many excellent detailed write-up exists for free. More problem areas are what parameters do you allow to be time dependent, under what measure are you evolving the short rate (spot, T-forward,...) and what instruments should you be calibrating your model to, given the instrument you are pricing.I hope my response assists you in clarifying your question and in turn get answers from more senior guys here.