# How to price zero coupon bonds with short term rates model?

I want to find the price of Zero coupon bond given a short rate model.

I think about Merton, Vasiceck, CIR, Ho & Lee models.

1) Given a simulation of $r_t$ how can I calculate $P(t,T) = \mathbb{E}^Q\left[\left. \exp{\left(-\int_t^T r_s\, ds\right) } \right| \mathcal{F}_t \right]$ ?

Using the simulations i think it would be easy to calculate the integral. But how to calculate the integral knowing $\mathcal{F}_t$ ? Am I supposed to find an expression of $r_s$ depending on $r_t$ ?

2) How to deal with the risk neutral probability here ?

3) Would this approach still be ok with a time dependant model ? (Hull White) Would this approach still be good with multiple factor model ? (Logstaff Schwartz)

• All good questions but i think youll find all the answers you need in a good fixed income book e.g brigo&mercurio – adelm May 17 '14 at 17:23
• I believe so for closed formulae. But i can't find anything about how to deal with the formula above in practice, how to deal with Q and Ft. – lcrmorin May 17 '14 at 17:35
• Regarding 2) All models depend on a set of parameters that you calibrate to observed market prices, once that is done you are using risk neutral probabilities. – Jonas K Jun 18 '14 at 21:19
• The solution quant.stackexchange.com/questions/15956/… may partially answered your question – Gordon Dec 22 '14 at 18:24

## 1 Answer

If you do not know anything about the dynamics of you short-rate $r_t$, then there is no way to express the price of the zero coupon bond better than what your already have:

$P(t,T) = \mathbb{E}^Q\left[\left. \exp{\left(-\int_t^T r_s\, ds\right) } \right| \mathcal{F}_t \right]$

You can use a model given in this page where you should be able to find close formulas for the zero coupon bond, if available, in their respective wiki pages or in FI books.

• I'd like to understand how to work with this formula given a short rate dynamic. (ie. how to find the closed formula when it is possible and how to use a simulated path of $r_t$ to calculate it when it is not possible.) – lcrmorin May 17 '14 at 16:17
• Regarding how to arrive at a closed form solution this answer is quite good: Use of Girsanov's theorem in bond pricing – Jonas K Jun 18 '14 at 21:25