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)

  • $\begingroup$ All good questions but i think youll find all the answers you need in a good fixed income book e.g brigo&mercurio $\endgroup$
    – adelm
    Commented May 17, 2014 at 17:23
  • $\begingroup$ 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. $\endgroup$ Commented May 17, 2014 at 17:35
  • $\begingroup$ 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. $\endgroup$
    – Jonas K
    Commented Jun 18, 2014 at 21:19
  • 1
    $\begingroup$ The solution quant.stackexchange.com/questions/15956/… may partially answered your question $\endgroup$
    – Gordon
    Commented Dec 22, 2014 at 18:24
  • $\begingroup$ Here's the solution for the Vasicek model: quant.stackexchange.com/q/47522/26559 $\endgroup$
    – jChoi
    Commented Nov 24, 2021 at 5:19

1 Answer 1


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.

  • $\begingroup$ 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.) $\endgroup$ Commented May 17, 2014 at 16:17
  • $\begingroup$ Regarding how to arrive at a closed form solution this answer is quite good: Use of Girsanov's theorem in bond pricing $\endgroup$
    – Jonas K
    Commented Jun 18, 2014 at 21:25

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