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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)

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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 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. –  lmorin May 17 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. –  klon Jun 18 at 21:19
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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.

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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.) –  lmorin May 17 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 –  klon Jun 18 at 21:25
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