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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)

SRKX
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Lucas Morin
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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 '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. – Lucas Morin 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
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    The solution http://quant.stackexchange.com/questions/15956/ho-and-lee-derivation-for-short-rates-model may partially answered your question – Gordon Dec 22 '14 at 18:24
  • Here's the solution for the Vasicek model: https://quant.stackexchange.com/q/47522/26559 – jChoi Nov 24 '21 at 05:19

1 Answers1

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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.

SRKX
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