# How to determine the prices computationally?

As in previous question mentioned, I attended a course in interest rate theory (I'm studying math). Now I have a question how one calculate this prices in reality. Suppose we assume a simple model for our interest rate, let's say the instantenous spot rate is modeled by the Vasicek model. Hence $r(t)$ has is a Gaussian process. The arbitrage free bond prices are given by

$$P(t,T)=E_Q[\exp{(-\int_t^Tr(u)du)}|\mathcal{F}_t]$$

where $Q$ is a equivalent (local) martingale measure. How do you compute this computationally? The conditional expectation is random variable, even more you integrate the Gaussian process, so how do you proceed in reality to compute such an conditional expectation?

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Several ways, some are applicable at certain times when others are not, sometimes you can chose freely which way to go.

a) You can attempt to solve using partial differential equations

b) If applicable you can find a discretization of the underlying model and run a monte carlo simulation

c) you can setup binary trees and work your way backward.

Here is a good paper (though a bit dated) that introduces binomial option pricing as part of numerical option pricing techniques:

http://finance.wharton.upenn.edu/~rlwctr/papers/9311.PDF

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 Thanks for your answer. I know the approaches a) and b). – hulik Jan 15 at 12:27 added a reference for c) – Matt Wolf Jan 15 at 12:40 Thanks for the pdf! I will check it carefully. – hulik Jan 15 at 12:56