# Girsanov's theorem and simulation of bond prices

Assume that we want to calculate the time $t=0$ price of a bond: $B(0,T) = E_P[\exp(-\int_0^T r_s ds)]$, where $r$ is the interest rate following the SDE $dr_t=k(\theta-r_t)dt+\sigma dB_t=b(r_t)dt+\sigma dB_t$.

I was shown that one could write the price as

$B(0,T) = E_{\hat{P}}[\exp(-\int_0^TB^{*}_sds)\exp(\int_0^Tb(B^{*}_s)dB^{*}s-\frac{1}{2}\int_0^Tb^2(B^{*}_s)ds)]$

where $B_t^{*}$ is the "new" Brownian motion from the Girsanov theorem.

However, when I try to implement it, it results in prices that are too low. Here is what I did:

Since $dr_t=b(r_t)dt+\sigma dB_t$, Girsanov's theorem gives a new Brownian motion with dynamics $dB_t^{*}=\frac{1}{\sigma}b(r_t)dt+dB_t$, and the dynamics of $r_t$ becomes $dr_t=\sigma dB_t^{*}$. So $r_t=r_0+\sigma B_t^{*}$ and $dB_t^{*}= \frac{1}{\sigma}b(r_0+\sigma B_t^{*})dt+dB_t$. With this last expression I tried to use Euler discretization to find $B_t^{*}$, then finally I approximated the three integrals as sums.

What am I doing wrong? Secondly, I also wonder what the starting point of $B_t^{*}$ should be, i.e. $B_0^{*}$.

• @ nothesharpestknife : IMO you should ask this question at the quant stack exchange quant.stackexchange.com Regards Mar 31, 2014 at 12:13