Chapter 8.P2, Problem 3P

Use the differential equation (4) to generate an ensemble of stock prices $S_N^{\left(k\right)}=S^{\left(k\right)}\left(N\Delta t\right),k=1,\dots ,M$ (where $T=N\Delta t$) and then use formula (6) to compare a Monte Carlo estimate of the value of a five-month call option $\left(T=\frac{5}{12}\text{years}\right)$ for the following parameter values: $r=0.06,\sigma =0.2$, and $K=\$50$. Find estimates corresponding to current stock prices of $S\left(0\right)=s=\$45,\$50$, and $\$55$. Use $N=200$ time steps for each trajectory and $M\cong 10,000$ sample trajectories for each Monte Carlo estimate. Check the accuracy of your results by comparing the Monte Carlo approximation with the value computed from the exact Black-Scholes formula

$C\left(s\right)=\frac{s}{2}\text{erfc}\left(-\frac{d_1}{\sqrt{2}}\right)-\frac{K}{2}{e}^{rT}\text{erfc}\left(-\frac{d_2}{\sqrt{2}}\right)$, (ii)

where

$\begin{array}{l}{d}_1=\frac{1}{\sigma \sqrt{T}}\left[\ln \left(\frac{s}{k}\right)+\left(r+\frac{{\sigma }^2}{2}\right)T\right],\\ d_2=d_1-\sigma \sqrt{T}\end{array}$

And $\text{erfc}\left(x\right)$ is the complementary error function,

$\text{erfc}\left(x\right)=\frac{2}{\sqrt{\pi }}{\displaystyle \underset{x}{\overset{\infty }{\int }}{e}^{-t^2}dt}$.

The difference equation (4) is given below:

$S_{n+1}^{\left(k\right)}=S_n^{\left(k\right)}+r{S}_n^{\left(k\right)}\Delta t+\sigma S_n^{\left(k\right)}{\in }_{n+1}^k\sqrt{\Delta t},S_0^k=s$