Chapter 2.P3, Problem 2P

Use the difference equation $\left(4\right)$ 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 $\left(6\right)$ to compute 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

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

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

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

Equation $4$: To estimate the price of a call option using a Monte Carlo method, an ensemble

$\left\{S_N^{\left(k\right)}=S^{\left(k\right)}\left(T\right),k=1,\mathrm{...},M\right\}$

of $M$ stock prices at expiration is generated using the difference equation

$\begin{array}{cc}{S}_{n+1}^{\left(k\right)}=S_n^{\left(k\right)}+r{S}_n^{\left(k\right)}\Delta t+\sigma S_n^{\left(k\right)}{\epsilon }_{n+1}^{\left(k\right)}\sqrt{\Delta t},& S_0^{\left(k\right)}\end{array}=s$. $\left(4\right)$

Formula $6$: Solving $\frac{1}{M}{\displaystyle \underset{k=1}{\overset{M}{\sum }}f\left(S_N^{\left(k\right)}\right)={\left(1+r\Delta t\right)}^N\hat{C}\left(s\right)}$ for $\hat{C}\left(s\right)$ yields the Monte Carlo estimate

$\hat{C}\left(s\right)={\left(1+r\Delta t\right)}^{-N}\left\{\frac{1}{M}{\displaystyle \underset{k=1}{\overset{M}{\sum }}f\left(S_N^{\left(k\right)}\right)}\right\}$

for the option price. Thus the Monte Carlo estimate $\hat{C}\left(s\right)$ is the present value of the average of the payoffs computed using the rules of compound interest.