Chapter 2.P3, Problem 3P

VarianceReduction by Antithetic Variates. A simple and widely used technique for increasing the efficiency and accuracy of Monte Carlo simulations in certain situations with little additional increase in computational complexity is the method of antithetic variates. For each $k=1,\dots ,M$, use the sequence $\left\{{\epsilon }_1^{\left(k\right)},\dots ,{\epsilon }_{N-1}^{\left(k\right)}\right\}$ Eq. $\left(4\right)$ to simulate a payoff $f\left(S_N^{\left(k+\right)}\right)$ and also use the sequence $\left\{-{\epsilon }_1^{\left(k\right)},\dots ,-{\epsilon }_{N-1}^{\left(k\right)}\right\}$ in Eq. $\left(4\right)$ to simulate an associated payoff $f\left(S_N^{\left(k-\right)}\right)$. Thus the payoffs are simulated in pairs $\left\{f\left(S_N^{\left(k+\right)}\right),f\left(S_N^{\left(k-\right)}\right)\right\}$. A modified Monte Carlo estimate is then computed by replacing each payoff $f\left(S_N^{\left(k\right)}\right)$ in Eq. $\left(6\right)$ by the average $\left[f\left(S_N^{\left(k+\right)}\right)+f\left(S_N^{\left(k-\right)}\right)\right]/2$,

${\hat{C}}_{AV}\left(s\right)=\frac{\frac{1}{M}{\displaystyle \underset{k=1}{\overset{M}{\sum }}\frac{f\left(S_N^{\left(k+\right)}\right)+f\left(S_N^{\left(k-\right)}\right)}{2}}}{{\left(1+r\Delta t\right)}^N}$ $\left(\text{iii}\right)$

Use the parameters specified in Problem $2$ to compute several (say, $20$ or so) option price estimates using Eq. $\left(6\right)$ and an equivalent number of option price estimates using $\left(\text{iii}\right)$. For each of the two methods, plot a histogram of the estimates and compute the mean and standard deviation of the estimates. Comment on the accuracies of the two methods.

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

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

Equation $6$: 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.

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,

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