Monte Carlo Simulation

The Problem

You want to estimate $\mathbb{E}[g(X)]$ where $X$ has a known distribution and $g$ is a payoff function (for instance, $g(x)=\max (x-K,0)$ for a call option with strike $K$).

1. Basic estimator. Write down the Monte Carlo estimator ${\hat{\mu }}_N$ based on $N$ i.i.d. samples $X_1,\dots ,X_N$. State its variance and the rate at which the standard error decays in $N$.

2. Variance reduction: antithetic variates. Suppose $X\sim \mathcal{N}(0,1)$. Instead of averaging $g(Z_i)$ for $i=1,\dots ,N$, average $\frac{1}{2}(g(Z_i)+g(-Z_i))$ over $N/2$ pairs. Derive the variance of this estimator and state when it is most effective.

3. Control variates. Suppose we know $\mathbb{E}[h(X)]={\mu }_h$ exactly (analytically) for some $h$ correlated with $g$. Construct a control-variate estimator with optimal coefficient ${\beta }^{\ast }$ and derive its variance relative to the basic estimator.

These are the three things every desk-quant asks in week one. Get the derivations down cold.