Quiz: Monte Carlo: Antithetic Variates, Control Variates, Quasi-MC

1. A Monte Carlo estimator uses $N = 10{,}000$ paths and achieves a standard error of $0.05$. How many paths are needed to reduce the standard error to $0.01$?

2. The antithetic variate estimator for a call option under GBM uses paired samples $Z_i$ and $-Z_i$ (where $Z_i \sim \mathcal{N}(0,1)$). Why is the antithetic pair negatively correlated?

3. The optimal control variate coefficient is $c^* = \mathrm{Cov}(f(X), Y) / \mathrm{Var}(Y)$. The resulting variance of the control-adjusted estimator is:

4. The Koksma-Hlawka inequality implies that quasi-Monte Carlo with a Sobol sequence always outperforms standard Monte Carlo for option pricing, regardless of the payoff function's smoothness.

5. In the Brownian bridge construction for quasi-Monte Carlo simulation of a GBM path, Sobol dimension 1 is assigned to $W_T$, dimension 2 to $W_{T/2}$, and so on recursively. Why is this preferred over the sequential construction (dimension $j$ assigned to time step $j$)?

6. A control variate estimator uses $Y = e^{-rT}S_T$ with known mean $\mu_Y = S_0$. For an at-the-money call ($S_0 = K$), the correlation $\rho_{f,Y}$ between the call payoff and the discounted stock is approximately 0.9. What variance reduction factor does this achieve?