Numerical / Monte Carlo Greek Estimation

You are pricing a European call option under geometric Brownian motion and need to estimate the Greeks -- delta and vega -- by Monte Carlo simulation. 1. **Pathwise derivative estimator for delta.** Write down the pathwise (infinitesimal perturbation analysis) estimator for $\Delta = \partial C / \partial S_0$. State the conditions under which this estimator is unbiased. 2. **Likelihood ratio estimator for vega.** Write down the likelihood ratio (score function) estimator for vega, $\mathcal{V} = \partial C / \partial \sigma$. How does its variance compare to the pathwise estimator? 3. **Variance reduction and confidence intervals.** Propose a combined estimator that uses the Black-Scholes closed-form Greek as a control variate. Explain how you would construct a confidence interval for each Greek estimate.