An undergraduate capstone paper on pricing options with Monte Carlo methods, with an emphasis on the mathematics behind variance reduction.
This capstone project explores how Monte Carlo simulation can be used to price financial options, and why naive simulation is often too slow to be practical. The paper builds up the theory from first principles and proves why several variance reduction techniques converge faster than plain Monte Carlo.
This is the core experiment from the paper, re-implemented in your browser: pricing a European call (S0 = 100, K = 90, r = 3%, T = 1 year) by simulation. Drag the volatility slider or re-run with fresh random numbers and watch the antithetic estimator lock onto the exact Black-Scholes price with far fewer samples than plain Monte Carlo.
The paper's emphasis is on proving why these techniques work, not just showing that they do. Three of the central results:
By the central limit theorem, the estimator's 95% confidence interval has half-width 1.96 σ/√n. One extra digit of accuracy costs 100× the samples, so the practical way forward is to shrink σ itself. That is what variance reduction does.
Each draw Z is paired with its mirror image −Z and the two payoffs are averaged. The paper proves that when the payoff is a monotone function of Z, the correlation ρ is at most zero, so an antithetic pair is guaranteed to beat two independent samples. The demo above shows exactly this guarantee at work.
If a quantity X correlated with the payoff has a known mean, the estimator can be corrected by the observed error in X. At the optimal coefficient β* = Cov(Y, X)/Var(X), the variance is multiplied by 1 − ρ²: a correlation of 0.95 cuts the required samples by roughly 10×.