Antithetic variates
Quantitative Finance glossary · Numerical Methods & Optimization
Definition
Variance-reduction technique in Monte Carlo: for every simulated path using boldsymbolZ, also evaluate the path using -boldsymbolZ, then average. Variance of the estimator becomes Var[(f(Z) + f(-Z))/2] = 1/2(Var,f + Cov(f(Z), f(-Z))); variance falls relative to plain MC whenever f is monotone in Z (negative covariance). Effective for vanilla European options; ineffective when payoff symmetry breaks (e.g. one-touch barriers).
Antithetic variates in plain English
When you can't solve a pricing problem with a formula, you can estimate the answer by simulating thousands of random "what if" scenarios and averaging them — that's Monte Carlo. The catch is that random guesses are noisy, so you need a huge number of runs to trust the average. Antithetic variates is a clever cost-saver: every time you roll a random shock Z, you also run the mirror-image scenario using -Z, then average the pair. It's like polling both an optimist and a pessimist on every question instead of two random strangers — their errors tend to cancel, so your average settles down faster. It works great when the outcome moves smoothly up or down with the shock, but it fails when the payoff isn't symmetric (like a bet that only pays if the price touches one specific level).
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