When there is no formula, simulate the future and average the payoff. Price a European option by drawing many risk-neutral geometric-Brownian paths (drift = the risk-free rate) and averaging the discounted payoffs. The random average walks onto the Black-Scholes number as the sample grows — and it is the only route for options that have no closed form.
source Phelim P. Boyle, Options: A Monte Carlo Approach, Journal of Financial Economics 4(3), 323–338 (1977). doi:10.1016/0304-405X(77)90005-8 Rendered, not quoted.
A pricing model with named assumptions, not investment advice.
Under the risk-neutral measure the stock is a geometric Brownian motion whose drift is the risk-free rate r, not the real-world drift:
ST = S0·exp((r − ½σ²)T + σ√T·Z), Z~N(0,1)
Draw many Z, take max(ST−K, 0), discount by e−rT, and average. The Law of Large Numbers pulls the average onto the true expectation; the Central Limit Theorem says the error shrinks like 1/√N.
the-monte-carlo method applied to the-risk-neutral expectation. Boyle (1977) took a tool built for physics integrals and pointed it at finance: an option price is an expectation, and any expectation can be estimated by sampling. This sphere is where the two neighbours meet — simulation becomes a pricer.
Where Black-Scholes gives a formula, Boyle gives a procedure that still works when the formula runs out: path-dependent, multi-asset, early-exercise payoffs.
Re-runs the convergence check against the live engine on every change. It confirms the Monte-Carlo price sits within tolerance of Black-Scholes — and flips red the instant the drift is tampered.
A fixed-seed PRNG (mulberry32) feeds Box-Muller to make standard normals Z. Contract fixed at S₀=K=100, r=5%, σ=20%, T=1. Determinism is the point: same seed, same paths, same audit.
Live: N risk-neutral paths simulated, payoffs discounted and averaged. Closed-form Black-Scholes shown for comparison.
| estimator | N | price | std err | |err vs BS| |
|---|
Proven result — the discounted sample average converges to the closed-form price:
Antithetic variates (pairing Z with −Z) reach the same answer with a smaller standard error at equal cost.
The estimate is only as honest as its measure and its model. Boyle’s method inherits every Black-Scholes assumption — constant volatility, log-normal prices, no jumps, frictionless, continuously-hedged markets — and adds one of its own: Monte-Carlo error never reaches zero. You buy accuracy at 1/√N, so 100× the paths buys only 10× the precision. Real markets have fat tails and gaps this Gaussian engine cannot see.
“More paths always means a better price.”
→ Only against the model. More paths shrink sampling error, not model error; a wrong σ stays wrong at any N.
“Simulate under the stock’s real growth rate μ.”
→ Fatal. Pricing uses the risk-neutral drift r. Using μ gives a number that is not a price — the exact bug the Tamper plants.
“The Monte-Carlo price is the true value.”
→ It is an estimate with a confidence band; report the standard error or it is theatre.
The disclosed planted void: swap the risk-neutral drift r for the real-world drift μ=10%. The paths grow too fast, payoffs inflate, and the price drifts systematically above Black-Scholes. The Witness (7) catches it live.