Monte Carlo vs closed form

A swarm of random draws, hunting the analytic answer.

Price a European call two ways. The Black-Scholes formula gives the exact answer instantly. Monte Carlo averages thousands of random payoffs — and the more draws you throw, the tighter it closes in. Error falls as 1/√N.

Black-Scholes (exact)

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Monte Carlo estimate

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|MC − BS|

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Std error ±1 SE

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Draws N

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discounted payoff of each draw → running mean closes on BS single run

MC estimate ± SE vs N (log scale)band ∝ 1/√N

Error vs N · log-logslope ≈ −½

Sample size

N — draws (log scale)10,000

Variance reduction Antithetic variatespair every Z with −Z · halves variance free Option (risk-neutral)

S spot100

K strike100

T years1.00

r rate0.050

σ volatility0.20

How to use

Drag N and the option parameters, then hit Re-seed & run. The big panel scatters every draw's discounted payoff and traces the running mean as it settles onto the gold Black-Scholes line. The lower-left chart logs each run's estimate with ±1 SE error bars; the lower-right shows error vs N on log-log axes — a straight line of slope −½ is the 1/√N law. Flip Antithetic variates to watch the error band shrink for the same N.

The Monte Carlo estimator

Under the risk-neutral measure, simulate N terminal prices

S_T = S₀·exp((r − σ²/2)·T + σ·√T·Z)

then average the discounted payoffs:

Ĉ = e^(−rT)·(1/N)·Σ max(S_T − K, 0)

Standard error is SE = e^(−rT)·sd(payoff)/√N. To halve the error you must quadruple N — that is the whole frustration and the whole beauty of Monte Carlo.

Why error falls as 1/√N

Each draw is an independent, unbiased estimate of the price with some variance v. Average N of them and the variance of the average is v/N, so the standard deviation — your typical error — is √(v/N) ∝ 1/√N.

That is the law of large numbers with its rate of convergence attached. On log-log axes, log(error) = const − ½·log(N) — a line of slope −½. The swarm never converges fast, but it converges reliably, in any dimension.

Antithetic variates — free variance

For every random draw Z, also use its mirror −Z. The two payoffs are negatively correlated, so their average has lower variance than two independent draws would. For a European call this cuts variance by roughly half (slightly less very deep in- or out-of-the-money).

It is exact: antithetic variates reduce variance, not bias. The estimator stays unbiased — you simply buy a tighter error band for the same number of draws.

Why this pattern matters

European calls have a closed form, so we don't need Monte Carlo here — it's pedagogical. But almost every real derivative — Asian, lookback, barrier, basket, autocallable — has no closed form. Monte Carlo is the workhorse for path-dependent payoffs.

The validation pattern shown here — use closed-form where it exists, trust MC where it doesn't — is how the entire derivatives industry checks its pricing engines. A new exotic gets priced two ways and the answers must agree.

Citations

Glasserman, P. (2003). Monte Carlo Methods in Financial Engineering. Springer. Ch. 1 (Foundations), Ch. 4 (Variance reduction).
Boyle, P. (1977). "Options: A Monte Carlo approach." Journal of Financial Economics, 4, 323–338. (The original paper applying MC to option pricing.)
Hammersley, J.M. & Handscomb, D.C. (1964). Monte Carlo Methods. Methuen. (Classical reference for the variance reduction tradition.)

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