Monte Carlo option pricing, made testable.

A numerical-finance engine that prices European calls and puts, validates simulation against Black–Scholes, calculates analytical sensitivities and measures the effect of antithetic variance reduction.

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2×variance reduction

600kevaluations compared

95%confidence intervals

01 / The question

How closely does simulation recover an analytical price—and how much uncertainty can a simple variance-reduction method remove?

The project begins with risk-neutral geometric Brownian motion and the closed-form Black–Scholes solution. Those analytical prices provide a benchmark for standard and antithetic Monte Carlo estimators.

Rather than reporting a single simulated price, the engine exposes standard errors and 95% confidence intervals. Tests cover analytical benchmarks, put–call parity, finite-difference Greek validation and the statistical treatment of independent antithetic pair averages.

02 / Method

One model. Three views of the price.

Analytical benchmark

Black–Scholes call and put prices establish the deterministic reference.

Standard simulation

Exact terminal GBM samples produce discounted payoff observations.

Antithetic estimator

Each normal draw is paired with its negative before payoffs are averaged.

03 / Results

Lower variance at the same evaluation budget.

Standard and antithetic estimators use equal numbers of terminal-price evaluations. The antithetic estimator consistently produces the lower standard error while retaining the expected N⁻¹ᐟ² scaling.

Spot price100

Strike100

Risk-free rate5%

Volatility20%

Maturity1 year

Evaluation budget600,000

Standard and antithetic Monte Carlo error comparison
Option	Standard SE	Antithetic SE	SE reduction	Variance reduction
Call	0.019017	0.013460	1.413x	1.996x
Put	0.011188	0.008564	1.306x	1.707x

Call and put price estimates from standard and antithetic Monte Carlo converging towards Black-Scholes prices — Standard and antithetic estimates converge towards the analytical call and put prices as the terminal-price evaluation budget increases.

Log-log comparison showing lower standard errors for antithetic call and put estimators — Antithetic standard errors remain below their standard Monte Carlo counterparts across every tested budget.

04 / Engineering

Designed as a reproducible numerical system.

Black–Scholes calls, puts and analytical Greeks
Standard and antithetic Monte Carlo estimators
Confidence intervals and equal-budget comparisons
Finite-difference validation of every Greek
Automated linting, tests and continuous integration
Signed version tags and reproducible analysis scripts

05 / Next

From research engine to interactive proof of concept.

The next version will expose the analysis through this website, allowing a visitor to change option parameters and compare prices, confidence intervals and Greeks without installing the Python package.

Explore the repository