Monte Carlo Option Pricing Simulator

Parameters

Spot price (S₀)100

10300

Strike (K)100

10300

Maturity (T, years)1.00

0.253

Risk-free rate (r)3.0%

00.1

Volatility (σ)20.0%

0.050.6

Number of simulations5000

20020000

Seed7

1100

Metrics

MC estimate

9.202

Black–Scholes price

9.413

Standard error

0.201

Convergence to the Black–Scholes price

95% confidence band (±1.96·SE)

MC estimateBlack–Scholes

Distribution of S_T and the paths

Select bins in the histogram (click or drag) to highlight the paths that ended there. Esc clears.

Path-dependent payoffs on the same paths

All four payoffs are evaluated on the same simulated paths (50 monitoring dates), so their differences come from the payoffs, not from sampling noise.

Drag the barrier B (or focus it and use the arrow keys). A path that touches it is knocked out: it turns grey from that moment and pays nothing.

Up-and-out call · MC price ± 1.96·SE

3.556 ± 0.178

European (Black–Scholes)

9.413

Paths knocked out

17.3%

Continuous monitoring (closed form)

3.203

Price against the barrier level

Up-and-out callEuropean (Black–Scholes)

Moving the barrier up raises the price from zero towards the European price. Discrete monitoring (50 dates) is slightly dearer than the continuous closed form, because it misses crossings between dates.

About this tool

Monte Carlo simulation prices an option by averaging its discounted payoff over many simulated outcomes. This tool simulates terminal stock prices under risk-neutral geometric Brownian motion and shows the estimate and its confidence interval converging towards the Black–Scholes price as the number of paths grows.

Frequently asked questions

How fast does a Monte Carlo estimate converge?

The standard error falls like 1/√N: four times as many simulations halve the error.

Why use Monte Carlo if Black–Scholes has a formula?

Here the formula is the benchmark that shows the method works. Monte Carlo becomes essential for path-dependent payoffs, many underlyings and models without closed-form prices.

About this tool

Frequently asked questions

How fast does a Monte Carlo estimate converge?

The standard error falls like 1/√N: four times as many simulations halve the error.

Why use Monte Carlo if Black–Scholes has a formula?

Here the formula is the benchmark that shows the method works. Monte Carlo becomes essential for path-dependent payoffs, many underlyings and models without closed-form prices.

Monte Carlo Pricing

Parameters

About this tool

Frequently asked questions

Monte Carlo Pricing

Parameters

About this tool

Frequently asked questions