Contract

$

Spot price (S)

$

Strike (K)

Volatility (σ)

Annualised, 0.20 = 20%

Time to maturity (T)

Years

Risk-free rate (r)

Annualised, 0.05 = 5%

Dividend yield (q)

Annualised, continuous

Call

Put

Exercise style

Try this

0 of 3 done

Halve the error

Simulate with 10,000 paths and note the standard error. Then use 40,000 paths.

What to look for: Four times the paths halves the standard error. Monte Carlo converges like one over the square root of the number of paths, which is why it is slow to get very precise.

Turn off variance reduction

Untick antithetic variates and the control variate, keep the same number of paths, and simulate again.

What to look for: The standard error grows several-fold for the same work. Variance reduction is the difference between a usable and an unusable estimate at this path count.

Read the running estimate

Look at the running-estimate chart while changing the seed a few times.

What to look for: The 95% band narrows like one over the square root of the path count, and the reference line stays inside the band most of the time. Occasionally it does not: that is what 95% means.

Simulation

Paths

Time steps

Used for American exercise and the path chart

Random seed

Same seed, same answer

Antithetic variates

Control variate (discounted spot)

European call estimate

$10.47

± 0.0085 standard error

95% confidence interval

10.4496 to 10.4829

Black-Scholes

10.4506

inside the interval

Run statistics

Paths	100,000
Standard error	0.008490
95% CI low	10.449605
95% CI high	10.482884
Control variate beta	2.536934
Variance reduction factor	14.782753
Seed	42
Compute time (price and Greeks)	425 ms

Greeks

Raw values are partial derivatives. The practitioner column rescales them to the units traders quote: per 1 volatility point, per 1% of rate, or per calendar day. Delta and Vega are pathwise estimates from the same simulation; the rest are bump-and-revalue with common random numbers, so Gamma and the second-order Greeks are noisy.

Greek	Definition	Raw	Practitioner	Meaning
Delta Δ	`∂V/∂S`	0.63645	0.63645 per $1 of spot	How much the option price moves for a $1 move in the underlying. Also the number of shares needed to hedge one option.
Gamma Γ	`∂²V/∂S²`	0.018753	0.018753 per $1 of spot	How fast Delta changes as the underlying moves. High Gamma means the hedge must be rebalanced often.
Vega ν	`∂V/∂σ`	37.558	0.37558 per 1 vol point	Sensitivity to implied volatility. Long options gain value when volatility rises.
Theta Θ	`∂V/∂t`	-6.4208	-0.017591 per calendar day	Time decay: how much value the option loses as one day passes with everything else unchanged.
Rho ρ	`∂V/∂r`	53.182	0.53182 per 1% of rate	Sensitivity to the risk-free rate. Calls gain and puts lose when rates rise.
Epsilon ε	`∂V/∂q`	-63.648	-0.63648 per 1% of yield	Sensitivity to the dividend yield. Higher expected dividends lower the forward price, hurting calls and helping puts.

Second-order Greeks

Greek	Definition	Raw	Practitioner	Meaning
Vanna	`∂²V/∂S∂σ`	-0.27283	-0.0027283 per 1 vol point	How Delta changes when volatility changes, or equally how Vega changes when spot moves.
Vomma	`∂²V/∂σ²`	9.1828	0.00091828 per 1 vol point²	Convexity in volatility: how Vega itself changes as volatility moves.
Charm	`∂Δ/∂t`	-0.065592	-0.0001797 per calendar day	Delta decay: how the hedge ratio drifts as one day passes.
Speed	`∂³V/∂S³`	-0.00079173	-0.00079173 per $1 of spot	How Gamma changes as the underlying moves.
Zomma	`∂Γ/∂σ`	-0.083148	-0.00083148 per 1 vol point	How Gamma changes when volatility changes.
Color	`∂Γ/∂t`	0.010607	2.9061E-05 per calendar day	Gamma decay: how Gamma changes as one day passes. Explodes for at-the-money options near expiry.

Watch the simulation

The first 200 of the run's 100,000 paths draw one by one, then in bursts, each coloured by whether it ends in the money. Beside them the running estimate and its 95% band close in on the reference price as the paths add up. It is the run priced above, so it ends on the same estimate.

Playback speed

Ends in the money Ends out of the money Strike

Estimate 95% bandBlack-Scholes

Sample paths

Thirty of the simulated price paths. The dashed line is the strike.

Distribution of the terminal price

Lognormal, as the model assumes: skewed with a long right tail.

Convergence

The running estimate as paths accumulate, with its 95% confidence band. The error shrinks with the square root of the path count: four times the paths halves the error.