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

Break the explicit scheme

Choose the explicit scheme with 200 spot steps and 200 time steps. Then read the stability limit in the diagnostics and raise the time steps above it.

What to look for: Below the limit the price surface is garbage with wild oscillations. Above it the price is sensible. The explicit scheme is only stable when the time step is small enough for the spot step.

Compare schemes on the same grid

Price with Crank-Nicolson and then with the fully implicit scheme, both at 200 by 200.

What to look for: Crank-Nicolson is closer to the closed form. It is second-order accurate in time; the implicit scheme is only first-order.

Read the surface

Look at the heatmap of the value surface from expiry back to today.

What to look for: The row at expiry is exactly the payoff with its kink at the strike. Rolling back in time smooths the kink away, which is what time value looks like on a grid.

Grid

Scheme

Averages the explicit and implicit schemes. Unconditionally stable and second-order accurate in time. The usual choice.

Spot steps

Time steps

European call value

$10.44

Finite difference (Crank-Nicolson)

Black-Scholes

$10.4506

Grid error

-0.0099

Grid details

Spot steps	200
Time steps used	200
Spot step (dS)	2.000000
Time step (dt)	0.005000
Max spot on grid	400.000000
Compute time (price and Greeks)	93 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, Gamma and Theta are read from the grid around the spot; the rest are bump-and-revalue.

Greek	Definition	Raw	Practitioner	Meaning
Delta Δ	`∂V/∂S`	0.63658	0.63658 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`	0.018791	0.018791 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`	37.578	0.37578 per 1 vol point	Sensitivity to implied volatility. Long options gain value when volatility rises.
Theta Θ	`∂V/∂t`	-6.4244	-0.017601 per calendar day	Time decay: how much value the option loses as one day passes with everything else unchanged.
Rho ρ	`∂V/∂r`	53.226	0.53226 per 1% of rate	Sensitivity to the risk-free rate. Calls gain and puts lose when rates rise.
Epsilon ε	`∂V/∂q`	-63.667	-0.63667 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`	-0.27997	-0.0027997 per 1 vol point	How Delta changes when volatility changes, or equally how Vega changes when spot moves.
Vomma	`∂²V/∂σ²`	9.2776	0.00092776 per 1 vol point²	Convexity in volatility: how Vega itself changes as volatility moves.
Charm	`∂Δ/∂t`	-0.065706	-0.00018002 per calendar day	Delta decay: how the hedge ratio drifts as one day passes.
Speed	`∂³V/∂S³`	-0.00018293	-0.00018293 per $1 of spot	How Gamma changes as the underlying moves.
Zomma	`∂Γ/∂σ`	-0.17855	-0.0017855 per 1 vol point	How Gamma changes when volatility changes.
Color	`∂Γ/∂t`	0.021135	5.7905E-05 per calendar day	Gamma decay: how Gamma changes as one day passes. Explodes for at-the-money options near expiry.

Value surface

Option value at every spot and time. Expiry is the top row, today the bottom.

Slices through the surface

Value against spot at several times. The kink at the strike smooths out as maturity grows.