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

Try this

0 of 3 done

Watch N(d2) collapse

Set the spot to 100, the strike to 150 and one year to maturity. Then shorten the maturity to 0.05.

What to look for: N(d2) in the diagnostics drops towards zero and the call price collapses: the option is very unlikely to finish in the money, so it is worth almost nothing.

Find the delta of an at-the-money call

Set the spot and the strike both to 100 and read Delta. Then raise the volatility to 0.60.

What to look for: Delta is a little above 0.5, not exactly 0.5, because the forward sits above the spot when r exceeds q. Higher volatility pushes it further above one half.

Check put-call parity by hand

Price a call, note it, then switch to a put with the same inputs and note that too.

What to look for: Call minus put equals S e^(-qT) minus K e^(-rT). With S = K = 100, r = 5%, q = 0 and T = 1 that is 100 minus 95.12, about 4.88.

Call value

$10.45

Intrinsic value

$0.00

Time value

$10.45

Working

d1	0.350000
d2	0.150000
N(d1)	0.636831
N(d2)	0.559618
Discount factor e^(-rT)	0.951229
Dividend factor e^(-qT)	1.000000
Compute time	0.004 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. All values here are analytic.

Greek	Definition	Raw	Practitioner	Meaning
Delta Δ	`∂V/∂S`	0.63683	0.63683 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`	0.018762	0.018762 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`	37.524	0.37524 per 1 vol point	Sensitivity to implied volatility. Long options gain value when volatility rises.
Theta Θ	`∂V/∂t`	-6.414	-0.017573 per calendar day	Time decay: how much value the option loses as one day passes with everything else unchanged.
Rho ρ	`∂V/∂r`	53.232	0.53232 per 1% of rate	Sensitivity to the risk-free rate. Calls gain and puts lose when rates rise.
Epsilon ε	`∂V/∂q`	-63.683	-0.63683 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`	-0.28143	-0.0028143 per 1 vol point	How Delta changes when volatility changes, or equally how Vega changes when spot moves.
Vomma	`∂²V/∂σ²`	9.8501	0.00098501 per 1 vol point²	Convexity in volatility: how Vega itself changes as volatility moves.
Charm	`∂Δ/∂t`	-0.065667	-0.00017991 per calendar day	Delta decay: how the hedge ratio drifts as one day passes.
Speed	`∂³V/∂S³`	-0.00051596	-0.00051596 per $1 of spot	How Gamma changes as the underlying moves.
Zomma	`∂Γ/∂σ`	-0.088885	-0.00088885 per 1 vol point	How Gamma changes when volatility changes.
Color	`∂Γ/∂t`	0.01053	2.885E-05 per calendar day	Gamma decay: how Gamma changes as one day passes. Explodes for at-the-money options near expiry.

Watch time decay

Everything stays as it is except the calendar. The value curve is redrawn 60 times from today to expiry, 365 days away, and it sinks onto the payoff. On the right, the value at today's spot of 100 falls towards its intrinsic value; the slope of that fall is theta, drawn as a tangent. Near the money it gets steeper as expiry comes close.

Playback speed

Value now Value today Earlier values Payoff at expiry Strike

Value at the spot Intrinsic value Theta: the slope now

Black-Scholes

Contract

Try this

$10.45

$0.00

$10.45

Greeks

Watch time decay