Discrete dividends

Real stocks pay cash on dates, not a continuous yield. On the ex-dividend date the spot drops by the dividend, which hurts calls, helps puts, and is the one reason to exercise an American call early. This page prices with the escrowed-dividend model: the present value of the dividends is set aside, and the rest of the spot follows Black-Scholes. The continuous yield in the contract editor is ignored here.

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

Find the early exercise

Load the one-large-dividend preset, set the spot to 120 and the strike to 100, and price a call.

What to look for: The American price exceeds the European and the boundary chart shows exercise only in the step before the ex-dividend date. Shrink the dividend to 1 and the premium disappears.

Yield versus cash

Compare the European price with the continuous-yield proxy in the summary panel.

What to look for: They match by construction for the European option. The equivalent yield is what most desks feed into Black-Scholes when they cannot be bothered with dates.

Puts like dividends

Switch to a put with the same dividend and compare against the no-dividend price.

What to look for: The put is worth more with the dividend, because the spot will drop. The American premium is small: waiting for the drop beats exercising before it.

Dividends

Cash dividends: time in years and amount
Time (years)	Amount	Remove
	$

Tree steps for the American price

European call

$6.09

Black-Scholes on the spot net of dividends

American, tree

$7.18

Early-exercise premium $1.0968

No dividends

$10.45

Dividends worth $7.80 today

Continuous-yield proxy

$6.09

q = 8.12% gives the same European price

Each point is the spot at which the tree exercises rather than waits. For a call, exercise only ever pays in the step before an ex-dividend date, when the dividend is about to leave the stock; a gap means the tree never exercises there. Dividend dates: 0.5.

The European price falls (call) or rises (put) as the dividends grow. The American call pulls away from the European one once the dividend is big enough to be worth capturing.

European Greeks and working

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. Analytic Greeks of the option on the reduced spot. Delta with respect to the quoted spot is the same, since the dividend value does not move with it.

Greek	Definition	Raw	Practitioner	Meaning
Delta Δ	`∂V/∂S`	0.4776	0.4776 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`	0.021601	0.021601 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`	36.723	0.36723 per 1 vol point	Sensitivity to implied volatility. Long options gain value when volatility rises.
Theta Θ	`∂V/∂t`	-5.5696	-0.015259 per calendar day	Time decay: how much value the option loses as one day passes with everything else unchanged.
Rho ρ	`∂V/∂r`	37.945	0.37945 per 1% of rate	Sensitivity to the risk-free rate. Calls gain and puts lose when rates rise.
Epsilon ε	`∂V/∂q`	-44.033	-0.44033 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`	0.51021	0.0051021 per 1 vol point	How Delta changes when volatility changes, or equally how Vega changes when spot moves.
Vomma	`∂²V/∂σ²`	2.6429	0.00026429 per 1 vol point²	Convexity in volatility: how Vega itself changes as volatility moves.
Charm	`∂Δ/∂t`	-0.1506	-0.0004126 per calendar day	Delta decay: how the hedge ratio drifts as one day passes.
Speed	`∂³V/∂S³`	-0.00016847	-0.00016847 per $1 of spot	How Gamma changes as the underlying moves.
Zomma	`∂Γ/∂σ`	-0.10645	-0.0010645 per 1 vol point	How Gamma changes when volatility changes.
Color	`∂Γ/∂t`	0.010342	2.8333E-05 per calendar day	Gamma decay: how Gamma changes as one day passes. Explodes for at-the-money options near expiry.