Merton jump-diffusion

Prices do not only diffuse; sometimes they gap. Merton adds Poisson-timed jumps of random size to Black-Scholes. The price becomes a weighted sum of Black-Scholes prices, one for each possible number of jumps, and the model explains something Heston cannot: the steep smile of short-dated options.

Contract

Spot price (S)

Strike (K)

Diffusion volatility (σ)

Volatility between jumps, annualised

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

Turn jumps off

Set the intensity λ to 0 and price.

What to look for: The Merton price equals the Black-Scholes price and the smile is flat. Without jumps the model is Black-Scholes.

Short-dated skew

Set λ = 1 and μJ = -0.15, then compare the implied-volatility curves for T = 0.1 and T = 2.

What to look for: The short-dated smile is steep and the two-year one is almost flat. A jump can strike at any moment, so it matters most when there is little time for diffusion to do anything.

Find the jumps

Set λ to 5 and look at the path chart.

What to look for: Vertical gaps appear in the paths and the summary counts them. Raise σJ and the gaps get bigger; make μJ positive and they tend to go up rather than down.

Jumps

Intensity (λ, jumps per year)

About 0.50 jumps expected over the life

Mean log jump (μJ)

Negative: crashes

Jump volatility (σJ)

Spread of jump sizes

Mean jump size k = exp(μJ + σJ²/2) − 1 = -7.69%. The drift is compensated so the model stays arbitrage-free.

Simulation

Paths

Time steps

Merton call

$12.16

Black-Scholes, no jumps

$10.45

Same diffusion volatility

Implied volatility

24.55%

vs σ = 20.00% between jumps

Simulation

$12.06

± 0.1235 standard error

A jump can move the spot a long way in an instant, so even a one-week option far from the money has real value. That is why the short-dated smile is the steepest, the opposite of Heston, where variance needs time to move.

5 jumps across these ten paths; about 5.0 expected.

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. Bump-and-revalue on the series. Vega is with respect to the diffusion volatility between jumps; the jump parameters are held fixed.

Greek	Definition	Raw	Practitioner	Meaning
Delta Δ	`∂V/∂S`	0.64786	0.64786 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`	0.015782	0.015782 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`	31.566	0.31566 per 1 vol point	Sensitivity to implied volatility. Long options gain value when volatility rises.
Theta Θ	`∂V/∂t`	-7.4015	-0.020278 per calendar day	Time decay: how much value the option loses as one day passes with everything else unchanged.
Rho ρ	`∂V/∂r`	52.63	0.5263 per 1% of rate	Sensitivity to the risk-free rate. Calls gain and puts lose when rates rise.
Epsilon ε	`∂V/∂q`	-64.794	-0.64794 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`	-0.31859	-0.0031859 per 1 vol point	How Delta changes when volatility changes, or equally how Vega changes when spot moves.
Vomma	`∂²V/∂σ²`	46.038	0.0046038 per 1 vol point²	Convexity in volatility: how Vega itself changes as volatility moves.
Charm	`∂Δ/∂t`	-0.054584	-0.00014955 per calendar day	Delta decay: how the hedge ratio drifts as one day passes.
Speed	`∂³V/∂S³`	-0.00047452	-0.00047452 per $1 of spot	How Gamma changes as the underlying moves.
Zomma	`∂Γ/∂σ`	-0.055885	-0.00055885 per 1 vol point	How Gamma changes when volatility changes.
Color	`∂Γ/∂t`	0.0097403	2.6686E-05 per calendar day	Gamma decay: how Gamma changes as one day passes. Explodes for at-the-money options near expiry.

Diagnostics

Expected jumps over the life (lambda T)	0.500000
Mean jump size k	-0.076884
Series terms used	13.000000
Black-Scholes without jumps	10.450584
Total variance per year incl. jumps	0.065000
Compute time (price and Greeks)	0.7 ms