Bates: stochastic volatility with jumps

Heston builds the smile up with maturity; Merton's jumps make it steep at the front. Real surfaces show both, and Bates simply combines the two: the variance mean-reverts and correlates with the spot, and on top the spot gaps at Poisson times. The characteristic function is the product of the two models', so pricing stays semi-analytic.

Contract

Spot price (S)

Strike (K)

Initial volatility (√v0)

Today's instantaneous volatility; v0 is its square

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

Two smiles in one

With the default parameters, compare the solid Bates and dashed Heston lines at 0.1 and 2 years.

What to look for: At the front the jumps lift the Bates wings well above Heston's; at two years the two nearly coincide. Each mechanism owns one end of the term structure.

Switch the jumps off

Set the intensity λ to 0 and recalculate.

What to look for: Every Bates line lands on its Heston twin and the Bates price equals the Heston-alone diagnostic. Without jumps, Bates is Heston.

Freeze the variance

Put λ back to 0.5, then set the vol of variance σ to 0.001 and √θ equal to √v0.

What to look for: The Heston lines go flat and the Bates lines become Merton's: steep at the front, flattening with maturity. The price matches the Merton-alone diagnostic.

Variance process

Mean reversion (κ)

Long-run volatility (√θ)

Vol of variance (σ)

Correlation (ρ)

Jumps

Intensity (λ, jumps per year)

Mean log jump (μJ)

Jump volatility (σJ)

Simulation paths

Time steps

Bates call

$11.67

Heston alone

$10.39

Merton alone $11.66

Black-Scholes at √v0

$10.45

Implied vol 23.23%

Simulation

$11.59

± 0.1019; 6 jumps on the ten kept paths

At short maturities the jumps dominate and Bates sits well above Heston in the wings; at long maturities the two converge as diffusion takes over.

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. Vega bumps today's instantaneous volatility √v0; the jump parameters are held fixed.

Greek	Definition	Raw	Practitioner	Meaning
Delta Δ	`∂V/∂S`	0.68293	0.68293 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`	0.014993	0.014993 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`	13.51	0.1351 per 1 vol point	Sensitivity to implied volatility. Long options gain value when volatility rises.
Theta Θ	`∂V/∂t`	-7.0647	-0.019355 per calendar day	Time decay: how much value the option loses as one day passes with everything else unchanged.
Rho ρ	`∂V/∂r`	56.636	0.56636 per 1% of rate	Sensitivity to the risk-free rate. Calls gain and puts lose when rates rise.
Epsilon ε	`∂V/∂q`	-68.304	-0.68304 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`	-0.12857	-0.0012857 per 1 vol point	How Delta changes when volatility changes, or equally how Vega changes when spot moves.
Vomma	`∂²V/∂σ²`	47.739	0.0047739 per 1 vol point²	Convexity in volatility: how Vega itself changes as volatility moves.
Charm	`∂Δ/∂t`	-0.047273	-0.00012951 per calendar day	Delta decay: how the hedge ratio drifts as one day passes.
Speed	`∂³V/∂S³`	-0.00064646	-0.00064646 per $1 of spot	How Gamma changes as the underlying moves.
Zomma	`∂Γ/∂σ`	-0.018269	-0.00018269 per 1 vol point	How Gamma changes when volatility changes.
Color	`∂Γ/∂t`	0.0086501	2.3699E-05 per calendar day	Gamma decay: how Gamma changes as one day passes. Explodes for at-the-money options near expiry.

Diagnostics

Heston alone (no jumps)	10.394219
Merton alone (constant variance v0)	11.661675
Black-Scholes at sqrt(v0)	10.450584
Expected jumps over the life	0.500000
Feller condition satisfied	1.000000
Compute time (price and Greeks)	46.1 ms