Heston stochastic volatility

Black-Scholes fixes one volatility. Heston lets the variance wander: it mean-reverts to a long-run level, has its own volatility, and is correlated with the spot. One set of five parameters then produces a whole smile and term structure. The price is semi-analytic, from a characteristic function integrated numerically, and a simulation shows the same model path by path.

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

Exercise style

Try this

0 of 4 done

Recover Black-Scholes

Set the vol of variance σ to 0.001 and the long-run volatility √θ equal to the initial volatility √v0, then price.

What to look for: The Heston price equals the Black-Scholes price at √v0 and the smile chart is flat. With no randomness in variance, Heston is Black-Scholes.

Lift the wings

Put σ back to 0.3 and then to 0.8, with ρ = 0.

What to look for: Both wings of the smile rise above the at-the-money level, more so for the short maturity. Random variance fattens both tails of the distribution.

Tilt the smile

Keep σ = 0.3 and move ρ from 0 to -0.7, then to +0.7.

What to look for: Negative correlation makes low strikes dearer, the equity skew. Positive correlation does the reverse. Watch the variance paths: with ρ < 0 they spike when the spot drops.

Exercise early under stochastic volatility

Set the exercise style to American with a put struck at 110, and price.

What to look for: A least-squares Monte Carlo panel appears with the American price, the European price on the same paths and the premium. The exercise chart shows when paths give up waiting; raise σ and fewer do, because a volatile variance makes waiting more valuable.

Variance process

Feller holds

Mean reversion (κ)

Speed of pull towards θ

Long-run volatility (√θ)

θ = 0.0400

Vol of variance (σ)

How much variance itself moves

Correlation (ρ)

Spot vs variance; negative gives a skew

Feller condition 2κθ > σ² keeps the variance away from zero. Violating it is allowed here and common in calibrated parameters; the simulation floors variance at zero.

Simulation

Paths

Time steps

Heston call

$10.39

Black-Scholes at √v0

$10.45

Same contract, constant volatility

Heston implied vol

19.85%

vs √v0 = 20.00%

Simulation

$10.48

± 0.0866 standard error, 20,000 paths

Each point is a Heston price inverted through Black-Scholes. Negative ρ tilts the smile down to the right; σ bends it; κ and θ set how the level changes with maturity.

The dashed line is the long-run variance θ. Paths are pulled towards it at speed κ.

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 and its relatives bump today's instantaneous volatility √v0, not a constant volatility, so they measure sensitivity to the starting level of variance.

Greek	Definition	Raw	Practitioner	Meaning
Delta Δ	`∂V/∂S`	0.69172	0.69172 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`	0.017411	0.017411 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`	15.685	0.15685 per 1 vol point	Sensitivity to implied volatility. Long options gain value when volatility rises.
Theta Θ	`∂V/∂t`	-6.4486	-0.017667 per calendar day	Time decay: how much value the option loses as one day passes with everything else unchanged.
Rho ρ	`∂V/∂r`	58.793	0.58793 per 1% of rate	Sensitivity to the risk-free rate. Calls gain and puts lose when rates rise.
Epsilon ε	`∂V/∂q`	-69.187	-0.69187 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`	-0.17451	-0.0017451 per 1 vol point	How Delta changes when volatility changes, or equally how Vega changes when spot moves.
Vomma	`∂²V/∂σ²`	48.148	0.0048148 per 1 vol point²	Convexity in volatility: how Vega itself changes as volatility moves.
Charm	`∂Δ/∂t`	-0.060832	-0.00016666 per calendar day	Delta decay: how the hedge ratio drifts as one day passes.
Speed	`∂³V/∂S³`	-0.00087609	-0.00087609 per $1 of spot	How Gamma changes as the underlying moves.
Zomma	`∂Γ/∂σ`	-0.030473	-0.00030473 per 1 vol point	How Gamma changes when volatility changes.
Color	`∂Γ/∂t`	0.010422	2.8554E-05 per calendar day	Gamma decay: how Gamma changes as one day passes. Explodes for at-the-money options near expiry.

Diagnostics

P1 (delta-like probability)	0.691867
P2 (probability of exercise)	0.618069
Black-Scholes at sqrt(v0)	10.450584
Feller condition satisfied	1.000000
Long-run volatility sqrt(theta)	0.200000
Compute time (price and Greeks)	11.8 ms