Heston Stochastic Volatility Option Pricing Lab

Simulate the Heston (1993) stochastic volatility model. Solve European options via characteristic function Fourier inversion, generate the market volatility smile and skew, verify CIR variance dynamics and Feller conditions, and compare against Black-Scholes.

Heston (1993) Stochastic Volatility Fourier Inversion (Gil-Pelaez) Volatility Smile / Skew CIR Variance Process

Calibrated Market Regimes:

1. Asset & Stochastic Vol Parameters

Contract Terms & Spot Asset

Spot Price ($S_0$)

Current underlying asset price.

Strike Price ($K$)

Exercise price of option.

Expiry ($T$, Yrs)

Time to maturity.

Risk-Free ($r$, %)

Interest rate.

Dividend ($q$, %)

Continuous yield.

CIR Variance Process ($v_t$)

Initial Vol (σ₀ = √v₀, %)

Initial variance v₀ = σ₀².

Long-Run Vol (σ̄ = √θ, %)

Equilibrium variance θ = σ̄².

Mean Rev (κ)

Rate of reversion.

Vol-of-Vol (ξ)

Diffusion of v_t.

Correlation (ρ)

Spot-vol correlation.

2. Heston Fourier Valuation & BS Comparison

FELLER SATISFIED

Heston Call Price

$6.03

BS: $6.08 (-0.05)

Heston Put Price

$4.55

BS: $4.59 (-0.04)

ATM Implied Vol

18.79%

Black-Scholes IV

Feller Ratio (2κθ/ξ²)

0.64

≥ 1.0 (Strictly v > 0)

Heston Integration Components: C = S₀ e^-qT P₁ - K e^-rT P₂

Component	Analytical Role	Value	Description
Risk-Neutral Prob P₁	`Asset measure probability S_T > K`	0.6605	Delta equivalent probability under the stock numeraire
Risk-Neutral Prob P₂	`Money-market probability S_T > K`	0.6092	Standard risk-neutral probability of finishing in-the-money
Variance Half-Life (t_1/2)	`ln(2) / κ`	0.35 yrs	Time for volatility shocks to mean-revert 50% toward θ
Put-Call Parity Check	`Call - Put ≡ S_0 e^{-qT} - K e^{-rT}`	$0.0000	Exact arbitrage-free parity verified to 4 decimal places

Volatility Smile Asymmetry: When ρ < 0, out-of-the-money puts trade at higher implied volatilities than out-of-the-money calls, producing the steep volatility skew ubiquitous in index options post-1987.

3. Implied Volatility Smile & Skew

Black-Scholes Implied Volatility Smile: Blue curve charts Heston-generated implied volatility across moneyness K/S₀. Dashed gray line indicates flat Black-Scholes benchmark volatility.

4. Strike Schedule & Implied Volatility

Inverted BS IV

Strike ($K$)	Moneyness	Heston Call	Heston Put	Implied Vol (σ_BS)

Implied volatilities solved via root inversion: C_BS(σ_implied) = C_Heston.

5. Sensitivity: Correlation (ρ) vs. Vol-of-Vol (ξ) on 90% OTM Put Price ($)

Shows how negative correlation ρ and higher vol-of-vol ξ exponentially inflate out-of-the-money downside put protection.

6. Sensitivity: Mean Reversion (κ) vs. Long-Run Vol (σ̄) on ATM Call ($)

Measures option price elasticity with respect to the central equilibrium variance anchor and pull speed.

7. Heston Stochastic Volatility Mathematics & Fourier Inversion

The Bivariate Diffusion Model

Published in The Review of Financial Studies in 1993 by Steven L. Heston, the model describes the simultaneous stochastic evolution of the stock price and its variance:

dS_t = (r - q) S_t dt + √v_t S_t dW_t^S
dv_t = κ (θ - v_t) dt + ξ √v_t dW_t^v
Corr(dW_t^S, dW_t^v) = ρ dt

Here, κ is the mean-reversion speed, θ is the long-term equilibrium variance, ξ is the volatility of variance (vol-of-vol), and ρ is the instantaneous correlation between asset returns and volatility shocks.

The Feller Condition for Variance Non-Negativity

Just like in the CIR interest rate model, William Feller's boundary condition determines whether variance can reach zero:

2 κ θ ≥ ξ² ⇔ (2 κ θ / ξ²) ≥ 1

When 2κθ ≥ ξ², the origin is inaccessible, ensuring v_t > 0 strictly. When violated, zero is an instantaneously reflecting boundary.

Gil-Pelaez Fourier Inversion Pricing

By applying the Feynman-Kac theorem to the risk-neutral valuation PDE, Heston demonstrated that the option price decomposes into two probabilities:

C(S_0, K, T) = S_0 e^{-qT} P_1 - K e^{-rT} P_2

Each probability P_j is computed by inverting the characteristic function f_j(φ; ln S₀, v₀) via the Gil-Pelaez (1951) inversion formula:

P_j = 1/2 + (1/π) ∫₀^∞ Re[ e^{-i φ ln K} f_j(φ) / (i φ) ] dφ

Using the numerically stable formulation of Albrecher et al. (2007) avoids multi-valued complex logarithm branch cuts, guaranteeing unconditionally stable numerical integration.

8. Heston Model Mastery Self-Assessment Quiz

9. Related Quantitative Finance & Derivatives Labs

Black-Scholes Calculator Lab

Benchmark constant-volatility analytical Greeks, European payoffs, and early exercise limits.

CRR Binomial Option Lab

Price American equity options, discrete dividend steps, and early exercise frontiers.

CIR Model Simulation Lab

Explore the identical square-root diffusion process used for Heston's variance equation.

10. Frequently Asked Questions (FAQ)

Introduced in 1993 by Steven L. Heston, the model relaxes Black-Scholes' assumption of constant volatility by modeling instantaneous variance v_t as a mean-reverting square-root diffusion process (CIR process) coupled with the asset price. It simultaneously accounts for fat tails (kurtosis), volatility clustering, and the asymmetric leverage effect (skewness), resolving the Black-Scholes smile mispricing.

The correlation parameter rho determines the skewness of asset returns. In equity markets, rho is strongly negative (typically -0.6 to -0.8), creating the 'leverage effect' where falling stock prices accompany rising volatility. This skews the distribution leftward, elevating the implied volatility of out-of-the-money puts. When rho is near zero, symmetric smiles emerge.

The Feller condition states that 2 * kappa * theta >= xi^2, where kappa is variance mean-reversion speed, theta is long-term variance, and xi is the volatility of volatility (vol-of-vol). When satisfied, the upward drift at zero is strong enough that variance v_t remains strictly positive (v_t > 0) with probability 1.

Because the joint transition density cannot be expressed in simple elementary functions, Heston solved the pricing PDE via characteristic functions. The call option price is C = S0 * exp(-q*T) * P1 - K * exp(-r*T) * P2, where P1 and P2 are risk-neutral probabilities evaluated via numerical integration of the characteristic function using Gil-Pelaez Fourier inversion.

Yes. This simulator provides a fully sanitized RFC 4180 compliant CSV export containing calibrated Heston parameters, Feller condition ratio, European call and put prices, Black-Scholes comparison values, strike-by-strike implied volatility smiles, and dual 5x5 sensitivity matrices with formula injection defense.