Heston Model

The variance (v_t) follows a Cox-Ingersoll-Ross (CIR) process, ensuring it reverts to a long-term mean (\theta) at a speed (\kappa). This prevents variance from becoming negative and captures volatility clustering observed in real markets.

• Key Parameters: (\kappa) (speed of reversion), (\theta) (long-run variance).
• Feller Condition: (2\kappa\theta > \sigma^2) ensures the process remains strictly positive.
• Behavior: Models the tendency of high volatility to eventually subside to normal levels.

The model incorporates a correlation coefficient (\rho) between the Brownian motions driving the asset price and its variance. A negative (\rho) captures the leverage effect—the empirical observation that asset prices tend to fall as volatility rises.

• Skew Control: (\rho < 0) generates the downward-sloping volatility skew typical in equity markets.
• Distribution Impact: Negative correlation introduces asymmetry into the return distribution, producing a fatter left tail.

Heston derived a quasi-closed-form solution for European option prices using characteristic functions. This avoids computationally expensive Monte Carlo simulation for calibration.

• Method: Prices are computed via Fourier inversion of the log-spot characteristic function.
• Numerical Integration: Requires a single numerical integration, making it fast enough for real-time calibration to market data.
• Advantage: Provides exact prices for vanilla options, serving as a benchmark for more complex models.

The parameter (\sigma) (vol-of-vol) governs the amplitude of fluctuations in the variance process itself. It directly controls the kurtosis of the return distribution and the convexity of the volatility smile.

• Smile Curvature: Higher (\sigma) produces a more pronounced volatility smile.
• Tail Behavior: Increasing vol-of-vol fattens both tails of the risk-neutral distribution.
• Calibration Sensitivity: This parameter is crucial for fitting short-dated, deep out-of-the-money options.

The model requires an initial variance (v_0) as a starting condition. This instantaneous variance drives the short-term implied volatility level and is typically calibrated from at-the-money options.

• Term Structure Impact: The path from (v_0) to the long-term mean (\theta) defines the volatility term structure.
• Market Fit: A high (v_0) relative to (\theta) creates a downward-sloping term structure, often seen during market stress.

The Heston model belongs to the class of affine jump-diffusion models. Its log-characteristic function is an affine (linear plus constant) function of the state variables, enabling tractable pricing.

• Tractability: This property allows for efficient Fourier-based pricing of European options.
• Extension: The affine structure permits extensions to include jumps in the price or variance process.
• Riccati ODEs: Pricing reduces to solving a system of ordinary differential equations.

The foundational concept that volatility itself is a random variable evolving over time, rather than a constant parameter. The Heston Model is a specific implementation where the variance follows a Cox-Ingersoll-Ross (CIR) process. This contrasts sharply with the Black-Scholes assumption of constant volatility.

• Captures the empirical observation of volatility clustering
• Generates the leptokurtic (fat-tailed) return distributions observed in real markets
• Essential for pricing long-dated options where the constant volatility assumption breaks down

A critical parameter in the Heston Model, typically denoted by ρ (rho), that captures the leverage effect. In equity markets, this correlation is strongly negative, meaning volatility rises as the asset price falls.

• Directly controls the steepness of the volatility skew
• A negative rho generates the asymmetric smile where out-of-the-money puts trade at a premium
• Mathematically, it couples the two driving Brownian motions: dW_S dW_v = ρ dt

Denoted by σ (sigma) in the Heston Model, this parameter governs the amplitude of fluctuations in the variance process. It is the primary driver of the volatility smile's curvature.

• A high vol-of-vol produces a pronounced smile (high kurtosis)
• A zero vol-of-vol collapses the model back to a deterministic volatility structure
• Critically, it must satisfy the Feller condition (2κθ > σ²) to ensure the variance process remains strictly positive

The variance process in Heston is mean-reverting, governed by the speed of reversion κ (kappa) and the long-run average variance θ (theta). This structure prevents volatility from exploding or collapsing to zero unrealistically.

• Kappa: Dictates how quickly volatility shocks decay; a high value implies a less persistent smile
• Theta: The anchor point for the variance term structure; it heavily influences the pricing of long-dated options
• This property is crucial for calibrating the model to the volatility term structure

Unlike Black-Scholes, the Heston Model does not have a simple closed-form price. Instead, it uses a semi-analytical formula based on characteristic functions and Fourier inversion.

• The price is computed as P = S*P1 - K*e^{-rT}*P2, where P1 and P2 are integrals of the characteristic function
• Requires numerical integration (e.g., Gauss-Lobatto quadrature) for each option price
• This computational overhead is a key consideration for real-time calibration and risk management

The practical application of the Heston Model involves finding the parameter set {κ, θ, σ, ρ, v0} that minimizes the error between model prices and liquid market quotes.

• Typically uses a non-linear least-squares optimizer (e.g., Levenberg-Marquardt)
• The initial variance v0 controls the short-term ATM level, while the other parameters fit the skew and term structure
• A well-calibrated model allows for consistent pricing of exotic options relative to the vanilla surface