Stochastic Volatility

Unlike constant volatility models, stochastic volatility frameworks assume variance reverts to a long-term average level over time. This captures the observed clustering behavior in financial markets where periods of high volatility are followed by calmer regimes.

• Key Parameter (κ): The speed of mean reversion dictates how quickly volatility shocks decay.
• Long-Run Variance (θ): The asymptotic level to which variance is pulled.
• Empirical Motivation: Addresses volatility clustering, a stylized fact absent in Black-Scholes dynamics.

A defining characteristic is the correlation coefficient (ρ) between the asset price process and the variance process. This parameter is crucial for reproducing the volatility skew observed in equity markets.

• Leverage Effect: Typically, ρ is negative for equities, meaning volatility rises as the asset price falls.
• Skew Control: The spot-vol correlation directly controls the steepness and asymmetry of the implied volatility smile.
• Heston Model: Explicitly parameterizes this correlation, allowing for semi-closed-form solutions for European options.

This parameter (often denoted ξ or σ_v) governs the amplitude of fluctuations in the variance process itself. It is the volatility of the volatility.

• Kurtosis Control: A higher vol-of-vol generates fatter tails in the return distribution, capturing extreme market moves better than log-normal assumptions.
• Smile Curvature: Directly influences the convexity of the implied volatility smile across strikes.
• Calibration Sensitivity: This parameter is critical for fitting short-dated options where the smile is most pronounced.

Stochastic volatility models define specific rules for how the implied volatility surface moves when the underlying spot price changes, moving beyond the fixed-surface assumption.

• Sticky-Strike vs. Sticky-Delta: The model's dynamics imply a specific mix of these regimes based on the correlation structure.
• Forward Volatility: Provides a consistent framework for pricing forward-starting options and cliquets, which depend on the evolution of future volatility.
• Vanna-Volga Connection: The model's parameters directly map to the vanna and volga exposures that exotic desks hedge.

Since volatility is not directly observable, stochastic volatility models require statistical filtering techniques to infer the current state of the variance process from discrete asset prices.

• Kalman Filter: Used for linear Gaussian approximations of the state-space model.
• Particle Filters: Employed for non-Gaussian extensions to handle the non-linear nature of variance dynamics.
• Practical Use: Essential for real-time parameter estimation and updating model forecasts based on incoming tick data.

A major advantage of affine stochastic volatility models (like Heston) is that they admit semi-closed-form solutions for European options using Fourier inversion, avoiding slow Monte Carlo simulations.

• Characteristic Function: The Fourier transform of the log-price distribution is known in closed form.
• Carr-Madan Formula: A widely used method to price options rapidly by integrating the dampened characteristic function.
• Computational Efficiency: Enables fast calibration to hundreds of market quotes, a necessity for real-time trading desks.

The foundational stochastic volatility model where variance follows a Cox-Ingersoll-Ross (CIR) mean-reverting square-root process. Key features:

• Captures the leverage effect via spot-vol correlation (ρ)
• Produces a semi-analytical pricing formula using characteristic functions
• Parameters: long-run variance (θ), mean reversion speed (κ), volatility of volatility (σ), initial variance (v₀), and correlation (ρ)
• Generates realistic volatility smiles and skews not possible under Black-Scholes

Denoted as σ (vol-of-vol) in stochastic volatility models, this parameter controls the amplitude of fluctuations in the variance process itself. Critical implications:

• Directly governs the kurtosis (fat-tailedness) of the return distribution
• Higher values produce more extreme volatility clustering and fatter tails
• In the Heston model, must satisfy the Feller condition (2κθ > σ²) to prevent variance from reaching zero
• Primary driver of volatility smile convexity in option markets

The correlation coefficient ρ between the asset price process and its variance process. This parameter is essential for capturing equity market asymmetry:

• Negative ρ (typically -0.7 to -0.9): Captures the leverage effect—volatility rises when prices fall, producing a downward-sloping skew
• Zero ρ: Produces symmetric volatility smiles, common in FX markets
• Positive ρ: Rare, occasionally observed in commodity markets where supply shocks drive both prices and volatility higher
• Directly controls the slope of the implied volatility skew across strikes

A defining characteristic of stochastic volatility models where variance tends to revert to a long-run mean level θ at a speed governed by κ. Properties:

• Models the observed volatility clustering phenomenon—periods of high volatility persist but eventually subside
• The half-life of volatility shocks equals ln(2)/κ, providing an intuitive measure of persistence
• Contrasts with local volatility models where variance is a deterministic function of price and time
• Essential for pricing path-dependent options and capturing term structure dynamics

The Stochastic Alpha-Beta-Rho (SABR) model, widely used in interest rate and FX derivatives. Distinct from Heston:

• Uses constant elasticity of variance (CEV) with parameter β to control the backbone of the smile
• Volatility follows a log-normal process rather than a square-root diffusion
• Provides closed-form asymptotic approximations for implied volatility, enabling extremely fast calibration
• Parameters: α (initial volatility level), β (CEV exponent), ρ (correlation), ν (vol-of-vol)
• Dominant model for managing swaption and cap/floor volatility surfaces

The compensation demanded by option sellers for bearing unhedgeable stochastic volatility risk. Key characteristics:

• Measured as the spread between implied volatility and subsequent realized volatility
• Empirically negative in equity markets—investors pay a premium to hedge downside risk, making implied vol exceed realized vol on average
• Exists because volatility is not a tradable asset; delta-hedging cannot eliminate variance risk
• Drives the profitability of variance risk premium harvesting strategies and explains why systematic option selling generates positive carry
• Directly linked to the market price of volatility risk (λ) in the risk-neutral drift adjustment