Stochastic Volatility Models

Why Stochastic Volatility?

The Black-Scholes model assumes constant volatility, but market data reveals that volatility is time-varying, clustered, and mean-reverting. The "volatility smile" — the pattern where out-of-the-money options trade at higher implied volatilities — demonstrates that constant volatility models are inadequate for realistic option pricing.

Stochastic volatility models address these limitations by treating volatility as a random process, allowing for more accurate pricing of vanilla and exotic derivatives, better risk management, and realistic simulation of market dynamics.

Empirical Stylized Facts

1. Volatility Clustering

High volatility periods are followed by high volatility, low volatility by low volatility.

Markets exhibit periods of calm and storm
GARCH effects in return data
Fat tails in return distributions

2. Mean Reversion

Volatility tends to revert to a long-run average level.

Very high volatility is unsustainable
Very low volatility eventually increases
Half-life typically ranges from weeks to months

3. Leverage Effect

Negative correlation between asset returns and volatility changes.

Stock price drops often coincide with volatility increases
Correlation typically ranges from -0.3 to -0.8
Creates asymmetric volatility surfaces

4. Volatility of Volatility

Volatility itself is volatile and exhibits clustering.

Second-order effects in volatility dynamics
Important for pricing volatility derivatives

The Heston Model

SDE Formulation

The Heston model is the most popular stochastic volatility model:

\begin{aligned} dS_t &= rS_t dt + \sqrt{V_t}S_t dW_t^{(1)} \\ dV_t &= \kappa(\theta - V_t)dt + \sigma_V \sqrt{V_t} dW_t^{(2)} \end{aligned}

with correlation $dW_t^{(1)} dW_t^{(2)} = \rho dt$ .

Parameters

$\kappa > 0$ : Mean-reversion speed of volatility
$\theta > 0$ : Long-run variance level
$\sigma_V > 0$ : Volatility of volatility ("vol-of-vol")
$\rho \in (-1,1)$ : Correlation between price and volatility
$V_0 > 0$ : Initial variance

Feller Condition

To ensure

V_t

remains positive, the Feller condition requires:

2\kappa\theta \geq \sigma_V^2

When violated, the process can reach zero but is instantaneously reflected.

Risk-Neutral Dynamics

Under the risk-neutral measure:

\begin{aligned} dS_t &= rS_t dt + \sqrt{V_t}S_t d\tilde{W}_t^{(1)} \\ dV_t &= \kappa(\theta - V_t)dt + \sigma_V \sqrt{V_t} d\tilde{W}_t^{(2)} \end{aligned}

The volatility process typically retains the same parameters (affine structure).

Option Pricing in the Heston Model

Characteristic Function

The Heston model has a semi-analytical solution via Fourier methods. The characteristic function of $\ln S_T$ is:

\phi(u) = \mathbb{E}[e^{iu \ln S_T}] = e^{iu \ln S_0 + C(T,u) + D(T,u)V_0}

where $C(T,u)$ and $D(T,u)$ are functions satisfying Riccati equations.

Fourier Inversion

European option prices are computed using:

C = S_0 P_1 - Ke^{-rT} P_2

where $P_1$ and $P_2$ are probabilities computed via inverse Fourier transform:

P_j = \frac{1}{2} + \frac{1}{\pi} \int_0^{\infty} \text{Re}\left[\frac{e^{-iu \ln K} f_j(u)}{iu}\right] du

Numerical Implementation

FFT methods: Fast computation for multiple strikes
Lewis formula: Alternative formulation avoiding singularities
Carr-Madan formula: Standard approach in practice

Alternative Stochastic Volatility Models

1. Hull-White Model

\begin{aligned} dS_t &= rS_t dt + \sigma_t S_t dW_t^{(1)} \\ d\sigma_t &= \kappa(\theta - \sigma_t)dt + \sigma_V \sigma_t dW_t^{(2)} \end{aligned}

Features:

Volatility (not variance) follows Ornstein-Uhlenbeck
Can become negative (requires careful handling)
Simpler than Heston but less realistic

2. SABR Model

The Stochastic Alpha Beta Rho model:

\begin{aligned} dS_t &= \sigma_t S_t^{\beta} dW_t^{(1)} \\ d\sigma_t &= \nu \sigma_t dW_t^{(2)} \end{aligned}

Features:

$\beta$ controls the elasticity of variance
$\beta = 0$ : Normal SABR, $\beta = 1$ : Log-normal SABR
Popular for interest rate derivatives

3. 3/2 Model

dV_t = \kappa(\theta - V_t)dt + \sigma_V V_t^{3/2} dW_t

Features:

Volatility of volatility proportional to $V_t^{3/2}$
Exhibits more realistic volatility clustering
More complex calibration and simulation

4. Scott Model

\begin{aligned} dS_t &= rS_t dt + e^{V_t/2} S_t dW_t^{(1)} \\ dV_t &= \kappa(\theta - V_t)dt + \sigma_V dW_t^{(2)} \end{aligned}

Features:

Log-volatility follows Ornstein-Uhlenbeck
Always positive volatility
Analytical tractability similar to Heston

Multi-Factor Stochastic Volatility

Two-Factor Models

\begin{aligned} dS_t &= rS_t dt + \sqrt{V_t}S_t dW_t^{(1)} \\ dV_t &= \kappa_1(V_t^{LR} - V_t)dt + \sigma_1 \sqrt{V_t} dW_t^{(2)} \\ dV_t^{LR} &= \kappa_2(\theta - V_t^{LR})dt + \sigma_2 \sqrt{V_t^{LR}} dW_t^{(3)} \end{aligned}

Features:

$V_t$ : Short-term volatility factor
$V_t^{LR}$ : Long-run volatility level
Better fit to volatility term structure

Bergomi Models

Bergomi (2005): Forward variance dynamics under specific measure Bergomi (2008): Multi-factor extension with rough volatility

Rough Stochastic Volatility

Rough Heston Model

\begin{aligned} dS_t &= rS_t dt + \sqrt{V_t}S_t dW_t^{(1)} \\ V_t &= V_0 + \frac{1}{\Gamma(H+1/2)} \int_0^t (t-s)^{H-1/2} \kappa(\theta - V_s)ds \\ &\quad + \frac{1}{\Gamma(H+1/2)} \int_0^t (t-s)^{H-1/2} \sigma_V \sqrt{V_s} dW_s^{(2)} \end{aligned}

where $H \in (0, 1/2)$ is the Hurst parameter.

Features:

$H < 1/2$ : Rough (more realistic) volatility paths
$H = 1/2$ : Reduces to standard Heston
Better matches volatility autocorrelation functions

Calibration to Market Data

Objective Functions

Implied volatility errors: $\min_{\theta} \sum_{i=1}^N w_i (\sigma_i^{\text{market}} - \sigma_i^{\text{model}}(\theta))^2$
Price errors: $\min_{\theta} \sum_{i=1}^N w_i (C_i^{\text{market}} - C_i^{\text{model}}(\theta))^2$

Challenges

Non-convex optimization: Multiple local minima
Computational cost: Each evaluation requires Fourier integration
Overfitting: Many parameters vs limited data
Stability: Small data changes can cause large parameter shifts

Regularization Techniques

Penalty methods: Add smoothness constraints
Bayesian approaches: Prior beliefs on parameters
Model averaging: Combine multiple calibrated models

Monte Carlo Simulation

Euler Scheme

\begin{aligned} S_{t+\Delta t} &= S_t + rS_t \Delta t + \sqrt{V_t}S_t \sqrt{\Delta t} Z_1 \\ V_{t+\Delta t} &= V_t + \kappa(\theta - V_t)\Delta t + \sigma_V \sqrt{V_t} \sqrt{\Delta t} (\rho Z_1 + \sqrt{1-\rho^2} Z_2) \end{aligned}

where $Z_1, Z_2 \sim \mathcal{N}(0,1)$ independent.

Issues with Standard Euler

Negative variance: $V_t$ can become negative
Bias: Systematic pricing errors
Instability: Requires very small time steps

Improved Schemes

Full Truncation: $V_{t+\Delta t} = \max(V_{t+\Delta t}, 0)$
Absorption: Set $V_t = 0$ when hitting zero
Reflection: Reflect negative values
Exact simulation: Exploit affine structure
QE scheme: Quadratic-exponential approximation

Volatility Surface Modeling

Local Volatility vs Stochastic Volatility

Local Volatility (Dupire):

$\sigma = \sigma(S,t)$ is deterministic function
Perfect fit to vanilla options
Poor forward volatility dynamics

Stochastic Volatility:

$\sigma_t$ is random process
Imperfect fit to vanilla options
Realistic forward volatility dynamics

Local Stochastic Volatility (LSV)

Combines both approaches:

dS_t = rS_t dt + L(S_t, t)\sqrt{V_t}S_t dW_t^{(1)}

where $L(S,t)$ is a local volatility component.

Benefits:

Perfect calibration to vanilla surface
Stochastic volatility for exotic pricing
Industry standard for equity derivatives

Greeks and Risk Management

Delta Hedging

In stochastic volatility models:

Delta: $\frac{\partial C}{\partial S}$
Vega: $\frac{\partial C}{\partial V}$ (sensitivity to variance)
Correlation sensitivity: $\frac{\partial C}{\partial \rho}$

Vega Hedging

Unlike Black-Scholes, options have exposure to volatility risk:

Long gamma positions: Typically long vega
Short gamma positions: Typically short vega
Vega hedging: Use volatility swaps or other options

Dynamic Hedging Performance

Stochastic volatility creates hedging errors because:

Incomplete markets: Cannot hedge volatility risk with stock alone
Model risk: True volatility process unknown
Transaction costs: Frequent rehedging expensive

Volatility Derivatives

Variance Swaps

Payoff:

N \times (\text{Realized Variance} - K_{\text{var}})

Replication: Static hedge using options across all strikes Fair Value: Model-independent under continuous trading

Volatility Swaps

Payoff:

N \times (\text{Realized Vol} - K_{\text{vol}})

More complex due to Jensen's inequality: $\mathbb{E}[\sqrt{X}] \neq \sqrt{\mathbb{E}[X]}$

VIX Options

Options on the CBOE Volatility Index:

Underlying: 30-day implied volatility
Settlement: Cash-settled based on VIX level
Complex dynamics: Multiple sources of randomness

Applications

1. Exotic Option Pricing

Barrier options: Volatility affects barrier hitting probability
Asian options: Path-dependent volatility important
Correlation products: Multi-asset stochastic volatility

2. Portfolio Optimization

Mean-variance: Volatility forecasting crucial
Dynamic allocation: Incorporate volatility timing
Risk parity: Volatility estimates drive allocations

3. Risk Management

VaR models: Stochastic volatility improves tail risk
Stress testing: Volatility scenarios
Model validation: Backtesting with realistic dynamics

Connection to Other Topics

Stochastic volatility models connect many areas:

Built on Stochastic Differential Equations
Use Itô's Lemma for derivatives
Extend Geometric Brownian Motion
Apply Martingale pricing theory
Connect to Jump-Diffusion for comprehensive models
Foundation for advanced option pricing beyond Black-Scholes
Enable sophisticated risk management techniques