Heston Model for Stochastic Volatility

Key features of the Heston model

The Heston model describes asset price dynamics using two correlated stochastic processes:

1. Asset price process: $dS_t = \mu S_t dt + \sqrt{v_t} S_t dW_t^1$

2. Variance process: $dv_t = \kappa(\theta - v_t)dt + \sigma\sqrt{v_t}dW_t^2$

Where:

• $S_t$ is the asset price
• $v_t$ is the variance (volatility squared)
• $\mu$ is the drift rate
• $\kappa$ is the mean reversion speed
• $\theta$ is the long-term variance
• $\sigma$ is the volatility of volatility
• $dW_t^1, dW_t^2$ are correlated Wiener processes with correlation $\rho$

Model advantages and improvements

The Heston model addresses several limitations of simpler models by incorporating:

1. Mean-reverting volatility behavior
2. Correlation between asset returns and volatility changes
3. More realistic modeling of implied volatility surfaces
4. Better capture of market skewness and kurtosis

Applications in derivatives pricing

The model is particularly valuable for:

1. Pricing exotic options with volatility dependence
2. Computing more accurate Greeks
3. Risk management of volatility-dependent positions
4. Calibration to market volatility surfaces

Calibration process

The calibration workflow involves:

1. Market data collection:

   • Option prices across strikes and maturities
   • Implied volatility surface

2. Parameter estimation: $\min_{\kappa,\theta,\sigma,\rho,v_0} \sum_{i=1}^N (C_i^{model} - C_i^{market})^2$

Where $C_i^{model}$ and $C_i^{market}$ are model and market option prices.

Implementation considerations

Key aspects of implementing the Heston model include:

1. Numerical methods:

   • Finite difference schemes
   • Monte Carlo simulation
   • Fourier transform techniques

2. Computational efficiency:

   • Parallel processing for calibration
   • Caching of intermediate calculations
   • Optimization of numerical routines

Market impact and adoption

The Heston model has become a standard tool in:

1. Options trading desks
2. Risk management systems
3. Systematic Trading platforms
4. Volatility Arbitrage Strategies

Limitations and extensions

While powerful, practitioners should be aware of:

1. Parameter stability challenges
2. Computational complexity
3. Model risk considerations
4. Need for regular recalibration

Modern extensions include:

1. Multi-factor versions
2. Jump components
3. Term structure modifications
4. Regime-switching variants

Integration with trading systems

The model interfaces with:

1. Real-Time Risk Assessment systems
2. Options Price Reporting Authority (OPRA) feeds
3. Pre-Trade Risk Analytics
4. Position Management Systems

This integration enables real-time pricing and risk monitoring across trading operations.