Stochastic Volatility Models

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Stochastic Volatility Models assume that the volatility of an underlying asset is a stochastic process driven by state variables such as the level of the asset price, the tendency of volatility to revert to a long-term mean, and the variance of the volatility process itself. By introducing randomness in volatility, these models address some limitations of the traditional Black-Scholes model.

Features of Stochastic Volatility Models

1. Addressing Limitations of the Black-Scholes Model:

   • The Black-Scholes model assumes constant volatility over the life of a derivative and no dependence on the underlying asset price. This assumption struggles to explain the smile and skew patterns observed in market-implied volatility surfaces.
   • Stochastic volatility models assume volatility is a stochastic process, providing a more accurate description of market volatility dynamics and the relationship between asset price movements and volatility.

2. Introduction of State Variables:

   • A set of latent state variables captures factors such as volatility risk, interest rate risk, correlation risk, and jump risk.
   • The cross-correlation between the volatility process and the asset price can explain the leverage effect, where volatility increases as prices decline.

3. Incomplete Market Models:

   • Stochastic volatility models are typically incomplete market models, meaning they cannot uniquely determine arbitrage-free option prices.
   • Unlike complete market models, incomplete markets do not allow the full replication of option risk exposures through self-financing trading strategies, leading to multiple equivalent martingale measures.

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Major Stochastic Volatility Models

Below is an overview of four classic stochastic volatility models:

1. Heston Model (1993)

   • Features:
     Assumes the underlying asset price follows geometric Brownian motion, while volatility follows a mean-reverting CIR (Cox-Ingersoll-Ross) process. The model allows for correlation between asset price and volatility, capturing the leverage effect.
   • Applications:
     Analyzing volatility smiles and skews, suitable for option pricing and volatility-related derivatives.
   • Advantages:
     The model is analytically tractable, with option prices computable via characteristic functions.
   • Limitations:
     Volatility does not depend on the level of the asset price, which may limit its accuracy in pricing products sensitive to asset price levels.

2. Bates Model (1996)

   • Features:
     Extends the Heston model by adding a jump process, allowing for discrete changes (jumps) in the underlying asset price.
   • Applications:
     Pricing derivatives sensitive to jump risk, such as short-term options.
   • Advantages:
     More comprehensively captures features in market-implied volatility surfaces, especially short-term volatility smiles.
   • Limitations:
     Increased model complexity and higher computational costs.

3. SABR Model (2002)

   • Features:
     Assumes both the log-returns of the underlying asset and volatility follow stochastic processes. Widely used in interest rate markets to capture asymmetric volatility smiles.
   • Applications:
     Pricing interest rate derivatives (e.g., swaptions, caplets) and fitting volatility surfaces.
   • Advantages:
     Simple analytical approximations, easy calibration, and ability to generate realistic implied volatility surfaces.
   • Limitations:
     Cannot capture long-term volatility dynamics, especially during volatility repricing.

4. Piterbarg Model (2005)

   • Features:
     Extends the stochastic volatility framework by incorporating correlations between interest rates and volatility.
   • Applications:
     Pricing interest rate derivatives and volatility-related products.
   • Advantages:
     Captures more complex market dynamics.
   • Limitations:
     High model complexity, difficult calibration, and significant computational costs.

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Introduction to the Heston Model

The Heston model is one of the most classic stochastic volatility models, proposed by Heston in 1993. It naturally extends the Black-Scholes model by introducing a mean-reverting stochastic volatility process.

Model Definition

1. Asset Price Dynamics:
   The dynamics of the underlying asset price $S(t)$ are:

   $$dS(t) = \mu S(t) dt + \sqrt{V(t)} S(t) dW_1,$$

   where $V(t)$ is the stochastic variance (square of volatility) at time $t$.

2. Stochastic Variance Process:
   The volatility dynamics follow a Cox-Ingersoll-Ross (CIR) process:

   $$dV(t) = k(\theta - V(t)) dt + \xi \sqrt{V(t)} dW_2,$$

   • $\theta$: Long-term variance mean.
   • $k$: Mean reversion speed.
   • $\xi$: Volatility of volatility.

3. Correlation of Brownian Motions:
   The two Brownian motions $W_1$ and $W_2$ are correlated:

   $$d\langle W_1, W_2 \rangle_t = \rho dt,$$

   where $\rho$ is the correlation coefficient between asset price and volatility.

4. Forward Price Dynamics:
   The model can be expressed in terms of the forward price $F(t)$:

   $$\begin{aligned} dF(t) &= F(t) \sqrt{V(t)} dW_1, \\ dV(t) &= k(\theta - V(t)) dt + \xi \sqrt{V(t)} dW_2, \\ d\langle W_1, W_2 \rangle_t &= \rho dt. \end{aligned}$$

Model Parameters

The Heston model has five parameters:

1. Initial variance: $V(0)$
2. Long-term variance mean: $\theta$
3. Mean reversion speed: $k$
4. Volatility of volatility: $\xi$
5. Correlation coefficient: $\rho$

These parameters can be calibrated using market data.

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Advantages and Limitations of the Heston Model

Advantages

1. Natural Extension of the Black-Scholes Model:
   Includes the Black-Scholes model as a special case (when volatility is constant).
2. Captures Market Features:
   • Asymmetric distributions: Accounts for skewness in asset returns.
   • Leverage effect: Captures the negative correlation between price and volatility.
   • Mean reversion: Volatility tends to revert to a long-term mean.
3. Analytical Tractability:
   Option prices can be computed quickly using characteristic functions.

Limitations

1. Complex Calibration:
   The model has five parameters, making calibration challenging.
2. Volatility Independent of Price Level:
   May struggle to accurately price derivatives sensitive to asset price levels.
3. Assumption Constraints:
   Assumes volatility follows a CIR process, which may not fit all market conditions.

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Numerical Pricing Methods for the Heston Model

Although the Heston model is analytically tractable, its analytical solutions often involve complex characteristic function calculations. For derivatives beyond European options (e.g., American or path-dependent options), numerical methods are required. Below are the commonly used numerical pricing methods for the Heston model:

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1. Characteristic Function Methods (Fourier Transform Methods)

Description

A key advantage of the Heston model is that European option prices can be expressed using characteristic functions. Fourier transform techniques, such as the Carr-Madan method or FFT method, can efficiently compute option prices by evaluating the characteristic function.

Key Formula

The European option price $C$ in the Heston model can be computed via inverse Fourier transform:

$$C(K) = e^{-rT} \int_{-\infty}^{\infty} e^{-i\phi \ln(K)} \frac{\hat{C}(\phi)}{i\phi} d\phi,$$

where $\hat{C}(\phi)$ is the characteristic function of the price, defined as:

$$\hat{C}(\phi) = \mathbb{E}\left[e^{i\phi \ln(S_T)}\right],$$

which can be derived analytically from the Heston model dynamics.

Numerical Implementation

1. Carr-Madan Method:
   • Transforms the integral to the complex domain and uses the Fast Fourier Transform (FFT) for efficient computation.
2. Numerical Integration:
   • Discretizes the integral range $\phi$ and applies numerical integration methods (e.g., trapezoidal or Simpson's rule).

Advantages and Disadvantages

• Advantages:
  1. Highly efficient for European options.
  2. No path simulation required, with controllable errors.
• Disadvantages:
  1. Limited to European options or simple derivatives.
  2. Sensitive to numerical calculations in the complex domain, potentially leading to oscillatory errors.

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2. Monte Carlo Simulation

Description

Monte Carlo simulation generates paths of the underlying asset price and volatility under the Heston model to compute the expected value of the option. It is suitable for complex derivatives (e.g., path-dependent or American options).

Numerical Steps

1. Simulate Asset Price Paths:
   • Use the Heston model dynamics:

     $$dS(t) = \mu S(t) dt + \sqrt{V(t)} S(t) dW_1,$$

     $$dV(t) = k(\theta - V(t)) dt + \xi \sqrt{V(t)} dW_2,$$

     and generate two correlated Brownian motions $W_1$ and $W_2$.
2. Time Discretization:
   • Discretize time steps $\Delta t$ using the Euler method or advanced methods (e.g., Milstein method).
   • Ensure variance $V(t)$ remains positive (e.g., using reflection or full truncation for the CIR process).
3. Compute Option Value:
   • Calculate the final payoff for each path, take the expectation, and discount to obtain the option price.

Improvements

1. Long Time Step Simulation:
   • Use higher-order discretization methods (e.g., Andersen QE method) to improve path simulation accuracy.
2. Variance Reduction Techniques:
   • Apply control variates or importance sampling to reduce variance and improve convergence.

Advantages and Disadvantages

• Advantages:
  1. Suitable for pricing complex derivatives (e.g., path-dependent or American options).
  2. Highly flexible and extendable to high-dimensional problems.
• Disadvantages:
  1. Slow convergence and high computational costs.
  2. Simulating the CIR process may introduce numerical errors.

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3. Finite Difference Method (FDM)

Description

The partial differential equation (PDE) of the Heston model can be solved using finite difference methods. By discretizing time and price space, a system of discrete equations is constructed and iteratively solved for the option price.

PDE for the Heston Model

The option price $C(t, S, V)$ in the Heston model satisfies the following two-dimensional PDE:

$$\frac{\partial C}{\partial t} + \frac{1}{2} V S^2 \frac{\partial^2 C}{\partial S^2} + \rho \xi S V \frac{\partial^2 C}{\partial S \partial V} + \frac{1}{2} \xi^2 V \frac{\partial^2 C}{\partial V^2} + r S \frac{\partial C}{\partial S} + k(\theta - V) \frac{\partial C}{\partial V} - rC = 0.$$

Numerical Implementation

1. Spatial Discretization:
   • Discretize the asset price $S$ and volatility $V$ ranges into grid points.
   • Use finite difference methods (e.g., central differences) to discretize second-order partial derivatives.
2. Temporal Discretization:
   • Use implicit methods or the Crank-Nicholson method (central differences in time) for time discretization.
3. Iterative Solution:
   • Start from the known terminal condition (e.g., payoff function for European options) and iteratively compute backward in time.

Advantages and Disadvantages

• Advantages:
  1. High accuracy and good error control for pricing problems.
  2. Can handle various boundary conditions (e.g., barrier options).
• Disadvantages:
  1. High computational complexity, especially for two-dimensional PDEs.
  2. Sensitive to grid point selection (requires high-resolution grids to avoid numerical errors).

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4. Tree Methods

Description

The Heston model can also be solved using extended binomial or trinomial tree methods. These methods construct a tree-like structure for asset price and volatility dynamics in discrete time and state spaces to compute option prices.

Numerical Steps

1. Construct Tree Structure:
   • Discretize asset price and volatility dynamics into upward or downward state transitions.
   • At each time step, volatility dynamics follow the CIR process, while asset price dynamics depend on the current volatility.
2. Backward Induction:
   • Start from the terminal payoff function and compute expected values at each node backward in time.
3. Parameter Calibration:
   • Ensure the discretized tree structure matches the Heston model dynamics.

Advantages and Disadvantages

• Advantages:
  1. Suitable for pricing simple derivatives (e.g., European options).
  2. Easy to implement and intuitive.
• Disadvantages:
  1. Low computational efficiency, especially for high-dimensional problems.
  2. Complexity increases rapidly with the number of time steps.

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5. Hybrid Methods

Description

In practice, numerical pricing for the Heston model often combines multiple methods. For example:

1. Fourier Methods + Monte Carlo:
   • Use characteristic functions to quickly compute European option prices and employ them as control variates to accelerate Monte Carlo simulations.
2. Finite Difference + FFT:
   • Handle the PDE part using finite difference methods and compute integral terms efficiently using FFT.

Advantages and Disadvantages

• Advantages:
  1. Combines the strengths of multiple methods for higher efficiency and accuracy.
  2. Suitable for handling complex boundary conditions or path dependencies.
• Disadvantages:
  1. Implementation complexity depends on the model and product type.

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Summary

Method	Advantages	Disadvantages	Use Cases
Characteristic Function Methods	Efficient for European options, controllable errors	Limited to European options, complex calculations in the complex domain	European options
Monte Carlo Simulation	Highly flexible, suitable for complex derivatives (e.g., path-dependent or American options)	Slow convergence, challenges in simulating the CIR process	Path-dependent or American options
Finite Difference Method	High accuracy, handles various boundary conditions	High computational complexity for two-dimensional PDEs	Barrier options, European options
Tree Methods	Simple and intuitive, easy to implement	Low efficiency for high-dimensional problems	Simple European or American options
Hybrid Methods	Balances efficiency and accuracy, adapts to complex boundary conditions	Implementation complexity	Efficient pricing of complex derivatives

Depending on the complexity of the pricing problem and requirements, an appropriate numerical method—or a combination of methods—can be chosen to achieve efficient and accurate pricing.

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Conclusion

The Heston model is a classic representation of stochastic volatility models, capable of capturing market features such as volatility smiles, leverage effects, and mean reversion while maintaining analytical tractability. However, its limitations necessitate model improvements (e.g., introducing jumps or more complex volatility structures) to address real-world market conditions. In modern financial markets, the Heston model often serves as a foundational framework for more complex models.