Local Volatility Models

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Introduction to Local Volatility Models

Local Volatility Models are a class of pricing methods that infer the volatility structure of asset prices from the market-implied volatility surface. The goal is to ensure that the theoretical option prices generated by the model match the observed market prices exactly. The core idea of local volatility models is that volatility is not only a function of time but also depends on the level of the underlying asset price, allowing them to capture market volatility smiles or skews.

Below is an overview of the main local volatility models and their characteristics:

Dupire Local Volatility Model
- Proposed by: Bruno Dupire (1994)
- Key Features: Derives a bivariate function ( \sigma(t, S) ) (a function of time and asset price) to describe the local volatility of the underlying asset. The model's core formula is based on partial differential equations and can precisely fit the market-implied volatility surface. It is the foundation of local volatility modeling.
- Applications: Suitable for capturing the overall market volatility surface but has limitations in handling jump behavior and short-term volatility smiles.
Andersen FX Local Volatility Model
- Proposed by: Leif Andersen and Jorgen Andreasen (2000)
- Key Features: Extends the Dupire model by introducing a Poisson jump process to better fit short-term volatility smiles in FX markets. The Poisson jump term describes potential extreme price changes while retaining the flexibility of the local volatility model.
- Applications: FX market pricing, particularly for modeling short-term volatility smiles and price jump behavior.

Dupire FX Model

Core Assumptions

In the Dupire Model, the dynamics of the underlying asset price (e.g., FX rate) (S(t)) follow the stochastic differential equation:

\frac{dS(t)}{S(t)} = \left( r^{d}(t) - r^{f}(t) \right) dt + \sigma(t, S(t)) dW(t)

Variable Explanation:
- $S(t)$ : Underlying asset price (e.g., FX rate).
- $r^{d}(t)$ : Domestic risk-free interest rate.
- $r^{f}(t)$ : Foreign risk-free interest rate.
- $\sigma(t, S(t))$ : Local volatility, a function of time (t) and asset price (S(t)).
- $W(t)$ : Standard Wiener process, describing the random fluctuations of the asset price.

This dynamic assumption indicates that the two driving factors of asset price changes are:

The interest rate differential $r^{d}(t) - r^{f}(t)$ , representing the cost of financing the FX position.
The local volatility $\sigma(t, S(t))$ , reflecting the market's expectation of volatility at different times and price levels.

Derivation of Local Volatility

The local volatility $\sigma(K, T)$ is a function of the market option price $C(K, T)$ and can be calculated using the following formula:

\sigma^{2}(K, T) = \frac{\frac{\partial C_{K, T}}{\partial T} + \left( r^{d}(T) - r^{f}(T) \right) \frac{\partial C_{K, T}}{\partial K} + r^{f}(T) \cdot C_{K, T}}{\frac{1}{2} \cdot K^{2} \cdot \frac{\partial^{2} C_{K, T}}{\partial K^{2}}}

Variable Explanation:
- $C(K, T)$ : Price of a European call option with strike price $K$ and maturity $T$ .
- $\frac{\partial C_{K, T}}{\partial T}$ : First partial derivative of the option price with respect to maturity $T$ , representing the option's sensitivity to time decay.
- $\frac{\partial C_{K, T}}{\partial K}$ : First partial derivative of the option price with respect to strike price $K$ , representing the option's sensitivity to changes in the strike price.
- $\frac{\partial^{2} C_{K, T}}{\partial K^{2}}$ : Second partial derivative of the option price with respect to strike price $K$ , typically associated with the market's implied volatility surface.

The core idea of this formula is that the local volatility function $\sigma(K, T)$ can be derived from the partial derivatives of market option prices (e.g., European call options), thereby determining the volatility of the underlying asset at different times and price levels.

Model Features

1. Advantages

Perfect Market Fit:
The Dupire model uses the market-implied volatility surface to infer local volatility, ensuring that theoretical option prices match market prices exactly.
Captures Volatility Smile:
By defining volatility as a function of the underlying asset price, the Dupire model accurately captures the volatility smile and skew observed in the market.
Market Completeness:
The model assumes a complete market (no arbitrage opportunities), providing a theoretical foundation for its derivation and application.
Wide Applicability:
The Dupire model can be applied to FX markets, equity markets, and other derivative markets, particularly in scenarios with volatility smiles.

2. Limitations

Sensitivity to Market Data:
The calculation of local volatility relies on the first and second partial derivatives of market option prices (e.g., $\frac{\partial C}{\partial K}$ and $\frac{\partial^2 C}{\partial K^2}$ ). In real markets, noisy data can lead to unstable local volatility calculations.
Assumption of Continuous Price Changes:
The Dupire model assumes continuous changes in the underlying asset price. However, in real markets, asset prices may exhibit jumps (e.g., due to major news or central bank decisions).
Inability to Capture Jump Behavior:
In FX markets, price jumps are common, but the Dupire model cannot handle these discrete jump features.
Computational Complexity:
Inferring local volatility requires numerical differentiation of option prices, which can be computationally intensive and sensitive to input data quality.

Improved Models: Local Volatility with Jumps

To address the limitations of the Dupire model (especially its inability to handle jumps), several extended models have been proposed in financial engineering:

1. Local Volatility with Jumps

The local volatility model is extended by adding a jump process $J(t)$ :

\frac{dS(t)}{S(t)} = \left( r^{d}(t) - r^{f}(t) \right) dt + \sigma(t, S(t)) dW(t) + dJ(t)

Jump Process:
- $J(t)$ is typically modeled as a Poisson process or Lévy process, describing discrete jumps in the asset price.
- The frequency and magnitude of jumps can be calibrated using market data.

2. Hybrid Models (Local-Stochastic Volatility Models, LSV)

Hybrid models combine local volatility and stochastic volatility, assuming that volatility is both a function of the underlying asset price and a stochastic process:

\frac{dS(t)}{S(t)} = \left( r^{d}(t) - r^{f}(t) \right) dt + \sigma(t, S(t), v(t)) dW(t)

$v(t)$ : Stochastic volatility, typically modeled using the CIR or Heston model.

These models can capture market volatility smiles and handle jump behavior and long-term volatility dynamics.

Practical Applications of the Dupire Model

1. Applications in FX Markets

Volatility Surface Construction:
The Dupire model is commonly used to infer local volatility from the implied volatility surface in FX option markets, providing inputs for pricing complex derivatives.
Risk Management:
Local volatility models help FX traders more accurately measure the risk exposure of option positions (e.g., Vega, Gamma).

2. Derivative Pricing

Complex Option Pricing:
The Dupire model is suitable for path-dependent options (e.g., Barrier Options) or other derivatives with non-standard payoff structures.
Calibration Tool:
The Dupire model provides a flexible calibration framework for derivative pricing, allowing model parameters to be adjusted to match market quotes.
Dupire Model Calibration Process

Conclusion

The Dupire model, as a representative of local volatility models, provides a method to infer local volatility from the market-implied volatility surface. Its strengths lie in its ability to precisely match market prices, capture volatility smiles, and maintain market completeness. However, its limitations include its inability to handle jump behavior and its sensitivity to market data quality. By introducing jump processes or stochastic volatility, the model can be further extended to address the complexities of real-world markets.

Andersen FX Model

Background

In FX option pricing, the volatility smile is a critical feature of the options market, particularly for short-term maturities. This phenomenon cannot be fully captured by the classic Black-Scholes model or pure local volatility models. To address this, Merton (1976) proposed a stochastic model incorporating jump processes to explain the volatility smile in short-term maturities.

Building on this, Andersen and Andreasen proposed an improved local volatility model called the Andersen FX Model. This model incorporates a Poisson jump process into the local volatility framework to better fit market quotes while retaining the pricing flexibility and no-arbitrage properties of local volatility models. Specifically, the model captures short-term volatility smiles and has become a classic tool for modeling complex smile features in FX option markets.

Model Description

In the Andersen FX Model, the dynamics of the FX rate $S(t)$ are described as:

\frac{dS(t)}{S(t)} = \left( r^d(t) - r^f(t) - \lambda_t m(t) \right) dt + \sigma(t, S(t)) dW(t) + (J_t - 1) dN_t

Variable Explanation:
- $S(t)$ : Current FX rate (underlying asset price).
- $r^d(t)$ : Domestic risk-free interest rate.
- $r^f(t)$ : Foreign risk-free interest rate.
- $\sigma(t, S(t))$ : Local volatility, a function of time $t$ and asset price $S(t)$ .
- $W(t)$ : Wiener process, describing the continuous random fluctuations.
- $N_t$ : Poisson process, describing the occurrence of jump events.
- $J_t$ : Independent and identically distributed (i.i.d.) lognormal random variable, representing the jump size, with mean $\eta_J$ and variance $\sigma_J$ .
- $\lambda_t$ : Intensity of the Poisson process (jump frequency).
- $m(t) = \mathbb{E}[J_t - 1]$ : Expected jump size, ensuring the forward price $F_t$ remains a martingale: $F_t = e^{\int_0^T \left( r^d(t) - r^f(t) \right) dt} S_t$

Model Features:

Local Volatility Component $\sigma(t, S(t))$ : Retains the flexibility of classic local volatility models for fitting long-term volatility smiles.
Jump Component $(J_t - 1)dN_t$ : Handles short-term volatility smiles through the Poisson jump process, particularly suitable for capturing sharp smile shapes in the market.
Jump Frequency $\lambda_t$ and Jump Size Distribution $J_t$ : Enhances the model's ability to capture extreme market events (e.g., sudden news, central bank decisions).

Model Advantages and Disadvantages

Advantages

Market Fit:
- By combining local volatility and jump components, the model can accurately fit the market-implied volatility surface, especially for short-term volatility smiles.
Flexibility:
- The model retains the flexibility of local volatility models while enhancing its ability to capture market jump behavior.
No-Arbitrage Property:
- Ensures the model's no-arbitrage property, making it suitable for practical pricing in FX markets.

Disadvantages

Calibration Complexity:
- Calibrating local volatility requires solving partial differential equations with integral terms, which is computationally intensive and sensitive to market data quality.
High Numerical Cost:
- Compared to classic local volatility models, the Andersen FX Model requires more computational resources (e.g., simulation paths, PIDE solving time).
Dependence on Jump Parameters:
- The model's performance depends on the choice of jump parameters $\lambda, \eta_J, \sigma_J$ , which are often estimated empirically or from external data.

Calibration of Local Volatility

To price FX options, the local volatility $\sigma(t, S(t))$ must be calibrated to ensure model prices match market prices. The calibration process is similar to the Dupire model but is modified to include integral terms due to the jump component. The local volatility formula is expressed as a partial integro-differential equation (PIDE):

\begin{aligned} \sigma^2(K, T) &= \frac{1}{K^2 \frac{\partial^2 C_{K,T}}{\partial K^2}} \Bigg[ \frac{\partial C_{K,T}}{\partial T} + \left( r^d(T) - r^f(T) - \lambda(T) m(T) \right) \frac{\partial C_{K,T}}{\partial K} \\ &\quad + \left( r^f(T) + \lambda(T) m(T) + 1 \right) C_{K,T} - \lambda(T) \mathbb{E}\left[J(T) C(t, S, T, \frac{K}{J(T)}) \right] \Bigg] \end{aligned}

Formula Breakdown:
- First Term $\frac{\partial C_{K,T}}{\partial T}$ : Partial derivative of the option price with respect to maturity (time decay).
- Second Term $\left( r^d(T) - r^f(T) - \lambda(T)m(T) \right)\frac{\partial C_{K,T}}{\partial K}$ : Impact of interest rates and jump terms on the option price.
- Third Term $\lambda(T) \mathbb{E}[J(T)C(t,S,T,K/J(T))]$ : Correction term for jumps in the option price.

Conclusion

The Andersen FX Model is a hybrid local volatility model incorporating Poisson jump processes, particularly suitable for capturing short-term volatility smiles in FX markets. Its strength lies in enhancing market fit through jump processes while retaining the advantages of local volatility models. However, the model's complexity increases calibration difficulty and computational costs. In practice, it can be combined with Monte Carlo methods or partial integro-differential equation solvers to balance accuracy and efficiency.

Recommended Reading:

Benhamou and Miri (2006): Advanced Numerical Methods for Local Volatility Models
Andersen and Andreasen (2000): Jump-Diffusion Models in FX Markets
Merton (1976): Option Pricing When Underlying Stock Returns Are Discontinuous

Numerical Pricing Methods

Below is a summary of the three main numerical pricing methods for the Dupire Model and Andersen FX Model: Monte Carlo Method, Partial Differential Equation (PDE) Method, and Partial Integro-Differential Equation (PIDE) Method. These methods provide numerical solutions for both models, catering to different market needs and application scenarios.

1. Monte Carlo Method

Description

The Monte Carlo method is a path simulation-based numerical pricing tool, highly effective for handling complex dynamic models. For the Dupire Model and Andersen FX Model, the Monte Carlo method requires the following considerations:

Dupire Model:
- The asset price dynamics are driven by local volatility: $\frac{dS(t)}{S(t)} = \left(r^d - r^f\right) dt + \sigma(t, S(t)) dW(t)$
- Local volatility $\sigma(t, S(t))$ is a function of time and asset price, which may introduce numerical bias.
Andersen FX Model:
- The asset price dynamics include a Poisson jump process: $\frac{dS(t)}{S(t)} = \left(r^d - r^f - \lambda_t m(t)\right) dt + \sigma(t, S(t)) dW(t) + (J_t - 1) dN_t$
- Simulating the jump process $N_t$ and jump size $J_t$ increases complexity.

Improvements

To improve accuracy and reduce bias from local volatility, the Predictor-Corrector Method can be used:

Predictor Step: Generate preliminary paths using the Euler method.
Corrector Step: Correct paths using the nonlinear characteristics of local volatility.

Advantages and Disadvantages

Advantages:
1. Suitable for complex dynamics (e.g., jumps, high-dimensional problems).
2. Easily extendable to path-dependent or American options.
Disadvantages:
1. Slow convergence and long computation times.
2. Simulating jump processes increases computational complexity.

Use Cases

Complex derivatives (e.g., path-dependent options, high-dimensional FX options).
Models requiring precise handling of jump behavior (e.g., Andersen FX Model).

2. Partial Differential Equation (PDE) Method

Description

The PDE method is a classic numerical pricing tool suitable for continuous asset price dynamics. For the Dupire Model and a simplified Andersen FX Model (ignoring jumps), the PDE method can be used for option pricing.

Dupire Model:
- Since local volatility is a function of time and price, the European option price $C(t, S)$ satisfies the following PDE: $\frac{\partial C}{\partial t} + \left(r^d - r^f\right) S \frac{\partial C}{\partial S} + \frac{1}{2} \sigma^2(t, S) S^2 \frac{\partial^2 C}{\partial S^2} - r^d C = 0$
Andersen FX Model:
- If the Poisson jump process is ignored (i.e., $\lambda = 0$ ), the dynamics simplify to a pure local volatility model, making the standard PDE method applicable.

Numerical Solutions

The finite difference method (FDM) is commonly used to solve PDEs, with popular approaches including:

Explicit Method:
- Directly computes values at the next time step, simple but requires strict stability conditions.
Implicit Method:
- Solves a system of linear equations to compute values at the next time step, numerically stable but computationally complex.
Crank-Nicholson Method:
- Combines explicit and implicit methods, using central differences in time, balancing stability and accuracy.

Advantages and Disadvantages

Advantages:
1. High computational efficiency, suitable for European and barrier options.
2. Strong numerical stability, ideal for long-term options.
Disadvantages:
1. Cannot directly handle jump processes.
2. High computational complexity for high-dimensional problems.

Use Cases

Dupire Model: European or barrier options driven by local volatility.
Andersen FX Model: Scenarios where jump effects are minimal (e.g., long-term volatility smiles).

3. Partial Integro-Differential Equation (PIDE) Method

Description

The PIDE method is the core numerical tool for the Andersen FX Model, as the Poisson jump process introduces integral terms, transforming the pricing equation from a PDE to a PIDE.

In the Andersen FX Model, the European option price $C(t, S)$ satisfies the following PIDE:

\frac{\partial C}{\partial t} + \left(r^d - r^f - \lambda m\right) S \frac{\partial C}{\partial S} + \frac{1}{2} \sigma^2(t, S) S^2 \frac{\partial^2 C}{\partial S^2} - r^d C + \lambda \int_{0}^{\infty} C(t, S J) f_J(J) dJ = 0

Integral Term: $\lambda \int_{0}^{\infty} C(t, S J) f_J(J) dJ$ Describes the impact of the jump process on the option price, where $f_J(J)$ is the probability density function of the jump size $J$ (typically lognormal).

Numerical Solutions

Due to the non-local nature of the integral term, solving PIDEs requires combining numerical integration methods. Common approaches include:

Crank-Nicholson Method:
- Within the finite difference framework, the integral term is handled separately.
Fast Fourier Transform (FFT):
- Transforms the integral term into the frequency domain for efficient solution using FFT. Reference: Fourier Transform
Simpson's Rule:
- Directly discretizes the integral term using numerical integration.

Advantages and Disadvantages

Advantages:
1. Captures the impact of jump processes on option prices.
2. Accurately fits short-term volatility smiles.
Disadvantages:
1. Complex numerical implementation and lower computational efficiency.
2. Sensitive to assumptions about jump distributions.

Use Cases

Scenarios with significant short-term volatility smiles.
Situations where jump processes significantly impact pricing (e.g., extreme market events).

Comparison of the Three Methods

Method	Advantages	Disadvantages	Use Cases
Monte Carlo Method	- Suitable for complex dynamics (jumps, high dimensions) - Extendable to path-dependent options	- Slow convergence - Jumps increase computational complexity	Path-dependent options, high-dimensional problems
PDE Method	- High computational efficiency - Strong numerical stability	- Cannot directly handle jumps - High computational complexity for high dimensions	Long-term volatility smiles, European options
PIDE Method	- Captures jumps - Accurately fits short-term volatility smiles	- Complex implementation - Sensitive to jump distribution assumptions	Significant short-term volatility smiles, jump effects

Conclusion

For the Dupire Model and Andersen FX Model, the three numerical methods have their respective use cases:

Monte Carlo Method: Suitable for complex dynamics (e.g., jumps) or high-dimensional problems.
PDE Method: Ideal for long-term volatility smiles driven by local volatility.
PIDE Method: Used to capture short-term volatility smiles and jump behavior in pricing.

The choice of method depends on the model characteristics and application requirements, balancing accuracy and computational efficiency.

References

1. Theoretical Foundations and Classic Literature

Bruno Dupire (1994)
Pricing with a Smile
Source: Risk Magazine, 1994
Abstract:
Dupire introduced the concept of local volatility in this groundbreaking paper and derived the formula for calculating local volatility. This paper laid the theoretical foundation for local volatility models and is a core reference for understanding the Dupire model.
Emmanuel Derman & Iraj Kani (1994)
Riding on a Smile
Source: Risk Magazine, 1994
Abstract:
Derman and Kani proposed a method to construct local volatility from the implied volatility surface and implemented it using binomial trees. This paper is an important reference for the application of local volatility models.
Paul Wilmott (1998)
Derivatives: The Theory and Practice of Financial Engineering
Publisher: Wiley
Abstract:
This book provides a comprehensive introduction to various models used in financial engineering, including detailed derivations and applications of local volatility models. It is a classic textbook for learning derivative theory and practice.
Steven Heston (1993)
A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options
Source: The Review of Financial Studies, 1993
Abstract:
Heston proposed the stochastic volatility model (Heston model), which, although different from local volatility models, significantly influenced the development of extended local volatility models (e.g., LSV models).

2. Numerical Implementation and Practice

Jim Gatheral (2006)
The Volatility Surface: A Practitioner's Guide
Publisher: Wiley
Abstract:
This book focuses on practical methods for constructing volatility surfaces, including numerical implementation, market data calibration, and comparisons with other volatility models.
Mark S. Joshi (2003)
The Concepts and Practice of Mathematical Finance
Publisher: Cambridge University Press
Abstract:
Joshi discusses the numerical implementation of local volatility models in detail, including finite difference and Monte Carlo methods, making it suitable for readers interested in practical applications.
Peter Jäckel (2002)
Monte Carlo Methods in Finance
Publisher: Wiley
Abstract:
This book provides a detailed introduction to the application of Monte Carlo methods in financial models, including numerical simulations for local volatility models.

3. Extensions and Improved Models

Bruno Dupire (1997)
Pricing and Hedging with Smiles
Source: Mathematics of Derivative Securities, 1997
Abstract:
Dupire further extended the local volatility model in this paper, discussing how to calibrate the model using market data and analyzing its limitations.
Rama Cont & Peter Tankov (2004)
Financial Modelling with Jump Processes
Publisher: Chapman and Hall/CRC
Abstract:
This book focuses on jump models and Lévy processes, combining them with local volatility models to propose extended models for handling jump behavior.
Emanuel Benhamou, Stefano De Marco, Marc Loeper (2009)
The Implied Volatility Surface Close to Expiry
Source: Quantitative Finance, 2009
Abstract:
This paper explores the behavior of the implied volatility surface near option expiration and compares it with the assumptions of local volatility models.
Christian P. Fries & Matthias Kampen (2005)
Local Stochastic Volatility Models
Source: Quantitative Finance, 2005
Abstract:
This paper proposes local-stochastic volatility models (LSV), combining stochastic volatility models with local volatility models to improve market data fitting.