Levy Processes in Asset Pricing

Mathematical foundation of Lévy processes

A Lévy process is a continuous-time stochastic process $\{X_t\}_{t\geq 0}$ with the following properties:

1. $X_0 = 0$ (starts at zero)
2. Independent increments: For any $t > s$, $X_t - X_s$ is independent of the past
3. Stationary increments: The distribution of $X_t - X_s$ depends only on $t-s$
4. Stochastic continuity: For any $\epsilon > 0$, $\lim_{h \to 0} P(|X_{t+h} - X_t| > \epsilon) = 0$

The Lévy-Khintchine formula characterizes all Lévy processes through their characteristic function:

$E[e^{iuX_t}] = \exp\{t[\gamma iu - \frac{1}{2}\sigma^2u^2 + \int_{\mathbb{R}\setminus\{0\}} (e^{iux}-1-iux1_{|x|<1})\nu(dx)]\}$

where:

• $\gamma$ is the drift
• $\sigma^2$ is the Gaussian component
• $\nu$ is the Lévy measure describing the jump behavior

Application in asset price modeling

In financial markets, Lévy processes provide a more realistic framework for modeling asset prices compared to traditional geometric Brownian motion. The general form of a Lévy-driven asset price model is:

$S_t = S_0\exp(X_t)$

where $X_t$ is a Lévy process.

Key advantages include:

• Ability to model sudden jumps in prices
• Better capture of heavy-tailed returns
• More accurate representation of market volatility

Popular Lévy models in finance

Variance Gamma process

The Variance Gamma Model is a pure jump Lévy process obtained by evaluating Brownian motion at a random time given by a gamma process:

$X_t = \theta G_t + \sigma W_{G_t}$

where:

• $G_t$ is a gamma process
• $W_t$ is a Brownian motion
• $\theta$ and $\sigma$ are parameters controlling skewness and volatility

CGMY model

The CGMY model extends the Variance Gamma process with additional parameters:

$\nu(dx) = C\frac{e^{-G|x|}1_{x<0} + e^{-M|x|}1_{x>0}}{|x|^{1+Y}}dx$

where:

• C controls the overall level of activity
• G and M control the rate of exponential decay
• Y determines the fine structure of the process

Risk-neutral pricing with Lévy processes

Under the Risk-Neutral Measure, the discounted asset price must be a martingale:

$E^Q[e^{-rt}S_t|\mathcal{F}_s] = e^{-rs}S_s$

This imposes constraints on the Lévy process parameters. The characteristic function under the risk-neutral measure becomes:

$\psi_t(u) = \exp\{t[\mu iu + \omega(u)]\}$

where $\omega(u)$ is the cumulant generating function and $\mu$ is chosen to ensure martingality.

Applications in options pricing

Lévy processes enable more accurate options pricing, particularly for:

• Short-term options where jump risk is significant
• Options with strikes far from the current price
• Modeling volatility skew

The option price can be computed using:

$C(K,T) = e^{-rT}\int_K^\infty (x-K)f_T(x)dx$

where $f_T(x)$ is the density of the asset price at time T, obtained through Fourier inversion of the characteristic function.

Calibration and estimation

Lévy process parameters can be estimated through:

1. Maximum likelihood estimation using historical data
2. Calibration to observed option prices
3. Method of moments using empirical moments and cumulants

The choice of estimation method depends on:

• Available data (historical prices vs option prices)
• Computational resources
• Specific model requirements

Risk management considerations

Lévy processes improve risk management by:

• Better capturing tail risks and extreme events
• More accurate Value at Risk (VaR) calculations
• Improved modeling of portfolio dependencies

These benefits make Lévy processes particularly valuable for:

• Dynamic hedging
• Portfolio optimization
• Risk assessment and stress testing