Chapter 8 Lévy Processes
8.1 The Ultimate Generalization
Lévy processes unify everything we’ve learned: random walks, Brownian motion, Poisson processes, and jump-diffusions are all special cases.
Definition: A process \(L(t)\) is a Lévy process if:
- \(L(0) = 0\)
- Independent increments: \(L(t) - L(s)\) independent of past for \(t > s\)
- Stationary increments: \(L(t) - L(s) \sim L(t-s)\) (distribution depends only on time difference)
- Stochastic continuity: \(\lim_{h \to 0} P(|L(t+h) - L(t)| > \epsilon) = 0\) for all \(\epsilon > 0\)
- Càdlàg paths: Right-continuous with left limits (allows jumps)
Lévy processes are the natural class of “time-homogeneous processes with independent increments.”
8.2 Examples We Know
All of these are Lévy processes:
- Brownian motion: Continuous paths, no jumps
- Poisson process: Jumps of size 1
- Compound Poisson: Random-sized jumps at Poisson times
- Merton jump-diffusion: Brownian + Compound Poisson
- Stable processes: Including Cauchy process
- Variance Gamma: Time-changed Brownian motion
- Normal Inverse Gaussian (NIG): Popular in finance
8.3 The Lévy-Khintchine Representation
Here’s the magic: Every Lévy process decomposes into three independent parts:
\[\boxed{L(t) = \gamma t + \sigma B(t) + J(t)}\]
where:
- \(\gamma t\): Linear drift (deterministic)
- \(\sigma B(t)\): Brownian component (continuous random)
- \(J(t)\): Pure jump component (discontinuous)
The jump component is characterized by a Lévy measure \(\nu(dy)\) which tells us: - How often jumps of size around \(y\) occur - Can have infinitely many jumps in finite time!
8.3.1 The Lévy-Khintchine Formula
The characteristic function (Fourier transform) is:
\[E[e^{i\theta L(t)}] = \exp\left[t\psi(\theta)\right]\]
where the Lévy symbol is:
\[\psi(\theta) = i\theta\gamma - \frac{1}{2}\sigma^2\theta^2 + \int_{-\infty}^{\infty}(e^{i\theta y} - 1 - i\theta y\mathbb{1}_{|y|<1})\nu(dy)\]
The Lévy triplet \((\gamma, \sigma^2, \nu)\) completely characterizes any Lévy process!
8.4 Types of Jump Behavior
The Lévy measure \(\nu\) determines the jump structure.
8.4.1 Finite Activity
\[\int \nu(dy) < \infty\]
Finitely many jumps in any finite interval. Example: Compound Poisson.
8.4.2 Infinite Activity
\[\int \nu(dy) = \infty\]
Infinitely many jumps, but most are tiny. Examples: Variance Gamma, NIG.
Intuition: Like continuous process, but with “microstructure” of tiny jumps.
8.5 Popular Lévy Processes in Finance
8.5.1 Variance Gamma (VG)
- Infinite activity, finite variation
- Time-changed Brownian motion: \(VG(t) = B(\Gamma(t))\) where \(\Gamma\) is a gamma process
- Lévy density: \(\nu(dy) = \frac{C}{|y|}e^{-\lambda|y|}dy\)
- Three parameters to fit: shape volatility smile
- Computationally tractable (FFT methods)
Use: Equity options, capturing skewness and kurtosis.
8.5.2 Normal Inverse Gaussian (NIG)
- Infinite activity, infinite variation
- Hyperbolic distribution for log-returns
- Lévy density involves modified Bessel function
- Four parameters: very flexible
- Good fit to empirical return distributions
Use: General modeling when need flexible, fat-tailed distribution.
8.5.3 CGMY (Carr-Geman-Madan-Yor)
- Generalizes many models
- Lévy density: \(\nu(dy) = C \frac{e^{-G|y|}}{|y|^{1+Y}}\) for \(y < 0\), similar for \(y > 0\)
- Parameter \(Y\) controls fine vs coarse structure:
- \(Y < 0\): Finite activity
- \(0 < Y < 1\): Infinite activity, finite variation
- \(1 < Y < 2\): Infinite activity, infinite variation
- \(Y = 0\): Reduces to Variance Gamma
Use: Ultimate flexibility in modeling jump behavior.
8.6 Why Lévy Processes Matter for Finance
8.6.1 Empirical Fit
Real asset return distributions exhibit:
- Fat tails: \(P(|R| > x)\) decays slower than Gaussian
- Skewness: Asymmetric (negative skew for equities)
- Excess kurtosis: More peaked at center, fatter tails
Lévy processes can match all three!
8.7 Pricing with Lévy Processes
For stock following:
\[\frac{dS}{S} = r \, dt + dL(t)\]
where \(L\) is a Lévy process under \(\mathbb{Q}\), the option price satisfies:
\[\frac{\partial V}{\partial t} + (r - \psi(-i))S\frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S^2\frac{\partial^2 V}{\partial S^2}\] \[+ \int_{-\infty}^{\infty}\left[V(t, Se^y) - V(t,S) - S(e^y - 1)\frac{\partial V}{\partial S}\right]\nu(dy) = rV\]
This PIDE is generally solved using:
- Fourier methods: Transform to frequency domain, solve algebraically, transform back
- Monte Carlo: Simulate Lévy process paths, average payoffs
- Finite differences: Discretize the PIDE on a grid
8.7.1 Characteristic Function Method
For European options, use:
\[C(K) = \frac{e^{-rT}}{2\pi}\int_{-\infty}^{\infty} e^{-i\omega \log K} \hat{g}(\omega) \, d\omega\]
where \(\hat{g}\) is the Fourier transform of the payoff function, computed using the characteristic function of \(L(T)\).
Fast Fourier Transform (FFT) makes this efficient for computing prices across many strikes simultaneously.
8.8 The Lévy Landscape
Lévy Processes
|
+----------------+----------------+
| |
Continuous Jumps
(Brownian) |
+---------------+---------------+
| |
Finite activity Infinite activity
(Compound Poisson) |
+---------------+---------------+
| |
Finite variation Infinite variation
(VG, some CGMY) (NIG, some CGMY)8.9 Subordination
Many Lévy processes arise through subordination: time-changing one process by another.
If \(X(t)\) is a Lévy process and \(T(t)\) is an increasing Lévy process (subordinator), then:
\[Y(t) = X(T(t))\]
is also a Lévy process.
Examples: - Variance Gamma: \(VG(t) = B(T(t))\) where \(T\) is gamma - NIG: Time-change Brownian motion by inverse Gaussian
This gives an intuitive way to build flexible processes.
8.10 Limitations and Extensions
8.10.1 What Lévy processes cannot capture
- Stochastic volatility: Volatility itself random and path-dependent
- Volatility clustering: High volatility periods persist
- Long-range dependence: Correlations decay slowly
8.10.2 Modern extensions
- Lévy-driven SDEs: \(dX = \mu(X) dt + \sigma(X) dL\)
- Time-changed Lévy processes: Random time changes
- Lévy copulas: Multivariate Lévy dependence
- Fractional Lévy processes: Long memory
- Rough volatility: Paths rougher than Brownian
These are active research areas combining stochastic calculus with other mathematical tools.
8.11 The Full Circle
We started with:
Random walks → (scaling) → Brownian motion → (SDEs) → Jump-diffusion → (generalization) → Lévy processes
Each step added:
- Brownian: Continuous randomness
- SDEs: State-dependent dynamics
- Jumps: Discontinuous shocks
- Lévy: Full generality of time-homogeneous randomness
All unified by: - PDEs/PIDEs (analytical methods) - Martingales (probabilistic methods) - Fourier methods (computational finance) - Itô calculus (the foundation)
This framework underlies modern quantitative finance, connecting pure mathematics, probability theory, and real-world markets in a beautiful and practical way.
8.12 Further Directions
Having mastered stochastic calculus and Lévy processes, you can explore:
- Stochastic volatility models (Heston, SABR)
- Local volatility and implied volatility surfaces
- Rough volatility and fractional Brownian motion
- Term structure models (interest rates, credit)
- Optimal stopping and American options
- Stochastic control and dynamic programming
- Filtering theory and partial information
- Malliavin calculus and Monte Carlo methods
The journey never ends - but you now have the foundation to explore any of these advanced topics. Congratulations!