Levy Processes in Finance
Diffusions make asset returns continuous, but markets reprice discontinuously around earnings, policy decisions, defaults, and liquidity shocks. A Levy process is a flexible building block for those moves: it has stationary, independent increments and starts at zero. Brownian motion is one special case; compound Poisson, variance gamma, and normal inverse Gaussian processes are others.
The important qualification is that a Levy process is a return driver, not a complete pricing model. A specification must still state the pricing measure, carry conventions, dependence with volatility, and calibration objective. The ability to fit a steep smile does not establish that a model will hedge it well.
The Levy-Khintchine representation
For a Levy process L_t, its characteristic function has the form:
E[exp(i u L_t)] = exp(t psi(u))
psi(u) = i b u - 0.5 sigma^2 u^2
+ integral_R (exp(i u x) - 1 - i u x 1_|x|<1) nu(dx)
The triplet (b, sigma, nu) identifies the process. The drift b governs deterministic movement, sigma is the continuous Brownian component, and the Levy measure nu specifies the arrival intensity and sizes of jumps. Its integrability condition prevents infinitely many large jumps in finite time.
| Component | Interpretation | Market implication |
|---|---|---|
| Brownian variance | continuous small shocks | central return dispersion |
| Finite jump activity | countable event jumps | gaps and event risk |
| Infinite jump activity | infinitely many small jumps | flexible short-horizon shape |
| Levy measure tails | frequency of large moves | wing option prices |
The subtraction term in the integral is technical but consequential: without it, the contribution of tiny jumps may not be integrable. Different truncation conventions shift b; compare parameterizations only after confirming the convention.
From a process to a tradable asset
Under a pricing measure Q, a common exponential model is:
S_t = S_0 exp((r - q - kappa) t + L_t)
kappa = log E_Q[exp(L_1)]
When the exponential moment exists, subtracting kappa makes the discounted total-return price a martingale under constant rates and dividend yield. This adjustment is not optional. Simulating a symmetric jump process and simply adding r-q usually creates a biased forward.
The moment condition itself limits model choice. Some heavy-tailed Levy laws lack E[exp(L_1)] over the required domain, so they cannot be inserted into an exponential equity model without modification. Check the moment-generating function analytically and then test simulated forwards numerically.
import numpy as np
def martingale_drift(r, q, log_mgf_at_one):
"""Annual log drift for S = S0 exp(drift*t + L_t)."""
return r - q - log_mgf_at_one
# log_mgf_at_one must use the calibrated risk-neutral Levy law.
Why characteristic functions matter
Independent increments imply a simple characteristic function over maturity T: phiT(u) = exp(T * psi(u)). Many European prices can therefore be computed with Fourier inversion rather than path simulation. Carr-Madan damping, COS expansions, and FFT grids transform the payoff so that an integral involving phiT yields a strip of strikes efficiently.
| Method | Best use | Main numerical control |
|---|---|---|
| Fourier inversion | many European strikes | damping and integration range |
| COS method | smooth characteristic functions | truncation interval |
| Monte Carlo | path dependence | jump sampling and variance |
| PIDE | low-dimensional early exercise | nonlocal integral discretization |
FFT output is not automatically reliable. Frequency spacing determines strike spacing; truncating the integration range produces ringing; an invalid damping parameter can make the transformed payoff non-integrable. Compare a subset of prices to adaptive quadrature and verify put-call parity and the forward.
Calibration identifies a surface, imperfectly
Jump intensity, asymmetry, and tail parameters can compensate for one another when fit to a sparse option chain. Weight residuals by bid-ask uncertainty and vega, exclude stale quotes, and enforce calendar and butterfly arbitrage checks on both market data and fitted output. A global optimizer can find a low error with economically implausible parameters, especially if the wings have few executable quotes.
Historical and risk-neutral jumps answer different questions. Realized returns identify the physical measure P; listed options embed Q, including compensation for downside jump risk. Treating a P estimate as a pricing parameter is a measure error, not a minor calibration shortcut. The distinction is central to risk-neutral pricing and martingales.
Extensions and limitations
Pure Levy models have independent increments, so they do not generate volatility clustering by themselves. Time changes, stochastic volatility, or regime switching can address that limitation. A useful benchmark is jump-diffusion models, where a diffusion supplies continuous variation and a finite-activity jump process supplies rare discontinuities. For equity-index surfaces, compare the hedge behavior with Heston stochastic volatility, not solely its fit error.
Path-dependent claims remain harder. Barrier crossing depends on jump overshoots, while American exercise leads to a partial integro-differential optimal-stopping problem. Discrete monitoring, dividends, and settlement conventions often dominate the apparent benefit of a more elaborate jump law.
Key takeaways
- A Levy triplet separates continuous risk, jump drift, and the distribution of jump sizes.
- Exponential asset models require an explicit martingale correction and moment check.
- Characteristic functions make European pricing fast, but Fourier settings need validation.
- Option calibration identifies risk-neutral jump premia, not physical event probabilities.
- Judge a jump model using arbitrage quality and hedging performance as well as fit.