Lévy Processes Overview. Why Normal Isn't Enough

Brownian motion is the canonical stochastic process in finance: Gaussian increments, continuous paths, a tractable theory. Textbooks put it first, and after a while that leaves an impression that it is the representative member of its class and everything else is a variation on it.

It is the other way around. Brownian motion is the one exception among Lévy processes, the only one whose paths are continuous. Every other Lévy process lives on jumps. Some jump rarely and large, some often and small, some at literally every instant. Those differences are not scattered across model definitions. They are all recorded in one ledger: the Lévy measure $\nu$.

Reading that ledger takes looking at two ends. The behavior of $\nu$ near zero determines how rough the paths are, meaning how densely it jumps and whether total variation is finite. The behavior of $\nu$ at infinity determines how fat the tails are and which moments exist. Nearly every difference among the models used in finance reduces to a trade-off between those two ends, and that is how this article is organized.

The Four Conditions

A Lévy process $X_t$ is a stochastic process satisfying:

1. Stationary increments. $X_{t+s} - X_t$ has the same distribution as $X_s$ for all $s, t \geq 0$.
2. Independent increments. $X_{t+s} - X_t$ is independent of $\mathcal{F}_t$ for all $s, t \geq 0$.
3. Stochastic continuity. $\lim_{s \to 0} \mathbb{P}(|X_{t+s} - X_t| > \epsilon) = 0$ for any $\epsilon > 0$.
4. $X_0 = 0$, with cadlag paths (right-continuous with left limits).

Four conditions, and the territory they mark out is large. Brownian motion is a Lévy process. So is the Poisson process, so are the stable processes, and so are the processes behind variance-gamma, CGMY, Merton jump-diffusion, normal inverse Gaussian, and the hyperbolic distributions.

The key fact is that the distribution of $X_1$ determines the entire process, through the Lévy-Khintchine representation. Fix that distribution and everything follows: tail behavior, jump structure, financial implications.

Lévy-Khintchine: Three Parameters, One of Them Interesting

For any Lévy process $X_t$, the characteristic function of $X_t$ has the form

This is the Lévy-Khintchine representation, with $b \in \mathbb{R}$ the linear part of the drift, $c \geq 0$ the Gaussian coefficient measuring the strength of the Brownian component, and $\nu$ a measure on $\mathbb{R} \setminus \{0\}$ satisfying $\int \min(x^2, 1) \, \nu(dx) < \infty$, called the Lévy measure.

The first two parameters are each a single number and there is not much to say about them. $\nu$ is where models actually differ. Read literally, $\nu(dx)$ is the intensity of jumps of size in $(x, x+dx)$ per unit time, which is to say it specifies how often this process intends to jump and by how much. The Brownian case is the boundary: $\nu = 0$, an empty ledger, nothing left but the Gaussian term $cu^2/2$.

The Zero End: How Densely It Jumps

Start with $\nu$ near the origin. It governs how many small jumps there are, and the number of small jumps governs how rough the path is. Three regimes.

Finite activity means $\nu(\mathbb{R}) < \infty$: finitely many jumps per unit time on average, with the Poisson process as the example. You can point at each jump and count it.

Infinite activity, finite variation means $\nu(\mathbb{R}) = \infty$ but $\int_{|x|<1} |x| \, \nu(dx) < \infty$: infinitely many jumps per unit time, small enough that total variation stays finite. VG and CGMY with $Y < 1$ live here.

Infinite activity, infinite variation means $\nu(\mathbb{R}) = \infty$ and $\int_{|x|<1} |x| \, \nu(dx) = \infty$: infinitely many jumps and divergent total variation. CGMY with $1 \leq Y < 2$, NIG, and the stable processes live here.

If $\nu$ blows up at the origin like a power law, $\nu(x) \sim |x|^{-1-Y}$, the three regimes become three ranges of $Y$. For $Y < 0$ the integral $\int_{|x|<1}\nu$ converges and activity is finite. For $0 \leq Y < 1$ activity is infinite but variation is finite. For $1 \leq Y < 2$ variation diverges as well. And $Y \geq 2$ is not allowed at all: it fails the basic integrability condition $\int \min(x^2,1)\nu(dx) < \infty$ that defines a Lévy measure. That is not a rougher process, it is a non-existent one.

That exponent reappears below as the $Y$ in CGMY. It is the same object.

The Infinity End: Tails and Moments

The other end governs large jumps, and large jumps govern moments. The criterion is clean: the moment $\mathbb{E}[|X_t|^p]$ exists if and only if

which splits two decay behaviors apart. If $\nu$ decays exponentially at infinity, $\nu(x) \sim e^{-G|x|}$, moments of all orders are finite. If $\nu$ decays as a power law, $\nu(x) \sim |x|^{-1-\alpha}$, every moment of order $\alpha$ and above diverges.

This is what decides what a model can be used for. In pricing you usually care about low-order moments, and option prices are expectations of bounded or near-bounded functions. In risk management variance is the workhorse, and infinite variance is fatal. Match the model to the application: if your risk system assumes finite variance, do not report risk out of a process that has none.

Why Finance Walks Away From the Gaussian

The pressure comes from a handful of empirical facts, and each maps onto one end of $\nu$. Prices jump (overnight gaps, flash crashes, earnings releases), and Brownian motion is continuous and cannot represent that, which requires $\nu \neq 0$. Empirical return distributions have fatter tails than Gaussian, which requires $\nu$ to decay slowly enough at infinity. Equity returns are negatively skewed, which requires $\nu$ to be asymmetric. Fourth, aggregational Gaussianity: stretch the horizon and returns look more Gaussian, a property most Lévy processes have for free.

The fifth, volatility clustering, is the one the plain Lévy framework cannot deliver, because it conflicts directly with stationary increments. Getting it in requires time-changed Lévy or stochastic volatility, which comes later.

Stable Distributions: Power Law at Both Ends

The stable distributions are a class of Lévy processes with power-law tails, characterized by four parameters: $\alpha \in (0, 2]$, the stability index controlling tail heaviness, with $\alpha = 2$ giving the Gaussian and $\alpha < 2$ giving heavy tails; $\beta \in [-1, 1]$, the skewness parameter, symmetric at $\beta = 0$; $\sigma > 0$, the scale parameter; and $\mu$, the location parameter. The characteristic function is

where $\Phi$ is a phase function with two common parameterizations, A and B. Stable distributions are the natural limit of normalized sums of i.i.d. random variables, the attractors of the central limit theorem in the heavy-tailed case. That is their theoretical appeal.

Their practical difficulty falls straight out of the two-end reading. A stable law’s $\nu$ is pure power law at infinity with nothing holding it down. So for $\alpha < 2$, every moment of order $\alpha$ and above is infinite, and the rule has no exceptions. $\alpha = 1.7$ has very heavy tails, has a mean, and has infinite variance. $\alpha = 1$ (Cauchy) has no mean at all, and $\alpha < 1$ is worse. Variance and kurtosis, and every risk measure built on them, are simply undefined.

For pricing the problem is harder still. Model the log price as an $\alpha$-stable process with $\alpha < 2$ and $\mathbb{E}[e^{X_t}]$ diverges, so the forward price is infinite and so is the call price. The only escape is to make the distribution fully left-skewed ($\beta = -1$), cutting off the right tail entirely, which is the route Carr & Wu (2003) take in the finite-moment log-stable model. Absent that, stable processes cannot be used for pricing directly.

Two engineering annoyances sit on top. The density has no closed form, so while the characteristic function is known the density needs numerical inversion; and estimation is difficult, since maximum likelihood is complicated by the missing density and moment-based methods are biased when the moments themselves fail to exist. Net result: stable distributions live mostly in theoretical work and stress-testing, where worst-case behavior is the point. Practical pricing goes to the two models below.

CGMY: Cut Off the Infinity End

If the stable law’s disease is an untamed right end, the obvious prescription is to multiply by an exponential decay factor, crushing the far tail while keeping the power-law behavior near zero. That is tempering, and that is CGMY.

Carr, Geman, Madan, and Yor (2002) proposed the CGMY model with Lévy measure

where $C > 0$, $G, M \geq 0$, and $Y < 2$. Each letter does one job. $C$ sets overall activity. $M$ and $G$ set the exponential decay rate of positive and negative jumps respectively, so $G \neq M$ produces skew. $Y$ controls small-jump behavior near zero and is exactly the exponent from two sections ago, so $Y < 0$ gives finite activity, $0 \leq Y < 1$ gives infinite activity with finite variation, and $1 \leq Y < 2$ gives infinite variation. One end, one set of parameters, no interference. That is why CGMY is pleasant to work with.

Read along those coordinates and several familiar models are just special cases. At $Y = -1$, $|x|^{1+Y} = 1$ and the Lévy density collapses to a pure exponential: Kou’s double-exponential jump model, finite activity. At $Y = 0$ you get the variance-gamma process. At $G = M = 0$ the exponential factor disappears, leaving only the power law, which is the symmetric stable process with $Y$ playing the role of the stability index $\alpha$. On this map the stable law is not a separate family; it is the corner where tempering goes to zero.

CGMY has four parameters and a closed-form characteristic function, flexible enough to fit any tail behavior up to the moment explosion threshold. The price is four parameters that all have to be estimated from the market.

Merton Jump-Diffusion: The Other Road

CGMY goes the pure-jump route. Merton (1976) went the other way: keep Brownian motion, add a sparse layer of log-normal jumps on top. The dynamics are

where $N_t$ is a Poisson process with intensity $\lambda$ and $J \sim \mathcal{N}(\mu_J, \sigma_J^2)$ is the jump size. The number of jumps in $[0, t]$ is Poisson with mean $\lambda t$, and each jump multiplies the price by $e^J$.

The beauty of this model is the closed form. Conditional on $n$ jumps the log price is still normal, so the option price is a string of Black-Scholes prices weighted by Poisson probabilities:

with $\sigma_n = \sqrt{\sigma^2 + n \sigma_J^2 / T}$. The sum converges fast, usually within 5-10 terms.

Merton is the workhorse of jump-diffusion modeling: simple, thoroughly understood, and a reasonable fit for assets with infrequent, well-defined jumps. It struggles with assets whose jumps are frequent and small, which is CGMY and VG territory. In the language of $\nu$, Merton’s jump measure is finite (a Gaussian density scaled by $\lambda$) with no activity at all near the origin, and all the fine-grained roughness is handed off to the Brownian component.

Pure Jump or Jump-Diffusion

Which raises a modeling choice. A pure jump Lévy process has $c = 0$, no Brownian component, and a non-zero $\nu$; examples are Poisson, VG, CGMY, NIG, and stable. A jump-diffusion has $c > 0$ and $\nu \neq 0$; examples are Merton, Kou, and Bates.

For an asset that trades continuously and has no true continuous path, pure jump is natural: the “jumps” are the tiny price moves that aggregate into what looks like continuous motion. For an asset that does move continuously with occasional large jumps on top, equity prices with earnings gaps for instance, jump-diffusion fits better.

The distinction is blurrier than it sounds. A pure-jump model with very small jumps can look continuous, and a jump-diffusion with a tiny Brownian component can look mostly jumpy. Empirical fit decides, not ontology.

Infinite Activity Sounds Pathological and Is Actually Normal

Infinitely many jumps per unit time reads like a mathematician being difficult. It is the empirically more correct side. Real prices look like they move continuously, but they move in tiny increments, and if you book those tiny increments as jumps you get infinitely many of them per second.

For a CGMY process with $0 < Y < 2$, the number of jumps in $[0, t]$ is infinite, while the sum of absolute jump sizes over any compact interval is finite for $Y < 1$ and infinite for $Y \geq 1$. VG is in the first camp, and VG paths have finite total variation.

In practice the distinction matters less than it sounds. The empirical fact is that prices are continuous but very jagged at high frequency, and an infinite-activity Lévy model captures that jaggedness with infinitely many small jumps. The actual number of “jumps” is a modeling choice, not a fact about the world.

So the Gaussian Is the Exception

Brownian motion with drift is the only Lévy process with continuous paths. Every other one jumps, some small, some large, some infinite in number. The Gaussian is also the only stable distribution with finite variance, and the only distribution arising as a limit of normalized sums of i.i.d. finite-variance random variables.

It is mathematically elegant and empirically insufficient. Financial returns have heavy tails (incompatible with Gaussian), skewness (incompatible with a symmetric distribution), jumps (incompatible with continuous paths), and volatility clustering (incompatible with stationary increments). Lévy processes with non-Gaussian $\nu$ handle the first three. The fourth needs time-changed Lévy or regime switching, and the plain framework cannot supply it.

Brownian motion is the simplest Lévy process, useful for pedagogy and limited for empirical work. It is the starting point, not the destination.

Carr-Madan: One Characteristic Function Fits All

Lévy models share a single pricing exit. Since Lévy-Khintchine hands you the characteristic function directly, use Fourier inversion to get back to prices. The Carr-Madan formula gives the call price as

where $k = \log K$ and $\alpha > 0$ is a damping factor, introduced to make the integrand integrable, with

and $\phi_X$ the characteristic function of $\log S_T$. The formula works for any Lévy process whose $\phi_X$ is known in closed form, with the integral computed by FFT or quadrature.

The engineering consequence is worth noticing. The only model-specific input is the characteristic function; everything else is numerical integration. Changing models means swapping one $\phi_X$ and leaving the pricing code alone. That, more than the merits of any individual model, is why Lévy models spread through institutions: they share a pipeline.

Calibration, and Where It Will Lie to You

Calibrating a Lévy model to option prices is three steps: choose a parametric family (VG, CGMY, Merton), compute model prices through the characteristic function, and minimize the sum of squared differences against market prices. With FFT pricing each iteration is sub-second, so speed is not the issue.

The trouble is elsewhere. The objective is non-convex, multiple parameter sets can produce near-identical fits, and which local optimum you land in depends on your starting values. Few parameters against many quotes (three or four against several hundred) looks safe from overfitting but is not; the model can still fit noise in the quotes. The most useful diagnostic is stability: recalibrate the same model day after day and the parameters should drift smoothly. If they jump around, you are fitting noise, not structure. Empirically VG and CGMY calibrate stably while Merton often does not, because $\sigma$ and the jump parameters are correlated and the same smile can be explained by “high diffusion, few jumps” or “low diffusion, many jumps.”

Take Merton as the worked case. With parameters $\sigma$, $\lambda$, $\mu_J$, $\sigma_J$, the price is

where $\tilde{\sigma}_n = \sqrt{\sigma^2 + n \sigma_J^2 / T}$ is the effective volatility given $n$ jumps. Calibrate by minimizing

with weights $w_i$ typically inverse-vega or inverse-bid-ask-spread. For liquid single-name equity options, Merton calibrations land in the neighborhood of a few tenths of a jump to one jump per year for $\lambda$, roughly 0.05 to 0.15 for $\sigma_J$, and roughly -0.05 to 0 for $\mu_J$, the slight negative jump drift being consistent with negative skew. These move with the sample period, the underlying, and the weighting scheme, so treat them as orders of magnitude.

import math

import numpy as np
from scipy.stats import norm

def merton_call(S0, K, r, T, sigma, lam, mu_j, sigma_j):
    """Merton jump-diffusion call price."""
    n_terms = 20
    # k = E[e^J] - 1, the mean relative jump size
    k = np.exp(mu_j + 0.5 * sigma_j**2) - 1
    price = 0.0
    for n in range(n_terms):
        poisson_prob = np.exp(-lam * T) * (lam * T)**n / math.factorial(n)
        sigma_n = np.sqrt(sigma**2 + n * sigma_j**2 / T)
        r_n = r - lam * k + n * np.log(1 + k) / T  # martingale correction
        d1 = (np.log(S0/K) + (r_n + 0.5*sigma_n**2)*T) / (sigma_n*np.sqrt(T))
        d2 = d1 - sigma_n*np.sqrt(T)
        price += poisson_prob * (S0*norm.cdf(d1) - K*np.exp(-r_n*T)*norm.cdf(d2))
    return price

The drift correction line is easy to get wrong. What you subtract is $\lambda k$, not $\lambda \mu_J$, where $k = \mathbb{E}[e^J] - 1 = e^{\mu_J + \sigma_J^2/2} - 1$. Substitute $\mu_J$ for $k$ and the martingale property breaks, biasing prices systematically. Wrap this function in a least-squares optimization and you have a working Merton calibrator.

Time-Changed Lévy: The Door to Volatility Clustering

There is a debt outstanding from earlier: stationary increments are incompatible with volatility clustering. The standard way to settle it is to run the Lévy process on a stochastic clock. Define

where $Y$ is a Lévy process and $\tau(t)$ is a stochastic, increasing process. When $\tau$ runs fast, $X$ is active; when $\tau$ runs slow, $X$ is quiet. Activity itself has become random, which is what volatility clustering means.

Take $\tau$ to be an integrated CIR process, $\tau(t) = \int_0^t v_s \, ds$ with $v_s$ following CIR, and take $Y$ to be VG, and you get variance-gamma with stochastic volatility: jumps and stochastic volatility present at once. This is the basis of the models in the Carr, Geman, Madan and Yor (2003) line and of many practitioner extensions. Implementation requires simulating both the time change and the underlying Lévy process, two layers of randomness, at moderate cost.

Pricing still runs through characteristic functions. Conditional on the time change, the characteristic function is that of the underlying Lévy process; the unconditional one is an integral over the time-change distribution, usually computed by Fourier methods too. The Carr-Madan pipeline still runs.

Self-Decomposability and the VG Relatives

A Lévy process is self-decomposable if for any $0 < c < 1$, $X_t$ has the same distribution as $c X_t + Z$ for some independent $Z$. Intuitively: shrink the distribution and what is left over is still a legitimate independent random variable. The distribution divides into itself.

The class includes Brownian motion (the Gaussian case), variance-gamma, normal inverse Gaussian, the hyperbolic and generalized hyperbolic distributions, the Lévy processes behind Student’s t, CGMY under certain parameter restrictions, and all stable processes. Self-decomposability implies infinite divisibility, meaning a distribution expressible as the sum of any number of i.i.d. copies, which is exactly the precondition for the Lévy-Khintchine representation. VG, NIG, and hyperbolic are all infinitely divisible.

These processes are popular in practice because their characteristic functions tend to be compact enough for Fourier inversion to run fast. That is an empirical observation, not a theorem, so don’t treat it as a guarantee.

Additive Processes and the Term Structure

There is one more natural way to loosen the definition: keep independent increments and drop stationarity. What you get is an additive process, which allows time-inhomogeneity. Term structure modeling requires this, because rates at different maturities genuinely have different dynamics.

The Heath-Jarrow-Morton (HJM) framework can be cast as an additive-process model for forward rates, with the drift pinned by no-arbitrage and time-dependent volatility. It is the most general framework for Gaussian term structure models. Swap the increments for non-Gaussian ones and you get extensions like the LIBOR market model with jumps, used in industry to price exotic rate derivatives with jump risk.

Multi-Asset: Lévy Copulas

The dependence structure of a multi-asset Lévy process is specified by a Lévy copula, describing how jumps in different assets co-occur. You cannot lift an ordinary copula here, because a Lévy measure is not a probability measure and its total mass can be infinite. The right construction is Sklar’s theorem transposed onto tail integrals. With marginal tail integrals $U_i(x) = \nu_i([x, \infty))$, the joint tail integral $U$ satisfies

with $\mathcal{C}$ the Lévy copula. It separates “how each asset jumps on its own” from “how they jump together” cleanly, so marginals and dependence can be specified independently.

One thing gets conflated constantly. A Lévy copula describes dependence between jump measures; an ordinary copula (Gaussian, t, Clayton) describes dependence in the terminal distribution at some maturity. They are different objects, even though a functional form like Clayton has a version on both sides. Pricing a basket option or an option on the minimum of a set of assets needs the joint distribution at expiry, which is generated by the jump-level dependence, and getting that wrong makes accurate pricing hopeless.

Which family you pick depends on what you are trying to capture. Gaussian dependence is simple and limited. A t-type gives heavier tail dependence, useful for joint crashes. A Clayton-type is asymmetric, with strong left-tail and weak right-tail dependence. Since “assets crash together and rise separately” is exactly the tail dependence pattern that shows up empirically, the latter two usually sit closer to the data.

A Bit of History

Lévy processes are named after Paul Lévy, the French mathematician who studied them in the 1930s and 1940s. The Lévy-Khintchine representation was developed independently by Lévy and Khintchine in the 1930s. Stable distributions go back further still, to Lévy (1925) and earlier work by Cauchy and others.

Finance came late. Madan and Seneta’s 1990 paper on the VG model was the turning point, and the 2002 CGMY paper nailed this line firmly into the derivatives pricing literature. Today every major bank runs some form of Lévy model in production.

How to Choose

Go back to the two ends of $\nu$ and the selection logic is short. For practical option pricing, start with Merton (simple, well-understood, closed form) and move to VG or CGMY when the fit is not good enough, since those are finer-grained at the zero end and can absorb high-frequency jaggedness. For tail-risk modeling, CGMY is the safe choice, and stable processes belong in stress-testing only, with full awareness that their variance does not exist. For exotics, simulation is often the only route, and the Lévy framework at least lets you switch models under one characteristic-function interface.

The cost is real: more parameters, non-Gaussian distributions, and counterintuitive regimes like infinite activity to reason about. What you buy is empirical fit. Whether that trade pays on your particular underlyings is a calculation you have to run yourself.

To read further, Cont and Tankov (2004), Financial Modelling with Jump Processes, chapters 2-4, is the standard practitioner reference; Sato (1999), Lévy Processes and Infinitely Divisible Distributions, is the rigorous mathematical treatment; and Kyprianou (2014), Fluctuations of Lévy Processes with Applications, sits between them. For the specific models above, Carr, Geman, Madan and Yor (2002) is the original CGMY paper, Carr & Wu (2003) handles the divergence problem in stable pricing, and Merton (1976) remains the best entry point to jump-diffusion.