In the theory of stochastic process

Stochastic process

In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...

es, a part of the mathematical theory of probability, the variance gamma process (VG), also known as Laplace motion, is a Lévy process

Lévy process

In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is any continuous-time stochastic process that starts at 0, admits càdlàg modification and has "stationary independent increments" — this phrase will be explained below...

 determined by a random time change. The process has finite moments

Moment (mathematics)

In mathematics, a moment is, loosely speaking, a quantitative measure of the shape of a set of points. The "second moment", for example, is widely used and measures the "width" of a set of points in one dimension or in higher dimensions measures the shape of a cloud of points as it could be fit by...

 distinguishing it from many Lévy processes. There is no diffusion

Diffusion

Molecular diffusion, often called simply diffusion, is the thermal motion of all particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size of the particles...

 component in the VG process and it is thus a pure jump process. The increments are independent and follow a Laplace distribution.

There are several representations of the VG process that relate it to other processes. It can for example be written as a Brownian motion

Wiener process

In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called standard Brownian motion, after Robert Brown...

  with drift subjected to a random time change which follows a gamma process  (equivalently one finds in literature the notation :


Since the VG process is of finite variation it can be written as the difference of two independent gamma processes :

where

Alternatively it can be approximated by a compound Poisson process

Compound Poisson process

A compound Poisson process is a continuous-time stochastic process with jumps. The jumps arrive randomly according to a Poisson process and the size of the jumps is also random, with a specified probability distribution...

 that leads to a representation with explicitly given (independent) jumps and their locations. This last characterization gives an understanding of the structure of the sample path with location and sizes of jumps.

On the early history of the variance-gamma process see Seneta (2000).

Option pricing

The VG process can be advantageous to use when pricing options since it allows for a wider modeling of skewness

Skewness

In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. The skewness value can be positive or negative, or even undefined...

 and kurtosis

Kurtosis

In probability theory and statistics, kurtosis is any measure of the "peakedness" of the probability distribution of a real-valued random variable...

 than the Brownian motion

Brownian motion

Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...

 does. As such the variance gamma model allows to consistently price options with different strikes and maturities using a single set of parameters. Madan and Seneta present a symmetric version of the variance gamma process. Madan, Carr and Chang extend the model to allow for an asymmetric form and present a formula to price European options under the variance gamma process.

Hirsa and Madan show how to price American options under variance gamma. Fiorani presents numerical solutions for European and American barrier options under variance gamma process. He also provides computer programming code to price vanilla and barrier European and American barrier options under variance gamma process.

Lemmens et al. construct bounds for arithmetic Asian option

Asian option

An Asian option is a special type of option contract. For Asian options the payoff is determined by the average underlying price over some pre-set period of time...

s for several Lévy models including the variance gamma model.

Applications to Credit Risk Modeling

The variance gamma process has been successfully applied in the modeling of credit risk

Credit risk

Credit risk is an investor's risk of loss arising from a borrower who does not make payments as promised. Such an event is called a default. Other terms for credit risk are default risk and counterparty risk....

 in structural models. The pure jump nature of the process and the possibility to control skewness and kurtosis of the distribution allow the model to price correctly the risk of default of securities having a short maturity, something that is generally not possible with structural models in which the underlying assets follow a Brownian motion. Fiorani, Luciano and Semeraro model credit default swap

Credit default swap

A credit default swap is similar to a traditional insurance policy, in as much as it obliges the seller of the CDS to compensate the buyer in the event of loan default...

s under variance gamma. In an extensive empirical test they show the overperformance of the pricing under variance gamma, compared to alternative models presented in literature.

Simulation

Monte Carlo methods for the variance gamma process are described by Fu (2000).
Algorithms are presented by Korn et al. (2010).

Simulating VG as Gamma time-changed Brownian Motion

• Input: VG parameters and time increments , where
• Initialization: Set X(0)=0.
• Loop: For i = 1 to N:

1. Generate independent gamma , and normal variates, independently of past random variates.
2. Return

Simulating VG as difference of Gammas

This approach is based on the difference of gamma representation , where are defined as above.

• Input: VG parameters ] and time increments , where
• Initialization: Set X(0)=0.
• Loop: For i = 1 to N:

1. Generate independent gamma variates independently of past random variates.
2. Return