Jump-diffusion models
Encyclopedia
Jump diffusion is a stochastic process
that involves jumps and diffusion
. It has important applications in condensed matter physics
and in option pricing.
typically consists of jumps between vacant lattice sites. On time and length scales that average over many single jumps, the net motion of the jumping atoms can be described as regular diffusion
.
Jump diffusion can be studied on a microscopic scale by inelastic neutron scattering
and by Mößbauer spectroscopy
. Closed expressions for the autocorrelation function have been derived for several jump(-diffusion) models:
, mixing a jump process
and a diffusion process
. Jump-diffusion models have been introduced by Robert C. Merton
as an extension of jump models. Due to their computational tractability, the special case of a basic affine jump diffusion is popular for some credit risk
and short-rate models.
Stochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...
that involves jumps and 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...
. It has important applications in condensed matter physics
Condensed matter physics
Condensed matter physics deals with the physical properties of condensed phases of matter. These properties appear when a number of atoms at the supramolecular and macromolecular scale interact strongly and adhere to each other or are otherwise highly concentrated in a system. The most familiar...
and in option pricing.
In physics
In crystals, atomic diffusionAtomic diffusion
Atomic diffusion is a diffusion process whereby the random thermally-activated movement of atoms in a solid results in the net transport of atoms. For example, helium atoms inside a balloon can diffuse through the wall of the balloon and escape, resulting in the balloon slowly deflating. Other air...
typically consists of jumps between vacant lattice sites. On time and length scales that average over many single jumps, the net motion of the jumping atoms can be described as regular 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...
.
Jump diffusion can be studied on a microscopic scale by inelastic neutron scattering
Inelastic neutron scattering
Inelastic neutron scattering is an experimental technique commonly used in condensed matter research to study atomic and molecular motion as well as magnetic and crystal field excitations....
and by Mößbauer spectroscopy
Mößbauer spectroscopy
Mössbauer spectroscopy is a spectroscopic technique based on the Mössbauer effect. This effect, discovered by Rudolf Mössbauer in 1957, consists in the recoil-free, resonant absorption and emission of gamma rays in solids....
. Closed expressions for the autocorrelation function have been derived for several jump(-diffusion) models:
- Singwi, Sjölander 1960: alternation between oscillatory motion and directed motion
- Chudley, Elliot 1961: jumps on a lattice
- Sears 1966, 1967: jump diffusion of rotational degrees of freedom
- Hall, Ross 1981: jump diffusion within a restricted volume
In economics and finance
In option pricing, a jump-diffusion model is a form of mixture modelMixture model
In statistics, a mixture model is a probabilistic model for representing the presence of sub-populations within an overall population, without requiring that an observed data-set should identify the sub-population to which an individual observation belongs...
, mixing a jump process
Jump process
A jump process is a type of stochastic process that has discrete movements, called jumps, rather than small continuous movements.In physics, jump processes result in diffusion...
and a diffusion process
Diffusion process
In probability theory, a branch of mathematics, a diffusion process is a solution to a stochastic differential equation. It is a continuous-time Markov process with continuous sample paths....
. Jump-diffusion models have been introduced by Robert C. Merton
Robert C. Merton
Robert Carhart Merton is an American economist, Nobel laureate in Economics, and professor at the MIT Sloan School of Management.-Biography:...
as an extension of jump models. Due to their computational tractability, the special case of a basic affine jump diffusion is popular for some 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....
and short-rate models.