Jump process
Stochastic process with discrete movements
A jump process is a type of stochastic process that has discrete movements, called jumps, with random arrival times, rather than continuous movement, typically modelled as a simple or compound Poisson process.[1]
In finance, various stochastic models are used to model the price movements of financial instruments; for example the Black–Scholes model for pricing options assumes that the underlying instrument follows a traditional diffusion process, with continuous, random movements at all scales, no matter how small. John Carrington Cox and Stephen Ross[2]: 145–166 proposed that prices actually follow a 'jump process'.
Robert C. Merton extended this approach to a hybrid model known as jump diffusion, which states that the prices have large jumps interspersed with small continuous movements.[3]
See also
- Poisson process, an example of a jump process
- Continuous-time Markov chain (CTMC), an example of a jump process and a generalization of the Poisson process
- Counting process, an example of a jump process and a generalization of the Poisson process in a different direction than that of CTMCs
- Interacting particle system, an example of a jump process
- Kolmogorov equations (continuous-time Markov chains)
References
- ^ Tankov, P. (2003). Financial modelling with jump processes (Vol. 2). CRC press.
- ^ Cox, J. C.; Ross, S. A. (1976). "The valuation of options for alternative stochastic processes". Journal of Financial Economics. 3 (1–2): 145–166. CiteSeerX 10.1.1.540.5486. doi:10.1016/0304-405X(76)90023-4.
- ^ Merton, R. C. (1976). "Option pricing when underlying stock returns are discontinuous". Journal of Financial Economics. 3 (1–2): 125–144. CiteSeerX 10.1.1.588.7328. doi:10.1016/0304-405X(76)90022-2. hdl:1721.1/1899.
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