Jump process explained

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] 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)

Notes and References

1. Tankov, P. (2003). Financial modelling with jump processes (Vol. 2). CRC press.
2. Cox . J. C. . John Carrington Cox. Ross . S. A. . Stephen Ross (economist). 10.1016/0304-405X(76)90023-4 . The valuation of options for alternative stochastic processes . Journal of Financial Economics. 3 . 1–2 . 145–166 . 1976 . 10.1.1.540.5486 .
3. Merton . R. C. . Robert C. Merton. 10.1016/0304-405X(76)90022-2 . Option pricing when underlying stock returns are discontinuous . Journal of Financial Economics. 3 . 1–2 . 125–144 . 1976 . 1721.1/1899. 10.1.1.588.7328 .