Jump-diffusion model

In jump diffusion model is a combination of events occurring continuously and rare events leading to jumps.

The continuous component of the change in the asset price is a Wiener process. The jump component is a Poisson-driven process.

The Brownian motion-based process is written as-

Since the process is continuous it does not allow and discontinuities i.e. rare events or news as such Poisson jump process is used-

Where,

Y_j – 1 =jump size

Nt = Jumps in the interval (0,t) given by parameter λ * t

The probability of a jump that occurs during a time interval of length   is written as-

P (the event occurs in given time interval (t, t+h) = λ * h

P (the event does not occurs in given time interval (t, t+h) = 1 - λ * h

If the rare event occurs in the time interval (t, t +h) causing a jump, the discontinuous change in the asset price will be S_t+h = S_t (Y – 1). The random variable Y-1, also called the jump size.

Since, the process is combination of two random processes. The two sources of randomness are independent of each other. Because of this adjustment is to be made in continuous random process, the formula can be written as-

where 

The expected change in S_t from the jump component dQ_t over the time interval dt is λ * µ₀ *dt . Therefore, if α_t denotes the total expected return (rate of change) on S_t , λ * µ₀ *dt  needs to be subtracted from the drift term of S_t: 

Merton's jump-diffusion Model the stochastic differential equation: