Jump – Diffusion Model
By - Madhuresh Rai, Jan 26th 2010.
The famous Black-Scholes model assumes a continuous change in the price of an asset. The prices change continuously to produce a lognormal distribution at any time in future. We know that there is enough evidence to suggest that the well-loved Geometric Brownian motion model for stock price behavior does not work well in practice. In particular, the financial instruments do not follow a lognormal random walk and the jumps can appear at any random time.
A model which has continuous price changes is known as Diffusion Model. When a diffusive model is overlaid with jumps, it becomes a Jump Diffusion Model (J-D from Scrubs!).
Merton’s Mixed J-D Model does the same combining jumps with continuous changes. It incorporates small day-to-day diffusive movements together with larger randomly occurring jumps. Such an inclusion of jumps allows for more realistic crash scenarios such as what happened on 19th & 20th Oct. 1987. In the J-D model, the stock price St follows the random process –
dSt / St = m dt + s dWt + (J-1) dN(t)
where, m is the drift rate, s is the volatility, dWt gives the random walk (Wiener Process Wt), and ‘J-1’ is a Jump function which produces a jump from S to SJ. E[J-1] gives the expected relative jump size.
Last term represents the jumps. J is the jump size; a multiple of the stock price while N(t) is the number of jump events that have occurred upto time t. N(t) is assumed to follow Poisson Process,
P [N(t) = k] = (λ t)k e– λt / k!
Where,
λ : average number of jumps per unit time.
k : average jump size measured as a % of asset price. It is assumed to be drawn from a probability distribution in the model
The probability of a jump in time ∆t = λ ∆t. Therefore, average growth rate in asset price from the jump = λ k. Using this, the differential equation above can be simply rewritten as:
dS / S = (rfr - λ k) dt + s dz + dp
The jump size may follow any distribution, but a common choice is a log-normal distribution. It should be noted that the arrival of a jump is random and through the J-D model, it becomes a part of the stochastic differential equation for S. The uncertainty is now produced by 2 sources:
· the already known s dz from the Brownian motion.
· (J-1)dN(t) corresponds to the exceptional and infrequent events
As we can see, when there are no jumps, the J-D model reduces to the Black Scholes Model, in which returns follow the Normal Distribution. A limitation of the J-D model assumes constant volatility in the diffusive component, which results in unrealistic-looking behavior when the jumps are large. Since in reality, s tends to increase dramatically around the time the large jumps occur!
References:
D. J. Duffy, Num. Analysis of Jump Diffusion Models using PDE, DataSim.
Wolfram Demonstration Projects, Distribution of Returns.
Wolfram Demonstration Projects, Option Prices in Merton's J-D Model.
J. C. Hull, Options, Futures, & Other Derivatives, Ed. 6th, Chap. 24.