Jump – Diffusion Model

Jump – Diffusion Model 

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 19^th & 20^th Oct. 1987. In the J-D model, the stock price S_t follows the random process – 

        dS_t / S_t  =  m dt  + s dW_t + (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. 

        The 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 is that it 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