WOLFRAM|DEMONSTRATIONS PROJECT

Option Prices in Merton's Jump Diffusion Model

​
option specs
option type
call
put
time to expiry (years)
0.5
general parameters
interest rate(% per year)
5.
stock volatility​% per
1/2
year
​
25.
jump parameters
average jumpfrequency (per year)
1
average jumpsize (multiplier)
0.9
jump volatility (%)
25.
call option price
Row[Transpose[{{
jump diffusion
,
Black-Scholes
},{}}], ]
The jump diffusion model, introduced in 1976 by Robert Merton, is a model for stock price behavior that incorporates small day-to-day "diffusive" movements together with larger, randomly occurring "jumps". The inclusion of jumps allows for more realistic "crash" scenarios and means that the standard dynamic replication hedging approach of the standard Black-Scholes model no longer works. This causes option prices to increase compared to the Black-Scholes model and to depend on the risk aversion of investors. This Demonstration explores how the price of European call and put options varies with the jump diffusion model parameters.