Details

In the jump diffusion model, the stock price follows the random process , which comprises, in order, drift, diffusive, and jump components. The jumps occur according to a Poisson distribution and their size follows a log-normal distribution. The model is characterized by the diffusive volatility , the average jump size (expressed as a fraction of ), the frequency of jumps , and the volatility of jump size .

In this Demonstration, you can vary the model parameters, along with the time to option expiry and the (constant) interest rate.

Note that, in addition to varying with the model parameters and strike/spot price ratio, the implied volatility is generally larger than the model diffusive volatility ; this is essentially the extra volatility introduced by the jumps.

Snapshot 1: volatility smile in a jump diffusion model with downward jumps ()

Snapshot 2: volatility smile with symmetric jumps ()

Snapshot 3: for long-dated options (here, time to expiry is four years) the smile is less pronounced

References:

R. Merton, Continuous‐Time Finance, Oxford: Blackwell, 1998.

M. Joshi, The Concepts and Practice of Mathematical Finance, Cambridge: Cambridge University Press, 2003.