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Adding Jumps to the Heston Stochastic Volatility Model

Article Quant Q&A · Author: nimbus3000
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Summary

The discussion outlines ways to extend Heston’s stochastic volatility framework with jumps in the asset price. It points to the broader affine class of models as a source of jump specifications and describes a jump-diffusion setup in which arrivals follow a Poisson process with an arrival rate independent of the stock price.

Jump sizes may be fixed or drawn from a distribution; the Merton model is cited as an example using lognormally distributed jump sizes. The material is introductory and mostly directs readers to references rather than deriving option pricing formulas or showing how a jump specification changes the volatility surface. It also cautions that random jump sizes prevent construction of a replicating portfolio, so the usual self-financing hedge is unavailable.

Key ideas

  • Heston belongs to the affine model class, which supports multiple choices for specifying jumps.
  • A Poisson process can represent jump arrivals at a rate assumed independent of the stock price.
  • Jump sizes can be fixed or sampled from a distribution, such as the lognormal jump sizes in Merton’s model.
  • Random jump sizes remove the replicating portfolio available under simpler assumptions.

Tags

OptionsVolatilityDerivatives pricingStatistics
Full text
# Heston model with Jumps


# Heston model with Jumps












Heston model can be used to find prices of options under stochastic volatility. How do I include jumps in the model, so that I end up with a different stochastic volatility curve? References to literature would be helpful.

## Answer by Allan Jonathan (score 1)

https://quant.stackexchange.com/a/16658

Yes!

Try this and this.

But if you don't know the black-scholes basics well consider to read the book "Paul Wilmott in Quantitative Finance" before to go to Stochastic Volatility models and models with jumps.

## Answer by Gabriele Pompa (score 1)

https://quant.stackexchange.com/a/16664

don't know If I understand well your question, but If you want to have a rather complete perspective about the affine class of models (to which Heston's model belongs), you better study Duffie et al. (2000). In this very important contribution you'll find many examples of jump specifications

## Answer by Will Gu (score 0)

https://quant.stackexchange.com/a/31533

You can start with Wilmott:

$dS_t = \mu S_t dt +\sigma S_t dZ_t + (J-1)S_t dq_t$

where the Poisson Process

\begin{equation} dq_t =\begin{cases} 0, & \text{with prob $1-\lambda(t)dt$}\\ 1, & \text{with prob $\lambda(t)dt$} \end{cases} \end{equation} Also assume that jump arrival rate $\lambda(t)$ is independent of stock price

Assuming a known jump size $J$ is simple but somewhat unrealistic, one can assume some distribution of jump size. For example, Merton's jump diffusion models assumes $J$ to be lognormally distributed.

Note however, that in the case of random jump size, there's no replicating portfolio so there is no self-financing hedge.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.