Pricing Heston Variance Options with Characteristic Functions

Summary

The document asks whether options on realized variance or volatility under the Heston model require Monte Carlo simulation or finite-difference methods, and whether the realized-variance distribution has an analytical density. The response points to the model’s closed-form characteristic function as a useful pricing tool, while clarifying that obtaining option prices still requires numerical integration. It also references work on characteristic-function methods and a paper addressing realized-variance options in a Heston setting with jumps in returns and volatility.

The cited approach is described as semi-analytical: pricing can be reduced to numerical integration rather than relying exclusively on simulation or a grid-based solver. The discussion does not derive a probability density for realized variance, give implementation details, or compare accuracy and computational cost across methods. Its guidance is therefore a starting point for further study, and the cited jump extension may involve assumptions beyond the basic Heston model.

Key ideas

• The Heston model has a closed-form characteristic function.
• Option prices based on that function still require numerical integration.
• Semi-analytical methods are presented as an alternative to relying only on simulation or finite differences.
• A cited study treats realized-variance options with jumps in returns and volatility.
• The discussion does not derive the realized-variance density or compare numerical methods.

Tags

# Options on realized volatility / variance


# Options on realized volatility / variance












If I'd like to price options on variance/volatility in the Heston model. Is MC simulation and/or finite difference the only way to do it? Or is there an analytical expression for the probability density function of realized variance in the Heston model?

Thanks!

## Answer by Ezy (score 2, accepted)

https://quant.stackexchange.com/a/42604

The characteristic function of the Heston model is known in closed form. To obtain the option prices you have to perform numerical integration though.

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.139.3204&rep=rep1&type=pdf

you can also check the already asked (and answered) question

Problem on Characteristic function in Heston model

which refers to this paper

Duffie, Pan, and Singleton (2000)

Now regarding your original question: here is a reference for options on variance which seems reasonable to me

Pricing Options on Realized Variance in Heston Model with Jumps in Returns and Volatility

they price a number of vol derivatives semi-analytically where only a numerical integration is necessary. This should help you

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