Calibrating the Heston Model to Determine Risk-Neutral Prices

Summary

The document addresses how to obtain risk-neutral Heston prices when the model is incomplete and real-world simulation alone does not determine a unique pricing measure. The response takes a practical calibration route: posit a risk-neutral variance process with mean-reverting square-root dynamics, then choose its parameters so model-generated option prices fit market quotes. This makes market option prices the input for specifying the risk-neutral model used for pricing.

It notes that Monte Carlo can generate Heston vanilla option prices, but recommends semi-analytical techniques for calibration because they are faster. Fourier methods based on the model’s characteristic function are offered as a direction to explore. The answer does not explain variance or volatility swaps, derive the market price of volatility risk, or establish a unique measure from the stated calibration. The result therefore gives a basic workflow, while leaving model specification, fitting choices, and incompleteness caveats largely untreated.

Key ideas

• Real-world Heston simulations do not by themselves specify risk-neutral option prices.
• The response proposes fitting a risk-neutral Heston variance process to observed option prices.
• Semi-analytical pricing methods can speed calibration compared with Monte Carlo.
• Fourier methods using the characteristic function are suggested for generating vanilla option prices.
• The response does not explain how variance or volatility swaps would determine the pricing measure.

Tags

# How to determine the risk-neutral measure in a Heston model?


# How to determine the risk-neutral measure in a Heston model?












To clarify, I'm quite familiar with the risk-neutral pricing framework, and I know one can efficiently Monte-Carlo a Heston model via the non-central $\chi^2$ distribution approach. But so far we're only playing with the real world probabilities, and we can never determine the risk-neutral measure because Heston model is incomplete. So even if we can Monte-Carlo the stock price paths under the real world probabilities, what then? We still cannot decide on the risk-neutral measure.

I also have read somewhere about using variance/vol swaps to make the market complete again, but haven't seen a good explanation (or at least the rough scheme/intuition etc) on how to use var/vol swaps to determine the risk-neutral measure.

Could anybody help? Thanks!

## Answer by user34971 (score 3)

https://quant.stackexchange.com/a/43886

First you assume that the Heston model under the risk neutral measure takes the following form:

$$ dv = \kappa (\theta - v) dt + \eta \sqrt{v} dW $$

Then you calibrate the model to the available options quoted in the market, i.e. find values for the Heston parameters $(\kappa, \theta, \eta)$ such that the options prices generated by the Heston model gives a good fit to the market.

Generating vanilla options prices with the Heston model can be done by MC, but for calibration purposes (and for vanilla options in general) there are semi-analytical methods which will greatly speed up your calibration procedure. Google for example "Heston" + "Fourier" + "Characteristic function".

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.