Pricing Arithmetic Asian Options Under Heston with a Geometric Control Variate

Summary

The discussion distinguishes geometric-average and arithmetic-average Asian options under the Heston stochastic volatility model. It reports that a closed-form pricing formula is available for the geometric Asian option, while an analogous closed form is not available for the arithmetic Asian option. The same limitation is noted for the Black-Scholes model, so arithmetic Asian valuation generally requires numerical methods.

For arithmetic Asian options, the proposed approach is Monte Carlo simulation. Because geometric and arithmetic Asian option prices are described as similar and highly correlated, the geometric option’s analytically available price can serve as a control variate. This reduces simulation error for a given number of paths by using the known geometric value to improve the arithmetic estimate. The answer mentions that analytic geometric pricing and Monte Carlo arithmetic pricing with this variance-reduction technique are implemented in QuantLib. It gives no derivation, accuracy figures, or detailed assumptions, and the effectiveness of the control variate depends on the instrument setup and simulation design.

Key ideas

• Geometric Asian options have a closed-form price under Heston in the discussed setting.
• Arithmetic Asian options generally require numerical valuation rather than a closed-form formula.
• Monte Carlo simulation can estimate arithmetic Asian option prices.
• A geometric Asian option price can serve as a control variate to reduce Monte Carlo error.
• The discussion provides no derivation or quantitative accuracy comparison for the method.

Tags

# Approximate Asian option price under Heston Model


# Approximate Asian option price under Heston Model












I am looking to see if there is a formula or a derivation at least of an approximation of an Asian (Average Price) option under the heston model of stochastic volatility.

Please advise

## Answer by StackG (score 3)

https://quant.stackexchange.com/a/55940

There is a closed-form pricing formula for the Geometric Asian Option in the Heston model (the only non-paywalled link I can find shows the double-Heston price, Heston is a special case of double-Heston), but not for the Arithmetic Asian option (this is also the case in Black-Scholes).

For Arithmetic Asian prices, a numerical technique like Monte-Carlo will be required.

However, the prices of Geometric and Arithmetic Asian options are very similar and highly correlated, so that the geometric price can be used as a control variate in the Monte Carlo simulation, vastly increasing the accuracy of the calculation for a set number of paths.

Update 28-07-2021:

For anyone looking for implementations, analytic pricing engines for geometric asian options under Heston are available in QuantLib, and the MC pricer for arithmetic asian options supports the use of the geometric asian option as a control variate

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.