Choosing Control Variates for Best-of-Assets Option Pricing

Summary

The note considers variance reduction when Monte Carlo simulation is used to price a best-of-assets payoff: the maximum of several asset forwards at expiry. A useful control variate should move closely with that maximum while having an expectation that can be calculated analytically. The response recommends choosing an average suited to the assumed distribution of the forwards.

For lognormal forwards, it suggests their geometric average, which is also lognormal and has an easily computed expectation. For normally distributed forwards, it suggests their arithmetic average, which remains Gaussian and likewise has a known expectation. These are candidate controls rather than a complete implementation: the note does not compare their correlations with the payoff, quantify variance reduction, or address calibration and simulation details. Their usefulness therefore depends on the joint distribution and how closely the selected average tracks the maximum in the pricing setup.

Key ideas

• A control variate can reduce Monte Carlo variance when it is correlated with the payoff and has a known expectation.
• For lognormal forwards, the geometric average is a suggested control because it is also lognormal.
• For normally distributed forwards, the arithmetic average is suggested because it remains Gaussian.
• The note gives candidate controls but no empirical comparison of their effectiveness.

Tags

# Control variate for pricing a best of assets option : $\mathop{{}\mathbb{E}}[ \max ( F^1_T,F^2_T, ...,F^N_T )]$


# Control variate for pricing a best of assets option : $\mathop{{}\mathbb{E}}[ \max ( F^1_T,F^2_T, ...,F^N_T )]$












I want to use Monte Carlo to price a best of assets derivative :

$$\mathop{{}\mathbb{E}}[ \max ( F^1_T,F^2_T, ...,F^N_T )]$$

where the $F^i_T$ is the forward of the ith asset observed at expiry time $T$ of the option.

What would be a good control variate to use for variance reduction?

I know that I have to look for a function (not necessairly a traded instrument) involving the underlyings that is :

- highly correletated with the payoff above

- has a known expectation

However i don't have enough experience with choosing control variates. Any suggestions, ideas?

thank you

## Answer by byouness (score 1)

https://quant.stackexchange.com/a/46587

If the $(F^i_T)_i$ are lognormal, I'd choose their geometric average $\left(\prod_{i=1}^N F^i_T\right)^{\frac{1}{N}}$ because it's lognormal as well and hence the expectation is easy to compute.

If they are normal, I'd choose the arithmetic average $\frac{1}{N}\sum_{i=1}^N F^i_T$, since it's gaussian as well.

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