Why Weighted Local Volatility Does Not Directly Price Basket Options

Summary

The document considers whether a two-asset European basket option can be reduced to a single underlying option by combining the assets’ local volatilities with their weights and a constant correlation. The response says the proposed expression is a variance relationship, so the quantities being combined should be variances. Even then, that relationship does not make the weighted basket behave like a single lognormal asset: a weighted sum of lognormal prices is generally not lognormal, including under the Black–Scholes special case.

As an alternative, the response proposes simulating the two asset price processes with dependence imposed through a Gaussian copula, pricing vanilla options on the resulting basket, and extracting an implied volatility surface for the basket. That surface can then be used to obtain basket local volatility. The answer is brief and does not specify calibration, numerical implementation, or validation details; its central caveat is that a variance aggregation formula alone does not justify a single-asset Black–Scholes representation.

Key ideas

• The weighted combination in the proposed formula is a variance relationship, not a volatility relationship.
• A weighted sum of lognormal asset prices is generally not itself lognormal.
• A constant correlation does not make a basket equivalent to a single Black–Scholes underlying.
• Simulation can generate basket prices, from which an implied volatility surface can be derived.
• The response does not provide implementation or calibration details for the simulation approach.

Tags

# Basket option: volatility surface


# Basket option: volatility surface












I would like to calculate a basket European option with Black Scholes local volatility model.

I want to simplified the basket option into a single underlying European option. Should we get the local volatility surface with single underlying instruments as usual and get volatility by the following formula?

Assume the correlation $\rho$ is constant.

(2-equal-weighted underlying for demonstration purposes) $\sigma_{basket}^2(S^a_{t},S^b_{t},t) = (\frac{1}{2}\sigma_{a}(S^a_{t},t))^2 + 2\rho(\frac{1}{2}\sigma_{a}(S^a_{t},t))(\frac{1}{2}\sigma_{b}(S^b_{t},t))+ (\frac{1}{2}\sigma_{b}(S^b_{t},t))^2$

## Answer by Quantuple (score 1, accepted)

https://quant.stackexchange.com/a/58224

Your equation holds if you replace $\sigma$ with $\text{var}$. But if you use it as such, even in the BS case ($\sigma(S,t) = \sigma$) then you have a problem, because you are manipulating a lognormal volatility and the (weighted) sum of lognormals is not lognormal. Now nothing prevents you from simulating the future price processes of assets $a$ and $b$ tying them by a Gaussian copula with parameter $\rho$, pricing vanillas on the basket and deriving an IVS for the basket. Strip this one to get the basket's LV.

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