Pricing Options on the Worst-of-Two-Assets Basket

Summary

The document describes a proposed dimension-reduction approach for valuing options on the minimum of two equity prices. The author starts with implied-volatility surfaces for each asset, computes option prices on the minimum across strikes and maturities, and approximates the minimum’s distribution as lognormal by matching moments. Black–Scholes is then used both to price those options and to infer a volatility surface for the worst-of basket.

The central unresolved issue is which forward price to use when converting the calculated option prices back into implied volatilities: a forward inferred from the reduced distribution, or the minimum of the two individual forwards. The text poses this as a question rather than giving a resolution, and supplies no numerical example or evidence that the lognormal approximation works. In practice, the minimum’s distribution depends on the joint behavior of the assets, including their dependence structure, so matching moments and selecting a forward require careful consistency with the intended pricing model.

Key ideas

• The proposed method approximates the minimum of two asset prices with a lognormal distribution matched to moments.
• Option values across strikes and maturities can be converted into an implied-volatility surface using Black–Scholes.
• The document leaves unresolved whether to use the reduced distribution’s forward or the minimum of the individual forwards.
• The approximation requires attention to dependence between the two assets and may not capture the full distribution.

Tags

# Dimension reduction for worst of basket on $min(S_1, S_2)$


# Dimension reduction for worst of basket on $min(S_1, S_2)$












Suppose we want to price an exotic equity which is a function of $min(S_1, S_2)$. To do this, I'm trying to compute an implied volatility surface for $min(S_1, S_2)$ and then price the option using that surface. It's essentially a reduction in dimension. In particular, we have the implied vol surfaces for $S_1$ and $S_2$ and we need the implied volatility surface for $min(S_1, S_2)$.

Here's what I've tried to do: For each $S_1$ and $S_2$, we have a forward and an implied volatility space which can be used to price options. For each strike and tenor, I am computing an option price (put for low strike, call for higher strike) on the $min(S_1, S_2)$ by matching moments, i.e. assume that $min(S_1, S_2)$ is lognormal and then analytically compute the moments of $min(S_1, S_2)$, imply the GBM params and price the option using the B-S equation.

Now I have option prices for $min(S_1, S_2)$ for each strike and tenor. Finally, I can imply the implied volatility surface using B-S. My question is which forward should I use in implying back out the implied volatility? I think I should use the zero-strike implied option price using the aforementioned dimension reduction ; however, I'm also thinking it might be correct to use the actual basket forward, i.e. literally compute the $min(F_1, F_2)$ at each tenor. Please let me know your thoughts. Really struggling with this.

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