How Correlation May Affect Basket Option Values

Summary

The document models two correlated stocks with geometric Brownian motions and asks how their correlation affects two call payoffs: one on the sum of the stocks and one on their difference. It proposes an intuitive direction: higher correlation may raise the value of the option on the sum while lowering the value of the option on the difference. The question seeks a rigorous proof or a reference for one.

The text supplies a useful setup for studying correlation sensitivity in multi-asset derivatives, but it does not give a proof, pricing formula, numerical example, or supporting reference. Its intuition is a conjecture rather than a demonstrated result. Any rigorous analysis would need to specify pricing assumptions such as the valuation measure, maturity, and relevant asset and strike conditions, then examine how the joint distribution affects each payoff. The document is therefore a focused mathematical question, not a complete derivation or general pricing rule.

Key ideas

• The setup considers two correlated stock prices following geometric Brownian motions.
• One option pays on the positive part of the stocks’ sum minus a strike.
• A second option pays on the positive part of their difference minus a strike.
• The author conjectures that higher correlation raises the first option’s value and lowers the second’s.
• No proof or pricing evidence is provided in the document.

Tags

# Correlation Sensitivity


# Correlation Sensitivity












Suppose I have 2 stocks $S_{1}$ and $S_{2}$: \begin{align} & dS_{1}=rS_{1}dt+\sigma_{1}S_{1}dB_{1}\\ & dS_{2}=rS_{2}dt+\sigma_{2}S_{2}dB_{2}\\ & dB_{1}dB_{2}=\rho dt \end{align} Then I have a option A with payoff $(S_{1}+S_{2}-K)^{+}$ and another option B with payoff $(S_{1}-S_{2}-K)^{+}$

Question: I want to know the rigorous proof of the relationship between option A/B and correlation $\rho$? Or you may tell me where I can find the proof?

Intuitively: when $\rho$ is increasing, will move aligned with each other, then we can think that $S_{1}+S_{2}$ will become larger, and $S_{1}+S_{2}$ will become smaller, the option A price will be bigger, Option B price will be smaller. So we think price of A is a increasing function of $\rho$ and price of $B$ is a decreasing function of $\rho$.

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