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Strike-Dependent Continuity of Correlation Between Two Call Payoffs

Article Quant Q&A · Author: user279687
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Summary

The document asks whether the correlation between two call-option payoffs on dependent underlyings varies continuously with their respective strike prices. It expands the covariance into conditional expectations, considering the joint event on which both calls finish in the money, and observes that the payoff products are algebraic expressions in the strikes on that event. This motivates the proposed continuity conclusion.

The derivation alone does not establish continuity of the correlation. The in-the-money event itself changes with the strikes, and continuity of the relevant expectations requires suitable integrability or distributional conditions. Correlation also divides covariance by both payoff standard deviations, so it may fail to be defined where either variance is zero and can behave poorly near such points. The document provides no formal proof or assumptions addressing these issues, making its conclusion a conjecture rather than a general result.

Key ideas

  • The question concerns correlation between two call payoffs as both strikes change.
  • The proposed argument expands the joint payoff product over the event where both calls finish in the money.
  • Strike-dependent events and conditional expectations require assumptions before continuity follows.
  • Correlation is undefined when either payoff has zero variance.

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# Continuity of a portfolio with two options with respect to the strikes


# Continuity of a portfolio with two options with respect to the strikes












Consider the covariance, evaluated at time $t$, between two call options written on two different but not independent underlyings $S_1$ and $S_2$ defined on the same (filtered) measure space $\left(\Omega,\mathbb{F},P,\mathbb{\bar{F}}\right)$: \begin{equation} E_t\left(\left(\left(S_{1,T}-k_1\right)^{+}-E_t\left(\left(S_{1,T}-k_1\right)^{+}|\mathbb{F}_t\right)\right)\left(\left(S_{2,T}-k_2\right)^{+}-E_t\left(\left(S_{2,T}-k_2\right)^{+}|\mathbb{F}_t\right)\right)|\mathbb{F}_t\right) \end{equation} Is the correlation continuous with respect to $k_1$ and $k_2$? Consider each component of the covariance and let $\tilde{\Omega}$ be the space of events that make the payoff of both call options positive: \begin{equation} \begin{aligned} E_t\left(\left(S_{1,T}-k_1\right)^{+}\left(S_{2,T}-k_2\right)^{+}|\mathbb{F}_t\right)&=E_{t,\tilde{\Omega}}\left(\left(S_{1,T}-k_1\right)\left(S_{2,T}-k_2\right)|\mathbb{F}_t\right)=\\ &= E_{t,\tilde{\Omega}}\left(S_{1,T}S_{2,T}-k_1S_{2,T}-S_{1,T}k_2+k_1k_2|\mathbb{F}_t\right)=\\ &=E_{t,\tilde{\Omega}}\left(S_{1,T}S_{2,T}|\mathbb{F}_t\right)-E_{t,\tilde{\Omega}}\left(S_{2,T}|\mathbb{F}_t\right)k_1+\\ &-E_{t,\tilde{\Omega}}\left(S_{1,T}\right)k_2+k_1k_2P\left(\tilde{\Omega}\right) \end{aligned} \end{equation} \begin{equation} \begin{aligned} E_t\left(\left(S_{1,T}-k_1\right)^{+}|\mathbb{F}\right)E_t\left(\left(S_{2,T}-k_2\right)^{+}|\mathbb{F}\right)&=E_{t,\tilde{\Omega}}\left(\left(S_{1,T}-k_1\right)|\mathbb{F}\right)E_{t,\tilde{\Omega}}\left(\left(S_{2,T}-k_2\right)|\mathbb{F}\right)\\ &=E_{t,\tilde{\Omega}}\left(\left(S_{1,T}\right)|\mathbb{F}\right)E_{t,\tilde{\Omega}}\left(\left(S_{2,T}\right)|\mathbb{F}\right)-E_{t,\tilde{\Omega}}\left(\left(S_{1,T}\right)|\mathbb{F}\right)k_2-E_{t,\tilde{\Omega}}\left(\left(S_{2,T}\right)|\mathbb{F}\right)k_1+k_1k_2P\left(\tilde{\Omega}\right) \end{aligned} \end{equation} Therefore I'd conclude that the correlation is continuous in both $k_1$ and $k_2$. Is this correct?

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