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

Quantitative Finance Asked by user279687 on December 15, 2020

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_tleft(left(left(S_{1,T}-k_1right)^{+}-E_tleft(left(S_{1,T}-k_1right)^{+}|mathbb{F}_tright)right)left(left(S_{2,T}-k_2right)^{+}-E_tleft(left(S_{2,T}-k_2right)^{+}|mathbb{F}_tright)right)|mathbb{F}_tright)
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_tleft(left(S_{1,T}-k_1right)^{+}left(S_{2,T}-k_2right)^{+}|mathbb{F}_tright)&=E_{t,tilde{Omega}}left(left(S_{1,T}-k_1right)left(S_{2,T}-k_2right)|mathbb{F}_tright)=\
&= 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}_tright)=\
&=E_{t,tilde{Omega}}left(S_{1,T}S_{2,T}|mathbb{F}_tright)-E_{t,tilde{Omega}}left(S_{2,T}|mathbb{F}_tright)k_1+\
&-E_{t,tilde{Omega}}left(S_{1,T}right)k_2+k_1k_2Pleft(tilde{Omega}right)
end{aligned}
end{equation}

begin{equation}
begin{aligned}
E_tleft(left(S_{1,T}-k_1right)^{+}|mathbb{F}right)E_tleft(left(S_{2,T}-k_2right)^{+}|mathbb{F}right)&=E_{t,tilde{Omega}}left(left(S_{1,T}-k_1right)|mathbb{F}right)E_{t,tilde{Omega}}left(left(S_{2,T}-k_2right)|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_2Pleft(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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