# 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 $$\left(\Omega,\mathbb{F},P,\mathbb{\bar{F}}\right)$$: $$$$E_t\left(\left(\left(S_{1,T}-k_1\right)^{+}-E_t\left(\left(S_{1,T}-k_1\right)^{+}|\mathbb{F}_t\right)\right)\left(\left(S_{2,T}-k_2\right)^{+}-E_t\left(\left(S_{2,T}-k_2\right)^{+}|\mathbb{F}_t\right)\right)|\mathbb{F}_t\right)$$$$
Is the correlation continuous with respect to $$k_1$$ and $$k_2$$? Consider each component of the covariance and let $$\tilde{\Omega}$$ be the space of events that make the payoff of both call options positive: \begin{aligned} E_t\left(\left(S_{1,T}-k_1\right)^{+}\left(S_{2,T}-k_2\right)^{+}|\mathbb{F}_t\right)&=E_{t,\tilde{\Omega}}\left(\left(S_{1,T}-k_1\right)\left(S_{2,T}-k_2\right)|\mathbb{F}_t\right)=\\ &= E_{t,\tilde{\Omega}}\left(S_{1,T}S_{2,T}-k_1S_{2,T}-S_{1,T}k_2+k_1k_2|\mathbb{F}_t\right)=\\ &=E_{t,\tilde{\Omega}}\left(S_{1,T}S_{2,T}|\mathbb{F}_t\right)-E_{t,\tilde{\Omega}}\left(S_{2,T}|\mathbb{F}_t\right)k_1+\\ &-E_{t,\tilde{\Omega}}\left(S_{1,T}\right)k_2+k_1k_2P\left(\tilde{\Omega}\right) \end{aligned} \begin{aligned} E_t\left(\left(S_{1,T}-k_1\right)^{+}|\mathbb{F}\right)E_t\left(\left(S_{2,T}-k_2\right)^{+}|\mathbb{F}\right)&=E_{t,\tilde{\Omega}}\left(\left(S_{1,T}-k_1\right)|\mathbb{F}\right)E_{t,\tilde{\Omega}}\left(\left(S_{2,T}-k_2\right)|\mathbb{F}\right)\\ &=E_{t,\tilde{\Omega}}\left(\left(S_{1,T}\right)|\mathbb{F}\right)E_{t,\tilde{\Omega}}\left(\left(S_{2,T}\right)|\mathbb{F}\right)-E_{t,\tilde{\Omega}}\left(\left(S_{1,T}\right)|\mathbb{F}\right)k_2-E_{t,\tilde{\Omega}}\left(\left(S_{2,T}\right)|\mathbb{F}\right)k_1+k_1k_2P\left(\tilde{\Omega}\right) \end{aligned} Therefore I'd conclude that the correlation is continuous in both $$k_1$$ and $$k_2$$. Is this correct?