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Suppose that you have an option portfolio composed by two plain vanilla call options. Each option has, as underlying, a different share following a different Brownian stochastic process. The two shares are correlated. Does it exist an analytical formula for this portfolio covariance?

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Let's work under Black-Scholes, with two correlated GBMs: $$ dX = \sigma X dW, \quad dY = \nu Y dZ, \quad dWdZ =\rho dt $$ I've taken interest rate is zero for simplicity, does not influence the covariation anyway.

Suppose $F$ is a claim on $X$ and $G$ is a claim on $Y$. Both satisfy the BS PDE, hence $$ dF = \left(\frac{\sigma X}{F}\frac{\partial F}{\partial X}\right) F dW = \sigma_F F dW $$ and $$ dG = \left(\frac{\nu Y}{G}\frac{\partial G}{\partial Y}\right) G dZ = \nu_G G dZ $$

The instantaneous correlation is therefore $$ \frac{dF}{F} \frac{dG}{G} = \rho_{FG} dt = \rho\sigma_F \nu_G dt $$ The instantaneous correlation between the two options is as you can see state-dependent, but for any $t$ you can in principle calculate it.

Generalization to stochastic volatility is similar, but there will be additional terms due to correlation between and with the stochastic instantaneous volas.

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To add to @ilovevolatility 's answer, in brevity no.

The covariance of a portfolio consisting of two options $O_1$ and $O_2$ on assets $S_1$ and $S_2$ is

$$ Cov=\mathrm{E}_\mathbb{P}\left[\left(O_1(S^{(1)}_t,t)-\mathrm{E}\left[O_1(S^{(1)}_t,t)\right ]\right)\left(O_2(S^{(2)}_t,t)-\mathrm{E}\left[O_2(S^{(2)}_t,t)\right ]\right)\right] $$

Let's have a look at the very first term when factoring the expectation: \begin{align} \mathrm{E}_\mathbb{P}\left[O_1(S^{(1)}_t,t)O_2(S^{(2)}_t,t)\right]=&\int_x\int_yO_1(S^{(1)}_0e^x,t)O_2(S^{(2)}_0e^y,t)f(x,y;t)dxdy\\ =&\int_x\int_y\mathrm{E}_\mathbb{Q}\left(e^{-r(T-t)}\phi_1\left(x,K_1\right)|x\right)\mathrm{E}_\mathbb{Q}\left(e^{-r(T-t)}\phi_2\left(y,K_2\right)|y\right)f(x,y;t)dxdy \end{align}

AFAIK, this four-dimensional integral is not easily solved in (semi)closed form. The 'usual' approximations, though, can still be applied.

  1. Monte Carlo: Simulate asset paths (under $\mathbb{P}$) and price the options.
  2. Approximation: Use first order ("Delta-Normal") and or first-and-second-derivatives ("Delta-Gamma-Normal")
  3. Valuation of the expectation $\mathrm{E}\left[(S_1-K_1)^+(S_2-K_2)^+\right]$ via a traffic light option (still very involved...)

HTH?

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