# Covariance of a simple option portfolio

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?

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.

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?