Covariance of Calls on Correlated Assets in Black–Scholes
Summary
The discussion considers two vanilla calls written on different, correlated assets under a Black–Scholes framework. For claims driven by correlated geometric Brownian motions, the instantaneous option returns inherit the underlying Brownian correlation, scaled by each option’s local sensitivity to its underlying. The resulting option volatilities, and therefore their instantaneous correlation exposure, depend on the current state. The argument assumes constant underlying volatilities and simplifies by setting interest rates to zero; it notes that stochastic volatility adds terms from volatility movements and their correlations.
A second answer distinguishes this instantaneous covariation from the covariance of option values over a finite horizon. That quantity requires an expectation of the product of the two option values under the joint distribution of the assets, alongside their individual expectations. The answer says the resulting integral is not readily available in semi-closed form and points toward approximations, including a traffic-light option valuation approach. Thus the local relationship is analytically tractable under the stated model, while finite-horizon portfolio covariance is more involved.
Key ideas
- Under correlated geometric Brownian motion, instantaneous option covariation depends on each option’s local sensitivity and volatility.
- The instantaneous dependence between option returns is state-dependent even when the underlying Brownian correlation is constant.
- Stochastic volatility introduces additional covariance contributions through volatility movements and their correlations.
- Finite-horizon covariance requires integrating the joint distribution of both option values and is not readily available in semi-closed form.
- Approximation methods can be used for finite-horizon option covariance, but the cited approach remains involved.
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Full text
# Covariance of a simple option portfolio
# 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?
## Answer by user34971 (score 3)
https://quant.stackexchange.com/a/58976
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.
## Answer by Kermittfrog (score 1)
https://quant.stackexchange.com/a/58995
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.
- 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?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.