# Correlation between assets used for valuing multi-asset options (Rainbow options, basket options etc.)

Is there an equivalent to implied volatility used when it comes to modelling correlation in option valuation for multi asset options such as rainbow options (best-of/worst-of calls/put), or is the best you can do to model the real world sample correlation of the time-series of the assets (or something other that I'm not considering)? Is there an industry standard to this?

Below is a sketch of how one can model multiple underlyings that are correlated together, in order to then price some options on them (i.e. spread options). I welcome any remarks that might improve this sketch (either feel free to comment or edit my post).

Suppose you have a multifactor model, with two assets, $$X_1$$ and $$X_2$$. Suppose the two assets follow GBM, where $$W_1$$ and $$W_1$$ are independent, as follows:

$$dX_1(t)=X_1(t)\mu_1 dt+X_1(t)\sigma_1 \lambda_{1,1} dW_1(t)+X_1(t)\sigma_1 \lambda_{1,2} dW_2(t)$$

$$dX_2(t)=X_2(t)\mu_2 dt+X_2(t)\sigma_2 \lambda_{2,1} dW_2(t)+X_2(t)\sigma_2 \lambda_{2,2} dW_2(t)$$

Above, $$\lambda_{i,j}$$ is the loading onto each Brownian motion (i.e. the source of risk). By Ito's lemma applied to $$ln(X_1)$$ and $$ln(X_2)$$, we then have:

$$dln(X_1)=\left(\mu_1-0.5\sigma_1^2(\lambda_{1,1}^2+\lambda_{2,1}^2)\right)dt+\sigma_1 \lambda_{1,1} dW_1(t)+\sigma_1 \lambda_{1,2} dW_2(t)$$

And

$$dln(X_2)=\left(\mu_2-0.5\sigma_2^2(\lambda_{2,1}^2+\lambda_{2,2}^2)\right)dt+\sigma_2 \lambda_{2,1} dW_2(t)+\sigma_2 \lambda_{2,2} dW_2(t)$$

The two assets would then be correlated as follows:

$$Cov(X_1,X_2)=\mathbb{E}\left[X_1X_2\right]-\mathbb{E}\left[X_1\right]\mathbb{E}\left[X_2\right]=f\left(\lambda_{1,1},\lambda_{1,2},\lambda_{2,1},\lambda_{2,2}\right)$$

I.e. the covariance is a function of the $$\lambda$$ parameters (with the $$\sigma$$ being calibrated separately to liquid option prices).

The correlation is then simply $$Corr=\frac{Cov(X_1,X_2}{Var(X_1)^{0.5}Var(X_2)^{0.5}}$$ (again, the Corr being a function of the $$\lambda$$ parameters).

So when valuing a multi-asset option, each underlying will typically have it's own volatility $$\sigma_i$$, which doesn't need to be constant: can be a deterministic function of $$t$$ or even a stochastic vol. The tricky part is then to use liquid options on the individual underlyings to calibrate each of the $$\sigma_i$$ so that the model above prices liquid options correctly.

The correlation can be even trickier: either a historical correlation can be computed and one can attempt to chose the parameters $$\lambda_{i,j}$$ so that the historical corr is equal the Corr quantity in the model above. Or (I have never seen this done, but apparently it's possible) one can try to extract the implied corr from liquid spread options.

Once the model is properly calibrated, it is then no issue to run a Monte-Carlo to price some payoff along the lines of $$\mathbb{E}^{\mathbb{Q}}\left[\left(X_1(t)-X_2(t)\right)^+\right]$$.