I have a payoff that is the worst of the returns two indices: S&P500 (SPX) and Euro Stoxx 50 (SX5E).

$$\pi = \min \left\{\left(\frac{\text{SPX}_\tau-\text{SPX}_0}{\text{SPX}_0}\right),\left(\frac{\text{SX5E}_\tau-\text{SX5E}}{\text{SX5E}_0}\right)\right\}$$

To compute $$\text{E}^{\mathbb Q}\left(\pi|\mathscr F\right)$$, I must include the following correlation pairs:

• SPX price to SX5E price
• SPX variance to SPX price
• SX5E variance to SX5E price
• SX5E price to EURUSD FX

I calibrate the Heston parameters for each index independently to benchmark option vols.

I will calibrate a simple model for the FX process (say CEV).

I will compute a 5x5 correlation matrix for Cholesky.

Question 1 : Can I use 5 correlated processes and univarate Heston + CEV? What about correlation between variance processes to FX?

• $$Z_1$$ for SPX variance process
• $$Z_2$$ for SPX price process
• $$Z_3$$ for SX5E variance process
• $$Z_4$$ for SX5E price process
• $$Z_5$$ for EURUSD FX evolution

Question 2 : Where and how do I apply the Quanto effect? Do I convert the EUR indices to USD at each step?

• as a side note, for the oter correlations you're missing, you can use the approximate formulae in this paper to fill it in. – will Jan 19 at 21:04