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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?

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