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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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  • $\begingroup$ as a side note, for the oter correlations you're missing, you can use the approximate formulae in this paper to fill it in. $\endgroup$ – will Jan 19 at 21:04
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You should have a correlation matrix with the following 5 parameters:
1. SPX price.
2. SPX variance.
3. SX5E price.
4. SX5E variance.
5. FX rate.
You can even consider that FX rate is not correlated to variances or if you have enough option prices for different dates compute the historical correlation between fx rates and calibrated variances.
Regarding the second question, you only have to simulate the EURUSD rate at maturity in order to compute the payoff then calculate the mean.

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