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Modeling a Quanto Worst-of Equity Basket Payoff

Article Quant Q&A · Author: Mike
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Summary

The document considers a worst-of return payoff on the S&P 500 and Euro Stoxx 50, valued under a risk-neutral measure. The proposed setup calibrates a separate Heston stochastic-volatility model to each index’s benchmark option volatilities and models EUR/USD with a process such as CEV. It identifies five risk drivers: each index’s price and variance, plus the FX rate, represented through a joint correlation matrix and Cholesky factorization.

The answer says correlations between FX and variance processes may be assumed absent or estimated historically if suitable option data are available. For the payoff, it recommends simulating the EUR/USD rate to maturity and using it in the calculation before averaging simulated outcomes. The note does not give model equations, calibration details, or numerical results, and its maturity-only FX instruction depends on the payoff’s currency definition and the precise quanto convention.

Key ideas

  • The payoff is the lower of the two indices’ returns, valued under a risk-neutral measure.
  • The proposed simulation uses separate Heston models for the indices and a stochastic process for EUR/USD.
  • A joint correlation structure can link both prices, both variances, and the FX rate.
  • FX correlations with variance processes may be omitted or estimated from historical data.
  • The answer proposes simulating FX at maturity when calculating the payoff expectation.

Tags

OptionsEquitiesDerivatives pricingVolatility
Full text
# Quanto basket payoff


# Quanto basket payoff












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?

## Answer by Valometrics.com (score 1, accepted)

https://quant.stackexchange.com/a/50753

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.

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.