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I would like to price equity quanto options with the Heston Local-Stochastic Volatility model (LSV) but I am having hard time understanding how to apply quanto adjustment in such complex setup.

When it comes to the pure Heston model, after the reading of the following paper: https://www.risk.net/derivatives/2171456/quanto-adjustments-presence-stochastic-volatility

I have an idea of how to apply the quanto drift adjustment:

  • calibrate the Heston Heston parameters to Vanillas
  • adjust the drift in the price process $S_t$ with the product of Equity/FX correlation, FX volatility and EQ volatility $(\rho_{S,FX}\sigma_S\sigma_{FX})$
  • adjust the drift in the volatility process $\nu_t$ by adjusting the long term mean $\theta$

However, in the LSV model there is local volatility entering the price process, which makes things even more complex and complicated. Unfortunately, I cannot find any resource about adjusting the LSV model for quuanto exotic options.

How one would approach the calibration of the LSV model for quanto options?

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  • $\begingroup$ Do you want to price only quanto vanilla or also quanto exotics? $\endgroup$
    – fwd_T
    Mar 29 at 20:12
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    $\begingroup$ Mainly quanto exotics. I want to retrieve the (quanto) vanillas under the LSV model to make sure that the model is properly calibrated. $\endgroup$
    – justLeito
    Mar 30 at 12:17

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You will need two LSV models. The first LSV model is calibrated to vanilla and potential exotics equity derivatives. With this you will be able to price any kind of vanilla and exotic equity derivatives under the domestic measure. But for quanto vanilla or exotic options you need the ability to price in a so-called foreign measure. This is why you will need a second LSV model modelling the FX rate. This second LSV model also has to be calibrated to the vanilla and also potentially to various exotic FX options. In addition to this, there will be a (potentially stochastic) correlation between the two spot processes. This is usually calibrated based on historical data, but if you have reliable data for various quanto vanillas, you can try to calibrate it and use it for example in Monte Carlo.

In conclusion, I would say this is not an easy exercise. It involves a lot of market calibration (two stochastic volatility processes and two leverage functions) and either the historical estimation or the calibration of the fifth stochastic process: a correlation between the two. Only after this will you be able to produce reliable equity quanto prices.

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  • $\begingroup$ Thank you for the answer. So you are suggesting to have one LSV model for equity underlying and the other LSV for FX process. Regarding the correlation between both price processes - I can make strong assumptions on correlations between (price) processes (the simpler the correlation structure and the more stable the better for me) and then easily correlate both price processes. What is unclear for me though is how to use both equity model and FX model to do actual pricing. Do I only need to multiply (pathwise) simulated equity prices with simulated FX rates, which are correlated? $\endgroup$
    – justLeito
    Mar 31 at 11:42
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    $\begingroup$ it's not as simple the multiplication is for a compo. There is a drift change due to the change of measure for the quanto case. Nothing specific to LSV, same theory as with LV. For SV, the difficulty will be your cross correlations and this is where you will want a simple structure. IMO it's overkill but may be interesting to evaluate the impact of the duak slv vs slv+lv or even slv+black. $\endgroup$
    – jherek
    Mar 31 at 18:37

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