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I have a Local Volatility model. I compute the LV surface $\sigma_{S}^{local}$ on vanilla option of $S$. Assume the vol of foreign exchange is constant and know, and the correlation equity/FX is known. I can know price quanto options as the model is fully calibrated.

But, if there are some quanto options liquid in the market, for some maturities. How can I make this model fit these prices of liquid quanto options as well? Up to know, I have only fit the prices of vanilla options on $S$. Is there some way to modify $\sigma_{S}^{local}$ in order to make it coherent with vanilla option prices and the some observed liquid quanto option prices?

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    $\begingroup$ You can't necessarily modify your vol surface as it will break the calibration to the non quanto options. The free parameter you have available to calibrate to the quanto is the correlation. If you're looking at longer quanto, it's worth noting that they are much more model dependent than most people think - have a look at Peter jaeckel's paper "quanto skew" and the follow up "quanto skew with stochastic volatility". $\endgroup$
    – will
    Commented Sep 19, 2021 at 11:46

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One way is to use a smile for the FX rate and another for compo options (the equity denominated in the other currency). FX smiles are liquid, so you only have freedom to choose the compo smile. Then use this model, Repricing the Cross Smile, to value quantos, and calibrate the compo smile to hit them. You can value quantos semi-analytically in that model, so it is quite tractable. You can then use the same model to value other quantos and other European options depending jointly on the equity and the FX.

If you do need to use local volatility because you are valuing exotics (eg a quanto barrier option), you could use the local correlation model, and solve a 2 factor PDE. The model uses Dupire local volatility for each of the three smiles, and backs out a local correlation function analytically using the triangle rule. Details are in Smile Pricing Explained.

Alternatively, you can use the joint probability distribution to integrate out the FX factor, and then solve an ordinary one dimensional local volatility PDE to value your exotic. I think this is pretty much what you are aiming for.

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