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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
    Sep 19 at 11:46

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