Fitting Quanto Options with Compo Smiles and Local Correlation
Summary
The note considers how to extend an equity local volatility model, initially calibrated to vanilla options, so it can also match liquid quanto option prices. It outlines an approach based on market smiles: use the liquid FX smile and calibrate a compo option smile, representing equity exposure denominated in the other currency, to observed quanto prices. A joint equity and FX model can then price quantos and other European claims that depend on both factors.
For exotic products that require local volatility, the response describes a local correlation model using Dupire local volatility for the equity, FX, and compo smiles, with a local correlation inferred through a triangle relationship and a two-factor PDE. It also suggests integrating out the FX factor using the joint distribution before solving a one-dimensional local volatility PDE. These are modeling routes rather than a full calibration recipe: the note gives no implementation details or numerical evidence, and the appropriate method depends on the products being valued.
Key ideas
- Vanilla equity calibration alone does not ensure that a model matches observed quanto prices.
- A liquid FX smile can be combined with a calibrated compo smile to fit quanto quotes.
- A local correlation model can combine three Dupire volatility surfaces for exotic pricing.
- A two-factor PDE is one proposed method for valuing quanto exotics under local correlation.
- Integrating out FX is an alternative route to a one-dimensional local volatility PDE.
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Full text
# Calibrate Local Volatility model to price quanto options
# Calibrate Local Volatility model to price quanto options
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?
## Answer by Peter A (score 1)
https://quant.stackexchange.com/a/70318
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.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.