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I was wondering why future volatility smile is important to path dependent option and American type option such as Bermudan swaption. It would be best if someone could provide a reference article as well.

Thanks!

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First of all, a Bermudan Swaption does not have to be of American type. Consider a "9NC2 Bermudan" (9 non call 2), basically a Bermudan swaption with final maturity in 9 years which is not exercisable for the first 2 years.

I have not worked at an exotic rates desk in a while (many years to be more precise) but from what I remember you need to use a lattice tree model to describe the evolution of interest rates. One can evolve such rates via a number different models such as HW, Gaussian 2-factor, Libor Market Model (possibly with SABR style stochastic volatility model), Black's model, among others. Some are short-rate models, others evolve discrete forward rates (such as LMM). I am not knowledgeable of newer approaches if they exist.

When you calibrate your model to match the prices of other swaptions (those that are co-terminal with your Bermudan) the choice of model will basically determine which parameters you calibrate for. In the example of the Libor Market Model you need to also choose how to model volatility (volatility of forward rates) and correlations between the driving Brownian Motions. This is where your volatility surface comes into play. The volatility surface you derive during your calibration exercise describes the volatility of different forward rates.

Once the calibration step is complete you can then iterate the lattice to price your Bermudan option.

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