Q1. Explain why a stochastic volatility model gives a smile? 



A stochastic volatility model takes the volatility to be driven by a random process.

It is then possible that the volatility will become large causing large movements in the underlying asset price.

The large moves give the distribution of the underlying asset fatter tails than in the BS model.

This means that the options away from the money will be priced more expensively than in the BS leading to a volatility smile. 


Another way to think about why the smile occurs is by looking at the second derivative of the option price w.r.t the volatility. 

In the BS, our vega sensitivity (how good our volatility hedge is) will be zero.

However, a stochastic volatility model will not necessarily have a non-zero derivative of vega, therefore there is vega convexity, which is related to the volatility smile. 

Q2. What differing models can be used to price exotic FX options consistently with market smiles? What are the pros and cons of each? 




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