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F# for Quantitative Finance by Johan Astborg

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Exploring the volatility smile

The volatility smile is a phenomenon frequently observed in the markets. This phenomenon is mostly explained by the assumptions made in the Black-Scholes formula. Black-Scholes assumes constant volatility throughout the life of an option. If the Black-Scholes formula was corrected for this behavior, by taking into account the nature of volatility being non-constant, we would end up with a flat volatility surface.

Further, the volatility smile describes the volatility for a certain price of the option relative to the strike price of the same. The volatility surface is often referring to a three-dimensional graph of the volatility smile, where time to maturity and moneyness.

Moneyness is the ratio between the spot price ...

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