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On the relations between implied and spot volatilities

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In the first part of the talk, we will present a result showing how one can compute the spot volatility dynamics from the implied volatility surface. Then, we will look at an application to foreign exchange options: we take the exchange rates EURUSD , USDJPY, and EURJPY and reconstruct the implied volatility smile of one exchange rate from the other two. In the third part of the talk, we study the convergence of at-the-money implied volatilities to the spot volatility in a general model with a Brownian component and a jump component of finite variation. This result is a consequence of the robustness of the Black-Scholes formula and of the central limit theorem for martingales.

This talk is part of the Probability series.

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