From spot volatilities to implied volatilities
Since the pioneering work of Dupire, the local volatility function can be derived from the implied volatility surface. Calculating implied volatilities from local volatilities is classically done by numerically solving the forward equation for call prices. Here, Julien Guyon and Pierre Henry-Labordère suggest new methods for calculating implied volatilities, based on a very general result that expresses the square of the implied volatility as an average over time and space of the square of the spot volatility
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From spot volatilities to implied volatilities
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