Abstract
We extend the widely used SABR model (Hagan et al (2002)) to include a general volatility function and a CEV power on the stochastic volatility process itself. Using a short time expansion we derive results for the Dupire local volatility which in turn is inserted into a single time step finite difference scheme to generate arbitrage free option prices. Our approach has a number of advantages over the standard SABR model: a. it eliminates arbitrage for low and high strikes, b. it allows for an exact fit to a set of discrete option quotes, and c. it gives more explicit control over the wings, both for low (and potentially negative) strikes and for very high strikes. All of this without sacrificing speed in the implementation.
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