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A Practical Guide to Implied and Local Volatility

81 Pages Posted: 20 Jan 2010 Last revised: 4 Mar 2010

Daniel Alexandre Bloch

Université Paris VI Pierre et Marie Curie

Date Written: January 18, 2010

Abstract

We consider a stochastic local volatility model with domestic and foreign stochastic interest rates such that the volatility decomposes into a deterministic local volatility plus some bias terms. Assuming a collapse process for the variance with the same random variable for all time and deterministic zero-coupon bond volatility functions, we are going to describe in detail the implementation of that model, focusing on the computation of a proper deterministic local volatility. To do so, we choose to generate an implied volatility surface without arbitrage in space and in time by parametrising a mixture of shifted lognormal densities under constraints and we use a Differential Evolution algorithm to calibrate the model's parameters to a finite set of option prices. We will therefore need to devise an evolutionary algorithm that handle constraints in a simple and efficient way. Using some of the improvements made to the DE algorithm combined with simple and robust constraints handling mechanisms we will propose a modified algorithm for solving our optimisation problem under constraints which greatly improves its performances.

Keywords: Stochastic Local Volatility Model, Implied Volatility, Stochastic Rates, Differential Evolution, Optimisation Problem Under Constraints

Suggested Citation

Bloch, Daniel Alexandre, A Practical Guide to Implied and Local Volatility (January 18, 2010). Available at SSRN: https://ssrn.com/abstract=1538808 or http://dx.doi.org/10.2139/ssrn.1538808

Daniel Alexandre Bloch (Contact Author)

Université Paris VI Pierre et Marie Curie ( email )

175 Rue du Chevaleret
Paris, 75013
France

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