A Practical Guide to Implied and Local Volatility
81 Pages Posted: 20 Jan 2010 Last revised: 4 Mar 2010
Date Written: January 18, 2010
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
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