Smiling at Evolution

Abstract: We generate a reliable implied volatility surface without arbitrage in space and in time by parameterising a mixture of shifted lognormal densities under constraints and use a Differential Evolution algorithm to calibrate the model's parameters to a finite set of option prices. It is used for marking options not directly visible as well as for computing a proper deterministic local volatility. To do so, we devise an evolutionary algorithm handling constraints in a simple and efficient way. Using some of the improvements made to the DE algorithm and taking advantage of the specific structure of our objective function, we use special operators to help satisfy the equality constraints together with feasibility rules to handle the inequality constraints. Finally, we propose a modified algorithm for solving our optimisation problem under constraints which, after testing on real market data, greatly improves its performances.

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