Thesis - Optimizing Smooth Local Volatility Surfaces with Power Utility Functions

University essay from Linköpings universitet/ProduktionsekonomiLinköpings universitet/Tekniska fakulteten; Linköpings universitet/ProduktionsekonomiLinköpings universitet/Tekniska fakulteten

Abstract:

The master thesis is focused on how a local volatility surfaces can be extracted by optimization with respectto smoothness and price error. The pricing is based on utility based pricing, and developed to be set in arisk neutral pricing setting. The pricing is done in a discrete multinomial recombining tree, where the timeand price increments optionally can be equidistant. An interpolation algorithm is used if the option that shallbe priced is not matched in the tree discretization. Power utility functions are utilized, where the log-utilitypreference is especially studied, which coincides with the (Kelly) portfolio that systematically outperforms anyother portfolio. A fine resolution of the discretization is generally a property that is sought after, thus a seriesof derivations for the implementation are done to restrict the computational encumbrance and thus allow finer discretization.

The thesis is mainly focused on the derivation of the method rather than finding optimal parameters thatgenerate the local volatility surfaces. The method has shown that smooth surfaces can be extracted, whichconsider market prices. However, due to lacking available interest and dividend data, the pricing error increasessymmetrically for longer option maturities. However, the method shows exponential convergence and robustnessto different initial (flat) volatilities for the optimization initiation.

Given an optimal smooth local volatility surface, an arbitrary payoff function can then be used to price thecorresponding option, which could be path-dependent, such as barrier options. However, only vanilla optionswill be considered in this thesis. Finally, we find that the developed

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