Qualitative and Quantitative Analysis of Black-Scholes Type Models of Pricing Derivatives on Assets with General Function of Volatility
Magdaléna Žitňanská
PhD thesis advisor: Daniel Ševčovič

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Summary
In the thesis we study and analyze the nonlinear models of Black-Scholes type, which are becoming more and more important since they take into account many efects that are not included in the linear model.
The main goals of the thesis can be summarized as follows:

• Review of existing nonlinear models. We review option pricing models of the Black--Scholes type with a general function of volatility. They provide more accurate values than the classical linear model by taking into account more realistic assumptions, such as transaction costs, the risk from an unprotected portfolio, large investor's preferences or illiquid markets.
• Novel nonlinear models. The main goals of the thesis is to derive models with variable transaction costs. We extend the models by two more new examples of realistic variable transaction costs that are decreasing with the amount of transactions. Using the Risk adjusted pricing methodology we derive a novel option pricing model under transaction costs and risk of the unprotected portfolio.
• Solving the model by Gamma equation. We show that the generalizations of the classical Black-Scholes model, including the novel model, can be solved by transformation of the fully nonlinear parabolic equation into a quasilinear parabolic equation for which one can construct an effective numerical scheme for approximation of the solution.
• Numerical scheme and experiments. The aim of this part is to propose an efficient numerical discretization of the Gamma equation, including, in particular, the model with variable transaction costs. The numerical scheme is based on the finite volume approximation of the partial derivatives entering the equation to be solved.

Related papers
[1] D. Ševčovič, M. Žitňanská: Analysis of the nonlinear option pricing model under variable transaction costs, Asia-Pacific Financial Markets, 23(2) 2016, 153-174.
PDF file    arXiv 1603.03874    DOI: 0.1007/s10690-016-9213-y