Option Pricing in Illiquid Markets with Jumps
José M. T. S. Cruz
PhD thesis advisors: Daniel Ševčovič and Maria do Rosário Grossinho

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PhD thesis - Full text

Abstract: In this project thesis we try to model feedback effects using Lévy Processes, therefore relaxing Black-Scholes’s model assumptions of market liquidity and completeness. The contribution would be to study option pricing in illiquid markets with jumps and the associated hedging strategy. The basic idea of this thesis is to extend the models already used in the literature and extend them using Lévy Processes. We arrive at a partial integro-differential equation which is nonlinear and where the solution, if it exists, should be the function representing the derivative’s security price. The objective is to study the existence and uniqueness of solution of that partial integro-differential equation and then develop numerical schemes to solve it and study its consistency, stability and convergence. Also we would like to study the equation when the influence of the large trader is small, in order to compare it to the already well established classical PIDE.

References
[1] J. Cruz and D. Ševčovič: On solutions of a partial integro-differential Black-Scholes equation in Bessel potential spaces, Japan Journal of Industrial and Applied Mathematics 37 (2020), 691-721.
[2] J. Cruz and D. Ševčovič: Option Pricing in Illiquid Markets with Jumps, Applied Mathematical Finance, 25(4), 2018, 389-409.