M.Sc Thesis

M.Sc StudentNayman Niv
SubjectStochastic Control For Shortfall Risk Minimization
under Constant Transaction Costs
DepartmentDepartment of Electrical and Computer Engineering
Supervisors PROF. Rami Atar
DR. Yan Dolinsky


The classical theory of mathematical finance assumes that the potential investor has no trading frictions. This idealized approach going back to Black and Scholes (1973) who considered an asset that can be traded continuously without incurring costs in such a way that the resulting profits or losses from trading completely eliminate any conceivable financial risk. It has long since emerged that if one negates the idealizations of the Black-Scholes framework, super-replication becomes prohibitively costly, hence prices and hedges can be obtained only after specifying preferences for residual risk. In this work, we deal with market frictions, which are given by constant transaction costs independent of the volume of the trade. The main question that we study is the minimization of shortfall risk in the Black-Scholes model under constraints on the initial capital. This problem does not have an analytic solution and so numerical schemes come into the picture. The Cox-Ross-Rubinstein (CRR) binomial models are an efficient tool to approximate the Black-Scholes (BS) model. In this thesis, we study in detail the CRR models with constant transaction costs. In particular, we introduce a Markov decision process (MDP) and provide a proof for the existence of optimal control/policy. We further suggest a dynamic programming algorithm for calculating the optimal hedging strategy and its corresponding shortfall risk. Moreover, we imply how the control is lifted to the Black-Scholes model and achieves the minimal shortfall risk asymptotically. In the absence of transaction costs, there is an analytical solution in both CRR and BS models, and so we used them for testing our algorithm and its convergence. Moreover, we point out various insights provided by our numerical results, for example regarding the change in the investor’s activity in the presence of friction.