In this work, we deal with market frictions which are given by fixed 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 (BS) model under constraints on the initial capital. This problem does not have an analytical solution and so numerical schemes come into the picture. The Cox–Ross–Rubinstein (CRR) binomial models are an efficient tool for approximating the BS model. In this paper, we study in detail the CRR models with fixed transaction costs. In particular, we construct an augmented state-action space forming 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. In the absence of transaction costs, there is an analytical solution in both CRR and BS models, and so we use 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.
On shortfall risk minimization for game options
