Lévy-type Black-Scholes-Merton optimal investment strategy in the financial market

Abstract

The study was motivated by searches for an optimal Lévy-type investment strategy in a Black-ScholesMerton complete financial market with no arbitrage. The paper stipulates shares of various financial instruments in an optimal portfolio. Their prices are described using Lévy processes, which are a generalised Wiener process. On top of that, an assumption was made about model indicators, which depend on Markov chains. This is an incomplete market meaning not every payment can be replicated using a certain investment strategy. In order to complete the market, jump financial instruments and power-jump assets have been added. Next, dynamic programming methods were deployed to determine an optimal investment strategy in this market. An optimal strategy is the one which maximises the expected utility of wealth accumulation at the end of a pre-determined period. The analysis was carried out for a logarithmic and power function of utility of the received payment.