Modification terms to the Black–Scholes model in a realistic hedging strategy with discrete temporal steps

Abstract

ABSTRACTOption pricing models generally require the assumption that stock prices are described by continuous-time stochastic processes. Although the time-continuous trading is easy to conceive theoretically, it is practically impossible to execute in real markets. One reason is because real markets are not perfectly liquid and purchase or sell any amount of an asset would change the asset price drastically. A realistic hedging strategy needs to consider trading that happens at discrete instants of time. This paper focuses on the impact and effect due to temporal discretization on the pricing partial differential equation (PDE) for European options. Two different aspects of temporal discretization are considered and used to derive the modification or correction source terms to the continuous pricing PDE. First the finite difference discretization of the standard Black–Scholes PDE and its modification due to discrete trading. Second the discrete trading leads to a discrete time re-balancing strategy that only cancels risks on average by using a discrete analogy of the stochastic process of the underlying asset. In both cases high order terms in the Taylor series expansion are used and the respective correction source terms are derived.