On stability of continuous time models of a financial market.
This note dwells on the following question. Consider the standard geometric Brownian motion model (the Black-Scholes-Merton model) for the evolution of a stock. Within this model continuous perfect hedging is possible and it determines the price of options. Suppose, nonetheless, that you decide to use the model above for managing an option and that you would like to upper hedge it, but that rebalancing the (upper-)hedging portfolio takes place only at times \(k(T/N)\) \((T\) is time to maturity) and that it is calculated as in a discrete version of the Black-Scholes-Merton model like the Cox-Ross-Rubinstein model. Let us call \(x(N)\) the minimum capital needed to assure this upper hedge with the strategy above. Does \(x(N)\) converge to the price of the option as calculated in the Black-Scholes-Merton model? This paper analyzes this question and obtains positive answers under certain assumptions.
scientific article; zbMATH DE number 1821727