On modifications of the Bachelier model

Abstract

Mathematically, stock prices described by a classical Bachelier model are sums of a Brownian motion and an absolute continuous drift. Hence, stock prices can take negative values, and financially, it is not appropriate. This drawback is overcome by Samuelson who has proposed the exponential transformation and provided the so-called Geometrical Brownian motion. In this paper, we introduce two additional modifications which are based on SDEs with absorption and reflection. We show that the model with reflection may admit arbitrage, but the model with an appropriate absorption leads to a better model. Comparisons regarding option pricing among the standard Bachelier model, the Black–Scholes model and the modified Bachelier model with absorption at zero are executed. Moreover, our main findings are also devoted to the Conditional Value-at-Risk based partial hedging in the framework of these models. Illustrative numerical examples are provided.