The Price Impact of a Nonlinear Feedback in the Black-Scholes Model

The linear Black-Scholes-Merton options pricing model has been (and is still) used since 1973 to attempt to estimate the option price for an underlying asset. In the last thirty years, researchers have introduced nonlinear feedback into this model to handle transaction costs, illiquidity, fluctuating markets and other market conditions. In this paper, we show that the Power Series Method (PSM) can be used to obtain accurate and straightforward results for certain nonlinear terms for several practical situations. From the presentation, one can ascertain how to modify the results demonstrated here for other types of nonlinear feedback. Risk management applications using comparative statics are natural extensions of the PSM framework. Both numeric and symbolic solutions using PSM are presented.