Extended Black and Scholes model under bankruptcy risk

In the Black-Scholes system, the underlying asset price model follows a geometric Brownian motion with constant volatility and no occurrence of bankruptcy. These two characteristics contradict real financial observations. In order to agree with the nonconstant feature of the volatility and take bankruptcy risk into consideration, we modify the Black and Scholes model and propose a new model based on the efficient market hypothesis. First, we present some probability properties of the bankruptcy risk by our model and demonstrate the statistical inference for the unknown parameters. We also study the European option pricing problem and propose its statistical computation method. In addition, via a real data analysis, we show that our model captures the trend of the stock prices much better than geometric Brownian motion and clarify that the bankruptcy is a crucial factor to be considered in the Black-Scholes system.