Exotic option pricing model of the Black–Scholes formula: a proactive investment strategy - Wu, Jingyu; Wang, Yaoping; Zhu, Minting; Zheng, Han; Li, Lingfei (Frontiers in Physics, 2023)
This paper studies an exotic option that extends the classic option to a broader range. Compared with the classic option theory, firstly, we suppose that the option holders act ahead of exercising the option since a wise investor usually has a good sense of the market and acts before the others. Secondly, we suppose that the option holders continuously trade the underlying assets according to a predetermined investment strategy. Based on the above assumptions, the pricing formula of the exotic option with a pre-designed nonlinear trading strategy is constructed by the Black-Scholes option pricing formula. Taking the European call option as an example, we first give some assumptions and define the loss function in terms of a logarithmic investment strategy. Then, the specific pricing expression of the exotic option is derived from the Black-Scholes option pricing formula. Moreover, numerical simulations are presented to visualize the mechanism of the exotic option in 2D and 3D dimensions by selecting appropriate parameters. Our numerical results indicate that the exotic option has a significant price advantage (up to 43.9% under specific parameter settings) over the classic option. The empirical results illustrate a perfect fit between 50ETF(issued by the Shanghai Stock Exchange) and our exotic option. The new proposed exotic option extends the Black-Scholes option theory from a no-trading condition (do not buy or sell underlying assets during the validity period) to a dynamic investment condition, and has important practical significance in real life. On the other hand, the exotic option should observe certain constraints, which means it is only valid for a small parametric range. Significantly, our new option shows to be positive in most domains while some other constructed options might be negative under certain parametric selections. Taken altogether, though the exotic option has a price advantage over the classical one, it narrows the range of applications.