American Barrier Option Pricing Formulas for Stock Model in Uncertain Environment

Rong Gao,Kaixiang Liu,Zhiguo Li,Rongjie Lv

doi:10.1109/access.2019.2928029

Abstract

In the foundation of uncertainty theory, uncertain stock model has been put forward to portray the price fluctuation of stocks in a market with uncertain information. In this paper, the model is depicted by uncertain differential equations involved by a Liu process that is a sequence of uncertain variables varying with time. According to this model, we mainly investigate the formulas to price the American barrier option for rights of buying or selling the stock with a set price in the uncertain financial market. Then, four new types of concepts are introduced that are, respectively, American call options, including both up-and-in and down-and-out, and American put options, including both down-and-in and up-and-out. Moreover, several formulas are derived for giving the price of the corresponding four types of options. At the same time, some examples are given.

Highlights

Brownian movement was firstly found in 1820s by Brown who is the Scottish botanist Robert
The stock price was independently investigated by Black and Scholes [1] and Metron [15] through applying random differential equation involved by Wiener process, where a famous option pricing formula named Black-Scholes was derived
The stock price was dealt with several other kinds of stochastic differential equations, which produced diverse option pricing formulas

Summary

INTRODUCTION

Brownian movement was firstly found in 1820s by Brown who is the Scottish botanist Robert. The stock price was independently investigated by Black and Scholes [1] and Metron [15] through applying random differential equation involved by Wiener process, where a famous option pricing formula named Black-Scholes was derived. R. Gao et al.: American Barrier Option Pricing Formulas for Stock Model in Uncertain Environment process called stationary and independent incremental process, where its increments are independent with each other and possess identically uncertainty distributions (i.i.d.). A kind of uncertain stock model was proposed in [10] by the aid of uncertain differential equations to portray the varying tendency of stock price, where several pricing formulas of European option were derived. We will consider the American barrier option for a stock modeled by uncertain differential equation, and the pricing formulas of corresponding options are going to be proved in this article.

PRELIMINARIES

CONCLUSION