Geometric Asian barrier option pricing formulas of uncertain stock model

https://doi.org/10.1016/j.chaos.2020.110178Get rights and content

Highlights

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    Geometric Asian barrier option based on uncertain stock model is studied.

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    Pricing formulas of four kinds of geometric Asian barrier options are derived.

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    The relation between some parameters and option price is analyzed.

Abstract

In the high-risk modern financial market, option is an effective tool to hedge the risks caused by uncertain demand, the fluctuation of price and foreign exchange rate, because the option can provide the holder with an entitlement to sell or purchase an asset with an exercise price. The acquisition of this entitlement requires the investor to pay option fee, which raises the option pricing issue. This article analyzes how to price Geometric Asian barrier option for uncertain stock model, where barrier option becomes activated or inactivated depending on whether a given barrier level is hit. Here, we suppose that stock price obeys an uncertain differential equation, based on which the pricing formulas of Geometric Asian barrier option are discovered. Furthermore, to express how to use the pricing formulas to calculate corresponding option prices, some numerical examples are given.

Introduction

Option is the inevitable product of avoiding risks and obtaining profits by leverage effect. The essence of the option is not an obligation but a right which allows the owner to sell or purchase an asset with an exercise price on or prior to the maturity date. The investor has to pay option fee in order to obtain this right. How to formulate reasonable option fee has been considered by scholars. The investigation of option pricing theory can be dated back to 1900 when Bachelier [2] proposed Brownian motion for depicting the fluctuation of stock price. However, the negative value of Brownian motion goes against the fact that the price of the stocks in greater than zero. Therefore, geometric Brownian motion was put forward by Samuelson [19]. Black and Scholes [3] derived the famous Black-Scholes formula on the basis of geometric Brownian motion. With the development of option pricing models, a large deal of literature concerning option pricing has been published, such as [23], [24], [31].

In 1987, an American bank introduced Asian option which is a kind of exotic option. It is different from European option where its return is related to the mean value of underlying asset over its lifetime. Sun and Chen [20] showed that Asian option can decrease the probability of market manipulation near the expiration date and lower option fee compared to standard options. Aimi et al. [1] provided a novel scheme for evaluating geometric Asian options with barriers. Mudzimbabwe et al. [18] discussed the price of Asian option by explicit and implicit finite difference methods. Kumar et al. [8] exploited the radial basis function approximation to describe the Asian option pricing issue. For more application of the finite difference method and radial basis function in the field of option pricing, interested readers may consult References [6], [21], [22].

The emergence of barrier option allows the investor to only consider price ranges that they believe may occur. The relationship between the trigger level and price of underlying asset determines whether barrier option is effective. Generally, it is classified into two sorts, knock-in option and knock-out option. The knock-in option becomes valid only if a certain level is reached, while the knock-out option is invalid in the event that the underlying asset price touches a given level. Zhang [29] asserted that barrier option improves convenience when the trading volume of stock option is quite low and has preferential price than standard options. Merton [17] proposed an explicit formula of down-and-out call option. Ballestra et al. [4] analyzed the barrier option pricing issue of the mixed fractional Brownian model.

Besides Wiener process, uncertain process can also depict the fluctuation of stock price. In the real financial market, many investors lack a large amount of data for reference due to the limited resources. They are more willing to believe experts in related fields. For properly handling belief degrees, Liu [11], [13] raised uncertainty theory based on subadditivity, duality, normality and product axioms. As the basic knowledge of uncertainty theory, uncertain variable was presented to portray uncertain quantity. For depicting the change of uncertain variable with time, the definition of uncertain process was put forward by Liu [10]. Moreover, Liu [15] investigated the independence and uncertainty distribution of uncertain processes.

As a particular kind of uncertain process, Liu process was devised by Liu [11], which is a stationary independent increment process. Liu [11] introduced the uncertain differential equation driven by Liu process and an uncertain stock model. Under the framework of uncertain stock model, many studies have been implemented. For example, Liu [11] first put forward the European option pricing formulas. Gao et al. [7] tackled the pricing problem of American barrier option. Yang et al. [28] obtained the Asian barrier option pricing formulas. Periodic dividends were introduced into uncertain stock model by Chen et al. [5]. The Asian option pricing formula was proposed by Sun and Chen [20]. The valuation formulas of geometric average Asian option were studied by Zhang and Liu [30]. For literature of option pricing under different models, interested readers may consult References [16], [25].

Geometric Asian barrier option is that when barrier option comes into effect, the gain of the option is related to the geometric mean of underlying asset price over a predetermined period. This paper considers that the stock price is an uncertain process which obeys the uncertain differential equation. Considering the above, we tackle the pricing issue of Geometric Asian barrier option and obtain the pricing formulas of Geometric Asian barrier option by rigorous deduction. The rest of this article is described below. A short review of some related properties and concepts of uncertainty theory is described in Section 2. The Geometric Asian barrier option pricing formulas in the knock-in and knock-out cases are derived in Section 3 and Section 4 by strict deduction, respectively. In Section 5, some concise conclusions are given.

Section snippets

Preliminaries

Definition 1

(Liu [11], [13]) Providing that (Γ, L) is a measurable space. For portraying the event Λ, an uncertain measure M is a set function defined on L and satisfied the following conditions

(i) M{Γ}=1 for the universal set Γ;

(ii) M{Λ}+M{Λc}=1 for each event Λ;

(iii) For any countable sequence of events Λ1,Λ2,…, we haveM{⋃i=1∞Λi}≤∑i=1∞M{Λi}.Product Axiom: If (Γk, Lk, Mk) (k=1,2,…) are uncertainty spaces, then the product measure M defined as the formula below is an uncertain measureM{∏k=1∞Λk}=⋀k=1∞Mk{Λk}

The knock-in case

Geometric Asian knock-in option is a contract in which the owner will receive a Geometric Asian option if a given barrier level is reached. In this section, we study two types of knock-in options, Geometric Asian up-and-in call option and Geometric Asian down-and-in put option. Moreover, the pricing formulas of up-and-in option and down-and-in option are obtained.

The knock-out case

Geometric Asian knock-out option is a contract in which the owner will lose a Geometric Asian option if a given barrier level is reached. In this section, we study two types of knock-out options, Geometric Asian up-and-out put option and Geometric Asian down-and-out call option. Moreover, the pricing formulas of up-and-out option and down-and-out option are obtained.

Conclusions

This article discussed the pricing issue of Geometric Asian barrier option for uncertain stock model. Under the theory of uncertain stock model, we proposed Geometric Asian knock-in option including both down-and-in put and up-and-in call option, and Geometric Asian knock-out option including both down-and-out call and up-and-out put option. Furthermore, we derived the pricing formulas of corresponding option by rigorous deduction. Finally, we used some numerical examples to indicate how to

CRediT authorship contribution statement

Rong Gao: Conceptualization, Methodology, Writing - review & editing. Wei Wu: Writing - original draft. Chao Lang: Software, Validation. Liying Lang: Data curation, Supervision.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

This work was supported by the Natural Science Foundation of Hebei Province (No. F2020202056), the Key Project of Hebei Education Department (No.ZD2020125) and the National Natural Science Foundation of China (No. 41804051).

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