Manuscript ID : AMFA-2401-1939 Visit : 95 Page: 1154 - 1165

Article Type: Original Research

An Uncertain Renewal Stock Model for Barrier Options Pricing with Floating Interest Rate

Subject Areas : Financial Mathematics

Behzad Abbasi ¹ (Department of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, P.O. 35195-363, Semnan, Iran.(PHD Student))
Kazem Nouri ² (Department of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, P.O. Box 35195-363, Semnan, Iran.(Professor))

Received: 2024-01-15 Accepted : 2024-04-23 Published : 2024-05-26

Keywords: Uncertain Process, Renewal Process, Barrier Options Pricing , Floating Interest Rate , Uncertain Differential Equation(UDE),

Abstract :

Option pricing is a main topic in contemporary financial theories, captivating the attention of numerous financial analysts and economists. Barrier option, classified as an exotic option, derives its value from the behavior of an underlying asset. The outcome of this option is based on whether or not the price of the underlying asset has reached a predetermined barrier level. Over the years, the stock price has been represented through continuous stochastic processes, with the prominent one being the Brownian motion process. Correspondingly, the widely used Black-Scholes model has been employed. Nevertheless, it has become evident that utilizing stochastic differential equations to characterize the stock price process is unsuitable and leads to a perplexing paradox. As a result, many researchers have turned to incorporating fuzzy or uncertain environments in such situations. This study presents a methodology for pricing barrier options on stocks in an uncertain environment, in which the interarrival times are uncertain variables. The approach employs the Liu process and renewal uncertain process, considering the interest rate as dynamic and floating. The pricing formulas for knock-in barrier options are derived using α-paths of uncertain differential equations with jumps.

References:

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Full-Text:

Original Research

1 Introduction

The pricing of options holds great importance in the financial markets, and it is a subject of considerable interest in mathematical finance. Nevertheless, barrier options and vanilla options share similarities, with the exception that barrier options are either activated or deactivated when the underlying asset price touches the barrier price before the maturity time. Barrier options have been traded in the over-the-counter (OTC) market since 1967 and have become the preferred choice among exotic options. Different pricing methods have been widely utilized in option pricing, including the Black-Scholes [1] and Merton's [2] option pricing theory, in which the price process for underlying assets follows the stochastic differential equations (SDEs). Merton [2] was the first to propose a theory for pricing rational options, focusing on down and out options. Rich, on the other hand, contributed to the pricing of barrier options. Subsequently, numerous researchers have explored various approaches for pricing such options. For example, Nouri, Abbasi, et al. [3, 4] introduced an enhanced Monte Carlo algorithm for pricing different types of barrier options. Additionally, [5] employed a Lie-algebraic method to determine the value of moving barrier options, and [6] conducted a study on the analytical valuation of American double barrier options. In 2013, Liu [7] argued that the application of stochastic differential equations to characterize the stock price process is unsuitable and leads to a perplexing paradox. This perspective is substantiated by empirical observations, which reveal that the peak of the distribution of underlying assets exceeds that of a normal probability distribution, accompanied by heavier tails. Numerous empirical studies have shown that the behavior of underlying asset prices does not conform to the principles of probability and randomness. So many researchers have applied fuzzy and uncertain environments to compute option pricing formulas [8-10]. Considering the influence of both randomness and human uncertainty on financial markets, it is evident that an investor's belief holds great importance in shaping market dynamics. As investors tend to base their decisions on their beliefs rather than solely on probabilities. In support of this, Kahneman [11] demonstrated that the degrees of beliefs exhibit a much wider range of variation compared to frequency. In 2004 Cont and Tankov [12] employed jump-diffusion models as an uncertain source and demonstrated the extensive structure these models possess for asset pricing. In 2007 Liu [13] established a theory of uncertainty within the framework of uncertain measure, focusing on the degree of belief. In 2008, Liu [14] introduced the concept of uncertain process to enhance the modelling of uncertain phenomena. Researchers in [15-17] have developed various methods for solving uncertain differential equations (UDEs) based on this work. Additionally, Yao [16] has proposed several numerical techniques for computing integration and differentiation, which can be applied to renewal uncertain processes. Furthermore, Chen and Liu [15] have demonstrated the existence and uniqueness theorem for the solutions of UDEs, and besides Liu [18] has proven the stability of UDEs. In 2009, Liu [18] developed several formulas for option pricing based on an uncertain stock model. Following that, researchers in [19- 23] extensively explored uncertain stock pricing models. Furthermore, Chen [24] introduced a formula to price American options in 2011. Meanwhile, Liu [13] highlighted the importance of uncertain renewal processes, specifically focusing on cases where the interarrival times are uncertain variables. Later on, Liu [25] proposed a renewal reward process that accounted for the uncertainty of interarrival times and rewards. In 2012, Yao [26] established a theory on uncertainty calculus specifically for renewal processes. Jia and Chen [27] conducted a study in 2020, uncovering noteworthy findings on pricing formulas for Knock-in barrier options within an uncertain stock pricing model featuring a floating interest rate. Additionally, Gao, et al. [28] investigated pricing American barrier option of currency model in uncertain environment. Section 2 of the paper begins with the necessary preliminaries. Subsequently, Section 3 presents the stock pricing model in uncertain space, which specifically focuses on real decision problems and incorporates a floating interest rate. Section 4, offers the proof for European knock-in options pricing formulas within the framework of the uncertain stock model. Finally, Section 5 concludes the paper by presenting a summary of the findings.

2 Preliminaries

Consider Γ denote a non-empty set, and define the σ-algebra L be a collection of all the events over Γ. We can define it as a function that assigns to each event the belief degree, which represents our confidence in the occurrence of. Liu [14] proposed five axioms to provide an axiomatic definition of uncertain measure to ensure that the number is not arbitrary and has special mathematical properties;

1: (Normality axiom) ;

2: (Monotonicity axiom) whenever ;

3: (Duality axiom) for every event ;

4: (subadditivity axiom) For each sequence of events , that can be

counted, we have

Article Info

Article history:

Received 2024-01-15

Accepted 2024-04-23

Keywords:

Uncertain Process

Renewal Process

Barrier Options Pricing

Floating Interest Rate

Uncertain Differential Equation(UDE)

Abstract

Option pricing is a main topic in contemporary financial theories, captivating the attention of numerous financial analysts and economists. Barrier option, classified as an exotic option, derives its value from the behavior of an underlying asset. The outcome of this option is based on whether or not the price of the underlying asset has reached a predetermined barrier level. Over the years, the stock price has been represented through continuous stochastic processes, with the prominent one being the Brownian motion process. Correspondingly, the widely used Black-Scholes model has been employed. Nevertheless, it has become evident that utilizing stochastic differential equations to characterize the stock price process is unsuitable and leads to a perplexing paradox. As a result, many researchers have turned to incorporating fuzzy or uncertain environments in such situations. This study presents a methodology for pricing barrier options on stocks in an uncertain environment, in which the interarrival times are uncertain variables. The approach employs the Liu process and renewal uncertain process, considering the interest rate as dynamic and floating. The pricing formulas for knock-in barrier options are derived using -paths of uncertain differential equations with jumps.

Definition 1. [18]. The set function which satisfies the above axioms, is called an uncertain measure.

Definition 2. [18]. Consider be a non-empty set, the -algebra , be a collection of all the events over and be an uncertain measure according to the above definition. Then the triple is called an uncertain space.

5: (Product Measure Axiom) [18]. Let the triple where = and ... be uncertainty space for , then product uncertain measure is an uncertain measure on the product -algebra satisfying the product uncertain measure is uncertain measure satisfying,

(1)

Where , are arbitrary chosen event from for , respectively.

Definition 3. [18]. The uncertainty distribution for an uncertain variable such as is defined by function that .

Definition 4. Following uncertainty distribution is called normal

(2)

If be an uncertain variable, in this case and are real numbers and it is shown by. The normal uncertainty distribution can be called standard, if and. So is the inverse uncertainty distribution of , if it exists. The expected value of an uncertain variable is defined as

(3)

Definition 5. [14] following UDE (uncertain differential equation),

(4)

Has an α-path (), if it solves the bellow corresponding ODE

(5)

Where is the inverse standard normal uncertainty distribution, i.e.,

(6)

Definition 6. [18]. Liu process is an uncertain process which have bellow properties

1- ;

2- has independent and stationary increments;

3- Almost all sample paths are Lipschitz continuous;

4- All increments - are normal uncertain variables with expected value and variance

Theorem 1. Let be the solution of the UDE eq. (5) and -path be the solution of ODE eq. (6). Then

(7)

Definition 7. [14] The uncertain process

	(8)

is called an uncertain renewal process, if be iid positive uncertain variables. Also and .

The uncertain renewal process has an expected value

(9)

Where denote the uncertainty distribution of s.

Definition 8. [29] Consider that indicate the interarrival times of sequential events. Hence, is the number of renewals in and is the total waiting time before the th event occurs. The relation between the fundamental formulas of an uncertain renewal process are as below:

(10)

Theorem 2. [29] Consider be an uncertain renewal process, if interarrival times have an uncertainty distribution , then has an uncertainty distribution