Abstract

Options play a very important role in the financial market, and option pricing has become one of the focus issues discussed by the scholars. This paper proposes a new uncertain mean-reverting stock model with floating interest rate, where the interest rate is assumed to be the uncertain Cox-Ingersoll-Ross (CIR) model. The European option and American option pricing formulas are derived via the -path method. In addition, some mathematical properties of the uncertain option pricing formulas are discussed. Subsequently, several numerical examples are given to illustrate the effectiveness of the proposed model.

1. Introduction

Previous studies of option pricing are based on the assumption that the underlying asset price follows the stochastic differential equation [1–4]. According to the viewpoint of behavioral finance, the change of underlying asset price is not completely random. In fact, investors’ belief degrees usually play an important role in real financial practice. So some scholars argued that stochastic differential equations may not be appropriate to describe the stock price process. Liu [5] founded a branch of axiomatic mathematics for modeling belief degrees. Liu [6] proposed the uncertain stock model and deduced the European option pricing formulas. Furthermore, the uncertainty theory is introduced into the financial field, then the uncertain financial theory is formed.

American option price formulas were derived by Chen [7]. Peng and Yao [8] proposed an uncertain stock model with mean-reverting process. Yao [9] gave the no-arbitrage determinant theorems on uncertain mean-reverting stock model in uncertain financial market. Zhang and Liu [10] investigated the pricing problem of geometric average Asian option. Yin et al. [11] gave the lookback option pricing formulas of uncertain exponential Ornstein-Uhlenbeck model, and Wang and Chen [12] derived Asian options pricing formulas in an uncertain stock model with floating interest rate. Zhang et al. [13] investigated the pricing problem of lookback options for uncertain financial market, and so on.

In this paper, we proposed a new uncertain stock model with floating interest rate. The European option and American option pricing formulas are investigated under the assumption that the underlying stock price follows an uncertain mean-reverting stock model, and the interest rate follows an uncertain CIR model.

2. Preliminaries

Uncertain measure is a real-valued set function on a -algebra over a nonempty set satisfying normality, duality, subadditivity, and product axioms [5].

Definition 1. (see [6]). An uncertain variable is a function from an uncertainty space to the set of real numbers. The uncertainty distribution of an uncertain variable follows asfor any real number . If the uncertainty distribution is a continuous and strictly increasing function with respect to at which , andthen is said to be a regular distribution, and the inverse function is called the inverse uncertainty distribution of .

Definition 2. (see [14]). Let be an uncertain variable. Then the expected value of is defined byprovided that at least one of the two integrals is finite.

Theorem 1 (see [5]). Let be an uncertain variable with uncertainty distribution . If the expected value exists, then

Theorem 2 (see [14]). Let be an uncertain variable with regular uncertainty distribution . Then

Definition 3. (see [6]). The uncertain variables are said to be independent iffor any Borel sets of real numbers.
An uncertain process is a sequence of uncertain variables indexed by a totally ordered set , which is used to model the evolution of uncertain phenomena.

Definition 4. (see [6]). An uncertain process is said to be a canonical Liu process if(i) and almost all sample paths are Lipschitz continuous;(ii) has stationary and independent increments;(iii)every increment is a normal uncertain variable with expected value 0 and variance , whose uncertainty distribution is

Definition 5. (see [6]). Suppose is a canonical Liu process, and are two real functions. Thenis called an uncertain differential equation with an initial value .

Definition 6. (see [15]). Let be a number with . An uncertain differential equationis said to have an -path if it solves the corresponding ordinary differential equationwhere is the inverse standard normal uncertainty distribution, i.e.,

Theorem 3 (see [15]). Assume that and are the solution and -path of the uncertain differential equationrespectively. Then

Theorem 4 (see [15, 16]). Let and be the solution and -path of the uncertain differential equationrespectively. Then, the solution has an inverse uncertainty distribution

Theorem 5 (see [16]). Let and be the solution and -path of the uncertain differential equationrespectively. Then, for any time and strictly increasing function , the supremumhas an inverse uncertainty distributionand the time integral has an inverse uncertainty distribution

Theorem 6 (see [16]). Let and be the solution and -path of the uncertain differential equationrespectively. Then, for any time and strictly decreasing function , the supremumhas an inverse uncertainty distributionand the time integral has an inverse uncertainty distribution

Liu [17] proposed that the uncertain processes are independent if, for any positive integer and any times , the uncertain vectorsare independent.

Theorem 7 (see [18]). Assume that are some independent uncertain processes derived from the solutions of some uncertain differential equations. If the function is strictly increasing with respect to and strictly decreasing with respect to , then the uncertain process has an -path

3. Uncertain Mean-Reverting Stock Model with Floating Interest Rate

In the real market, the interest rate is an important economic indicator, which is always affected by some uncertain factors. To meet the needs of actual financial markets, Yao [18] assumed that both the interest rate and the stock price follow uncertain differential equations and presented an uncertain stock model with floating interest rate as follows,where and are the drift and diffusion of the interest rate, respectively, and are the drift and diffusion of the stock price, respectively, and and are independent canonical Liu processes. Considering the long-term fluctuations of the stock price and the changing of the interest rate from over time, Sun and Su [19] proposed an uncertain mean-reverting stock model with floating interest rate to describe the stock price and interest rate.

In Sun and Su’s model, the interest rate model was assumed to be the uncertain Vasicek model. There is no doubt that the Vasicek model could bring a negative value to the interest rate. However, the CIR model can overcome the problem, and it can ensure that the interest rate remains positive all the time.

In this paper, we will make some improvements to the stock models (27). To ensure that the interest rate is always positive, we assume the interest rate process to be the uncertain CIR model and introduce a new uncertain mean-reverting stock model with floating interest rate,where represents the rate of adjustment of , represents the average interest rate, represents the interest rate diffusion, are constants, and and are independent canonical Liu processes.

4. European Option Pricing Formulas

4.1. European Call Option

A European call option offers the holder the right without the obligation to buy a certain asset at an expiration time with a strike price , and is the stock price of the time . The payoff of the European call option is given by .

Definition 7. Assume European call option has a strike price and an expiration time . Then the European call option price is

Theorem 8. Assume European call option for the uncertain stock model (3) has a strike price and an expiration time . Then the European call option price iswhere solves the following the ordinary differential equation

Proof. According to Theorem 3, we can get the -path of the stock price :Similarly, we also get that satisfies the differential equation,It follows from Theorem 5 that the -path of is .
Since is strictly decreasing with respect to , from Theorem 6, the discount ratehas an -pathSince is an increasing function with respect to , it has an -pathTherefore, the present value of the optionhas an -pathaccording to Theorem 7. We have the price of the European call optionaccording to Theorems 2 and 4. The theorem is proved.

Theorem 9. Let be the European call option price of the uncertain stock model (28). Then(1) is an increasing function of ;(2) is an increasing function of ;(3) is a decreasing function of .

Proof. According to Theorem 8,where solves the following the ordinary differential equation(1)Since , the function is increasing with respect to and the European call option price is increasing with respect to the initial stock price . This means that the higher the initial stock price, the higher the European call option price.(2)Since , , the function is increasing with respect to and the European call option price is increasing with respect to the parameter .(3)Since the functionis decreasing with respect to and the European call option price is decreasing with respect to the strike price , this means that the higher the strike price, the lower the European call option price.

Example 1. Assume the parameters of the interest rate are , and the parameters of the stock price are , the strike price and the expiration time . Then a European call option price is  = 1.2859.

4.2. European Put Option

Suppose that a European put option has a strike price and an expiration time , and is the stock price of the time . The payoff of the European put option is given by .

Definition 8. Assume a European option has a strike price and an expiration time . Then the European put option price is

Theorem 10. Assume a European option for the uncertain stock model (28) has a strike price and an expiration time . Then the European put option price iswhere solves the following the ordinary differential equation:

Proof. According to the proof of Theorem 8, we can get that the discount ratehas an -pathSince is an decreasing function with respect to , it has an -pathTherefore, the present value of the optionhas an -pathaccording to Theorem 7. We have the price of the European put optionAccording to Theorems 2 and 4. The theorem is proved.

Theorem 11. Let be the European put option price of the uncertain stock model (28). Then(1) is a decreasing function of (2) is a decreasing function of (3) is an increasing function of

Proof. According to Theorem 10,(1)Since , the function is decreasing with respect to and the European put option price is decreasing with respect to the initial stock price . This means that the higher the initial stock price, the lower the European put option price.(2)Since , , the function is decreasing with respect to and the European put option price is decreasing with respect to the parameter .(3)Since the functionis increasing with respect to and the European put option price is increasing with respect to the strike price , this means that the higher the strike price, the higher the European put option price.

Example 2. Assume the parameters of the interest rate are , and the parameters of the stock price are , the strike price and the expiration time . Then a European put option price is  = 1.0362.

5. American Option Pricing Formulas

5.1. American Call Option

The American call option gives the holder the right, without obligation, to buy an agreed quantity of stock at any time before the expiration date with a strike price . Apparently, the best choice for the holder is to exercise the right at the supreme value, so the payoff of the American call option is given by .

Definition 9. Assume that the American call option has a strike price and an expiration time . Then the American call option price is

Theorem 12. Assume that the American call option for the uncertain stock model (28) has a strike price and an expiration time . Then the American call option price iswhere solves the following the ordinary differential equation:

Proof. It follows from Theorem 8 that the uncertain processhas an -pathAccording to Theorem 5, the uncertain processhas an -pathWe have the price of the American call optionAccording to Theorems 2 and 4. The theorem is proved.