Option Pricing Formulas in a New Uncertain Mean-Reverting Stock Model with Floating Interest Rate

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 
 
 α
 
 -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.


Introduction
Previous studies of option pricing are based on the assumption that the underlying asset price follows the stochastic differential equation [1][2][3][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.
e 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.

Preliminaries
Uncertain measure M is a real-valued set function on a σ-algebra L over a nonempty set Γ satisfying normality, duality, subadditivity, and product axioms [5].
Definition 1 (see [6]). An uncertain variable is a function ξ from an uncertainty space (Γ, L, M) to the set of real numbers. e uncertainty distribution Φ of an uncertain variable ξ follows as for any real number x. If the uncertainty distribution Φ(x) is a continuous and strictly increasing function with respect to x at which 0 < Φ(x) < 1, and then Φ(x) is said to be a regular distribution, and the inverse function Φ − 1 (α) is called the inverse uncertainty distribution of ξ.
Definition 2 (see [14]). Let ξ be an uncertain variable. en the expected value of ξ is defined by provided that at least one of the two integrals is finite.
Theorem 1 (see [5]). Let ξ be an uncertain variable with uncertainty distribution Φ. If the expected value exists, then Theorem 2 (see [14]). Let ξ be an uncertain variable with regular uncertainty distribution Φ. en Definition 3 (see [6]). e uncertain variables ξ 1 , ξ 2 , . . . , ξ n are said to be independent if for any Borel sets B 1 , B 2 , . . . , B n of real numbers. An uncertain process is a sequence of uncertain variables indexed by a totally ordered set T, which is used to model the evolution of uncertain phenomena.
Definition 4 (see [6]). An uncertain process C t is said to be a canonical Liu process if (i) C 0 � 0 and almost all sample paths are Lipschitz continuous; (ii) C t has stationary and independent increments; (iii) every increment C s+t − C t is a normal uncertain variable with expected value 0 and variance t 2 , whose uncertainty distribution is Definition 5 (see [6]). Suppose C t is a canonical Liu process, f and g are two real functions. en is called an uncertain differential equation with an initial value X 0 .
Definition 6 (see [15]). Let α be a number with 0 < α < 1. An uncertain differential equation is said to have an α-path X α t if it solves the corresponding ordinary differential equation where Φ − 1 (α) is the inverse standard normal uncertainty distribution, i.e., Theorem 3 (see [15]). Assume that X t and X α t are the solution and α-path of the uncertain differential equation respectively. en Theorem 4 (see [15,16]). Let X t and X α t be the solution and α-path of the uncertain differential equation respectively. en, the solution X t has an inverse uncertainty distribution Theorem 5 (see [16]). Let X t and X α t be the solution and α-path of the uncertain differential equation respectively. en, for any time s > 0 and strictly increasing function J(x), the supremum has an inverse uncertainty distribution and the time integral s 0 J(X t )dt has an inverse uncertainty distribution Theorem 6 (see [16]). Let X t and X α t be the solution and α-path of the uncertain differential equation 2 Discrete Dynamics in Nature and Society respectively. en, for any time s > 0 and strictly decreasing function J(x), the supremum has an inverse uncertainty distribution and the time integral s 0 J(X t )dt has an inverse uncertainty distribution Liu [17] proposed that the uncertain processes X 1t , X 2t , . . . , X nt are independent if, for any positive integer k and any times t 1 , t 2 , . . . , t k , the uncertain vectors are independent.
Theorem 7 (see [18]). Assume that X 1t , X 2t , . . . , X nt are some independent uncertain processes derived from the solutions of some uncertain differential equations. If the . , x m and strictly decreasing with respect to x m+1 , x m+2 , . . . , x n , then the uncertain process X t � f(X 1t , X 2t , . . . , X nt ) has an α-path

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 r t and the stock price X t follow uncertain differential equations and presented an uncertain stock model with floating interest rate as follows, where μ 1 and σ 1 are the drift and diffusion of the interest rate, respectively, μ 2 and σ 2 are the drift and diffusion of the stock price, respectively, and C 1t and C 2t 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.
ere 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 meanreverting stock model with floating interest rate, where a 1 represents the rate of adjustment of r t , b 1 represents the average interest rate, σ 1 represents the interest rate diffusion, a 1 , b 1 , σ 1 , a 2 , b 2 , σ 2 are constants, and C 1t and C 2t are independent canonical Liu processes.

European Call Option.
A European call option offers the holder the right without the obligation to buy a certain asset at an expiration time T with a strike price K, and X t is the stock price of the time t. e payoff of the European call option is given by (X T − K) + .
Definition 7. Assume European call option has a strike price K and an expiration time T. en the European call option price is

Theorem 8 Assume European call option for the uncertain stock model (3) has a strike price K and an expiration time T. en the European call option price is
where r 1− α s solves the following the ordinary differential equation Proof. According to eorem 3, we can get the α-path of the stock price X t : Discrete Dynamics in Nature and Society 3 Similarly, we also get that r 1− α s satisfies the differential equation, It follows from eorem 5 that the α-path of T 0 r s ds is T 0 r α s ds.
Since y � exp(− x) is strictly decreasing with respect to x, from eorem 6, the discount rate has an α-path Since (X T − K) + is an increasing function with respect to X T , it has an α-path erefore, the present value of the option has an α-path according to eorem 7. We have the price of the European call option according to eorems 2 and 4. e theorem is proved. □

Theorem 9. Let f Ec be the European call option price of the uncertain stock model (28). en
(1) f Ec is an increasing function of X 0 ; (2) f Ec is an increasing function of a 2 ; (3) f Ec is a decreasing function of K.
Proof. According to eorem 8, where r 1− α s solves the following the ordinary differential equation is increasing with respect to X 0 and the European call option price f Ec is increasing with respect to the initial stock price X 0 . is means that the higher the initial stock price, the higher the European call option price.
is increasing with respect to a 2 and the European call option price f Ec is increasing with respect to the parameter a 2 . (3) Since the function is decreasing with respect to K and the European call option price f Ec is decreasing with respect to the strike price K, this means that the higher the strike price, the lower the European call option price.

European Put Option.
Suppose that a European put option has a strike price K and an expiration time T, and X t is the stock price of the time t. e payoff of the European put option is given by (K − X T ) + .
Definition 8. Assume a European option has a strike price K and an expiration time T. en the European put option price is where r α s solves the following the ordinary differential equation: 4 Discrete Dynamics in Nature and Society Proof. According to the proof of eorem 8, we can get that the discount rate has an α-path Since (K − X T ) + is an decreasing function with respect to X T , it has an α-path erefore, the present value of the option has an α-path according to eorem 7. We have the price of the European put option According to eorems 2 and 4. e theorem is proved. □ Theorem 11. Let f Ep be the European put option price of the uncertain stock model (28). en (1) f Ep is a decreasing function of X 0 (2) f Ep is a decreasing function of a 2 (3) f Ep is an increasing function of K Proof. According to eorem 10, (1) Since exp(− b 2 T) > 0, the function − X 0 exp(− b 2 T) is decreasing with respect to X 0 and the European put option price f Ep is decreasing with respect to the initial stock price X 0 . is means that the higher the initial stock price, the lower the European put option price.
is decreasing with respect to a 2 and the European put option price f Ep is decreasing with respect to the parameter a 2 . (3) Since the function is increasing with respect to K and the European put option price f Ep is increasing with respect to the strike price K, this means that the higher the strike price, the higher the European put option price.

American Call Option.
e American call option gives the holder the right, without obligation, to buy an agreed quantity of stock at any time before the expiration date T with a strike price K. 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 sup 0≤t≤T (X t − K) + .
Definition 9. Assume that the American call option has a strike price K and an expiration time T. en the American call option price is Discrete Dynamics in Nature and Society