European Spread Option Pricing with the Floating Interest Rate for Uncertain Financial Market

In this paper, we investigate the pricing problems of European spread options with the floating interest rate. In this model, uncertain differential equation and stochastic differential equation are used to describe the fluctuation of stock price and the floating interest rate, respectively. We derive the pricing formulas for spread options including the European spread call option and the European spread put option. Finally, numerical algorithms are provided to illustrate our results.


Introduction
Since financial derivatives have the function of hedging and risk aversion, they are widely used in the financial market. Options are special financial derivatives, and they are used not only in the adjustment of the national debt but also in the leveraged investment of enterprises and the hedging of commodities. e value of an option depends not only on the price of its underlying asset but also on other factors such as the interest rate. Since Black and Scholes [1] established the B-S model, option pricing has become an important issue in financial mathematical research.
However, the expected return of a stock is different from the assumption of the B-S model in the financial market. In order to better match with the real market, scholars have improved the B-S model. Merton [2] proposed a jump-diffusion model and gave the European option pricing formula. Hull and White [3] studied the pricing problem for the European option under stochastic volatility. Heston [4] investigated the option pricing under stochastic interest rate and studied the pricing problems of the bond and currency option. Stock prices are easily affected by the social environment and investor belief degrees, but the existing models and theories are difficult to give a reasonable explanation. For dealing with the uncertainty of human behavior, Liu [5] founded the uncertainty theory based on normality, duality, subadditivity, and product axioms. To describe uncertain dynamic systems, Liu [6] introduced the concept of uncertain processes and proposed uncertain differential equations driven by canonical Liu process. Liu [7] proposed an uncertain stock model in which stock price was described by an uncertain differential equation. Subsequently, Chen [8] and Zhang and Liu [9] gave the pricing formulas for the European option, American option, and geometric Asian option, respectively. Considering the uncertain fluctuations of the interest rate, Chen and Gao [10] firstly studied the term structure of the uncertain interest rate. Yao [11] proposed an uncertain stock model with floating interest rate. Zhang et al. [12] derived the pricing formulas of interest rate ceiling and interest rate floor. Sun and Su [13] studied the pricing problems for the European and American option under the mean-reverting stock model. Gao et al. [14] discussed the pricing formulas of lookback options based on the uncertain exponential Ornstein-Uhlenbeck model. e latest research on the applications of uncertainty theory and probability theory was given by Gao [15], Lu and Zhu [16], Li et al. [17], Yu et al. [18,19], and Zhang and Sun [20].
In this paper, we investigate the pricing problems for European spread options. It is well known that spread options are path-dependent exotic options whose returns depend on the spread of two or more assets. Although spread options are widely traded in different financial markets, it is still difficult to price such options. At present, there are relatively few research studies on the pricing of spread options. It is well known that interest rate is an important factor influencing the price of financial derivatives, and it is often fluctuated by stochastic factors such as economy and policy. Two stocks issued newly are considered in this paper, so the lack of historical data leads to the unsuccessful investigation on the price of spread options based on probability theory. us, basing uncertainty theory and probability theory (socalled chance theory), we study the pricing of spread options with a stochastic interest rate under uncertain environment. We introduce some basic knowledge of uncertainty theory and chance theory in Section 2. en, we derive European spread call option and put option pricing formulas with stochastic interest rates and also give some numerical algorithms to calculate the prices in Section 3. Finally, a brief conclusion is given in Section 4.

Preliminaries
is section mainly introduces uncertainty theory and chance theory. Uncertainty theory is an effective tool for dealing with reliability issues related to human uncertainty, while chance theory is a basic tool to deal with complex systems with randomness and uncertainty.
is section provides some basic definitions and results on uncertainty theory and chance theory. For more details, please see Liu's latest book [21].

Uncertain Variable
Definition 1 (see Liu [5,7]). Let L be a σ-algebra on a nonempty set Γ. A set function M: L ⟶ [0, 1] is called an uncertain measure if it satisfies the following axioms: Axiom 4 (product axiom): let (Γ k , L k , M k ) be uncertainty spaces for k � 1, 2, . . .. e product uncertain measure M is an uncertain measure satisfying where Λ k are arbitrarily chosen events from L k for k � 1, 2, . . ., respectively.
Definition 2 (see Liu [5]). An uncertain variable is a function from an uncertainty space (Γ, L, M) to the set of real numbers; for any Borel set B of real numbers, the set is an event.
is said to be regular if it is a continuous and strictly increasing function with respect to x at which 0 < Φ(x) < 1, and

(4)
If ξ has a regular uncertainty distribution Φ(x), then the inverse function Φ − 1 (α) is called the inverse uncertainty distribution of ξ.
Definition 3 (see Liu [7]). e uncertain variables for any Borel sets B 1 , B 2 , . . . , B m of real numbers. Liu [22] proposed the operation law of uncertain variables and calculated the inverse uncertainty distribution of strictly monotone function of uncertain variables.
Theorem 1 (see Liu [22]). Let ξ 1 , ξ 2 , . . . , ξ n be independent uncertain variables with uncertainty distributions . , x m and strictly decreasing with x m+1 , x m+2 , . . . , x n , then is an uncertain variable with inverse uncertainty distribution Definition 4 (see Liu [5]). e expected value of an uncertain variable ξ is defined by provided that at least one of the two integrals exists.
For an uncertain variable ξ with an uncertainty distribution Φ(x), if its expected value exists, Liu [5] showed that 2 Mathematical Problems in Engineering Theorem 2 (see Liu [22]). Assume the uncertain variable ξ has a regular uncertainty distribution Φ; then, 2.1.2. Uncertain Differential Equations. Liu [6] proposed the concept of uncertain process and defined the time integral of uncertain process.
Definition 5 (see Liu [6]). Let T be an index set, and let (Γ, L, M) be an uncertainty space. An uncertain process is a measurable function from T × (Γ, L, M) to the set of real numbers; for each t ∈ T and any Borel set B, is an event.
An uncertain process X t is said to have independent increments if X t 0 , X t 1 − X t 0 , X t 2 − X t 1 , . . . , X t k − X t k− 1 are independent uncertain variables, where t 0 is the initial time and t 1 , t 2 , . . . , t k are any times with t 0 < t 1 < · · · < t k . An uncertain process X t is said to have stationary increments if for any given t > 0, the increments X s+t − X s are identically distributed uncertain variables for all s > 0.
Definition 6 (see Liu [7]). An uncertain process C t is said to be a Liu process if (i) C t � 0, and almost all sample paths are Lipschitz continuous. (ii) C t has stationary and independent increments. (iii) Every increment C s+t − C s is a normal uncertain variable with expected value 0 and variance t 2 , whose uncertainty distribution is Definition 7 (see Liu [5]). Let X t be an uncertain process, and let C t be a Liu process. For any partition of closed interval [a, b] with a � t 1 < t 2 < · · · < t k+1 � b, the mesh is written as en, Liu integral of X t with respect to C t is defined as provided that the limit exists almost surely and is finite. In this case, the uncertain process is said to be integrable.
Definition 8 (see Liu [6]). Suppose C t is a Liu process, and f and g are two functions. en, is called an uncertain differential equation. A solution is an uncertain process X t that satisfies the equation identically in t.
Definition 9 (see Yao and Chen [23]). 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 uncertainty distribution of the standard normal uncertain variable Theorem 3 (see Yao and Chen [23]). Let X t and X α t be the solution and α-path of the uncertain differential equation respectively. en,
en, the chance measure of Θ is defined as Theorem 4 (see Liu [24]). Let ξ be an uncertain random variable on the chance space (Γ, L, M) × (Ω, F, P), and let B be a Borel set of real numbers. en, ξ ∈ B { } is an uncertain random event with chance measure Definition 12 (see Liu [24]). Let ξ be an uncertain random variable. en, its chance distribution is defined by for any x ∈ R.
Definition 13 (see Liu [24]). Let ξ be an uncertain random variable. en, its expected value is defined by provided that at least one of the two integrals is finite.

Corollary 1.
has an expected value Proof. From eorem 7, the expectation of the uncertain random variable ξ can be given by the following equation: □

European Spread Option Pricing Formulas
In this section, we discuss the pricing of spread options under the stochastic interest rate environment where stock prices follow uncertain exponential Ornstein-Uhlenbeck process and uncertain log-normal process, respectively. Firstly, we assume interest rate r, one stock price X, and another stock price Y which satisfy the following differential equation, respectively: where a, b, σ 0 , μ 1 , c, σ 1 , μ 2 , σ 2 are some positive real numbers, W t is a Brownian motion, and C 1t and C 2t are independent canonical Liu processes. en, the inverse uncertainty distribution of X t , Y t is given by Dai et al. [26] and Yao and Chen [23]. e specific form is as follows: In the next sections, the pricing formulas of the European spread call option and the European spread put option are derived.

European Spread Call
Option. Suppose X t and Y t are two asset price processes. en, the European spread call option with maturity date T and strike price K is the contract that pays Considering time value of the stock return, the present value of the payoff is European spread call option should be the expectation of the discounted value of the stock return. So, the European spread call option has a price Proof. According to Corollary 1, European spread call option is given by the following form: Since X t , Y t are independent uncertain processes, their α-path are, respectively, given by formulas (36) and (37). According to eorem 1, the value of investor's return difference at the maturity date T, has the α-path So, the expectation of the uncertain variable (X T − Y T − K) + can be derived easily: us, the price of the European spread call option is (46) e pricing formula of the European spread call option is derived.
From eorem 8, the algorithm designed for calculating European spread call option price is divided by two parts. In the first procedure, E[exp(− T 0 r t dt)] is calculated by Monte Carlo simulation. In the second procedure, E[(X T − Y T − K) + ] can be calculated by using the property of inverse distribution. In the final procedure, European spread call option price is derived by virtue of eorem 8.
In the first procedure, we will calculate E[exp(− T 0 r t dt)]. We will first generate some sample trajectories by stochastic simulations and calculate the average of the terminal trajectories.
Step 5: calculate the value of the stochastic interest rate at the time t j : where ε k j ∼ N(0, 1) are generated by stochastic simulations.
Step 6: calculate the discount rate: Step In the second procedure, we will calculate E[(X T − Y T − K) + ].
Step 3: calculate the inverse uncertainty distribution of the stock processes: Mathematical Problems in Engineering