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

Authors: Zhaopeng Liu
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## 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 theα-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.


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## Body

## 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 measureℳ 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. The uncertainty distribution Φ of an uncertain variable ξ follows as(1)Φx=ℳξ≤x,for any real number x. If the uncertainty distribution Φx is a continuous and strictly increasing function with respect to x at which 0<Φx<1, and(2)limx⟶−∞Φx=0,limx⟶+∞Φx=1,then Φx is said to be a regular distribution, and the inverse function Φ−1α is called the inverse uncertainty distribution of ξ.Definition 2.
(see [14]). Let ξ be an uncertain variable. Then the expected value of ξ is defined by(3)Eξ=∫0+∞ℳξ≥xdx−∫−∞0ℳξ≤xdx.
provided that at least one of the two integrals is finite.Theorem 1 (see [5]).
Letξ be an uncertain variable with uncertainty distribution Φ. If the expected value exists, then(4)Eξ=∫0+∞1−Φxdx−∫−∞0Φxdx.Theorem 2 (see [14]).
Letξ be an uncertain variable with regular uncertainty distribution Φ. Then(5)Eξ=∫01Φ−1αdα.Definition 3.
(see [6]). The uncertain variables ξ1,ξ2,…,ξn are said to be independent if(6)ℳ∩i=1nξi∈Bi=∧k=1nℳξi∈Bi.
for any Borel setsB1,B2,…,Bn of real numbers.
An uncertain process is a sequence of uncertain variables indexed by a totally ordered setT, which is used to model the evolution of uncertain phenomena.Definition 4.
(see [6]). An uncertain process Ct is said to be a canonical Liu process if(i)
C0=0 and almost all sample paths are Lipschitz continuous;(ii)
Ct has stationary and independent increments;(iii)
every incrementCs+t−Ct is a normal uncertain variable with expected value 0 and variance t2, whose uncertainty distribution is(7)Φx=1+exp−πx3t−1.Definition 5.
(see [6]). Suppose Ct is a canonical Liu process, f and g are two real functions. Then(8)dXt=ft,Xtdt+gt,XtdCt.
is called an uncertain differential equation with an initial valueX0.Definition 6.
(see [15]). Let α be a number with 0<α<1. An uncertain differential equation(9)dXt=ft,Xtdt+gt,XtdCt.
is said to have anα-path Xtα if it solves the corresponding ordinary differential equation(10)dXtα=ft,Xtαdt+gt,XtαΦ−1αdt,X0α=X0,where Φ−1α is the inverse standard normal uncertainty distribution, i.e.,(11)Φ−1α=3πlnα1−α.Theorem 3 (see [15]).
Assume thatXt and Xtα are the solution and α-path of the uncertain differential equation(12)dXt=ft,Xtdt+gt,XtdCt,respectively. Then(13)ℳXt≤Xtα,∀t=α,ℳXt>Xtα,∀t=1−α.Theorem 4 (see [15, 16]).
LetXt and Xtα be the solution and α-path of the uncertain differential equation(14)dXt=ft,Xtdt+gt,XtdCt,respectively. Then, the solution Xt has an inverse uncertainty distribution(15)Ψt−1α=Xtα.Theorem 5 (see [16]).
LetXt and Xtα be the solution and α-path of the uncertain differential equation(16)dXt=ft,Xtdt+gt,XtdCt,respectively. Then, for any time s>0 and strictly increasing function Jx, the supremum(17)sup0≤t≤sJXt,has an inverse uncertainty distribution(18)Ψs−1α=sup0≤t≤sJXtα,and the time integral ∫0sJXtdt has an inverse uncertainty distribution(19)Ψs−1α=∫0sJXtαdt.Theorem 6 (see [16]).
LetXt and Xtα be the solution and α-path of the uncertain differential equation(20)dXt=ft,Xtdt+gt,XtdCt,respectively. Then, for any time s>0 and strictly decreasing function Jx, the supremum(21)sup0≤t≤sJXt,has an inverse uncertainty distribution(22)Ψs−1α=sup0≤t≤sJXt1−α,and the time integral ∫0sJXtdt has an inverse uncertainty distribution(23)Ψs−1α=∫0sJXt1−αdt.Liu [17] proposed that the uncertain processes X1t,X2t,…,Xnt are independent if, for any positive integer k and any times t1,t2,…,tk, the uncertain vectors(24)ξi=Xit1,Xit2,…,Xitk,i=1,2,…,n,are independent.Theorem 7 (see [18]).
Assume thatX1t,X2t,…,Xnt are some independent uncertain processes derived from the solutions of some uncertain differential equations. If the function fx1,x2,…,xn is strictly increasing with respect to x1,x2,…,xm and strictly decreasing with respect to xm+1,xm+2,…,xn , then the uncertain process Xt=fX1t,X2t,…,Xnt has an α-path(25)Xtα=fX1tα,…,Xmtα,Xm+1,t1−α,…,Xnt1−α.


## 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 rt and the stock price Xt follow uncertain differential equations and presented an uncertain stock model with floating interest rate as follows,(26)drt=μ1rtdt+σ1rtdC1t,dXt=μ2Xtdt+σ2XtdC2t,where μ1 and σ1 are the drift and diffusion of the interest rate, respectively, μ2 and σ2 are the drift and diffusion of the stock price, respectively, and C1t and C2t 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.(27)drt=m1−a1rtdt+σ1dC1t,dXt=m2−a2Xtdt+σ2dC2t.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,(28)drt=a1b1−rtdt+σ1rtdC1t,dXt=a2−b2Xtdt+σ2dC2t,where a1 represents the rate of adjustment of rt, b1 represents the average interest rate, σ1 represents the interest rate diffusion, a1,b1,σ1,a2,b2,σ2 are constants, and C1t and C2t 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 timeT with a strike price K, and Xt is the stock price of the time t. The payoff of the European call option is given by XT−K+.Definition 7.
Assume European call option has a strike priceK and an expiration time T. Then the European call option price is(29)fEc=Eexp−∫0TrsdsXT−K+.Theorem 8.
Assume European call option for the uncertain stock model (3) has a strike price K and an expiration time T. Then the European call option price is(30)fEc=∫01exp−∫0Trs1−αdsXTα−K+dα,where rs1−α solves the following the ordinary differential equation(31)drs1−α=a1b1−rs1−αds+3σ1πln1−ααrs1−αds,XTα=X0exp−b2T+1−exp−b2Ta2b2+σ2b23πlnα1−α.Proof.
According to Theorem3, we can get the α-path of the stock price Xt:(32)Xtα=X0exp−b2t+1−exp−b2ta2b2+σ2b23πlnα1−α.
Similarly, we also get thatrs1−α satisfies the differential equation,(33)drs1−α=a1b1−rs1−αds+3σ1πln1−ααrs1−αds.
It follows from Theorem5 that the α-path of ∫0Trsds is ∫0Trsαds.
Sincey=exp−x is strictly decreasing with respect to x, from Theorem 6, the discount rate(34)exp−∫0Trsds,has an α-path(35)exp−∫0Trs1−αds.
SinceXT−K+ is an increasing function with respect to XT, it has an α-path(36)XTα−K+=X0exp−b2T+1−exp−b2Ta2b2+σ2b23πlnα1−α−K+.
Therefore, the present value of the option(37)exp−∫0TrsdsXT−K+,has an α-path(38)exp−∫0Trs1−αdsXTα−K+,according to Theorem 7. We have the price of the European call option(39)fEc=∫01exp−∫0Trs1−αdsXTα−K+dα,according to Theorems 2 and 4. The theorem is proved.Theorem 9.
LetfEc be the European call option price of the uncertain stock model (28). Then(1)
fEc is an increasing function of X0;(2)
fEc is an increasing function of a2;(3)
fEc is a decreasing function of K.Proof.
According to Theorem8,(40)fEc=∫01exp−∫0Trs1−αdsX0exp−b2T+1−exp−b2Ta2b2+σ2b23πlnα1−α−K+dα,where rs1−α solves the following the ordinary differential equation(41)drs1−α=a1b1−rs1−αds+3σ1πln1−ααrs1−αds.(1)
Sinceexp−b2T>0, the function X0exp−b2T>0 is increasing with respect to X0 and the European call option price fEc is increasing with respect to the initial stock price X0. This means that the higher the initial stock price, the higher the European call option price.(2)
Since1−exp−b2T>0, b2>0, the function 1−exp−b2Ta2/b2 is increasing with respect to a2 and the European call option price fEc is increasing with respect to the parameter a2.(3)
Since the function(42)X0exp−b2T+1−exp−b2Ta2b2+σ2b23πlnα1−α−K,is decreasing with respect to K and the European call option price fEc is decreasing with respect to the strike price K, this means that the higher the strike price, the lower the European call option price.Example 1.
Assume the parameters of the interest rate area1=0.05,b1=2,σ1=0.04,r0=0.03, and the parameters of the stock price are a2=6,b2=1,σ2=3,X0=4, the strike price K=5 and the expiration time T=1. Then a European call option price is fEc = 1.2859.


### 4.2. European Put Option

Suppose that a European put option has a strike priceK and an expiration time T, and Xt is the stock price of the time t. The payoff of the European put option is given by K−XT+.Definition 8.
Assume a European option has a strike priceK and an expiration time T. Then the European put option price is(43)fEp=Eexp−∫0TrsdsK−XT+.Theorem 10.
Assume a European option for the uncertain stock model (28) has a strike price K and an expiration time T. Then the European put option price is(44)fEp=∫01exp−∫0TrsαdsK−XTα+dα,where rsα solves the following the ordinary differential equation:(45)drsα=a1b1−rsαds+3σ1πlnα1−αrsαds,XTα=X0exp−b2T+1−exp−b2Ta2b2+σ2b23πlnα1−α.Proof.
According to the proof of Theorem8, we can get that the discount rate(46)exp−∫0Trsds,has an α-path(47)exp−∫0Trs1−αds.
SinceK−XT+ is an decreasing function with respect to XT, it has an α-path(48)K−XT1−α+=K−X0exp−b2T+1−exp−b2Ta2b2+σ2b23πln1−αα+.
Therefore, the present value of the option(49)exp−∫0TrsdsK−XT+,has an α-path(50)exp−∫0Trs1−αdsK−XT1−α+,
according to Theorem7. We have the price of the European put option(51)fEp=∫01exp−∫0Trs1−αdsK−XT1−α+dα=∫01exp−∫0TrsαdsK−XTα+dα.
According to Theorems2 and 4. The theorem is proved.Theorem 11.
LetfEp be the European put option price of the uncertain stock model (28). Then(1)
fEp is a decreasing function of X0(2)
fEp is a decreasing function of a2(3)
fEp is an increasing function of KProof.
According to Theorem10,(52)fEp=∫01exp−∫0TrsαdsK−X0exp−b2T+1−exp−b2Ta2b2+σ2b23πlnα1−α+dα.(1)
Sinceexp−b2T>0, the function −X0exp−b2T is decreasing with respect to X0 and the European put option price fEp is decreasing with respect to the initial stock price X0. This means that the higher the initial stock price, the lower the European put option price.(2)
Since1−exp−b2T>0, b2>0, the function 1−1−exp−b2Ta2/b2 is decreasing with respect to a2 and the European put option price fEp is decreasing with respect to the parameter a2.(3)
Since the function(53)K−X0exp−b2T+1−exp−b2Ta2b2+σ2b23πlnα1−α,is increasing with respect to K and the European put option price fEp is increasing with respect to the strike price K, this means that the higher the strike price, the higher the European put option price.Example 2.
Assume the parameters of the interest rate area1=0.05,b1=2,σ1=0.04,r0=0.03, and the parameters of the stock price are a2=6,b2=1,σ2=3,X0=4, the strike price K=5 and the expiration time T=1. Then a European put option price is fEp = 1.0362.


## 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 timeT with a strike price K, and Xt is the stock price of the time t. The payoff of the European call option is given by XT−K+.Definition 7.
Assume European call option has a strike priceK and an expiration time T. Then the European call option price is(29)fEc=Eexp−∫0TrsdsXT−K+.Theorem 8.
Assume European call option for the uncertain stock model (3) has a strike price K and an expiration time T. Then the European call option price is(30)fEc=∫01exp−∫0Trs1−αdsXTα−K+dα,where rs1−α solves the following the ordinary differential equation(31)drs1−α=a1b1−rs1−αds+3σ1πln1−ααrs1−αds,XTα=X0exp−b2T+1−exp−b2Ta2b2+σ2b23πlnα1−α.Proof.
According to Theorem3, we can get the α-path of the stock price Xt:(32)Xtα=X0exp−b2t+1−exp−b2ta2b2+σ2b23πlnα1−α.
Similarly, we also get thatrs1−α satisfies the differential equation,(33)drs1−α=a1b1−rs1−αds+3σ1πln1−ααrs1−αds.
It follows from Theorem5 that the α-path of ∫0Trsds is ∫0Trsαds.
Sincey=exp−x is strictly decreasing with respect to x, from Theorem 6, the discount rate(34)exp−∫0Trsds,has an α-path(35)exp−∫0Trs1−αds.
SinceXT−K+ is an increasing function with respect to XT, it has an α-path(36)XTα−K+=X0exp−b2T+1−exp−b2Ta2b2+σ2b23πlnα1−α−K+.
Therefore, the present value of the option(37)exp−∫0TrsdsXT−K+,has an α-path(38)exp−∫0Trs1−αdsXTα−K+,according to Theorem 7. We have the price of the European call option(39)fEc=∫01exp−∫0Trs1−αdsXTα−K+dα,according to Theorems 2 and 4. The theorem is proved.Theorem 9.
LetfEc be the European call option price of the uncertain stock model (28). Then(1)
fEc is an increasing function of X0;(2)
fEc is an increasing function of a2;(3)
fEc is a decreasing function of K.Proof.
According to Theorem8,(40)fEc=∫01exp−∫0Trs1−αdsX0exp−b2T+1−exp−b2Ta2b2+σ2b23πlnα1−α−K+dα,where rs1−α solves the following the ordinary differential equation(41)drs1−α=a1b1−rs1−αds+3σ1πln1−ααrs1−αds.(1)
Sinceexp−b2T>0, the function X0exp−b2T>0 is increasing with respect to X0 and the European call option price fEc is increasing with respect to the initial stock price X0. This means that the higher the initial stock price, the higher the European call option price.(2)
Since1−exp−b2T>0, b2>0, the function 1−exp−b2Ta2/b2 is increasing with respect to a2 and the European call option price fEc is increasing with respect to the parameter a2.(3)
Since the function(42)X0exp−b2T+1−exp−b2Ta2b2+σ2b23πlnα1−α−K,is decreasing with respect to K and the European call option price fEc is decreasing with respect to the strike price K, this means that the higher the strike price, the lower the European call option price.Example 1.
Assume the parameters of the interest rate area1=0.05,b1=2,σ1=0.04,r0=0.03, and the parameters of the stock price are a2=6,b2=1,σ2=3,X0=4, the strike price K=5 and the expiration time T=1. Then a European call option price is fEc = 1.2859.


## 4.2. European Put Option

Suppose that a European put option has a strike priceK and an expiration time T, and Xt is the stock price of the time t. The payoff of the European put option is given by K−XT+.Definition 8.
Assume a European option has a strike priceK and an expiration time T. Then the European put option price is(43)fEp=Eexp−∫0TrsdsK−XT+.Theorem 10.
Assume a European option for the uncertain stock model (28) has a strike price K and an expiration time T. Then the European put option price is(44)fEp=∫01exp−∫0TrsαdsK−XTα+dα,where rsα solves the following the ordinary differential equation:(45)drsα=a1b1−rsαds+3σ1πlnα1−αrsαds,XTα=X0exp−b2T+1−exp−b2Ta2b2+σ2b23πlnα1−α.Proof.
According to the proof of Theorem8, we can get that the discount rate(46)exp−∫0Trsds,has an α-path(47)exp−∫0Trs1−αds.
SinceK−XT+ is an decreasing function with respect to XT, it has an α-path(48)K−XT1−α+=K−X0exp−b2T+1−exp−b2Ta2b2+σ2b23πln1−αα+.
Therefore, the present value of the option(49)exp−∫0TrsdsK−XT+,has an α-path(50)exp−∫0Trs1−αdsK−XT1−α+,
according to Theorem7. We have the price of the European put option(51)fEp=∫01exp−∫0Trs1−αdsK−XT1−α+dα=∫01exp−∫0TrsαdsK−XTα+dα.
According to Theorems2 and 4. The theorem is proved.Theorem 11.
LetfEp be the European put option price of the uncertain stock model (28). Then(1)
fEp is a decreasing function of X0(2)
fEp is a decreasing function of a2(3)
fEp is an increasing function of KProof.
According to Theorem10,(52)fEp=∫01exp−∫0TrsαdsK−X0exp−b2T+1−exp−b2Ta2b2+σ2b23πlnα1−α+dα.(1)
Sinceexp−b2T>0, the function −X0exp−b2T is decreasing with respect to X0 and the European put option price fEp is decreasing with respect to the initial stock price X0. This means that the higher the initial stock price, the lower the European put option price.(2)
Since1−exp−b2T>0, b2>0, the function 1−1−exp−b2Ta2/b2 is decreasing with respect to a2 and the European put option price fEp is decreasing with respect to the parameter a2.(3)
Since the function(53)K−X0exp−b2T+1−exp−b2Ta2b2+σ2b23πlnα1−α,is increasing with respect to K and the European put option price fEp is increasing with respect to the strike price K, this means that the higher the strike price, the higher the European put option price.Example 2.
Assume the parameters of the interest rate area1=0.05,b1=2,σ1=0.04,r0=0.03, and the parameters of the stock price are a2=6,b2=1,σ2=3,X0=4, the strike price K=5 and the expiration time T=1. Then a European put option price is fEp = 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 dateT 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 sup0≤t≤TXt−K+.Definition 9.
Assume that the American call option has a strike priceK and an expiration time T. Then the American call option price is(54)fAc=Esup0≤t≤Texp−∫0trsdsXt−K+.Theorem 12.
Assume that the American call option for the uncertain stock model (28) has a strike price K and an expiration time T. Then the American call option price is(55)fAc=∫01sup0≤t≤Texp−∫0trs1−αdsXtα−K+dα,where rs1−α solves the following the ordinary differential equation:(56)drs1−α=a1b1−rs1−αds+3σ1πln1−ααrs1−αds,Xtα=X0exp−b2t+1−exp−b2ta2b2+σ2b23πlnα1−α.Proof.
It follows from Theorem8 that the uncertain process(57)exp−∫0trsdsXt−K+,has an α-path(58)exp−∫0trs1−αdsXtα−K+.
According to Theorem5, the uncertain process(59)sup0≤t≤Texp−∫0trsdsXt−K+,has an α-path(60)sup0≤t≤Texp−∫0trs1−αdsXtα−K+.
We have the price of the American call option(61)fAc=∫01sup0≤t≤Texp−∫0trs1−αdsXtα−K+dα.
According to Theorems2 and 4. The theorem is proved.Theorem 13.
LetfAc be the American call option price of the uncertain stock model (28). Then(1)
fAc is an increasing function of X0(2)
fAc is an increasing function of a2(3)
fAc is a decreasing function of KProof.
According to Theorem12,(62)fAc=∫01sup0≤t≤Texp−∫0trs1−αdsX0exp−b2t+1−exp−b2ta2b2+σ2b23πlnα1−α−K+dα.(1)
Sinceexp−b2t>0, the function X0exp−b2t>0 is increasing with respect to X0 and the American call option price fAc is increasing with respect to the initial stock price X0. This means that the higher the initial stock price, the higher the American call option price.(2)
Since1−exp−b2t>0, b2>0, the function 1−exp−b2ta2/b2 is increasing with respect to a2 and the American call option price fAc is increasing with respect to the parameter a2.(3)
Since the function(63)X0exp−b2t+1−exp−b2ta2b2+σ2b23πlnα1−α−K,is decreasing with respect to K and the American call option price fAc is decreasing with respect to the strike price K. This means that the higher the strike price, the lower the American call option price.Example 3.
Assume the parameters of the interest rate area1=0.05,b1=2,σ1=0.04,r0=0.03, and the parameters of the stock price are a2=6,b2=1,σ2=3,X0=4, the strike price K=5 and the expiration time T=1. Then an American call option price is fAc = 2.3762.


### 5.2. American Put Option

The American put option gives the holder the right, without obligation, to sell an agreed quantity of stock at any time before the expiration dateT 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 put option is given by sup0≤t≤TK−Xt+.Definition 10.
Assume that the American put option has a strike priceK and an expiration time T. Then the American put option price is(64)fAp=Esup0≤t≤Texp−∫0trsdsK−Xt+.Theorem 14.
Assume that the American put option for the uncertain stock model (28) has a strike price K and an expiration time T. Then the American put option price is(65)fAp=∫01sup0≤t≤Texp−∫0trsαdsK−Xtα+dα,where rsα solves the following the ordinary differential equation:(66)drsα=a1b1−rsαds+3σ1πlnα1−αrsαds,Xtα=X0exp−b2t+1−exp−b2ta2b2+σ2b23πlnα1−α.Proof.
It follows from Theorem10 that the uncertain process(67)exp−∫0trsdsK−Xt+,has an α-path(68)exp−∫0trs1−αdsK−Xt1−α+.
According to Theorem6, the uncertain process(69)sup0≤t≤Texp−∫0trsdsK−Xt+,has an α-path(70)sup0≤t≤Texp−∫0trs1−αdsK−Xt1−α+.
So, we have the price of the American put option(71)fAp=∫01sup0≤t≤Texp−∫0trs1−αdsK−Xt1−α+dα=∫01sup0≤t≤Texp−∫0trsαdsK−Xtα+dα,according to Theorems 2 and 4. The theorem is proved.Theorem 15.
LetfAp be the American put option price of the uncertain stock model (28). Then(1)
fAp is a decreasing function of X0(2)
fAp is a decreasing function of a2(3)
fAp is an increasing function of KProof.
According to Theorem14,(72)fAp=∫01sup0≤t≤Texp−∫0trsαdsK−X0exp−b2t+1−exp−b2ta2b2+σ2b23πlnα1−α+dα,(1)
Sinceexp−b2t>0, the function −X0exp−b2t<0 is decreasing with respect to X0 and the American put option price fAp is decreasing with respect to the initial stock price X0. This means that the higher the initial stock price, the lower the American put option price.(2)
Since1−exp−b2t>0, b2>0, the function −1−exp−b2ta2/b2 is decreasing with respect to a2 and the American put option price fAp is decreasing with respect to the parameter a2.(3)
Since the function(73)K−X0exp−b2t+1−exp−b2ta2b2+σ2b23πlnα1−α,is increasing with respect to K and the American put option price fAp is increasing with respect to the strike price K, this means that the higher the strike price, the higher the American put option price.Example 4.
Assume that the parameters of the interest rate area1=0.05,b1=2,σ1=0.04,r0=0.03, and the parameters of the stock price are a2=6,b2=1,σ2=3,X0=4, the strike price K=5 and the expiration time T=1. Then an American put option price is fAp=1.6712.


## 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 dateT 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 sup0≤t≤TXt−K+.Definition 9.
Assume that the American call option has a strike priceK and an expiration time T. Then the American call option price is(54)fAc=Esup0≤t≤Texp−∫0trsdsXt−K+.Theorem 12.
Assume that the American call option for the uncertain stock model (28) has a strike price K and an expiration time T. Then the American call option price is(55)fAc=∫01sup0≤t≤Texp−∫0trs1−αdsXtα−K+dα,where rs1−α solves the following the ordinary differential equation:(56)drs1−α=a1b1−rs1−αds+3σ1πln1−ααrs1−αds,Xtα=X0exp−b2t+1−exp−b2ta2b2+σ2b23πlnα1−α.Proof.
It follows from Theorem8 that the uncertain process(57)exp−∫0trsdsXt−K+,has an α-path(58)exp−∫0trs1−αdsXtα−K+.
According to Theorem5, the uncertain process(59)sup0≤t≤Texp−∫0trsdsXt−K+,has an α-path(60)sup0≤t≤Texp−∫0trs1−αdsXtα−K+.
We have the price of the American call option(61)fAc=∫01sup0≤t≤Texp−∫0trs1−αdsXtα−K+dα.
According to Theorems2 and 4. The theorem is proved.Theorem 13.
LetfAc be the American call option price of the uncertain stock model (28). Then(1)
fAc is an increasing function of X0(2)
fAc is an increasing function of a2(3)
fAc is a decreasing function of KProof.
According to Theorem12,(62)fAc=∫01sup0≤t≤Texp−∫0trs1−αdsX0exp−b2t+1−exp−b2ta2b2+σ2b23πlnα1−α−K+dα.(1)
Sinceexp−b2t>0, the function X0exp−b2t>0 is increasing with respect to X0 and the American call option price fAc is increasing with respect to the initial stock price X0. This means that the higher the initial stock price, the higher the American call option price.(2)
Since1−exp−b2t>0, b2>0, the function 1−exp−b2ta2/b2 is increasing with respect to a2 and the American call option price fAc is increasing with respect to the parameter a2.(3)
Since the function(63)X0exp−b2t+1−exp−b2ta2b2+σ2b23πlnα1−α−K,is decreasing with respect to K and the American call option price fAc is decreasing with respect to the strike price K. This means that the higher the strike price, the lower the American call option price.Example 3.
Assume the parameters of the interest rate area1=0.05,b1=2,σ1=0.04,r0=0.03, and the parameters of the stock price are a2=6,b2=1,σ2=3,X0=4, the strike price K=5 and the expiration time T=1. Then an American call option price is fAc = 2.3762.


## 5.2. American Put Option

The American put option gives the holder the right, without obligation, to sell an agreed quantity of stock at any time before the expiration dateT 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 put option is given by sup0≤t≤TK−Xt+.Definition 10.
Assume that the American put option has a strike priceK and an expiration time T. Then the American put option price is(64)fAp=Esup0≤t≤Texp−∫0trsdsK−Xt+.Theorem 14.
Assume that the American put option for the uncertain stock model (28) has a strike price K and an expiration time T. Then the American put option price is(65)fAp=∫01sup0≤t≤Texp−∫0trsαdsK−Xtα+dα,where rsα solves the following the ordinary differential equation:(66)drsα=a1b1−rsαds+3σ1πlnα1−αrsαds,Xtα=X0exp−b2t+1−exp−b2ta2b2+σ2b23πlnα1−α.Proof.
It follows from Theorem10 that the uncertain process(67)exp−∫0trsdsK−Xt+,has an α-path(68)exp−∫0trs1−αdsK−Xt1−α+.
According to Theorem6, the uncertain process(69)sup0≤t≤Texp−∫0trsdsK−Xt+,has an α-path(70)sup0≤t≤Texp−∫0trs1−αdsK−Xt1−α+.
So, we have the price of the American put option(71)fAp=∫01sup0≤t≤Texp−∫0trs1−αdsK−Xt1−α+dα=∫01sup0≤t≤Texp−∫0trsαdsK−Xtα+dα,according to Theorems 2 and 4. The theorem is proved.Theorem 15.
LetfAp be the American put option price of the uncertain stock model (28). Then(1)
fAp is a decreasing function of X0(2)
fAp is a decreasing function of a2(3)
fAp is an increasing function of KProof.
According to Theorem14,(72)fAp=∫01sup0≤t≤Texp−∫0trsαdsK−X0exp−b2t+1−exp−b2ta2b2+σ2b23πlnα1−α+dα,(1)
Sinceexp−b2t>0, the function −X0exp−b2t<0 is decreasing with respect to X0 and the American put option price fAp is decreasing with respect to the initial stock price X0. This means that the higher the initial stock price, the lower the American put option price.(2)
Since1−exp−b2t>0, b2>0, the function −1−exp−b2ta2/b2 is decreasing with respect to a2 and the American put option price fAp is decreasing with respect to the parameter a2.(3)
Since the function(73)K−X0exp−b2t+1−exp−b2ta2b2+σ2b23πlnα1−α,is increasing with respect to K and the American put option price fAp is increasing with respect to the strike price K, this means that the higher the strike price, the higher the American put option price.Example 4.
Assume that the parameters of the interest rate area1=0.05,b1=2,σ1=0.04,r0=0.03, and the parameters of the stock price are a2=6,b2=1,σ2=3,X0=4, the strike price K=5 and the expiration time T=1. Then an American put option price is fAp=1.6712.


## 6. Conclusions

In this paper, we presented a new uncertain mean-reverting stock model with floating interest rate, in which the interest rate model was the uncertain CIR model. Based on the new model, we derived the pricing formulas of the European option and American option. In addition, some properties of the pricing formulas were also studied. Furthermore, some numerical examples were given. Further research could consider the lookback option pricing and power option pricing problems of the stock model.



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