1 Introduction

An option, as one of the most well-known derivatives, gives its owner the right to trade a certain quantity of underlying assets at a certain price on or before a given date. The call option gives the right of buying the assets, while the put option gives the right of selling. As a result, the value of an option largely depends on the prices of the underlying assets. Many researchers have been devoted to exploring the inner correlation between the option price and the asset price, especially the stock price. In 1900, the Brownian motion was firstly used in modeling the fluctuation of stock price. In order to insure the positive value of stock, the geometric Brownian motion was put forward to describe the change of stock price over time instead of the Brownian motion. Then, Black and Scholes (1973) as well as Merton (1973) constructed the Black–Scholes formula to price the European options.

Although the Black–Scholes formula is generally accepted and applied in estimating the inner-value of derivatives, its rationality relies largely on the estimation of some parameters like the price diffusion and yield rate of the underlying asset which ask for the sufficiency of historical data. But actually in many practical situations, there are not enough historical data available, which means it is unreliable to insist on considering the pricing problems within the framework of the probability theory. An alternative solution is to take some experts’ advice to get a reasonable prediction of the underlying assets. However, from the view of Bayesianism, the belief degree cannot be simply dealt with via probability under situations with few samples available. On account of this circumstance, the uncertainty theory was founded by Liu (2007) and refined by Liu (2009) based on normality, duality, subadditivity and product axioms. Nowadays, it has become a branch of axiomatic mathematics to model human’s uncertain behavior. In order to describe uncertain variables, Liu (2007) employed the definitions of uncertainty distribution, expected value and variance. Liu and Ha (2010) and Yao (2015) presented some formulas to calculate the expected value and variance of an uncertain variable. Sheng and Kar (2015) generalized these results to the moments of uncertain variables.

The uncertain process, as a sequence of uncertain variables, was presented to put the uncertainty theory into the area of dynamic analysis. Similar to the Brownian motion, the Liu process which is a stationary and independent increment process with normal increments was designed to found the theories of uncertain calculus and uncertain differential equation. The existence and uniqueness of the solution of an uncertain differential equation was verified by Chen and Liu (2010) under the linear growth condition and Lipschitz condition, and the stability of the equation was investigated by Yao et al. (2013), Tao and Zhu (2016), Yang et al. (2017) and Liu and Zhang (2020). Most importantly, Yao and Chen (2013) put forward the Yao–Chen formula, which reveals the characteristic of the solution of an uncertain differential equation by ordinary differential equations and constructs the base of many numerical algorithms to solve the pricing problems in uncertain finance.

Based on an assumption that the stock price follows a geometric Liu process, Liu (2009) firstly introduced the uncertainty theory into finance and proposed an uncertain stock model. Then, Chen (2011) and Sun and Chen (2015) derived the pricing formulas for American options and Asian options, respectively. However, Liu’s stock model is so simplified that it neglects many characteristics during the fluctuation of stock price, which inspired Dai et al. (2017) to construct a stock model with the reversion behavior by means of an uncertain counterpart of the exponential Ornstein–Uhlenbeck process. There are some other works about uncertain stock models and their derivatives such as Yu and Ning (2019), Gao et al. (2019), Lu et al. (2019), Yang et al. (2019), Zhang et al. (2019) and Wang and Chen (2020). However, all these models fail to take the dividends payments, a benefit to stock owners, into consideration. Theoretically, dividends payment may exert an influence on the yield rate or diffusion of a stock because it reflects some information about the listed company. According to Howatt et al. (2009) and Hussainey et al. (2016), there is a clearly positive relationship between dividend yield and stock price. Hence, the neglect of dividends in these models will lead to an inaccurate estimation of the stock price.

Chen et al. (2013) developed Liu’s stock model by taking periodic dividends into consideration. However, the modified model remains failed to describe the stock price in long run, as the mean-reversion behavior is still neglected. For this reason, we aim to present and investigate a new stock model in this paper which considers both the periodic dividends and the reversion property of the stock price. The rest of the paper is organized as follows. In the next section, some preliminaries about uncertain differential equations and uncertain stock models are introduced. Then, the uncertain exponential Ornstein–Uhlenbeck model with periodic dividends is presented in Sect. 3, whose European call and put option pricing formulas are derived in Sects. 4 and 5, respectively. Numerical experiments are also performed in these two sections. Finally, some conclusions are made in Sect. 6.

2 Preliminary

The uncertainty theory is a branch of axiomatic mathematics and has been put into practice for modeling people’s belief degrees. In this section, we introduce some concepts about uncertain differential equations and stock models.

2.1 Uncertain differential equation

Definition 1

(Liu 2007, 2009) Let be a \(\sigma \)-algebra on a non-empty set \(\Gamma \). A set function : is called an uncertain measure if it satisfies the following axioms:

Axiom 1: (normality axiom) for the universal set \(\Gamma \).

Axiom 2: (duality axiom) for any event \(\Lambda \).

Axiom 3: (subadditivity axiom) For every countable sequence of events \(\Lambda _1, \Lambda _2,\ldots \), we have

Axiom 4: (product axiom) Let be uncertainty spaces for \(k=1,2,\ldots \). The product uncertain measure is an uncertain measure satisfying

where

$$\begin{aligned} \prod _{k=1}^{\infty }\Lambda _k=\{(\gamma _1,\gamma _2,\ldots )\,|\,\gamma _k\in \Lambda _k \text{ for } k=1,2,\ldots \}. \end{aligned}$$

An uncertain variable \(\xi \) is a measurable function from an uncertainty space to the set of real numbers. The uncertainty distribution \(\Phi (x)\) of \(\xi \) is defined by for any real number x, and its inverse function \(\Phi ^{-1}(\alpha )\) is called the inverse uncertainty distribution. For example, a normal uncertain variable \(\xi \sim \mathcal{N}(e,\sigma )\) has an uncertainty distribution

$$\begin{aligned} \Phi (x)=\left( 1+\exp {\left( \frac{\pi \left( e-x \right) }{\sqrt{3} \sigma }\right) } \right) ^{-1} \end{aligned}$$

and an inverse uncertainty distribution

$$\begin{aligned} \Phi ^{-1}(\alpha )=e+\frac{\sqrt{3}\sigma }{\pi }\ln {\frac{\alpha }{1-\alpha }}. \end{aligned}$$

Theorem 1

(Liu 2007, 2010) The expected value of an uncertain variable \(\xi \) is

provided that at least one of the two integrals exists and is finite.

In order to model the evolution of uncertain phenomena, the concept of uncertain process is proposed as a sequence of uncertain variables indexed by the time.

Definition 2

(Liu 2009) An uncertain process \(C_t\) is called a canonical Liu process if

  1. (i)

    \(C_0=0\) and almost all sample paths are Lipschitz continuous,

  2. (ii)

    \(C_t\) has stationary and independent increments,

  3. (iii)

    every increment \(C_{s+t}-C_s\) is a normal uncertain variable with an uncertainty distribution

    $$\begin{aligned} \Phi _t(x)=\left( 1+\exp \left( \frac{-\pi x}{\sqrt{3}t}\right) \right) ^{-1}. \end{aligned}$$

Let f(t, x) and g(t, x) be two real functions. Then, given an initial value \(X_0\), the differential equation

$$\begin{aligned} \mathrm{d}X_t=f(t,X_t)\mathrm{d}t+g(t,X_t)\mathrm{d}C_t \end{aligned}$$

is called an uncertain differential equation. Its \(\alpha \)-path \((0< \alpha < 1)\) is a deterministic function \(X_{t}^{\alpha }\) with respect to t that solves the corresponding ordinary differential equation

$$\begin{aligned} \mathrm{d}X_{t}^{\alpha }=f(t, X_{t}^{\alpha })\mathrm{d}t+\arrowvert g(t, X_{t}^{\alpha })\arrowvert \Phi ^{-1}(\alpha )\mathrm{d}t,\quad X_0^{\alpha }=X_0, \end{aligned}$$

where

$$\begin{aligned} \Phi ^{-1}(\alpha )=\frac{\sqrt{3}}{\pi }\ln \frac{\alpha }{1-\alpha },\qquad 0<\alpha <1 \end{aligned}$$

is the inverse uncertainty distribution of a standard normal uncertain variable.

Theorem 2

(Yao-Chen formula, Yao and Chen 2013) Assume that \(X_t\) and \(X_t^\alpha \) are the solution and \(\alpha \)-path of the uncertain differential equation

$$\begin{aligned} \mathrm{d}X_t=f(t,X_t)\mathrm{d}t+g(t,X_t)\mathrm{d}C_t \end{aligned}$$

with a crisp initial value. Then, we have

Furthermore, the solution \(X_t\) has an inverse uncertainty distribution

$$\begin{aligned} \Psi _s^{-1}(\alpha )=X_t^{\alpha }. \end{aligned}$$

2.2 Uncertain stock models

The uncertainty theory was firstly applied by Liu (2009) to simulating stock price. Denoting the stock price with \(X_t\) and the bond price with \(Y_t\), Liu’s stock model is formulated as

$$\begin{aligned} \left\{ \begin{array}{l} \mathrm{d}X_t=\mu X_t \mathrm{d}t + \sigma X_t \mathrm{d}C_t\\ \mathrm{d}Y_t=r Y_t \mathrm{d}t, \end{array}\right. \end{aligned}$$
(1)

where \(\mu \) is the stock drift, \(\sigma \) is the stock diffusion, \(C_t\) is a canonical Liu process and r is the risk-free interest rate. Later on, Dai et al. (2017) presented an uncertain counterpart of the exponential Ornstein–Uhlenbeck stock model as follows:

$$\begin{aligned} \left\{ \begin{array}{l} \mathrm{d}X_t=\mu (1-c\ln X_t)X_t \mathrm{d}t + \sigma X_t \mathrm{d}C_t\\ \mathrm{d}Y_t=r Y_t \mathrm{d}t, \end{array}\right. \end{aligned}$$
(2)

where \(\mu \), c,  \(\sigma \) and r are some positive constants. This nonlinear stock model is more rational in long run as it describes the reversion behavior of stock price, that is, the stock price fluctuates around some value.

3 Exponential Ornstein–Uhlenbeck model with dividends

Based on the assumption in the model of Dai et al. (2017), that is, the stock price follows an uncertain exponential Ornstein–Uhlenbeck process, we construct a new model with the consideration of periodic dividends as follows:

$$\begin{aligned} \left\{ \begin{array}{l} \mathrm{d}X_t=\mu _i (1-c_i\ln X_t)X_t \mathrm{d}t \\ \quad + \sigma _i X_t \mathrm{d}C_t,\quad t_i \le t<t_{i+1}, i=0,1,2,3,\ldots \\ X_{t_{i+1}}=\left( 1-\delta \right) \lim \nolimits _{\ t \rightarrow t_{i+1}^- }{X_t},\quad i=0,1,2,3,\ldots \\ \mathrm{d}Y_t=r Y_t \mathrm{d}t, \end{array}\right. \end{aligned}$$
(3)

in which \(t_i\) is the time to pay the ith dividend, \(\delta \) is the proportion of the dividend, \(\mu _i\), \(c_i\) and \(\sigma _i\) are some parameters related to the stock price after the ith dividend and r is the risk-free interest rate.

Now, we discuss the solution and the \(\alpha \)-path of the uncertain exponential Ornstein–Uhlenbeck model (3) with dividends.

Theorem 3

Suppose that the stock price \(X_t\) follows the model

$$\begin{aligned} \left\{ \begin{array}{l} \mathrm{d}X_t=\mu _i (1-c_i \ln X_t)X_t \mathrm{d}t + \sigma _i X_t \mathrm{d}C_t,\quad t_i \le t<t_{i+1} \\ X_{t_{i+1}}=\left( 1-\delta \right) \lim \nolimits _{\ t \rightarrow t_{i+1}^- }{X_t} \end{array}\right. \nonumber \\ \end{aligned}$$
(4)

with an initial price \(X_0\), in which \(t_1, t_2, t_3,\ldots \) are the times to pay dividends. Then,

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle X_t=\exp \left( \frac{1}{c_i}+\exp \left( -\mu _i c_i \left( t-t_i\right) \right) \left( \ln X_{t_i}-\frac{1}{c_i}\right) \right. \\ \quad \left. +\sigma _i \int _{t_i}^{t}\exp \left( \mu _i c_i s-\mu _i c_i t\right) \mathrm{d}C_s \right) ,\ t_i \le t < t_{i+1} \\ X_{t_{i+1}}=\left( 1-\delta \right) \lim \nolimits _{\ t \rightarrow t_{i+1}^- }{X_t}. \end{array}\right. \end{aligned}$$
(5)

Proof

From the stock model (4), we have

$$\begin{aligned} \frac{\mathrm{d}X_t}{X_t}=\mu _i (1-c_i \ln X_t)\mathrm{d}t + \sigma _i \mathrm{d}C_t,\quad t_i\le t<t_{i+1}, \end{aligned}$$

which means

$$\begin{aligned} \mathrm{d}\ln X_t=\mu _i (1-c_i \ln X_t)\mathrm{d}t + \sigma _i \mathrm{d}C_t,\quad t_i\le t<t_{i+1}. \end{aligned}$$

Substituting \(\ln X_t\) with \(Z_t\), we get the uncertain differential equation

$$\begin{aligned} \mathrm{d}Z_t=\mu _i (1-c_i Z_t)\mathrm{d}t + \sigma _i \mathrm{d}C_t,\quad t_i \le t<t_{i+1} \end{aligned}$$

whose solution is

$$\begin{aligned} Z_t= & {} \frac{1}{c_i}+\exp \left( -\mu _i c_i (t-t_i)\right) \left( Z_{t_i} - \frac{1}{c_i}\right) \\&+\, \sigma _i\int _{t_i}^{t}\exp \left( \mu _i c_i s - \mu _i c_i t\right) \mathrm{d}C_s,\quad t_i \le t < t_{i+1}. \end{aligned}$$

As \(X_t=\exp (Z_t)\), we have

$$\begin{aligned} X_t= & {} \exp \left( \frac{1}{c_i}+\exp \left( -\mu _i c_i \left( t-t_i\right) \right) \left( \ln X_{t_i}-\frac{1}{c_i}\right) \right. \\&\left. +\,\sigma _i \int _{t_i}^{t}\exp \left( \mu _i c_i s-\mu _i c_i t\right) \mathrm{d}C_s \right) ,\ t_i \le t < t_{i+1}. \end{aligned}$$

The theorem is proved. \(\square \)

Theorem 4

Suppose that the stock price \(X_t\) follows the model (4). Then, the \(\alpha \)-path of \(X_t\) is

$$\begin{aligned} \left\{ \begin{array}{l} X_t^\alpha =\exp \left( \ln X_{t_i}^\alpha \cdot \exp \left( -\mu _i c_i\left( t-t_i\right) \right) \right. \\ \quad \left. +\gamma _i \left( 1-\exp \left( -\mu _i c_i \left( t-t_i\right) \right) \right) \right) ,\ t_i \le t < t_{i+1} \\ X_{t_{i+1}}^{\alpha }=\left( 1-\delta \right) \lim \nolimits _{\ t \rightarrow t_{i+1}^- }{X_t^\alpha } \end{array}\right. , \end{aligned}$$
(6)

where

$$\begin{aligned} \gamma _i=\left( \frac{1}{c_i}+\frac{\sigma _i}{\mu _i c_i}\cdot \frac{\sqrt{3}}{\pi }\ln \frac{\alpha }{1-\alpha }\right) . \end{aligned}$$

Proof

From the stock model (4), we have

$$\begin{aligned} {\mathrm{d}X_t^ \alpha }= & {} \mu _i (1-c_i \ln X_t^\alpha )X_t^\alpha \mathrm{d}t\\&+ \sigma _i X_t^\alpha \Phi ^{-1}(\alpha )\mathrm{d}t,\quad t_i\le t<t_{i+1}, \end{aligned}$$

which means

$$\begin{aligned} \frac{\mathrm{d}X_t^\alpha }{X_t^\alpha }=\mu _i (1-c_i \ln X_t^\alpha ) \mathrm{d}t + \sigma _i \Phi ^{-1}(\alpha )\mathrm{d}t, \end{aligned}$$

where

$$\begin{aligned} \Phi ^{-1}(\alpha )=\frac{\sqrt{3}}{\pi }\ln \frac{\alpha }{1-\alpha }. \end{aligned}$$

Substituting \(\ln X_t^\alpha \) into \(Z_t^\alpha \), we have

$$\begin{aligned} \mathrm{d}Z_t^{\alpha }=\mu _i (1-c_i Z_t^\alpha ) \mathrm{d}t + \sigma _i \Phi ^{-1}\left( \alpha \right) \mathrm{d}t,\quad t_i \le t<t_{i+1}, \end{aligned}$$

whose solution is

$$\begin{aligned} Z_t^\alpha= & {} Z_{t_i}^\alpha \cdot \exp \left( -\mu _i c_i\left( t-t_i\right) \right) \\&+\,\gamma _i \left( 1-\exp \left( -\mu _i c_i \left( t-t_i\right) \right) \right) ,\ t_i \le t < t_{i+1} \end{aligned}$$

with

$$\begin{aligned} \gamma _i=\left( \frac{1}{c_i}+\frac{\sigma _i}{\mu _i c_i}\cdot \frac{\sqrt{3}}{\pi }\ln \frac{\alpha }{1-\alpha }\right) . \end{aligned}$$

As \(X_t^\alpha =\exp (Z_t^\alpha )\), we have

$$\begin{aligned} X_t^\alpha= & {} \exp \left( \ln X_{t_i}^\alpha \cdot \exp \left( -\mu _i c_i\left( t-t_i\right) \right) \right. \\&\left. +\,\gamma _i\left( 1-\exp \left( -\mu _i c_i \left( t-t_i\right) \right) \right) \right) ,\ t_i \le t < t_{i+1}. \end{aligned}$$

Obviously,

$$\begin{aligned} X_{t_{i+1}}^{\alpha }=\left( 1-\delta \right) \lim \limits _{\ t \rightarrow t_{i+1}^- }{X_t^\alpha }. \end{aligned}$$

The theorem is proved. \(\square \)

Example 1

Assume the initial value of the stock price is \(X_0=25\), and in the next periods, the dividends will be paid by 5 % for 3 times at \(t_1=0.3\), \(t_2=0.6\), \(t_3=0.9\). The other parameters of this stock are \(\mu _0=0.5\), \(\mu _1=0.7\), \(\mu _2=1\), \(\mu _3=1.1\), \(c_0=0.01\), \(c_1=0.03\), \(c_2=0.05\), \(c_3=0.07\), and \(\sigma _0=0.4\), \(\sigma _1=0.4\), \(\sigma _2=0.3\), \(\sigma _3=0.3\). Then three \(\alpha \)-paths of the possible stock price with \(\alpha _1=0.1\), \(\alpha _2=0.5\) and \(\alpha _3=0.9\), respectively, are shown in Fig. 1.

Fig. 1
figure 1

Three \(\alpha \)-paths of \(X_t\)

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4 European call option pricing formula

A European call option is a kind of financial derivatives that can give its buyer the right rather than the obligation to buy a certain kind of financial assets for a specified price at a specified time. Consider a European call option with its underlying stock following

$$\begin{aligned} \left\{ \begin{array}{l} \mathrm{d}X_t=\mu _i (1-c_i \ln X_t)X_t \mathrm{d}t + \sigma _i X_t \mathrm{d}C_t,\quad t_i \le t<t_{i+1} \\ X_{t_{i+1}}=\left( 1-\delta \right) \lim \nolimits _{\ t \rightarrow t_{i+1}^- }{X_t}. \end{array}\right. \end{aligned}$$
(7)

Suppose the strike price and the expiration time of this option are K and T, respectively. Then, its expected payoff at the expiration time is \(E\left[ (X_T-K)^+ \right] \). Assume the risk-free interest rate remains r during this period. Then, based on the assumption of continuous compounding, the price of a European call option is the present value of its expected payoff, i.e.,

$$\begin{aligned} C=\exp \left( -rT \right) E\left[ (X_T-K)^+ \right] . \end{aligned}$$
(8)

Theorem 5

(European call option pricing formula) Suppose a European call option with an underlying stock following model (7) has a strike price K and an expiration time T. Then, the pricing formula for this option is

$$\begin{aligned} C=\exp (-rT) \int _{0}^{1} (X_T^\alpha -K)^+ \mathrm{d}\alpha , \end{aligned}$$
(9)

where

$$\begin{aligned} X_T^\alpha= & {} \exp \left( \ln X_{t_i}^\alpha \cdot \exp \left( -\mu _i c_i\left( T-t_i\right) \right) \right. \\&\left. +\,\gamma _i \left( 1-\exp \left( -\mu _i c_i \left( T-t_i\right) \right) \right) \right) , \text{ if } t_i \le T < t_{i+1} \end{aligned}$$

with

$$\begin{aligned} \gamma _i=\left( \frac{1}{c_i}+\frac{\sigma _i}{\mu _i c_i}\cdot \frac{\sqrt{3}}{\pi }\ln \frac{\alpha }{1-\alpha }\right) \end{aligned}$$

and \(X_{t_i}^\alpha \) is determined by Eq. (6).

Proof

Since \(X_t\) is an uncertain process with \(\alpha \)-path represented by Eq. (6), the inverse uncertainty distribution of \(X_T\) is \(X_T^\alpha \). So, the payoff of this option has an inverse uncertainty distribution \((X_T^\alpha -K)^+\). Hence,

$$\begin{aligned} {E}\left[ (X_T-K)^+\right] =\int _{0}^{1}(X_T^\alpha -K)^+\,\mathrm{d}\alpha . \end{aligned}$$

It follows from Eq. (8) that

$$\begin{aligned} C=\exp (-rT) \int _{0}^{1} (X_T^\alpha -K)^+\, \mathrm{d}\alpha . \end{aligned}$$

The pricing formula of European call option is verified. \(\square \)

According to Theorem 5, the algorithm to calculate the European call option price of stock model (3) is designed as follows:

Fig. 2
figure 2

European call option price with respect to expiration time T in Example 2

Full size image

Step 0 Choose a large number N according to the desired precision degree. Set \(\alpha _k = k/N\), \(k = 1, 2,\ldots , N-1.\)

Step 1 Set \(k = 0\).

Step 2 Set \(k \leftarrow k+1\).

Step 3 Find the maximum m such that \(t_m \le T\), and calculate the stock price at dividend payment times \(t_1, t_2,\ldots , t_m\) successively via

$$\begin{aligned} \left\{ \begin{aligned} X_{t_1}^{\alpha _k}&=\exp \left( \ln X_0\cdot \exp \left( -\mu _0 c_0 t_1\right) \right. \\&\quad \left. +\gamma _0^{\alpha _k} \left( 1-\exp \left( -\mu _0 c_0 t_1 \right) \right) \right) \cdot (1-\delta ) \\ X_{t_2}^{\alpha _k}&= \exp \left( \ln X_{t_1}^{\alpha _k}\cdot \exp \left( -\mu _1 c_1\Delta t_2\right) \right. \\&\quad \left. +\gamma _1^{\alpha _k} \left( 1-\exp \left( -\mu _1 c_1 \Delta t_2\right) \right) \right) \cdot (1-\delta ) \\ \cdots \\ X_{t_m}^{\alpha _k}&=\exp \left( \ln X_{t_{m-1}}^{\alpha _k}\cdot \exp \left( -\mu _{m-1} c_{m-1}\Delta t_m \right) \right. \\&\quad \left. +\gamma _{m-1}^{\alpha _k} \left( 1-\exp \left( -\mu _{m-1} c_{m-1}\Delta t_m\right) \right) \right) \cdot (1-\delta ), \end{aligned}\right. \end{aligned}$$

in which

$$\begin{aligned} \Delta t_i=t_i-t_{i-1} \end{aligned}$$

and

$$\begin{aligned} \gamma _i^{\alpha _k}=\left( \frac{1}{c_{i}}+\frac{\sigma _{i}}{\mu _{i} c_{i}}\cdot \frac{\sqrt{3}}{\pi }\ln \frac{\alpha _k}{1-\alpha _k}\right) . \end{aligned}$$

If \(T=t_m\), then go to Step 5 .

Step 4 Calculate the stock price at time T, that is,

$$\begin{aligned}&X_{T}^{\alpha _k}=\exp \left( \ln X_{t_m}^{\alpha _k}\cdot \exp \left( -\mu _m c_m\left( T-t_m\right) \right) \right. \\&\quad \left. +\gamma _m \left( 1-\exp \left( -\mu _m c_m \left( T-t_m\right) \right) \right) \right) , \end{aligned}$$

where

$$\begin{aligned} \gamma _m^{\alpha _k}=\left( \frac{1}{c_{m}}+\frac{\sigma _{m}}{\mu _{m} c_{m}}\cdot \frac{\sqrt{3}}{\pi }\ln \frac{\alpha _k}{1-\alpha _k}\right) . \end{aligned}$$

Step 5 Calculate the payoff of the European call option

$$\begin{aligned} (X_T^{\alpha _k}-K)^+=\max (0,X_T^{\alpha _k}-K). \end{aligned}$$

If \(k<N-1\), then return to Step 2.

Step 6 Calculate the price of the European call option

$$\begin{aligned} C \leftarrow \frac{1}{N-1}\exp (-rT)\sum _{i=1}^{N-1}(X_T^{\alpha _k}-K)^+. \end{aligned}$$

Example 2

Assume a European call option has an underlying stock following model (7), and the parameters are the same as Example 1. The strike price is \(K=30\) and the risk-free interest rate is \(r=0.03\). Then, the price of this European call option is \(C=14.53\) if the expiration time is \(T=1\). Figure 2 shows the price of this European call option with respect to the expiration time from \(T=0\) to 1, and it is obvious that the option price C is an increasing function with respect to the expiration time T during the period between two successive dividend payment times.

Fig. 3
figure 3

European put option price with respect to time T in Example 3

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5 European put option pricing formula

A European put option is a kind of financial derivatives that can give its buyer the right rather than the obligation to sell a certain kind of financial assets for a specified price at a specified time. Consider a European put option with its underlying stock following

$$\begin{aligned} \left\{ \begin{array}{l} \mathrm{d}X_t=\mu _i (1-c_i \ln X_t)X_t \mathrm{d}t + \sigma _i X_t \mathrm{d}C_t,\quad t_i \le t<t_{i+1} \\ X_{t_{i+1}}=\left( 1-\delta \right) \lim \nolimits _{\ t \rightarrow t_{i+1}^- }{X_t}. \end{array}\right. \end{aligned}$$
(10)

Suppose the strike price and the expiration time of a European put option are K and T, respectively. Then, its expected payoff at the expiration time is \(E\left[ (K-X_T)^+ \right] \). Assume the risk-free interest rate remains r during this period. Then, based on the assumption of continuous compounding, the price of a European put option is the present value of its expected payoff, i.e.,

$$\begin{aligned} P=\exp \left( -rT \right) E\left[ (K-X_T)^+ \right] . \end{aligned}$$
(11)

Theorem 6

(European put option pricing formula) Suppose a European put option with an underlying stock following model (10) has a strike price K and an expiration time T. Then, the pricing formula for this option is

$$\begin{aligned} P=\exp (-rT) \int _{0}^{1} (K-X_T^{\alpha })^+ \mathrm{d}\alpha , \end{aligned}$$
(12)

where

$$\begin{aligned} X_T^{\alpha }= & {} \exp \left( \ln X_{t_i}^{\alpha }\cdot \exp \left( -\mu _i c_i\left( T-t_i\right) \right) \right. \\&\left. +\,\gamma _i \left( 1-\exp \left( -\mu _i c_i \left( T-t_i\right) \right) \right) \right) , \text{ if } t_i \le T < t_{i+1} \end{aligned}$$

with

$$\begin{aligned} \gamma _i=\left( \frac{1}{c_i}+\frac{\sigma _i}{\mu _i c_i}\cdot \frac{\sqrt{3}}{\pi }\ln \frac{\alpha }{1-\alpha }\right) \end{aligned}$$

and \(X_{t_i}^\alpha \) is determined by Eq. (6).

Proof

Since \(X_t\) is an uncertain process with \(\alpha \)-path represented by Eq. (6), the inverse uncertainty distribution of \(X_T\) is \(X_T^\alpha \). As the payoff \((K-X_T)^+\) is a decreasing function of \(X_T\), it has an inverse uncertainty distribution \((K-X_T^{1-\alpha })^+\). Hence,

$$\begin{aligned} { E}[(K-X_T)^+]=\int _{0}^{1}(K-X_T^{1-\alpha })^+\,\mathrm{d}\alpha . \end{aligned}$$

It follows from Eq. (12) that

$$\begin{aligned} P= & {} \exp (-rT) \int _{0}^{1} (K-X_T^{1-\alpha })^+\, \mathrm{d}\alpha \\= & {} \exp (-rT) \int _{0}^{1} (K-X_T^{\alpha })^+ \mathrm{d}\alpha . \end{aligned}$$

The pricing formula of European put option is verified. \(\square \)

According to Theorem 6, the algorithm to calculate the European put option price of stock model (3) is designed as follows:

Step 0 Choose a large number N according to the desired precision degree. Set \(\alpha _k = k/N\), \(k = 1, 2,\ldots , N-1.\)

Step 1 Set \(k = 0\).

Step 2 Set \(k \leftarrow k+1\).

Step 3 Find the maximum m such that \(t_m\le T\), and calculate the stock price at dividend payment times \(t_1, t_2,\ldots , t_m\) successively via

$$\begin{aligned} \left\{ \begin{aligned} X_{t_1}^{\alpha _k}&=\exp \left( \ln X_0\cdot \exp \left( -\mu _0 c_0 t_1\right) \right. \\&\quad \left. +\gamma _0^{\alpha _k} \left( 1-\exp \left( -\mu _0 c_0 t_1 \right) \right) \right) \cdot (1-\delta ) \\ X_{t_2}^{\alpha _k}&= \exp \left( \ln X_{t_1}^{\alpha _k}\cdot \exp \left( -\mu _1 c_1\Delta t_2 \right) \right. \\&\quad \left. +\gamma _1^{\alpha _k} \left( 1-\exp \left( -\mu _1 c_1 \Delta t_2 \right) \right) \right) \cdot (1-\delta ) \\ \cdots \\ X_{t_m}^{\alpha _k}&=\exp \left( \ln X_{t_{m-1}}^{1-\alpha _k}\cdot \exp \left( -\mu _{m-1} c_{m-1}\Delta t_m \right) \right. \\&\quad \left. +\gamma _{m-1}^{\alpha _k} \left( 1-\exp \left( -\mu _{m-1} c_{m-1} \Delta t_m \right) \right) \right) \cdot (1-\delta ), \end{aligned}\right. \end{aligned}$$

in which

$$\begin{aligned} \Delta t_i=t_i-t_{i-1} \end{aligned}$$

and

$$\begin{aligned} \gamma _i^{\alpha _k}=\left( \frac{1}{c_{i}}+\frac{\sigma _{i}}{\mu _{i} c_{i}}\cdot \frac{\sqrt{3}}{\pi }\ln \frac{\alpha _k}{1-\alpha _k}\right) . \end{aligned}$$

If \(T=t_m\), then go to Step 5 .

Step 4 Calculate the stock price at time T

$$\begin{aligned} X_{T}^{\alpha _k}= & {} \exp \left( \ln X_{t_m}^{\alpha _k}\cdot \exp \left( -\mu _m c_m\left( T-t_m\right) \right) \right. \\&\left. +\,\gamma _m \left( 1-\exp \left( -\mu _m c_m \left( T-t_m\right) \right) \right) \right) , \end{aligned}$$

where

$$\begin{aligned} \gamma _m^{\alpha _k}=\left( \frac{1}{c_m}+\frac{\sigma _m}{\mu _m c_m}\cdot \frac{\sqrt{3}}{\pi }\ln \frac{\alpha _k}{1-\alpha _k}\right) . \end{aligned}$$

Step 5 Calculate the payoff of the European put option

$$\begin{aligned} (K-X_T^{\alpha _k})^+=\max (0,K-X_T^{\alpha _k}). \end{aligned}$$

If \(k<N-1\), then return to Step 2.

Step 6 Calculate the price of the European put option

$$\begin{aligned} P \leftarrow \frac{1}{N-1}\exp (-rT)\sum _{i=1}^{N-1}(K-X_T^{\alpha _k})^+. \end{aligned}$$

Example 3

Assume the European put option has an underlying stock following model (10), and the parameters are the same as Example 1. The strike price is \(K=30\) and the risk-free interest rate is \(r=0.03\). Then, the price of this European put option is \(P=0.79\) if the expiration time is \(T=1\). Figure 3 shows the price of this European put option price with respect to the expiration time from \(T=0\) to 1, and it is obvious that the option price P is a decreasing function with respect to the expiration time T during the period between two successive dividend payment times.

6 Conclusions

In this paper, we modeled the stock price using the exponential Ornstein–Uhlenbeck process under uncertain financial market and took dividends payment, a general stock behavior, into consideration. Based on the stock model, the pricing formulas of European call options and European put options were derived. In addition, some algorithms to calculate the option prices numerically were designed by using the \(\alpha \)-path method, and the relationships between the option price and the expiration time were also discussed based on the numerical experiments. Further research may consider the American and Asian option pricing formulas of the presented stock model. Besides, how to price various types of options for the exponential Ornstein–Uhlenbeck stock models with dividends and floating interest rate is also an interesting and challenging problem.