Original articlesMonotonicity theorem for the uncertain fractional differential equation and application to uncertain financial market
Introduction
Options are a type of financial products derived from the demands of securities traders, such as risk control, cost containment, and income enhancement. Thanks to the flexibility of contract regulations, options became widely adopted by investors to guide financial transactions such as capital management, mergers and acquisitions, financing decisions, and so on. However, owing to the occlusion of transaction information and narrow circulation channels, options did not receive sufficient development space in the initial stage. It was not until the Chicago Board Options Exchange opened in 1973 that the trading of options contracts was unified and standardized. Up to now, options have already occupied a significant share in the financial market, which has attracted the attention of many investors and scholars.
Over the last few decades, as the problem of option pricing has been extensively studied, many numerical and analytical methods have been developed. Options can be divided into European options and American options according to the execution time. European options contracts require the holder to perform the contract only on the expiration date, which is one or two days after the performance. Therefore, European options are mainly priced by closed pricing formula. In 1999, Carr and Madan [3] used the Fourier transform and the fast Fourier transform algorithm to speed up the calculation of European option prices. Fang and Oosterlee [9] proposed a European option pricing method based on Fourier cosine series expansion in 2009. Afterwards, the pricing method of European Asian options was derived under the index Lévy model [10], [43]. American option contracts can be executed at any time before or on the maturity date, and the settlement date is one or two days after the performance date. Therefore, unlike the pricing formula of European options, the pricing of American options is complicated by its exercise contingency. Except for Alghalith’s [1] calculating American options without dividends, other American options have no closed solution, for example, binomial lattice model of Cox et al. [6], finite difference method of Schwartz [29], Brennan and Schwartz [2], Cen & Chen [4], least-square method of Stentoft [32], etc. Other newest least-squares methods for option pricing, refer to [30], [31]. Meanwhile, Mehdi and Somayeh [7] presented two different methods to approximate the solution of the Black–Scholes equation for the valuation of barrier options.
In particular, early option pricing models tended to incorporate relevant knowledge of probability theory. It is undeniable that as an important tool for dealing with random information, probability theory provides important theoretical support for the early research on option pricing. However, the application of probability theory sometimes is not very optimistic because it requires a large amount of historical, reliable, and real data, but the actual data and reliable data sources are few. Domain experts’ belief degrees have since been used to measure the possibility that an event may occur. Nevertheless, due to the belief degree is easily influenced by the subjective preference of domain experts [15], it cannot accurately replace the real probability distribution.
In order to simulate non-probabilistic events, Liu [19] innovatively presented the uncertainty theory. In 2009, to describe the uncertain dynamic system, Liu [21] designed the Liu process and established the theory of uncertain calculus based on it, giving the definitions of uncertain differential equations (UDEs), extreme values, first hitting time [23], and time integral [20] of the uncertain process. In 2009, Liu [21] put forward an uncertain stock model after applying the uncertainty theory in the financial field for the first time. Since then, UDEs have been widely used in the financial field, which is usually influenced by fiscal balances and economic policies, especially in option pricing. Among them, Chen [5] derived American option pricing formulas based on UDEs. Sun & Chen [33] obtained the pricing formulas for the Asian option. Peng & Yao [27] developed a different uncertain stock model of mean reversion in uncertain financial markets. Furthermore, Yao & Chen formula [41] revealed a relationship between the ordinary differential equation and UDE. -path is essentially the solution’s inverse uncertainty distribution (IUD). A special uncertain process was defined as the contour process by Yao [40] in 2015, by which an uncertain stock model with floating interest rate was presented successfully and proved its effectiveness. Also, Sun & Su [34] presented a mean-reverting stock model with a floating interest rate and derived European options and American options pricing formulas accordingly. In 2019, Tian et al. [35], [36] investigated the barrier options and equity warrants pricing issues for the mean-reverting stock model. Tian et al. [37] also studied the lookback options pricing issue for an uncertain stock model in 2020. Furthermore, Yang et al. [39] investigated the pricing of Asian barrier options whose asset price obeys UDEs, Li et al. [18] investigated an uncertain portfolio optimization problem under a minimax risk measure and obtained the set of analytic Pareto optimal solutions in 2019. Moments estimations were proposed by Yao and Liu [42], which were extended to minimum cover estimations by Yang et al. [38]. Also, Liu [24] proposed the generalized moment estimation for unknown parameters in uncertain differential equations. The uncertain differential equation has many more important applications in other areas, such as the epidemic spread and the population growth [17], [44].
Fractional-order differential operators [28] possess memorability and heritability [16], enabling a more fine-grained portrayal of real financial markets with global relevance. Hence, the difference between UFDEs and UDE models is that UDE only has the character of representing the instantaneous rate of change of variables, while UFDEs are characterized by memory and heritability. In 2015, Zhu [45] defined the Riemann–Liouville type and Caputo type of UFDEs for the first time. Later on, Lu and Zhu [25] formulated the European options pricing model by solving the UFDE with the Caputo type. To give the numerical solution for such a UFDE, Lu and Zhu [26] presented the -path, which was the fractional counterpart of UDE’s -path. Furthermore, Jin et al. [12], [13] presented the extreme values and time integral theorems of the solution about UFDEs for Caputo type, of which the explicit expression of the American options’ price and the zero-coupon bond price was derived, respectively. To our knowledge, there is no study on the monotone function of UFDEs’ solution. Based on the previous work, this paper investigates the monotone function of UFDEs’ solution and proposes a novel uncertain fractional mean-reverting model (UFMM) with a floating interest rate. In view of the differences between European options and American options, we calculate them separately under this model.
This paper consists of five sections. In Section 2, we recall some facts in uncertainty theory and UFDEs. The monotone function theorems of UFDE’s solution are proposed in Section 3. Section 4 presents a novel uncertain fractional model with a floating interest rate where the monotone function’s IUD is used to formulate the option price. The European option pricing and American option pricing formulas are deduced under the proposed model in Sections 5 European option pricing formulas, 6 American option pricing formula, respectively. Meanwhile, numerical schemes are designed to compute option pricing formulas, and numerical experiments are illustrated concerning different parameters. A conclusion is drawn in the final section.
Section snippets
Preliminary
All the required concepts and conclusions in uncertainty theory are revisited here. For more information, please refer to [20], [21], [22].
Monotone function about solutions for UFDE
Besides the extreme value and integral, the IUDs of the monotone functions of the solutions to UFDEs (1) were also worth studying. Based on this motivation and the previous work, the monotone functions of solutions for UFDEs with the Caputo derivative will be studied, and IUDs will be derived accordingly in this section.
Theorem 3.1 Assume that are real numbers satisfying , where are positive integers, and are defined on . Let be
UFMM with floating interest rate
Floating interest rates, containing uncertain factors, are important economic variables that interfere with the final pricing of options and the risk–return structure of investment portfolios. Based on Liu’s [21] stock model, Yao [40] discussed the corresponding model of floating interest rates and formulated the option prices for it. Later on, Sun & Su [34] studied the mean-reverting model, which had a floating interest rate and gave European and American options’ pricing formulas. Ulteriorly,
European option pricing formulas
European options, whose values depend on interest rates, are widely adopted by domestic banks in dealing with foreign exchange business. Especially, European options can only be exercised on the expiration date. This section focuses on European call options and put options, and gives a practical formula for calculating the price of UFMM (26) with floating interest rates.
Consider a European call (put) option, which enables the holder to purchase (sell) at an agreed lower (higher) price on the
American option pricing formula
American options, whose expiration time is different from European options, are also meaningful and urgent to be discussed. Therefore, useful formulas for computing their prices about UFMM (26) with floating interest rate are given in this section.
Consider an American call (put) option, which enables the holder to buy (sell) at an agreed lower (higher) price any day before the maturity date, even if the price of the underlying asset rises (decreases). Denote as the strike price, as the
Simulation
The previous part mainly explained and analyzed the assumptions and models in the uncertain market from a theoretical point of view, and gave the pricing formulas for European and American options, respectively. This section is mainly from the perspective of empirical application, specifically considering the realization of the proposed UFMM (26).
Example 7.1 European Call Option Assume a UFMM (26) with floating interest rate has , , , initial value , , and , , , initial value
Conclusion
This paper extended the monotone function theorem for a solution, which is governed by a UFDE with the Caputo type. IUD about the monotone function was given. A UFMM of floating interest rate was introduced, which was more compatible with the real uncertain finance market. The pricing formulas of European options and American options were obtained for the proposed model by using the monotone function theorem. Obviously, some numerical algorithms were designed, and some numerical examples were
Acknowledgments
This work is supported by the National Natural Science Foundation of China (No. 12071219), the Program for Young Excellent Talents in UIBE, China (No. 18YQ06) and the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD), China .
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