Asian rainbow option pricing formulas of uncertain stock model.

Asian rainbow option is option on the minimum or the maximum of several average prices. In modern financial market, Asian rainbow option is an effective instrument for asset allocation and risk management. The investor with Asian rainbow option enjoys an entitlement to select a max or min from multiple assets with an exercise price at maturity date. The investor has to defray fee to acquire this right, which raises the option pricing issue. This paper mainly explores the pricing of Asian rainbow option in the uncertain financial environment, in which the underlying assets prices are treated as uncertain processes. Here, the pricing formulas of Asian rainbow option are derived under the condition that stock prices obey uncertain differential equations driven by independent Liu processes. Furthermore, some numerical examples are designed to compute the prices of these options.

Introduction

In a complex financial market, option is often used as a tool to gain high returns at small costs. The owner of the option may wield or waive the right to trade an asset at a certain price on a specified date. The investor has to pay fee to obtain the option, which causes the issue of option pricing. Based on the stochastic differential equation, the famous Black–Scholes formula was presented by Black and Scholes (1973), which was a major breakthrough in option pricing theory. Since then, the stock price was depicted by various stochastic differential equations. Merton (1976) put forward a jump-diffusion model in which the continuous and jump processes determine the return of the underlying stock. Cox and Ross (1976) proposed the constant elasticity of variance model. The advancement of option pricing theory greatly promotes the derivation of option types.

The theoretical study of Asian option started with the work of Ingersoll (1987). The payoff of Asian option is determined by the average price of underlying asset over its lifetime. Benefited from the averaging feature, Asian option can decrease the volatility of underlying asset price and the risk brought by market manipulation occurring near the expiry date. Sun and Chen (2015) contended that Asian option has a lower premium than standard option. In the Black–Scholes setting, Willems (2019) derived orthogonal polynomial expansion for the price of a continuous arithmetic Asian option. Under general stochastic asset model, Fusai and Kyriakou (2016) developed the pricing method of continuous and discrete arithmetic Asian options. For pricing discrete arithmetic Asian option, a transform-based algorithm was presented by Corsaro et al. (2019). By means of two finite difference methods, the pricing formulas of Asian option were deduced by Mudzimbabwe et al. (2012).

Asian rainbow option raised by Wu and Zhang (1999) is a mixture of Asian option and rainbow option, which is widely used in incentive contracts design and risk management. Due to the characteristic of rainbow option, Asian rainbow option is a type of multi-asset option. For the literature on multi-asset option pricing, interested readers can refer to references (Johnson 1987; Margrabe 1978; Ouwehand and West 2006; Stulz 1982). Thus, the payoff of Asian rainbow option is related to the averages of the prices of several underlying assets over certain time interval. Based on the arbitrage-free principle and the Ito lemma, Bin and Fei (2009) provided the pricing formula of geometric Asian rainbow option with known dividends. Within a Black–Scholes environment, Zhan and Cheng (2010) came up with a simple method to price Asian rainbow option on dividend-paying assets. By using fractional Brownian motion to portray the underlying assets prices, Wang et al. (2018) studied the pricing problem of Asian rainbow option. Under the mixed fractional Brownian motion, Ahmadian and Ballestra (2020) deduced the closed-form solution of geometric Asian rainbow options.

As well known, the traditional option pricing model is established on the basis of probability theory, which regards underlying asset price as the Wiener process. As can be seen from the law of large numbers, the application of probability theory requires that the cumulative probability distribution approximates the real frequency very much. Therefore, it is essential to obtain enough samples. Nevertheless, it is sometimes difficult to acquire adequate samples for each indeterministic events. For example, during the coronavirus outbreak, we do not know the demand for masks in the harder-hit areas. We need to invite the domain experts to evaluate the belief degree, which is largely determined by individual awareness and preferences for the indeterministic factors. For processing belief degrees mathematically, uncertainty theory was pioneered by Liu (2009, 2012). Uncertain variable is the basic tool in uncertainty theory, which was proposed to depict uncertain quantity. For describing the change of uncertain variable with time, Liu (2008) came up with the conception of uncertain process.

As a special uncertain process, Liu process provided by Liu (2009) can be considered as a counterpart of the Wiener process. For the differential equation driven by the Wiener process, its noise term is generally understood as a normal random variable which has expected value 0 and variance . Then, the stock price has infinite changes at any moment, which goes against the reality. Hence, it is more reasonable to use Liu process to portray the stock price (Liu 2013). Driven by Liu process, uncertain differential equation was presented by Liu (2009). By employing uncertain differential equation, Liu (2009) proposed an uncertain stock model and discovered the European option pricing formula. Following uncertain stock model, the pricing formulas of Asian barrier option and American barrier option were deduced by Yang et al. (2019) and Gao et al. (2019), respectively. Lu et al. (2019) brought forward a new uncertain stock model and explored the pricing of European option. Tian et al. (2019) studied the pricing problem of European barrier option under the uncertain mean-reverting stock model.

In this paper, the average prices of underlying assets is calculated by the arithmetic mean and geometric mean methods. We regard the underlying assets prices as uncertain processes and provide the pricing formulas of Asian rainbow option based on the uncertain stock model. The structure of the paper is organized below. Section 2 will briefly state some related theorems and definitions of uncertainty theory. Section 3 will explore the pricing of arithmetic Asian put 2 and call 1 option, and geometric Asian put 2 and call 1 option. In Sect. 4, Asian rainbow call options will be investigated, including Asian rainbow call on max options and Asian rainbow call on min options. And their option pricing formulas are obtained by rigorous deductions. Four types of Asian rainbow put options will be analyzed in Sect. 5. Similar to Sect. 4, the option pricing formulas in the above four cases are strictly deduced. Some brief conclusions are given in Sect. 6.

Preliminaries

Definition 1

(Liu 2009, 2012) Providing that () is a measurable space. A set function : is called an uncertain measure if it satisfies the following conditions,

Axiom 1. (Normality Axiom) for the universal set ;

Axiom 2. (Duality Axiom) for any event ;

Axiom 3. (Subadditivity Axiom) For every countable sequence of events , we haveAxiom 4. (Product Axiom) Let represent uncertainty spaces. The product uncertain measure is an uncertain measure satisfyingwhere are arbitrary events chosen from .

Theorem 1

(Liu 2010a, Monotonicity Theorem) For any events and with , if the uncertain measure is a set function with monotone increasing feature, then we have

Theorem 2

(Liu 2013) A function is an inverse uncertainty distribution if and only if it is a continuous and strictly increasing function concerning .

Theorem 3

(Liu 2007) Assume that there is uncertainty distribution for an uncertain variable . If exists, thenFurthermore if is regular, then we also have

Definition 2

(Liu 2009) An uncertain process is called a Liu process if

and almost all sample paths are Lipschitz continuous,

has stationary and independent increments,

each increment is a normal uncertain variable owning expected value 0 and variance , whose uncertainty distribution is

Theorem 4

(Yao and Chen 2013) Suppose that there are the solution and -path for an uncertain differential equationThen,where possesses an inverse uncertainty distribution

Theorem 5

(Yao 2013) Suppose that there are the solution and -path for an uncertain differential equationThen, the time integral possesses an inverse uncertainty distributionwhere Y(x) is a function owning strictly increasing feature.

Theorem 6

(Liu 2010b) Let be independent uncertain variables with regular uncertainty distributions respectively. If f is strictly increasing with respect to and strictly decreasing with respect to , thenhas an inverse uncertainty distribution

Asian put 2 and call 1 option

Asian put 2 and call 1 option provides the owner with an entitlement to exchange Asset 2 for Asset 1 on the expiration date. The gain of the option is related to the mean of two underlying assets prices at a predetermined time in the future. The means of underlying assets prices are calculated by the arithmetic average and geometric average methods. In this section, we study arithmetic Asian put 2 and call 1 option and geometric Asian put 2 and call 1 option. Moreover, the pricing formulas of corresponding option are obtained by strict deduction.

Arithmetic Asian put 2 and call 1 option

Arithmetic Asian put 2 and call 1 option is a kind of exchange option, in which market prices of Asset 1 and Asset 2 are calculated by arithmetic average method. The payoff of the option relies on the arithmetic mean of Asset 1 and Asset 2 market prices within an agreed period of time. The arithmetic mean of N numbers is defined as . When the underlying asset price changes continuously from time zero to time T, the arithmetic mean of can be written as .

Considering that the underlying asset prices of arithmetic Asian put 2 and call 1 option with a maturity time T obey different uncertain differential equations, the following model is proposed:where is the bond price, represents the price of Asset 1, is the price of Asset 2, and are independent Liu processes, and are, respectively, the log-diffusion of and , and are, respectively, the log-drift of and , and is the riskless interest rate.

Since and are independent Liu process, the changes of and are also independent of each other.

The solution of Model (1) iswhose inverse uncertainty distribution iswhere and are, respectively, inverse uncertainty distribution of and .

Firstly, from the holder’s perspective, the payoff at time T isLet denote the price of arithmetic Asian put 2 and call 1 option. Thus, at the initial moment, the holder of the option owns the net returnSecondly, from the seller’s perspective, the payoff at time T isThen, at the initial moment, the net return of the seller can be expressed:Both should own uniform expected payoff by the fair price principle, that isFrom the above, we conclude that the price of arithmetic Asian put 2 and call 1 option is equal to the present value of the expected payoff.

Definition 3

Suppose that an arithmetic Asian put 2 and call 1 option for Model (1) has a maturity time T. Then, its price is defined by

Theorem 7

Suppose that an arithmetic Asian put 2 and call 1 option for Model (1) has a maturity time T. Then, its price is

Proof

We show that the uncertain variablehas an inverse uncertainty distributionwhereThe uncertain processes and own inverse uncertainty distributions which are represented by Equation (2). By using Theorem 4 and Theorem 5, we derive that the inverse uncertainty distributions of the two time integralsarerespectively.

It follows from Theorem 6 that the uncertain variablehas an inverse uncertainty distributionHence, the pricing formula isThe proof is completed.

Example 1

In Model (1), we assume that , , , , . Consider that there are , , for an arithmetic Asian put 2 and call 1 option. Then, by Theorem 7.

Figure 1 demonstrates that the price is an increasing function of maturity time T if other parameters are invariant.

Fig. 1

price concerning maturity time T in Example 1

Geometric Asian put 2 and call 1 option

Geometric Asian put 2 and call 1 option is a kind of exchange option, in which market prices of Asset 1 and Asset 2 are calculated by geometric average method. The payoff of the option relies on the geometric mean of Asset 1 and Asset 2 market price within an agreed period of time. The geometric mean of N numbers is defined as , namely . When the underlying asset price changes continuously from time zero to time T, the geometric mean of can be written as .

Assume that geometric Asian put 2 and call 1 option for Model (1) has a maturity time T.

Firstly, from the holder’s perspective, the payoff at time T isLet denote the price of geometric Asian put 2 and call 1 option. Thus, at the initial moment, the holder of the option owns the net returnSecondly, from the seller’s perspective, the payoff at time T isThen, at the initial moment, the net return of the seller can be expressed:Both should own uniform expected payoff by the fair price principle, that isFrom the above, we conclude that the price of geometric Asian put 2 and call 1 option is equal to the present value of the expected payoff.

Definition 4

Suppose that a geometric Asian put 2 and call 1 option for Model (1) has a maturity time T. Then, its price is defined by

Theorem 8

Suppose that a geometric Asian put 2 and call 1 option for Model (1) has a maturity time T. Then, its price is

We show that the uncertain variablehas an inverse uncertainty distributionwhereThe uncertain processes and own inverse uncertainty distributions which are represented by Equation (2). Since is a continuous and strictly increasing function concerning , the inverse uncertainty distributions of the two time integralsarefollowing from Theorem 4 and Theorem 5, respectively.

By using Theorem 6, we derive that the uncertain variablehas an inverse uncertainty distributionHence, the pricing formula isThe proof is completed.

Example 2

In Model (1), we assume that , , , , . Consider that there are , , for a geometric Asian put 2 and call 1 option. Then by Theorem 8.

Figure 2 demonstrates that the price is an increasing function of maturity time T if other parameters are invariant.

Fig. 2

price concerning maturity time T in Example 2

Asian rainbow call option

Asian rainbow call option is a contract that provides the holder with a right to purchase the maximum or minimum asset at a given price at maturity time. The gain of the option depends on the mean of the maximum or minimum underlying asset price over its lifetime. The means of underlying assets prices are calculated by the arithmetic average and geometric average methods. In this section, we study arithmetic Asian rainbow call on max option, geometric Asian rainbow call on max option, arithmetic Asian rainbow call on min option and geometric Asian rainbow call on min option. Furthermore, we obtain the pricing formulas of Asian rainbow call on max option and Asian rainbow call on min option by strict deduction.

Arithmetic Asian rainbow call on max option

Arithmetic Asian rainbow call on max option signifies that the holder can purchase maximum asset at the exercise price at the time of execution. The payoff of the option relies on the arithmetic mean of maximum asset price within an agreed period of time.

Considering that the underlying asset prices of arithmetic Asian rainbow call on max option with a maturity time T and an exercise price K obey different uncertain differential equations, the following model is proposed:where is the bond price with riskless interest rate , is the price of Asset i at time t, are independent Liu processes, are the log-diffusion of , and are the log-drift of , .

Since are independent Liu process, the changes of are also independent of each other.

The solution of Model (3) iswhose inverse uncertainty distribution iswhere are inverse uncertainty distribution of .

Firstly, from the holder’s perspective, the payoff at time T isLet be the price of arithmetic Asian rainbow call on max option. Thus, at the initial moment, the holder of the option owns the net returnSecondly, from the seller’s perspective, the payoff at time T isThen, at the initial moment, the net return of the seller can be expressed:Both should own uniform expected payoff by the fair price principle, that isFrom the above, we conclude that the price of arithmetic Asian rainbow call on max option is equal to the present value of the expected payoff.

Definition 5

Suppose that an arithmetic Asian rainbow call on max option for Model (3) has a maturity time T and an exercise price K. Then, its price is defined by

Theorem 9

Suppose that an arithmetic Asian rainbow call on max option for Model (3) has a maturity time T and an exercise price K. Then, its price is

We show that the uncertain variablehas an inverse uncertainty distributionwhereSince is an increasing function with regard to x, we deriveandFrom Theorem 1 and Theorem 4, we haveandWe can obtainby using duality axiom.

It follows from Eqs. (4)–(6) thatThus, the uncertain variablehas an inverse uncertainty distributionHence, the pricing formula isThe proof is completed.

Example 3

Consider that there are , , , for an arithmetic Asian rainbow call on max option with two underlying stocks. In Model (3), we assume that , , , , . Then, by Theorem 9.

Figure 3 demonstrates that the price is an increasing function of maturity time T if other parameters are invariant.

Fig. 3

price concerning maturity time T in Example 3

Geometric Asian rainbow call on max option

Geometric Asian rainbow call on max option signifies that the holder can purchase maximum asset at the exercise price at the time of execution. The payoff of the option relies on the geometric mean of maximum asset price within an agreed period of time.

Assume that geometric Asian rainbow call on max option possesses a maturity time T and an exercise price K for Model (3).

Firstly, from the holder’s perspective, the payoff at time T isLet be the price of geometric Asian rainbow call on max option. Thus, at the initial moment, the holder of the option owns the net returnSecondly, from the seller’s perspective, the payoff at time T isThen, at the initial moment, the net return of the seller can be expressed:Both should own uniform expected payoff by the fair price principle, that isFrom the above, we conclude that the price of geometric Asian rainbow call on max option is equal to the present value of the expected payoff.

Definition 6

Suppose that a geometric Asian rainbow call on max option for Model (3) has a maturity time T and an exercise price K. Then, its price is defined by

Theorem 10

Suppose that a geometric Asian rainbow call on max option for Model (3) has a maturity time T and an exercise price K. Then, its price is

We show that the uncertain variablehas an inverse uncertainty distributionwhereSince is an increasing function with regard to , we deriveandFrom Theorem 1 and Theorem 4, we haveandWe can obtainby using duality axiom.

It follows from Eqs. (7)–(9) thatThus, the uncertain variablehas an inverse uncertainty distributionHence, the pricing formula isThe proof is completed.

Example 4

Consider that there are , , , for a geometric Asian rainbow call on max option with two underlying stocks. In Model (3), we assume that , , , , . Then, by Theorem 10.

Figure 4 demonstrates that the price is an increasing function of maturity time T if other parameters are invariant.

Fig. 4

price concerning maturity time T in Example 4

Arithmetic Asian rainbow call on min option

Arithmetic Asian rainbow call on min option signifies that the holder can purchase minimum asset at the exercise price at the time of execution. The payoff of the option relies on the arithmetic mean of minimum asset price within an agreed period of time.

Assume that arithmetic Asian rainbow call on min option possesses a maturity time T and an exercise price K for Model (3).

Firstly, from the holder’s perspective, the payoff at time T isLet be the price of arithmetic Asian rainbow call on min option. Thus, at the initial moment, the holder of the option owns the net returnSecondly, from the seller’s perspective, the payoff at time T isThen, at the initial moment, the net return of the seller can be expressed:Both should own uniform expected payoff by the fair price principle, that isFrom the above, we conclude that the price of arithmetic Asian rainbow call on min option is equal to the present value of the expected payoff.

Definition 7

Suppose that an arithmetic Asian rainbow call on min option for Model (3) has a maturity time T and an exercise price K. Then, its price is defined by

Theorem 11

Suppose that an arithmetic Asian rainbow call on min option for Model (3) has a maturity time T and an exercise price K. Then, its price is

We show that the uncertain variablehas an inverse uncertainty distributionwhereSince is an increasing function with regard to x, we deriveandFrom Theorem 1 and Theorem 4, we haveandWe can obtainby using duality axiom.

It follows from Eqs. (10)–(12) thatThus, the uncertain variablehas an inverse uncertainty distributionHence, the pricing formula isThe proof is completed.

Example 5

Consider that there are , , , for an arithmetic Asian rainbow call on min option with two underlying stocks. In Model (3), we assume that , , , , . Then, by Theorem 11.

Figure 5 demonstrates that the price is an increasing function of maturity time T if other parameters are invariant.

Fig. 5

Price concerning maturity time T in Example 5

Geometric Asian rainbow call on min option

Geometric Asian rainbow call on min option signifies that the holder can purchase min asset at the exercise price at the time of execution. The payoff of the option relies on the geometric mean of minimum asset price within an agreed period of time.

Assume that geometric Asian rainbow call on min option possesses a maturity time T and an exercise price K for Model (3).

Firstly, from the holder’s perspective, the payoff at time T isLet be the price of geometric Asian rainbow call on min option. Thus, at the initial moment, the holder of the option owns the net returnSecondly, from the seller’s perspective, the payoff at time T isThen, at the initial moment, the net return of the seller can be expressed:Both should own uniform expected payoff by the fair price principle, that isFrom the above, we conclude that the price of geometric Asian rainbow call on min option is equal to the present value of the expected payoff.

Definition 8

Suppose that a geometric Asian rainbow call on min option for Model (3) has a maturity time T and an exercise price K. Then, its price is defined by

Theorem 12

Suppose that a geometric Asian rainbow call on min option for Model (3) has a maturity time T and an exercise price K. Then, its price is

We show that the uncertain variablehas an inverse uncertainty distributionwhereSince is an increasing function with regard to , we deriveandFrom Theorem 1 and Theorem 4, we haveandWe can obtainby using duality axiom.

It follows from Eqs. (13)–(15) thatThus, the uncertain variablehas an inverse uncertainty distributionHence, the pricing formula isThe proof is completed.

Example 6

Consider that there are , , , for a geometric Asian rainbow call on min option with two underlying stocks. In Model (3), we assume that , , , , . Then, by Theorem 12.

Figure 6 demonstrates that the price is an increasing function of maturity time T if other parameters are invariant.

Fig. 6

Price concerning maturity time T in Example 6

Asian rainbow put option

Asian rainbow put option is a contract that provides the holder with a right to sell the maximum or minimum asset at a given price at maturity time. The gain of the option depends on the mean of the maximum or minimum underlying asset price over its lifetime. The mean of underlying assets prices is calculated by the arithmetic average and geometric average methods. In this section, we study arithmetic Asian rainbow put on max option, geometric Asian rainbow put on max option, arithmetic Asian rainbow put on min option and geometric Asian rainbow put on min option. Furthermore, we obtain the pricing formulas of Asian rainbow put on max option and Asian rainbow put on min option by strict deduction.

Arithmetic Asian rainbow put on max option

Arithmetic Asian rainbow put on max option signifies that the holder can purchase maximum asset at the exercise price at the time of execution. The payoff of the option relies on the arithmetic mean of maximum asset price within an agreed period of time.

Assume that arithmetic Asian rainbow put on max option possesses a maturity time T and an exercise price K for Model (3).

Firstly, from the holder’s perspective, the payoff at time T isLet be the price of arithmetic Asian rainbow put on max option. Thus, at the initial moment, the holder of the option owns the net returnSecondly, from the seller’s perspective, the payoff at time T isThen, at the initial moment, the net return of the seller can be expressed:Both should own uniform expected payoff by the fair price principle, that isFrom the above, we conclude that the price of arithmetic Asian rainbow put on max option is equal to the present value of the expected payoff.

Definition 9

Suppose that an arithmetic Asian rainbow put on max option for Model (3) has a maturity time T and an exercise price K. Then, its price is defined by

Theorem 13

Suppose that an arithmetic Asian rainbow put on max option for Model (3) has a maturity time T and an exercise price K. Then, its price is

We show that the uncertain variablehas an inverse uncertainty distributionwhereSince is an increasing function with regard to x, we deriveandFrom Theorem 1 and Theorem 4, we haveandWe can obtainby using duality axiom.

It follows from Eqs. (16)–(18) thatThus, the uncertain variablehas an inverse uncertainty distributionHence, the pricing formula isThe proof is completed.

Example 7

Consider that there are , , , for an arithmetic Asian rainbow put on max option with two underlying stocks. In Model (3), we assume that , , , , . Then, by Theorem 13.

Figure 7 demonstrates that the price is an increasing function of maturity time T if other parameters are invariant.

Fig. 7

Price concerning maturity time T in Example 7

Geometric Asian rainbow put on max option

Geometric Asian rainbow put on max option signifies that the holder can sell maximum asset at the exercise price at the time of execution. The payoff of the option relies on the geometric mean of maximum asset price within an agreed period of time.

Assume that geometric Asian rainbow put on max option possesses a maturity time T and an exercise price K for Model (3).

Firstly, from the holder’s perspective, the payoff at time T isLet be the price of geometric Asian rainbow put on max option. Thus, at the initial moment, the holder of the option owns the net returnSecondly, from the seller’s perspective, the payoff at time T isThen, at the initial moment, the net return of the seller can be expressed:Both should own uniform expected payoff by the fair price principle, that isFrom the above, we conclude that the price of geometric Asian rainbow put on max option is equal to the present value of the expected payoff.

Definition 10

Suppose that a geometric Asian rainbow put on max option for Model (3) has a maturity time T and an exercise price K. Then, its price is defined by

Theorem 14

Suppose that a geometric Asian rainbow put on max option for Model (3) has a maturity time T and an exercise price K. Then, its price is

We show that the uncertain variablehas an inverse uncertainty distributionwhereSince is an increasing function with regard to , we deriveandFrom Theorem 1 and Theorem 4, we haveandWe can obtainby using duality axiom.

It follows from Eqs. (19)–(21) thatThus, the uncertain variablehas an inverse uncertainty distributionHence, the pricing formula isThe proof is completed.

Example 8

Consider that there are , , , for a geometric Asian rainbow put on max option with two underlying stocks. In Model (3), we assume that , , , , . Then, by Theorem 14.

Figure 8 demonstrates that the price is an increasing function of maturity time T if other parameters are invariant.

Fig. 8

Price concerning maturity time T in Example 8

Arithmetic Asian rainbow put on min option

Arithmetic Asian rainbow put on min option signifies that the holder can sell minimum asset at the exercise price at the time of execution. The payoff of the option relies on the arithmetic mean of minimum asset price within an agreed period of time.

Assume that arithmetic Asian rainbow put on min option possesses a maturity time T and an exercise price K for Model (3).

Firstly, from the holder’s perspective, the payoff at time T isLet be the price of arithmetic Asian rainbow put on min option. Thus, at the initial moment, the holder of the option owns the net returnSecondly, from the seller’s perspective, the payoff at time T isThen, at the initial moment, the net return of the seller can be expressed:Both should own uniform expected payoff by the fair price principle, that isFrom the above, we conclude that the price of arithmetic Asian rainbow put on min option is equal to the present value of the expected payoff.

Definition 11

Suppose that an arithmetic Asian rainbow put on min option for Model (3) has a maturity time T and an exercise price K. Then, its price is defined by

Theorem 15

Suppose that an arithmetic Asian rainbow put on min option for Model (3) has a maturity time T and an exercise price K. Then, its price is

We show that the uncertain variablehas an inverse uncertainty distributionwhereSince is an increasing function with regard to x, we deriveandFrom Theorem 1 and Theorem 4, we haveandWe can obtainby using duality axiom.

It follows from Eqs. (22)–(24) thatThus, the uncertain variablehas an inverse uncertainty distributionHence, the pricing formula isThe proof is completed.

Example 9

Consider that there are , , , for an arithmetic Asian rainbow put on min option with two underlying stocks. In Model (3), we assume that , , , , . Then, by Theorem 15.

Figure 9 demonstrates that the price is an increasing function of maturity time T if other parameters are invariant.

Fig. 9

Price concerning maturity time T in Example 9

Geometric Asian rainbow put on min option

Geometric Asian rainbow put on min option signifies that the holder can sell min asset at the exercise price at the time of execution. The payoff of the option relies on the geometric mean of minimum asset price within an agreed period of time.

Assume that geometric Asian rainbow put on min option possesses a maturity time T and an exercise price K for Model (3).

Firstly, from the holder’s perspective, the payoff at time T isLet be the price of geometric Asian rainbow put on min option. Thus, at the initial moment, the holder of the option owns the net returnSecondly, from the seller’s perspective, the payoff at time T isThen, at the initial moment, the net return of the seller can be expressed:Both should own uniform expected payoff by the fair price principle, that isFrom the above, we conclude that the price of geometric Asian rainbow put on min option is equal to the present value of the expected payoff.

Definition 12

Suppose that a geometric Asian rainbow put on min option for Model (3) has a maturity time T and an exercise price K. Then, its price is defined by

Theorem 16

Suppose that a geometric Asian rainbow put on min option for Model (3) has a maturity time T and an exercise price K. Then, its price is

We show that the uncertain variablehas an inverse uncertainty distributionwhereSince is an increasing function with regard to , we deriveandFrom Theorem 1 and Theorem 4, we haveandWe can obtainby using duality axiom.

It follows from Eqs. (25)–(27) thatThus, the uncertain variablehas an inverse uncertainty distributionHence, the pricing formula isThe proof is completed.

Example 10

Consider that there are , , , for a geometric Asian rainbow put on min option with two underlying stocks. In Model (3), we assume that , , , , . Then, by Theorem 16.

Figure 10 demonstrates that the price is an increasing function of maturity time T if other parameters are invariant.

Fig. 10

Price concerning maturity time T in Example 10

Conclusions

By using uncertain differential equations to depict the underlying assets prices, this article explored the pricing of Asian rainbow option under the uncertain stock model. We investigated three types of Asian rainbow options which are Asian put 2 and call 1 option, Asian rainbow call option and Asian rainbow put option. Furthermore, we derived the pricing formulas of corresponding option by rigorous deduction. Finally, we used some numerical examples to indicate how to utilize the option pricing formulas.