American Barrier Option Pricing Formulae for Uncertain Stock Model

American Barrier Option Pricing Formulae for Uncertain Stock Model Rong Gao School of Economics and Management, Heei University of Technology, Tianjin 341, China gaor14@tsinghua.org.cn Astract Uncertain stock models have een proposed as applications of uncertainty theory to descrie the price fluctuation of the stock in uncertain market. This paper mainly studies the American arrier option of stock model in uncertain finance market. Then we put forward four new types of concepts which are American up-and-in call option, American down-and-in put option, American up-and-out put option and American down-and-out call option. Furthermore, we prove some pricing formulas to calculate the corresponding options. Keywords: uncertain finance; arrier option; stock model. 1 Introduction Brownian motion was first discovered y Scottish otanist Roert Brown in 182s, and then it was descried y Wiener 2] through giving a mathematical definition of Wiener process in 1923. Wiener process is type of stochastic process which possesses independent and stationary increments. On the asis of Wiener process, Ito 6, 7] pioneered stochastic calculus theory and estalished stochastic differential equation. Samuelson 16, 17] asserted that geometric Wiener process is a valid tool for modeling stock prices. Then Black and Scholes 1] and Metron 14] independently used Wiener process to descrie the stock price and derived the famous Black-Scholes option pricing formulas y using stochastic differential equation. Following that, some other types of stochastic differential equations were applied into stock market, which resulted in various option pricing formulas. In addition to random factors, there is human uncertainty appearing in real life. For coping with such uncertainty, Liu 8, 1] founded uncertainty theory which is a ranch of mathematical system on the asis of normality, duality, suadditivity and product axioms. In this framework, the elief degree is descried y uncertain measure, and the uncertain quantity is modeled y uncertain variale. For descriing and ranking an uncertain variale, concepts of uncertainty distriution and expected value were put forward in 8], respectively. As an almost perfect mathematical system, uncertainty theory has een applied into many fields, such as uncertain programming Wang et al. 19], uncertain reliaility Liu 11], Gao and Yao 5], and uncertain portfolio selection Qin 15]. For modeling the uncertain quantity varying with time, Liu 9] first introduced a concept of uncertain process in 28. As a tool to descrie the uncertain process, uncertainty distriution was defined y Liu 13]. Meanwhile, Liu proved a sufficient and necessary condition for the uncertainty distriution of an uncertain process in 13]. Besides, Liu proposed the independence of uncertain processes, ased on which the operational law for uncertain processes was put forward y Liu 13] and a concept of stationary independent increment process was introduced y Liu 9]. Then Liu 1] proposed a canonical Liu process, 1

which is an uncertain process with stationary and independent normal uncertain increments. Meanwhile, uncertain calculus was estalished y Liu 1] to deal with the integral and differential of an uncertain process in regard to Liu process. Based on uncertain calculus, Liu 9] first uilt uncertain differential equation which is a type of differential equations driven y Liu process. Then Chen and Liu 3] discussed the solution of the uncertain differential equation. Gao 4] used Milne method to design an algorithm to get its numerical solution. With the development of uncertain differential equation, it has een widely used in the financial market in uncertain environment such as Yao 21]. Liu 1] presented uncertain stock model y using uncertain differential equation to characterize the trend of price, and otained some pricing formulae for European options his model. Through Liu s model, Chen 2] investigated American options and derived some pricing formulae of such options. Sun and Chen 18] discussed American options and derived their pricing formulae. Option pricing is the core content of modern finance. American option is widely accepted y investors for its flexiility of exercising time. In this paper, an American arrier option will e discussed and corresponding pricing formulae will e derived for uncertain financial market. The rest of the paper is organized as follows. Some fundamental concepts and properties of uncertain process and uncertain differential equation are recalled in Section 2. American knock-in and knock-out option pricing formulae are derived and some properties of them are studied in Sections 3 and 4, respectively. Finally, a rief summary is given in Section 5. 2 Preliminaries In this section, we will introduce some fundamental concepts and properties concerning uncertain variales, uncertain processes, and uncertain differential equations. 2.1 Uncertain Differential Equation Let Γ e a nonempty set, and L a σ-algera over Γ. Each element Λ in L is called an event and assigned a numer MΛ to indicate the elief degree with which we elieve Λ will happen. In order to deal with elief degrees rationally, Liu 8] suggested the following three axioms: Axiom 1. Normality Axiom MΓ 1 for the universal set Γ; Axiom 2. Duality Axiom MΛ + MΛ c 1 for any event Λ; Axiom 3. Suadditivity Axiom For every countale sequence of events Λ 1, Λ 2,, we have M Λ i MΛ i. i1 i1 Definition 1. Liu 8] The set function M is called an uncertain measure if it satisfies the normality, duality, and suadditivity axioms. Theorem 1. The uncertain measure is a monotone increasing set function, i.e., for any events Λ 1 and Λ 2 with Λ 1 Λ 2, we have MΛ 1 MΛ 2. 2

The triplet Γ, L, M is called an uncertainty space. Furthermore, the product uncertain measure on the product σ-algera L was defined y Liu 1] as follows: Axiom 4. Product Axiom Let Γ k, L k, M k e uncertainty spaces for k 1, 2,. The product uncertain measure M is an uncertain measure satisfying M Λ k M k Λ k where Λ k are aritrary events chosen from L k for k 1, 2,, respectively. k1 Definition 2. Liu 8] An uncertain variale is a measurale function ξ from an uncertainty space Γ, L, M to the set of real numers, i.e., for any Borel set B of real numers, the set is an event. k1 ξ B γ Γ ξγ B Theorem 2. Let ξ 1, ξ 2,, ξ n e uncertain variales, and f a real-valued measurale function. Then ξ fξ 1, ξ 2,, ξ n is an uncertain variale defined y ξγ fξ 1 γ, ξ 2 γ,, ξ n γ, γ Γ. Definition 3. Liu 8] Suppose ξ is an uncertain variale. Then the uncertainty distriution of ξ is defined y for any real numer x. Φx M ξ x Definition 4. Liu 1] The uncertain variales ξ 1, ξ 2,, ξ n are said to e independent if n n M ξ i B i M ξ i B i for any Borel sets B 1, B 2,, B n. i1 Theorem 3. Liu?] Let ξ 1, ξ 2,, ξ n distriutions Φ 1, Φ 2,, Φ n, respectively. i1 e independent uncertain variales with regular uncertainty If the function fx 1, x 2,, x n is strictly increasing with respect to x 1, x 2,, x m and strictly decreasing with respect to x m+1, x m+2,, x n, then the uncertain variale has an inverse uncertainty distriution ξ fξ 1, ξ 2,, ξ n Φ 1 fφ 1 1,, Φ 1 m, Φ 1 m+1 1, Φ 1 n 1. For ranking uncertain variales, the concept of expected value was proposed y Liu 8] as follows: Definition 5. Liu 8] Let ξ e an uncertain variale. Then the expected value of ξ is defined y Eξ] + provided that at least one of the two integrals is finite. Mξ xdx Mξ xdx 3

Theorem 4. Liu 8] Let ξ e an uncertain variale with uncertainty distriution Φ. If the expected value exists, then Eξ] + xdφx. If the uncertainty distriution Φ is regular, then we also have Eξ] Φ 1 d. An uncertain process is essentially a sequence of uncertain variales indexed y time. The study of uncertain process was started y Liu 9] in 28. Definition 6. Liu 9] Let T e an index set and let Γ, L, M e an uncertainty space. An uncertain process is a measurale function from T Γ, L, M to the set of real numers such that X t B is an event for any Borel set B for each t. Definition 7. Liu 1] An uncertain process C t is said to e a Liu process if i C and almost all sample paths are Lipschitz continuous, ii C t has stationary and independent increments, iii every increment C s+t C s is a normal uncertain variale with expected value and variance t 2, whose uncertainty distriution is Φx 1 πx 1 + exp, x R. 3t Definition 8. Liu 9] Suppose that C t is a Liu process, f and g are continuous functions. Given an initial value X, the uncertain differential equation dx t ft, X t dt + gt, X t dc t is called an uncertain differential equation with an initial value X. Definition 9. Yao and Chen 22] The -path < < 1 of an uncertain differential equation dx t ft, X t dt + gt, X t dc t with initial value X is a deterministic function X t with respect to t that solves the corresponding equation dx t ft, X t dt + gt, X t Φ 1 dt where Φ 1 is the inverse uncertainty distriution of standard normal uncertain variale, i.e., 3 Φ 1 π ln 1, < < 1. Theorem 5. Yao and Chen 22] Assume ft, x and gt, x are continuous and Xt is the -path of the uncertain differential equation dx t ft, X t dt + gt, X t dc t, t 1, s]. Then we get MX t X t, t, MX t > X t, t 1. From Theorem 5, many properties of -path have een derived. Firstly, Chen and Yao 22] found the inverse uncertainty distriution of X s, that is Ψ 1 X s, < < 1. 4

3 Knock-in Options We apply this section to study the knock-in option which is one type of arrier option. Knock-in option means that the option is activated only when the market price of the underlying asset touches the trigger point and the right is valid for the period of the option. investigated in this section. Furthermore, its pricing formulas are also Next, we define an indictor function which will e used in the main results as follows 1, if L x I L, if L > x where L, a given numer, represents the trigger point. Firstly, we consider up-and-in option that ecomes activated only when the market price of the underlying asset starts elow the trigger point L and moves up efore the expiration date. Look at the American up-and-in call option with a strike price K, an expiration date T and a trigger point L for some stocks in the uncertain financial market as follows dxt rx t dt dz t uz t dt + vz t dc t 1 where X t represents the ond price, Z t represents the stock price, r is riskless interest rate, u and v are respectively the drift and diffusion, and C t is Liu process. Let fui c denote the price of contract. The investor pays fui c to uy the contract at initial time, and has a present value of the pay off as follows exp rti L Z s Z t K +. Then the net return of the investor at initial time is fui c + exp rti L X s X t K +. On the other hand, the ank receives fui c due to selling the contract at initial time, and should pay exp rti L Z s Z t K +. Then the net return of the ank at initial time is fui c exp rti L Z s Z t K +. The fair price of this contract should lead to an identical expected return of the investor and the ank, that is, fui+e c exp rti L ] Z s Z t K + fui E c exp rti L Thus the American up-and-in call option price is just the expected present value of the payoff. Z s Z t K + ]. Definition 1. Suppose there is an American up-and-in call option with a strike price K, an expiration time T and trigger point L. Then the American up-and-in call option price is ] fui c E exp rti L Z s Z t K +. 5

Theorem 6. Suppose there is an American up-and-in call option with a strike price K, an expiration time T and trigger point L. Then the American up-and-in call option price is determined y 1 ] + fui c exp rt Z exp ut + π ln K d 1 where 1 πut + ln Z ln L 1 + exp. Proof. Firstly, we show that the uncertain variale exp rti L has an inverse uncertainty distriution where Noting that Φ 1 Z t Z exp exp rti L ut + Z s Z t K + π ln. 1 Z t K + exp rti L Z s Z t K + exp rti L Z s, Z t Zt, t, T ] Z t Z t, t, T ] Z t K + and exp rti L Z s Z t K + > Z s, Z t Zt, t, T ] Z t > Z t, t, T ], exp rti L Z t K + we have M exp rti L MZ t Z t, t, T ] Z s Z t K + exp rti L Z t K + 2 and M exp rti L MZ t > Z t, t, T ] Z s Z t K + > exp rti L Z t K + 1 3 6

from Theorems 1 and 5. It follows from duality axiom that M exp rti L Z s Z t K + exp rti L M exp rti L According to 2-4, we otain that M exp rti L That is, the uncertain variale Z s Z t K + > Z s Z t K + has an inverse uncertainty distriution From Theorem 4, we have Moreover, the equation holds if and only if Φ 1 f c ui exp rti L exp rti L exp rti L I L Z us + exp rti L exp rti L Z s Z t K + 1 3vs ln π holds for any t, T ]. Due to Z < L, 9 equals to Z ut + π ln L 1 Z t K +. Z t K + d. Z t K + + Z t K + 1. 4 Z t K +. L 5 1 which indicates that 1 πut + ln Z ln L 1 + exp. Therefore, the price of American up-and-in call option is fui c exp rti L Thus the proof is finished. exp rtz t exp rt ut + K + d Z t K + d ] + π ln 1 K d. 7

Secondly, we consider down-and-out option that ecomes void only when the market price of the underlying asset starts up the trigger point L and moves down efore the expiration date. Look at the American down-and-in put option with a strike price K, an expiration date T and a trigger point L for some stocks in the uncertain financial market as defined y 1 Let f p di denote the price of contract. The investor pays f p di to uy the contract at initial time, and has a present value of the pay off as follows exp rt 1 I L Z s K Z t +. Then the net return of the investor at initial time is 1 I L f p di + exp rt Z s K Z t +. On the other hand, the ank receives f p di due to selling the contract at initial time, and should pay exp rt 1 I L Z s K Z t +. Then the net return of the ank at initial time is 1 I L f p di exp rt Z s K Z t +. The fair price of this contract should lead to an identical expected return of the investor and the ank, that is, ] f p di + E exp rt 1 I L Z s K Z t + ] f p di E exp rt 1 I L Z s K Z t +. Thus the American down-and-in put option price is just the expected present value of the payoff. Definition 11. Suppose there is an American down-and-in put option with a strike price K, an expiration time T and trigger point L. Then the American down-and-in put option price is ] f p di E exp rt 1 I L Z s K Z t +. Theorem 7. Suppose there is an American up-and-in put option with a strike price K, an expiration time T and trigger point L. Then the American down-and-in put option price is determined y ] + f p di exp rt K Z exp ut + π ln d 1 where 1 πut + ln Z ln L 1 + exp. Proof. Firstly, we show that the uncertain variale 1 I L exp rt has an inverse uncertainty distriution Φ 1 exp rt 1 I L 8 Z s Z1 s K Z t + K Z 1 t +

where Noting that and we have Z t Z exp ut + π ln. 1 exp rt 1 I L Z s K Z t + Φ 1 Z s 1, Z t Zt, t, T ] Z t Z 1 t, t, T ] exp rt 1 I L s K Z t + > Φ 1 Z s < 1, Z t Zt, t, T ] Z t < Z 1 t, t, T ], M exp rt 1 I L Z s K Z t + Φ 1 MZ t Z 1 t, t, T ] 6 and M exp rt 1 I L Z s K Z t + > Φ 1 MZ t < Z 1 t, t, T ] 1 7 from Theorems 1 and 5. It follows from duality axiom that M exp rt 1 I L Z s K Z t + Φ 1 + M exp rt 1 I L Z s K Z t + > Φ 1 1. 8 According to 6-8, we otain that M exp rt 1 I L Z s K Z t + Φ 1. That is, the uncertain variale exp rt 1 I L Z s K Z t + has an inverse uncertainty distriution Φ 1 exp rt 1 I L 9 Z1 s K Z 1 t +.

From Theorem 4, we have Moreover, the equation holds if and only if f p di exp rt 1 I L exp rt 1 I L I L Z s Z1 s Z s K Zt 1 + d K Z t + d. 3vs Z s Z us + ln < L 9 π 1 holds for any t, T ]. Due to Z L, 9 equals to Z ut + π ln < L 1 which indicates that < 1 πut + ln Z ln L 1 + exp. Therefore, the price of American down-and-in put option is f p di exp rt 1 I L Thus the proof is finished. exp rtk Z t + d exp rt Z s K Z exp K Z t + d ] + ut + π ln d 1 4 Knock-out Options We apply this section to study the knock-out option which is one type of arrier option. Knock-out option means that the option is void only when the market price of the underlying asset touches the trigger point and the right is valid for the period of the option. Furthermore, its pricing formulas are also investigated in this section. Firstly, we consider up-and-out option that ecomes void only when the market price of the underlying asset starts elow the trigger point L and moves up efore the expiration date. Look at the American up-and-out put option with a strike price K, an expiration date T and a trigger point L for some stocks in the uncertain financial market defined y stockmodel. Let f p uo denote the price of contract. The investor pays fuo p to uy the contract at initial time, and has a present value of the pay off as follows 1 I L exp rt Then the net return of the investor at initial time is 1 I L fuo p + exp rt 1 Z s K Z t +. Z s K Z t +.

On the other hand, the ank receives fuo c due to selling the contract at initial time, and should pay 1 I L exp rt Then the net return of the ank at initial time is 1 I L fuo p exp rt Z s K Z t +. Z s K Z t +. The fair price of this contract should lead to an identical expected return of the investor and the ank, that is, fuo p + E exp rt fuo p E exp rt 1 I L 1 I L ] Z s K Z t + ] Z s K Z t +. Thus the American up-and-out put option price is just the expected present value of the payoff. Definition 12. Suppose there is an American up-and-out put option with a strike price K, an expiration time T and trigger point L. Then the American up-and-out put option price is ] fuo p E exp rt 1 I L Z s K Z t +. Theorem 8. Suppose there is an American up-and-out put option with a strike price K, an expiration time T and trigger point L. Then the American up-and-out put option price is determined y ] + fuo p exp rt K Z exp ut + π ln d 1 where 1 πut + ln Z ln L 1 + exp. Proof. Firstly, we show that the uncertain variale 1 I L exp rt has an inverse uncertainty distriution Φ 1 exp rt 1 I L Z s K Z t + 1 K Z 1 t + where Noting that Z t Z exp ut + π ln. 1 exp rt 1 I L Z s K Z t + Φ 1 Z s 1, Z t Zt 1, t, T ] Z t Z 1 t, t, T ] 11

and we have exp rt 1 I L Z s K Z t + > Φ 1 Z s < 1, Z t < Zt 1, t, T ] Z t < Z 1 t, t, T ], M exp rt 1 I L MZ t Z 1 t, t, T ] Z s K Z t + Φ 1 1 and M exp rt 1 I L MZ t < Z 1 t, t, T ] Z s K Z t + > Φ 1 1 11 from Theorems 1 and 5. It follows from duality axiom that M exp rt 1 I L M exp rt 1 I L Z s K Z t + Φ 1 Z s K Z t + > Φ 1 + According to 1-12, we otain that M exp rt 1 I L Z s K Z t + Φ 1. That is, the uncertain variale exp rt has an inverse uncertainty distriution Φ 1 From Theorem 4, we have f p uo exp rt 1 I L 1 I L exp rt 1 I L exp rt 1 I L Z s K Z t + 1 Z s K Z 1 t +. 1 K Zt 1 + d K Z t + d. 1 12 Moreover, the equation I L 12

holds if and only if Z us + 3vs ln π holds for any t, T ]. Due to Z L, 9 equals to Z ut + π ln < L 1 < L 13 1 which indicates that < 1 πut + ln Z ln L 1 + exp. Therefore, the price of American up-and-out put option is f p uo Thus the proof is finished. exp rt 1 I L exp rtk Z t + d exp rt K Z exp K Z t + d ] + ut + π ln d. 1 Secondly, we consider down-and-out option that ecomes void only when the market price of the underlying asset starts up the trigger point L and moves down efore the expiration date. Look at the American down-and-out call option with a strike price K, an expiration date T and a trigger point L for some stocks in the uncertain financial market defined y 1. Let fdo c denote the price of contract. The investor pays fdo c to uy the contract at initial time, and has a present value of the pay off as follows exp rti L Z t K +. Z s Then the net return of the investor at initial time is fdo c + exp rti L X s X t K +. On the other hand, the ank receives fdo c due to selling the contract at initial time, and should pay exp rti L Z s Z t K +. Then the net return of the ank at initial time is fdo c exp rti L Z s Z t K +. The fair price of this contract should lead to an identical expected return of the investor and the ank, that is, fdo+e c exp rti L Z s Z t K + ] fdo E c exp rti L Z s Z t K + ]. Thus the American down-and-out call option price is just the expected present value of the payoff. 13

Definition 13. Suppose there is an American down-and-out call option with a strike price K, an expiration time T and trigger point L. Then the American down-and-out call option price is ] fdo c E exp rti L Z t K +. Z s Theorem 9. Suppose there is an American down-and-out call option with a strike price K, an expiration time T and trigger point L. Then the American down-and-out call option price is determined y 1 ] + fdo c exp rt Z exp ut + π ln K d 1 where 1 πut + ln Z ln L 1 + exp. Proof. Firstly, we show that the uncertain variale exp rti L has an inverse uncertainty distriution Φ 1 Z s exp rti L Z t K + Z s Z t K + where Z t Z exp ut + π ln. 1 Noting that exp rti L Z s Z t Z t, t, T ] Z s Z t K + exp rti L Z s, Z t Zt, t, T ] Z s Z t K + and exp rti L Z s Z t > Z t, t, T ], Z s Z t K + > exp rti L Z s, Z t Zt, t, T ] Z s Z t K + we have M exp rti L MZ t Z t, t, T ] Z s Z t K + exp rti L Z s Z t K + 14 14

and M exp rti L MZ t > Z t, t, T ] Z s Z t K + > exp rti L Z s Z t K + 1 15 from Theorems 1 and 5. It follows from duality axiom that M exp rti L Z s Z t K + exp rti L M exp rti L Z s According to 14-16, we otain that M exp rti L That is, the uncertain variale Z s has an inverse uncertainty distriution From Theorem 4, we have Moreover, the equation holds if and only if Φ 1 f c do Z t K + > Z t K + exp rti L Z s exp rti L exp rti L I L Z s exp rti L exp rti L Z s Z t K + Z s Z s 1 Z s Z s Z t K +. Z t K + d. Z t K + + Z t K + 1. 16 Z t K +. 3vs Z s Z us + ln L 17 π 1 holds for any t, T ]. Due to Z > L, 17 equals to Z ut + π ln L 1 which indicates that 1 πut + ln Z ln L 1 + exp. Therefore, the price of American down-and-out call option is fdo c exp rti L Thus the proof is finished. exp rtz t exp rt ut + Z s K + d Z t K + d ] + π ln 1 K d. 15

5 Conclusions This paper aimed at studying the American arrier option of stock model in uncertain finance market. Four new types of concepts were presented which are American up-and-in call option, American upand-in call option, American down-and-in put option, American up-and-out put option and American down-and-out call option. Furthermore, pricing formulas were provided for computing the corresponding options. Acknowledgments This work was ported y National Natural Science Foundation of China Grant No.6157321. References 1] Black F, and Scholes M, The pricing of option and corporate liailities, Journal of Political Economy, Vol.81, 637-654, 1973. 2] Chen XW, American Option Pricing Formula for Uncertain Financial Market, International Journal of Operations Research, Vol.8, No.2, 32-37, 211. 3] Chen XW, and Liu B, Existence and uniqueness theorem for uncertain differential equations, Fuzzy Optimization and Decision Making, Vol.9, No.1, 69-81, 21. 4] Gao R, Milne method for solving uncertain differential equations, Applied Mathematics and Computation, Vol.274, 774-785, 216. 5] Gao R, and Yao K, Importance index of component in uncertain random reliaility system, Knowledge-Based Systems, Vol.19, 28-217, 216. 6] Ito K, Stochastic integral, Proceedings of the Japan Academy, Ser. A, Mathematical Sciences, Vol.2, No.8, 519-524, 1944. 7] Ito K, On stochastic differential equations, Memoirs of the American mathematical Society, Vol.4, 1-51, 1951. 8] Liu B, Uncertainty Theory, 2nd ed., Springer-Verlag, Berlin, 27. 9] Liu B, Fuzzy process, hyrid process and uncertain process, Journal of Uncertain Systems, Vol.2, No.1, 3-16, 28. 1] Liu B, Some research prolems in uncertainty theory, Journal of Uncertain Systems, Vol.3, No.1, 3-1, 29. 11] Liu B, Uncertain risk analysis and uncertain reliaility analysis, Journal of Uncertain Systems, Vol.4, No.3, 163-17, 21. 12] Liu B, Uncertain random graph and uncertain random network, Journal of Uncertain Systems, Vol.8, No.1, 3-12, 214. 13] Liu B, Uncertainty distriution and independence of uncertain processes, Fuzzy Optimization and Decision Making, Vol.13, No.3, 259-271, 214. 14] Merton R, Theory of rational option pricing, Bell Journal Economics & Management Science, Vol.4, No.1 141-183, 1973. 15] Qin ZF, Mean-variance model for portfolio optimization prolem in the simultaneous presence of random and uncertain returns, European Journal of Operational Research, Vol.245, 48-488, 215. 16] Samuelson PA, Proof that properly anticipated prices fluctuate randomly, Industrial Manage Review, Vol.6, 41-5, 1965. 16

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