Barrier Option Pricing Formulae for Uncertain Currency Model

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1 Barrier Option Pricing Formulae for Uncertain Currency odel Rong Gao School of Economics anagement, Hebei University of echnology, ianjin 341, China Abstract Option pricing is the core issue of modern finance. Barrier option is a type of path-dependent exotic option which is similar to the ordinary option in some ways. It becomes activated or null void only when the underlier reaches a predetermined level barrier. For the uncertain foreign exchange market, this paper mainly studies European barrier options, specifically they are European up--in call option, European down--in put option, European up--out put option European down--out call option. Furthermore, we prove some pricing formulae to calculate the corresponding option pricing. Keywords: pricing; uncertain differential equation; barrier option; currency model. 1 Introduction With the rapid growth of trading volume in foreign exchange markets, the currency option pricing becomes a widely discussed issue. A currency option is a contract that gives the owner the right instead of the obligation to buy or sell the indicated amount of foreign currency at a strike, where European option can be exercised at the maturity time American option can be exercised at anytime before the maturity time. In financial market, we often use currency option to avoid risks obtain profits, so the option pricing formula is an extremely significant subject in the financial field. In the framework of probability theory, the study for the option pricing aroused scholars interest extensively. In 1973, Black Scholes [1], independently erton [22] derived an option pricing formula which is said to be the famous Black-Scholes formula. In 1983, Garman Kohlhagen [11] built G-K model to determine the price of the European currency option under the assumption that the dynamics of the spot exchange rate is governed by a geometric Wiener process with constant drift diffusion. Since then, as modifications of the G-K model, many methodologies for pricing currency options were investigated, such as Hilliard et al. [12], elino urnbull [21], Sarwar Krehbiel [23], Bollen Rasiel [2], Carr Wu [3], Xiao et al. [28]. Clearly the currency models interest rate models were established based on probability theory in the above literature. Probability theory is used to model the rom phenomenon which is a type of indeterminacy associated with the frequency. According to law of large numbers, if we want to apply probability, we should assume that the long-run cumulative probability distribution is close enough to the frequency. hat is, we should possess lots of samples. However, not all indeterminacies have adequate samples or even some of indeterminacies have not samples. For example, we cannot know the carrying weight of a bridge being used. For modeling the indeterminacy without samples, Liu [14, 16] pioneered uncertainty theory based on normality, duality, subadditivity product axioms. In this theory, uncertain measure is used to describe human belief degree, uncertain variable is used to describe 1

2 uncertain quantity, uncertainty distribution expected value are respectively used to describe rank the uncertain variable. Up to now, the uncertainty theory has been applied into many fields such as uncertain programming Liu [17], uncertain risk reliability analysis Liu [19], Gao Yao [9, 1]. o characterize a dynamic uncertain system, Liu [15] proposed the concept of uncertain process which is a sequence of uncertain variables varying with time. And uncertain field, multi-dimensional uncertain process, was put forward by Liu [15], its properties were studied by Gao Chen [8]. As a special type of uncertain process, Liu process was proposed by Liu [16] which has independent stationary increments. Furthermore, uncertain calculus was introduced by Liu [16] on the basis of Liu process. Following the appearance of uncertain calculus, Liu [16] established uncertain differential equation driven by Liu process. hen Chen Liu [5] provided a sufficient condition for an uncertain differential equation having a unique solution. With the development of the uncertain differential equation, a new tool appeared in the complicated financial market. Liu [16] first thought that stock price follows an uncertain differential equation presented uncertain stock model. By using Liu s stock model, European option pricing formulae American option pricing formulae were derived by Liu [16] Chen [4], respectively. hen Chen Gao [6] first assumed that interest rate follows a uncertain differential equation, proposed three different uncertain interest rate models which are the counterparts of the Ho-Lee model [13], Vasicek model [25] Cox-Ingersoll-Ross model [7], respectively. Zhu [32] recently presented another uncertain interest rate model obtained the zero-coupon bond price. Zhang et al. [31] studied the option pricing formulae of the interest rate ceiling. Foreign currency is an active instrument in capital market with a large percentage. Currency option is one of the most classic useful options one of the core content of modern finance. Liu et al. [2] assumed that the exchange rate follows an uncertain differential equation proposed an uncertain currency model. oreover, they also derived the option pricing formulae by using their model including European option American option. Shen Yao [24] presented mean-reverting currency model obtained its European American option pricing formulae. Wang Ning [26] put forward a currency model with floating interest rate proved its option pricing formulae relative to European option American option. In this paper, barrier option pricing formulae are derived for the Liu-Chen-Ralescu s foreign currency model. he rest of the paper is organized as follows: Some preliminary concepts of uncertain processes are recalled in Section 2. Section 3 Section 4 are used to derive European barrier option pricing formulae in the knock-in case in the knock-out case, respectively. Finally, a brief summary is given in Section 5. 2 Preliminaries In this section, we will introduce some fundamental concepts properties concerning uncertain variables, uncertain processes, uncertain differential equations. 2.1 Uncertain Differential Equation Let Γ be a nonempty set, let L be a σ-algebra over Γ. Each element Λ in L is called an event assigned a number Λ to indicate the belief degree with which we believe Λ will happen. In order to deal with belief degrees rationally, Liu [14] suggested the following three axioms: 2

3 Axiom 1. Normality Axiom Γ = 1 for the universal set Γ; Axiom 2. Duality Axiom Λ + Λ c = 1 for any event Λ; Axiom 3. Subadditivity Axiom For every countable sequence of events Λ 1, Λ 2,, we have Λ i Λ i. i=1 i=1 Definition 1. Liu [14] he set function is called an uncertain measure if it satisfies the normality, duality, subadditivity axioms. heorem 1. he uncertain measure is a monotone increasing set function, i.e., for any events Λ 1 Λ 2 with Λ 1 Λ 2, we have Λ 1 Λ 2. he triplet Γ, L, is called an uncertainty space. Furthermore, the product uncertain measure on the product σ-algebra L was defined by Liu [16] as follows: Axiom 4. Product Axiom Let Γ k, L k, k be uncertainty spaces for k = 1, 2,. he product uncertain measure is an uncertain measure satisfying Λ k = k Λ k k=1 k=1 where Λ k is an event arbitrarily chosen from L k for each k k = 1, 2,, respectively. Definition 2. Liu [14] An uncertain variable is a measurable function ξ from an uncertainty space Γ, L, to the set of real numbers, i.e., for any Borel set B of real numbers, the set ξ B = γ Γ ξγ B is an event. heorem 2. Let ξ 1, ξ 2,, ξ n be uncertain variables, f a real-valued measurable function. hen ξ = fξ 1, ξ 2,, ξ n is an uncertain variable defined by ξγ = fξ 1 γ, ξ 2 γ,, ξ n γ, γ Γ. Definition 3. Liu [14] Suppose ξ is an uncertain variable. hen the uncertainty distribution of ξ is defined by Φx = ξ x for any real number x. Definition 4. Liu [16] he uncertain variables ξ 1, ξ 2,, ξ n are said to be independent if n n ξ i B i = ξ i B i i=1 i=1 for any Borel sets B 1, B 2,, B n of real numbers. 3

4 heorem 3. Liu [18] Let ξ 1, ξ 2,, ξ n be independent uncertain variables with regular uncertainty distributions Φ 1, Φ 2,, Φ n, respectively. If the function fx 1, x 2,, x n is strictly increasing with respect to x 1, x 2,, x m strictly decreasing with respect to x m+1, x m+2,, x n, then the uncertain variable 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 variables, the concept of expected value was proposed by Liu [14] as follows: Definition 5. Liu [14] Let ξ be an uncertain variable. hen the expected value of ξ is defined by E[ξ] = + provided that at least one of the two integrals is finite. ξ xdx ξ xdx heorem 4. Liu [14] Let ξ be an uncertain variable with uncertainty distribution Φ. If the expected value exists, then E[ξ] = + xdφx. If the uncertainty distribution Φ is regular, then we also have E[ξ] = 1 Φ 1 d. An uncertain process is essentially a sequence of uncertain variables indexed by time. he study of uncertain process was started by Liu [15] in 28. Definition 6. Liu [15] Let be an index set let Γ, L, be an uncertainty space. An uncertain process is a measurable function from Γ, L, to the set of real numbers such that X t B is an event for any Borel set B for each t. Definition 7. Liu [16] An uncertain process C t is said to be a Liu process if i C = almost all sample paths are Lipschitz continuous, ii C t has stationary independent increments, iii every increment C s+t C s is a normal uncertain variable with expected value variance t 2, whose uncertainty distribution is Φx = 1 πx 1 + exp, x R. 3t Definition 8. Liu [15] Suppose that C t is a Liu process, f 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. 4

5 Definition 9. Yao Chen [3] he -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 distribution of stard normal uncertain variable, i.e., Φ 1 = 3 π ln 1, < < 1. heorem 5. Yao Chen [3] Assume ft, x gt, x are continuous X t the uncertain differential equation is the -path of dx t = ft, X t dt + gt, X t dc t, t [1, s]. hen we get X t X t, t =, X t > Xt, t = 1. From heorem 5, many properties of -path have been derived. Firstly, Chen Yao [3] found the inverse uncertainty distribution of X t, that is Ψ 1 = X t, < < 1. 3 Knock-in Options We apply this section to study the knock-in option which is one type of barrier option. Knock-in option means that the option is activated only when the market price of the underlying asset touches the trigger point 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 be used in the main results as follows 1, if L x I L =, if L > x where L is a given number to represent the trigger point. Firstly, we consider up--in option that becomes activated only when the exchange rate starts below the trigger point L moves up before the expiration date. Look at the European up--in call option with a strike price K, an expiration date a trigger point L for some exchange rate in the uncertain financial market as follows dx t = X t dt dy t = βx t dt 1 dz t = uz t dt + vz t dc t where X t indicates the domestic currency with domestic interest rate, Y t indicates the foreign currency with foreign interest rate β, Z t indicates the exchange date which is domestic currency price of one unit of foreign currency at time t, u v are respectively the drift diffusion, C t is a Liu process. 5

6 Suppose that fui c is the price of the contract in domestic currency. he investor pays f ui c to buy the contract at initial time, has a present value of the pay off in domestic currency as follows exp I L sup Z t Z K +. hen the net return of the investor at initial time is fui c + exp I L sup Z t Z K +. On the other h, the bank receives fui c due to selling the contract at initial time, receives 1 K/Z + in foreign currency at expiration date. So it should pay exp β I L sup Z t Z 1 K/Z +. hen the net return of the bank at initial time is fui c exp β I L sup Z t Z 1 K/Z +. he fair price of this contract should lead to an identical expected return of the investor the bank, that is, ] ] fui c + E [exp I L sup Z t Z K + = fui c E [exp β I L sup Z t Z 1 K/Z + hus the European up--in option price is given by the definition below. Definition 1. Suppose there is a European up--in call option with a strike price K, an expiration time trigger point L. hen the European up--in call option price for model 1 is fui c = 1 [ ] 2 exp E I L sup Z s Z t K ] 2 exp β Z E [I L sup Z t 1 K/Z +. heorem 6. Suppose there is a European up--in call option with a strike price K, an expiration time trigger point L. hen the European up--in call option price is determined by f c ui = 1 2 exp exp β 1 b 3v Z exp u + π b Z K/ exp u + ln 1 3v ln π K + d + d 1 where b = 1 πu + ln Z ln L 1 + exp. 3v Proof. First, we show that the uncertain variable exp I L sup Z t Z K + Φ 1 = exp I L Z K + 6

7 where Noting that we have Z t = Z exp ut + 3vt π ln. 1 exp I L sup Z t Z K + exp I L sup sup Z t, Z Z Z t Z t, t [, ] Zt exp I L sup Z t Z K + > exp I L sup sup Z t >, Z > Z Z t > Z t, t [, ], exp I L sup Z t Z K + exp I L Z t Z t, t [, ] Zt Z K + Z K + Z K + = 2 exp I L sup Z t Z K + > exp I L Z t > Z t, t [, ] Z K + = 1 3 from heorems 1 5. It follows from duality axiom that exp I L sup Z t Z K + exp I L exp I L sup Z t Z K + > exp I L According to 2-4, we obtain that exp I L sup Z t Z K + exp I L sup hat is, the uncertain variable exp I L sup Φ 1 = exp I L Z t Z K + sup Zt Z t Z K +. Z K + + Z K + = 1. 4 Z K + =. 7

8 From heorem 4, we have f c ui = 1 exp I L Second, we show that the uncertain variable exp β I L where sup Ψ 1 = exp β I L Z t = Z exp ut + sup Z t Z K +. Z t 1 K/Z + 1 K/Z + 3vt π ln. 1 Noting that exp β I L sup Z t 1 K/Z + exp β I L sup Z t, Z Z Z t Z t, t [, ] 1 K/Z + we have exp β I L sup Z t 1 K/Z + > exp β I L sup sup Z t >, Z > Z Z t > Z t, t [, ], exp β I L sup Z t 1 K/Z + exp β I L Z t Z t, t [, ] Zt 1 K/Z + 1 K/Z + = 5 exp β I L sup Z t 1 K/Z + > exp β I L Z t > Z t, t [, ] 1 K/Z + = 1 6 from heorems 1 5. It follows from duality axiom that exp β I L sup Z t 1 K/Z + exp β I L exp β I L sup Z t Z 1 K/Z + > exp β I L 8 1 K/Z K/Z + = 1. 7

9 According to 5-7, we obtain that exp β I L sup Z t 1 K/Z + exp β I L sup Zt 1 K/Z + =. hat is, the uncertain variable exp β I L sup Z t Z 1 K/Z + Ψ 1 = exp β I L 1 K/Z +. From heorem 4, we have f c ui = 1 exp β I L 1 K/Z +. oreover, the equation holds if only if I L sup Zs s t = sup Z ut + = 1 holds for any t [, ]. Due to Z < L, 8 equals to 3v Z u + ln L π 1 3vt π ln L 8 1 which indicates that 1 πu + ln Z ln L 1 + exp b. 3v herefore, the price of European up--in option is fui c = 1 1 exp I L exp β I L sup 2 = hus the proof is finished. b Zt exp Z K + d Z = 1 2 exp exp β 1 b Z exp u + b Z K/ exp Z K + d Z 1 K/Z + 1 b exp β 1 K/Z + d 3v ln π 1 3v u + ln π K + d + d. 1 Secondly, we consider down--in option that becomes void only when the market price of the underlying asset starts above the trigger point L moves down before the expiration date. Look at the European down--in put option with a strike price K, an expiration date a trigger point L for some stocks in the uncertain financial market as defined by 1. 9

10 Let f p di denote the price of contract. he investor pays f p di to buy the contract at initial time, has a present value of the pay off as follows exp 1 I L Z t K Z t +. hen the net return of the investor at initial time is f p di + exp 1 I L Z t K Z t +. On the other h, the bank receives f p di due to selling the contract at initial time, should pay exp β 1 I L Z t Z K/Z 1 +. hen the net return of the bank at initial time is f p di exp β 1 I L Z t Z K/Z 1 +. he fair price of this contract should lead to an identical expected return of the investor the bank, that is, ] f p di [exp + E 1 I L Z t K Z t + ] = f p di [ E exp β 1 I L Z t Z K/Z 1 + hus the European down--in put option price is given by below. Definition 11. Suppose there is a European down--in put option with a strike price K, an expiration time trigger point L. hen the European down--in put option price is f p di = 1 [ ] 2 exp E 1 I L Z t K Z t ] 2 exp β Z E [1 I L Z t K/Z 1 +. heorem 7. Suppose there is a European down--in put option with a strike price K, an expiration time trigger point L. hen the American down--in put option price is determined by f p di = 1 b + 3v 2 exp K Z exp u + ln d π b + 3v 2 exp β K/ exp u + ln Z d π 1 where b = 1 πu + ln Z ln L 1 + exp. 3v Proof. First, we show that the uncertain variable exp 1 I L Z t K Z t + Φ 1 = exp 1 I L Z1 t Z 1 K + 1

11 where Z 1 t = Z exp ut + 3vt π ln 1. Noting that exp 1 I L Z t K Z t + exp Z t Z1 t, Z Z 1 Z t Z 1 t, t [, ] 1 I L Z1 t Z 1 K + exp 1 I L Z t K Z t + > exp Z t < Z1 t, Z < Z 1 Z t < Z 1 t, t [, ], 1 I L Z1 t Z 1 K + we have exp 1 I L Z t K Z t + exp 1 I L Z t Z 1 t, t [, ] Z1 t Z 1 K + = 9 exp 1 I L Z t K Z t + > exp 1 I L Z t < Z 1 t, t [, ] Z1 t Z 1 K + = 1 1 from heorems 1 5. It follows from duality axiom that exp 1 I L Z t K Z t + exp 1 I L 1 I L exp 1 I L Z t K Z t + > exp According to 9-11, we obtain that exp 1 I L Z t K Z t + exp 1 I L Z1 t Z1 t Z1 t Z 1 K + + Z 1 K + = Z 1 K + =. hat is, the uncertain variable exp 1 I L Z t K Z t + 11

12 Φ 1 = exp 1 I L From heorem 4, we have f p di = 1 exp 1 I L Z1 t Z1 t Z 1 K +. Z 1 K +. Second, we show that the uncertain variable exp β 1 I L Z t K/Z 1 + Ψ 1 = exp β 1 I L Z1 t K/Z where Noting that we have Z 1 t = Z exp ut + 3vt π ln 1. exp β 1 I L Z t K/Z 1 + Ψ 1 Z t Z1 t, Z Z 1 Z t Z 1 t, t [, ] exp β 1 I L Z t K/Z 1 + > Ψ 1 Z t < Z1 t, Z < Z 1 Z t < Z 1 t, t [, ] exp β 1 I L Z t K/Z 1 + Ψ 1 Z t Z 1 t, t [, ] = 12 exp β 1 I L Z t K/Z 1 + > Ψ 1 Z t < Z 1 t, t [, ] =

13 from heorems 1 5. It follows from duality axiom that exp β 1 I L Z t K/Z 1 + Ψ 1 + exp β 1 I L Z t K/Z 1 + > Ψ 1 = According to 12-14, we obtain that exp β 1 I L Z t Z K/Z 1 + Ψ 1 =. hat is, the uncertain variable exp β 1 I L Z t K/Z 1 + Ψ 1 = exp β 1 I L From heorem 4, we have oreover, the equation holds if only if f p di = 1 Due to Z L, 15 equals to which indicates that exp β 1 I L I L Z1 t Z1 t Z1 t = K/Z Z K/Z vt Z1 t = Z ut + π ln 1 < L. 15 < 3v Z u + ln < L π 1 1 πu + ln Z ln L 1 + exp b. 3v herefore, the price of European down--in put option is f ui = 1 1 exp 1 I L 2 Z1 t I L 2 Z1 t = 1 2 b exp K Z 1 + d Z = 1 2 exp b hus the proof is finished exp β b K Z exp u + K/ exp u + Z K/Z 1 K Z 1 + d b 1 + d exp β K/Z 1 + 3v ln d π 1 3v ln π 1 Z + d 1 + d 13

14 4 Knock-out Options We apply this section to study the knock-out option which is one type of barrier option. Knock-out option means that the option is void only when the market price of the underlying asset touches the trigger point the right is valid for the period of the option. Furthermore, its pricing formulas are also investigated in this section. First, we consider up--out option that becomes void only when the market price of the underlying asset starts below the trigger point L moves up before the expiration date. Look at the European up--out put option with a strike price K, an expiration date a trigger point L for some stocks in the uncertain financial market as defined by 1. Let f p uo represent the price of contract. he investor pays f p uo to buy the contract at initial time, has a present value of the pay off as follows exp 1 I L sup Z t K Z t +. hen the net return of the investor at initial time is fuo p + exp 1 I L sup Z t K Z t +. On the other h, the bank receives f p di due to selling the contract at initial time, should pay exp β 1 I L sup Z t Z K/Z 1 +. hen the net return of the bank at initial time is fuo p exp β 1 I L sup Z t Z K/Z 1 +. he fair price of this contract should lead to an identical expected return of the investor the bank, that is, [ fuo p + E exp 1 I L [ = fuo p E exp β 1 I L ] sup Z t K Z t + hus the European down--in put option price is given by below. ] sup Z t Z K/Z 1 +. Definition 12. Suppose there is a European up--out put option with a strike price K, an expiration time trigger point L. hen the European up--out put option price is fuo p = 1 [ 2 exp E 1 I L sup Z t K Z t ]+ + 1 ] 2 exp β Z E [1 I L sup Z t K/Z 1 +. heorem 8. Suppose there is a European up--out put option with a strike price K, an expiration time trigger point L. hen the European up--out put option price is determined by fuo p = 1 b + 3v 2 exp K Z exp u + ln d π b + 3v 2 exp β K/ exp u + ln Z d π 1 where b = 1 πu + ln Z ln L 1 + exp. 3v 14

15 Proof. First, we show that the uncertain variable exp 1 I L sup Z t K Z t + Φ 1 = exp 1 I L 1 Z 1 K + where Z 1 t = Z exp ut + 3vt π ln 1. Noting that exp 1 I L sup Z t K Z t + exp sup Z t sup Z t Z 1 t, t [, ] Zt 1, Z Z 1 1 I L 1 Z 1 K + exp 1 I L sup Z t K Z t + > exp sup Z t < sup Z t < Z 1 t, t [, ], Zt 1, Z < Z 1 1 I L 1 Z 1 K + we have exp 1 I L sup Z t K Z t + exp 1 I L 1 Z 1 K + Z t Z 1 t, t [, ] = 16 exp 1 I L sup Z t K Z t + > exp Z t < Z 1 t, t [, ] 1 I L 1 Z 1 K + = 1 17 from heorems 1 5. It follows from duality axiom that exp 1 I L sup Z t K Z t + exp exp 1 I L sup Z t K Z t + > exp 1 I L 1 I L 1 1 Z 1 K + + Z 1 K + =

16 According to 16-18, we obtain that exp 1 I L sup Z t K Z t + exp 1 I L 1 Z 1 K + =. hat is, the uncertain variable exp 1 I L sup Z t K Z t + Φ 1 = exp 1 I L 1 Z 1 K +. From heorem 4, we have f p di = 1 exp 1 I L 1 Z 1 K +. Second, we show that the uncertain variable exp β 1 I L sup Z t K/Z 1 + Ψ 1 = exp β 1 I L 1 K/Z where Noting that we have Z 1 t = Z exp ut + 3vt π ln 1. exp β 1 I L sup Z t K/Z 1 + Ψ 1 sup Z t sup Z t Z 1 t, t [, ] Zt 1, Z Z 1 exp β 1 I L sup Z t K/Z 1 + > Ψ 1 sup Z t < sup Z t < Z 1 t, t [, ] Zt 1, Z < Z 1 exp β 1 I L sup Z t K/Z 1 + Ψ 1 Z t Z 1 t, t [, ] = 19 16

17 exp β 1 I L sup Z t K/Z 1 + > Ψ 1 Z t < Z 1 t, t [, ] = 1 2 from heorems 1 5. It follows from duality axiom that exp β 1 I L exp β 1 I L sup Z t K/Z 1 + Ψ 1 sup Z t K/Z 1 + > Ψ 1 + According to 19-21, we obtain that exp β 1 I L sup Z t Z K/Z 1 + Ψ 1 =. hat is, the uncertain variable exp β 1 I L sup Z t K/Z 1 + Ψ 1 = exp β 1 I L 1 K/Z From heorem 4, we have oreover, the equation holds if only if f p di = 1 Due to Z < L, 22 equals to which indicates that exp β 1 I L 1 I L Z1 t = Z K/Z = vt 1 = sup Z ut + π ln 1 < L. 22 < 3v Z u + ln < L π 1 1 πu + ln Z ln L 1 + exp b. 3v 17

18 herefore, the price of European up--out put option is f p uo = 1 2 = b exp 1 I L 1 1 I L sup 1 exp K Z 1 + d Z = 1 2 exp b hus the proof is finished exp β b K Z exp u + K/ exp u + Zt 1 K Z 1 + d Z K/Z 1 b 1 + d exp β K/Z 1 + 3v ln d π 1 3v ln π 1 Z + d. 1 + d Second, we consider down--out option that becomes void only when the market price of the underlying asset starts above the trigger point L moves up before the expiration date. Look at the European down--out call option with a strike price K, an expiration date a trigger point L for some currencies in the uncertain financial market defined by 1. Suppose that fdo c is the price of the contract in domestic currency. he investor pays f do c to buy the contract at initial time, has a present value of the pay off in domestic currency as follows exp I L Z K +. Z t hen the net return of the investor at initial time is fdo c + exp I L Z t Z K +. On the other h, the bank receives fdo c due to selling the contract at initial time, receive 1 K/Z + in foreign currency at expiration date. So it should pay exp β I L Z 1 K/Z +. Z t hen the net return of the bank at initial time is fdo c exp β I L Z t Z 1 K/Z +. he fair price of this contract should lead to an identical expected return of the investor the bank, that is, ] fdo c + E [exp I L Z t Z K + = fdo c E [exp β I L hus the European up--in option price is given by the definition below. Z t Z 1 K/Z + ] Definition 13. Suppose there is a European down--out option with a strike price K, an expiration time trigger point L. hen the European down--out option price for model 1 is fdo c = 1 [ ] 2 exp E I L Z s Z t K ] 2 exp β Z E [I L Z t 1 K/Z +. 18

19 heorem 9. Suppose there is a European down--out option with a strike price K, an expiration time trigger point L. hen the European down--out option price is determined by fdo c = v 2 exp Z exp u + ln K d b π v 2 exp β Z K/ exp u + ln d π 1 where b = Proof. First, we show that the uncertain variable exp I L b 1 πu + ln Z ln L 1 + exp. 3v Z t Φ 1 = exp I L Z K + Z t Z K + where Noting that we have Z t = Z exp ut + 3vt π ln. 1 exp I L Z t Z K + exp I L Z t Z t, Z Z Z t Z t, t [, ] exp I L Z t Z K + > exp I L Z t > Z t, Z > Z Z t > Z t, t [, ], exp I L Z t Z t Z t, t [, ] Z K + exp I L Z t Z t Z t Z K + Z K + Z K + = 23 exp I L Z t Z t > Z t, t [, ] Z K + > exp I L Z t Z K + =

20 from heorems 1 5. It follows from duality axiom that exp I L Z t Z K + exp I L exp I L Z t Z K + > exp I L According to 23-25, we obtain that exp I L Z t Z K + exp I L Z t Z t Z t Z K + + Z K + = Z K + =. hat is, the uncertain variable exp I L Z t Z K + Φ 1 = exp I L Z t Z K +. From heorem 4, we have f c do = 1 exp I L Z t Z K +. Second, we show that the uncertain variable exp β I L Z t Ψ 1 = exp β I L 1 K/Z + Z t 1 K/Z + where Z t = Z exp ut + 3vt π ln. 1 Noting that exp β I L Z t Z t Z t, Z Z Z t Z t, t [, ] 1 K/Z + exp β I L Z t 1 K/Z + exp β I L Z t 1 K/Z + > exp β I L Z t > Z t, Z > Z Z t > Z t, t [, ], Z t 1 K/Z + 2

21 we have exp β I L Z t Z t Z t, t [, ] 1 K/Z + exp β I L Z t 1 K/Z + = 26 exp β I L Z t Z t > Z t, t [, ] 1 K/Z + > exp β I L Z t 1 K/Z + = 1 27 from heorems 1 5. It follows from duality axiom that exp β I L Z t 1 K/Z + exp β I L exp β I L Z t Z 1 K/Z + > exp β I L According to 26-28, we obtain that exp β I L Z t 1 K/Z + exp β I L Z t Z t Z t 1 K/Z K/Z + = K/Z + =. hat is, the uncertain variable exp β I L Z t 1 K/Z + From heorem 4, we have oreover, the equation holds if only if Ψ 1 = exp β I L f c do = 1 exp β I L I L Z t Z t Z t = 1 1 K/Z +. 1 K/Z +. 3vt Z t = Z ut + π ln L 29 1 holds for any t [, ]. Due to Z L, 29 equals to 3v Z u + ln L π 1 which indicates that 1 πu + ln Z ln L 1 + exp b. 3v 21

22 herefore, the price of European up--in option is fdo c = 1 1 exp I L 2 Z t exp β I L 2 = hus the proof is finished. b Z t exp Z K + d Z = 1 2 exp exp β 1 b 3v u + π b Z K/ exp Z K + d Z 1 K/Z + 1 b + ln 1 K d 3v u + ln π exp β 1 K/Z + d + d 1 5 Conclusions his paper aimed at studying the European barrier option of foreign currency model in uncertain finance market. Four number of new types of concepts were presented which are European up--in call option, European down--in put option, European up--out put option European down--out call option, respectively. Furthermore, pricing formule were provided for computing the corresponding options. Acknowledgments his work was supported by National Natural Science Foundation of China Grant No References [1] Black F, Scholes, he pricing of option corporate liabilities, Journal of Political Economy, Vol.81, , [2] Bollen N, Rasiel E, he performance of alternative valuation models in the OC currency options market, Journal of International oney Finance, Vol.22, No.1, 33-64, 23. [3] Carr P, Wu L, Stochastic skew in currency options, Journal of Financial Economics, Vol.86, No.1, , 27. [4] Chen XW, American Option Pricing Formula for Uncertain Financial arket, International Journal of Operations Research, Vol.8, No.2, 32-37, 211. [5] Chen XW, Liu B, Existence uniqueness theorem for uncertain differential equations, Fuzzy Optimization Decision aking, Vol.9, No.1, 69-81, 21. [6] Chen XW, Gao JW, Uncertain term structure model of interest rate, Soft Computing, Vol.17, No.4, , 213. [7] Cox J, Ingersoll J, Ross S, An intertemporal general equilibrium model of asset prices, Econometrica Vol.53, ,

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