Pricing Currency Option Based on the Extension Principle and Defuzzification via Weighting Parameter Identification

Abstract

We present a fuzzy version of the Garman-Kohlhagen (FG-K) formula for pricing European currency option based on the extension principle. In order to keep consistent with the real market, we assume that the interest rate, the spot exchange rate, and the volatility are fuzzy numbers in the FG-K formula. The conditions of a basic proposition about the fuzzy-valued functions of fuzzy subsets are modified. Based on the modified conditions and the extension principle, we prove that the fuzzy price obtained from the FG-K formula for European currency option is a fuzzy number. To simplify the trade, the weighted possibilistic mean (WPM) value with a weighting function is adopted to defuzzify the fuzzy price to a crisp price. The numerical example shows our method makes the α-level set of fuzzy price smaller, which decreases the fuzziness. The example also indicates that the WPM value has different approximation effects to real market price by taking different values of weighting parameter in the weighting function. Inspired by this example, we provide a method, which can identify the optimal parameter.

1. Introduction

With the fast growing of the trading volume in the foreign exchange market, the trading of currency option has increased. It is well know that currency option manages the risk of the foreign exchange market. Hence, an appropriate formula for pricing currency option is becoming extremely significant. Garman and Kohlhagen (1983) [1] derived the closed-form formula (G-K formula) for pricing European currency option by the method of Black and Scholes (1973) [2]. There are six variables in the G-K formula, that is, the spot exchange rate, the volatility of spot exchange rate, the domestic and foreign risk-free interest rate, the strike price, and the time to maturity date. However, some variables in the G-K formula are assumed as constants, which is inconsistent with the empirical phenomena, such as the volatility smile. According to these empirical phenomena, many researchers have been devoted to modify the G-K model, such as Chesney and Scott (1989) [3], Amin and Jarrow (1991) [4], Heston (1993) [5], Bates (1996) [6], Sarwar and Krehbiel (2000) [7], and Carr and Wu (2007) [8].

In general, the data, for instance, the spot exchange rate, the domestic or foreign risk-free interest rate, cannot be recorded or collected precisely. Usually, these financial tools may have different values in different commercial banks and financial institutions. Meanwhile, there are differences between the buy (bid) and the sale (ask) price for these financial tools, which are similar to bid-ask spreads in dealer markets for the stock. The existence of bid-ask spreads implies that we cannot precisely get the true market price. The bid-ask spreads are considered to be natural bands to represent the uncertainty around the market price, which is similar to fuzzy number. Furthermore, expert opinions or statistical estimators of market parameters may be expressed as fuzzy numbers. Therefore, the fuzzy set theory proposed by Zadeh (1965) [9] is useful to model such uncertainty. In a parallel development, the fuzzy set theory is applied to option pricing. Yoshida (2003) [10, 11] discussed the valuation of European call and put options in fuzzy environment. Muzzioli and Torricelli (2004) [12] discussed the pricing of European options in a multiperiod binomial model. Wu (2007) [13] obtained the fuzzy pattern of Black-Scholes formula using the extension principle. Xu et al. (2009) [14] and Zhang et al. (2012) [15] investigated a jump-diffusion option pricing model under fuzzy environment. Liu (2009) [16] applied fuzzy approach in [17] to the G-K formula of currency option. Xu et al. (2010) [18] and Guerra et al. (2011) [19] calculated the Greek letters of currency option and stock option under uncertainty environment, respectively, where the Greek letters are useful tools for managing option risk for an option writer. But the division operation of fuzzy numbers in Liu [16], Xu et al. [18] is invalid occasionally. In this paper, we adopt the extension principle to obtain the fuzzy version of G-K formula for European currency option.

Owing to the vague of the currency market data as we just described above, some input variables, especially the interest rate, the spot exchange rate, and the volatility in the G-K formula, cannot always be expected in a precise sense. Therefore, it is reasonable to assume these input variables in the G-K formula are fuzzy numbers. Then, the fuzzy version of the G-K formula for pricing European currency option is obtained via the extension principle. It is found that one of the conditions of the basic proposition in [13] is not verified. Actually, it is difficult to be verified. So we modify the conditions such that they can be satisfied easily. Based on the modified conditions and the extension principle, we prove that the fuzzy price obtained from the FG-K formula for European currency option is a fuzzy number. Meanwhile, we obtain the explicit expressions of the endpoints of every 𝛼-level set (the closed interval) of the fuzzy price for European currency option. In so doing, the investor can pick any value from some closed interval (𝛼-level set) of the fuzzy price with an acceptable belief degree 𝛼 as European currency option trading price. To simplify the trade convenient, the weighted possibilistic mean (WPM) value with weighting function [20] is adopted to defuzzify the fuzzy price. Usually, the process of defuzzification is to find a crisp number that synthesizes the fuzzy price. Compared with the methods in [16, 18], the numerical example shows our method makes the 𝛼-level set of fuzzy price smaller, which decreases the fuzziness. The reason of the differences is also discussed. The example also indicates that the WPM value has different approximation effects to real market price by taking different values of weighting parameter in the weighting function. Inspired by this example, we provide a method, which can identify the the optimal parameter.

The rest of the paper is organized as follows. In Section 2, some terminologies, notations of fuzzy number, and the main results of [16, 18] are introduced. The fundamental theories about the fuzzy-valued functions of fuzzy subsets are proposed in Section 3. In Section 4, the fuzzy version of the G-K formula for European currency option is obtained by the extension principle. About the fuzzy price for European currency option, defuzzification via weighting parameter identification and numerical analysis are discussed in Section 5. Finally, the conclusions are stated in Section 6.

2. Preliminaries

In this section, we will introduce some definitions and propositions that will be used in the sequel. In order to weaken the conditions of the basic proposition in [13], we modify the definition of fuzzy number by another definitions of the 0-level set and the support set of fuzzy subset. Let 𝑋 be the universe of discourse. Throughout this paper, the universe set 𝑋 is assumed to be the set of all real number (ℝ) endowed with a usual topology.

Definition 1 (see [9]). A fuzzy subset 𝐴 of 𝑋 is a set of ordered pairs 𝐴={(𝑥,𝜇_𝐴(𝑥))∣𝑥∈𝑋}, where 𝜇_𝐴∶𝑋→[0,1] is called membership function of 𝐴.

The concept of fuzzy subset was first introduced by [9]. The value 𝜇_𝐴(𝑥) can be interpreted as the membership degree of a point 𝑥 to the set 𝐴.

Definition 2 (see [9]). A fuzzy subset 𝐴 of 𝑋 is called normal if there exists at least one element 𝑥∈𝑋 such that 𝜇_𝐴(𝑥)=1.

Definition 3 (see [21]). Let 𝐴 be a fuzzy subset of 𝑋. The support of 𝐴, denoted by Supp(𝐴), is the crisp set cl{𝑥∈𝑋∣𝜇_𝐴(𝑥)>0}.

Definition 4 (see [9]). Let 𝐴 be a fuzzy subset of 𝑋. The 𝛼-level set of 𝐴, denoted by 𝐴_𝛼, is the crisp set {𝑥∈𝑋∣𝜇_𝐴(𝑥)≥𝛼}, where 𝛼∈[0,1].

Definition 5 (see [9]). A fuzzy subset 𝐴 of 𝑋 is called convex if for all 𝛼∈[0,1] its 𝛼-level sets 𝐴_𝛼 are (crisp) convex sets.

Alternatively, a fuzzy subset 𝐴 of 𝑋 is a convex fuzzy subset if and only if 𝜇_𝐴(𝜆𝑥+(1−𝜆)𝑦)≥min{𝜇_𝐴(𝑥),𝜇_𝐴(𝑦)} for all 𝜆∈[0,1]; that is, 𝜇_𝐴(𝑥) is a quasi-concave function.

Definition 6 (see [22]). A real-valued function 𝑓∶ℝ→ℝ is called upper semicontinuous at a point 𝑥∈ℝ if 𝑓(𝑥)≥limsup_𝑘→∞𝑓(𝑥_𝑘) for every sequence {𝑥_𝑘}⊂ℝ that converges to 𝑥. If 𝑓∶ℝ→ℝ is upper semicontinuous at every point in ℝ, we say that 𝑓(𝑥) is upper semicontinuous.

A real-valued function 𝑓(𝑥) is upper semicontinuous if and only if {𝑥∣𝑓(𝑥)≥𝛼} is closed for all 𝛼∈ℝ [22].

Definition 7 (fuzzy number). Let 𝑎 be a fuzzy subset of ℝ. Then 𝑎 is called a fuzzy number if the following conditions are satisfied:(i)𝑎 is a normal and convex fuzzy subset,(ii)its membership function 𝜇_𝑎(𝑥) is upper semicontinuous,(iii)the Supp(𝑎) is bounded.

Note that if 𝑎 is a fuzzy number, then the 𝛼-level set 𝑎_𝛼, for all 𝛼∈(0,1], is a compact (closed and bounded in ℝ) and convex set; that is, 𝑎_𝛼 is a bounded and closed interval for all 𝛼∈(0,1]. Then, the 𝛼-level set of 𝑎 is denoted as 𝑎_𝛼=[𝑎^𝐿_𝛼,𝑎^𝑈_𝛼] for all 𝛼∈(0,1], and 𝑎₀=ℝ especially. For convenience, we consider the fuzzy input variables as triangular fuzzy numbers; that is, the graph of the membership function looks like triangles.

Definition 8. A fuzzy number 𝑎∈ℱ is a triangular fuzzy number if its membership function 𝜇_𝑎(𝑥) is characterized as follows: 𝜇_𝑎(𝑥)=𝑥−𝑎₁𝑎₂−𝑎₁𝑎₁≤𝑥≤𝑎₂,𝑥−𝑎₃𝑎₂−𝑎₃𝑎₂≤𝑥≤𝑎₃,0otherwise. (1) Here, [𝑎₁,𝑎₃] is the support set and the membership function has a peak at 𝑎₂. The triangular fuzzy number 𝑎 usually is denoted as 𝑎=(𝑎₁,𝑎₂,𝑎₃).

Definition 9 (see [20]). Let 𝑎∈ℱ be a fuzzy number with 𝑎_𝛼=[𝑎^𝐿_𝛼,𝑎^𝑈_𝛼], 𝛼∈[0,1]. A function 𝑓∶[0,1]→ℝ is said to be a weighting function if 𝑓(𝑥) is nonnegative, monotone increasing and satisfies the normalization condition: ¹₀𝑓(𝛼)d𝛼=1. (2) We define the 𝑓-weighted possibilistic mean (WPM) value of the fuzzy number 𝑎 as 𝑀_𝑓(𝑎)=¹₀𝑎^𝐿_𝛼+𝑎^𝑈_𝛼2𝑓(𝛼)d𝛼. (3)

Next, the decomposition theory is introduced which establishes an important connection between fuzzy sets and crisp sets. That is, we can get the expression of membership function of a fuzzy set if all its 𝛼-level sets are available.

Proposition 10 (decomposition theorem [23–25]). Let 𝐴 be a fuzzy subset with membership function 𝜇_𝐴(𝑥) and 𝛼-level set 𝐴_𝛼={𝑥∣𝜇_𝐴(𝑥)≥𝛼}. Then 𝜇_𝐴(𝑥)=sup_0≤𝛼≤1𝐴𝛼⋅1_𝛼(𝑥), (4) where 1𝐴_𝛼(𝑥) is an indicator function of set 𝐴_𝛼, that is, 1𝐴_𝛼(𝑥)=1 if 𝑥∈𝐴_𝛼 and 1𝐴_𝛼(𝑥)=0 if 𝑥∉𝐴_𝛼.

Let ℱ denote all the fuzzy subsets of ℝ. Assume 𝐴_𝑖∈ℱ, 𝑖=1,2,…,𝑛. Let 𝐴≜𝐴₁×𝐴₂×⋯×𝐴_𝑛 and Supp(𝐴)≜Supp(𝐴₁)×Supp(𝐴₂)×⋯×Supp(𝐴_𝑛), where × is Cartesian product. A fruitful and powerful tool of fuzzy set theory for calculating the fuzzy-valued functions on fuzzy sets is the extension principle.

Proposition 11 (extension principle [23–25]). Let 𝑓∶ℝ^𝑛→ℝ be a real-valued function and let 𝐴₁,𝐴₂,…,𝐴_𝑛 be fuzzy subsets of ℝ. Then one can induce a fuzzy-valued function 𝑓∶ℱ^𝑛→ℱ according to the real-valued function 𝑓(𝑥₁,𝑥₂,…,𝑥_𝑛). In fact, 𝑓(𝐴₁,𝐴₂,…,𝐴_𝑛) is a fuzzy subset of ℝ. Then the membership function of 𝑓(𝐴₁,𝐴₂,…,𝐴_𝑛) is characterized as follows: 𝜇_𝑓(𝐴₁^,𝐴₂^,…,𝐴_𝑛⁾(𝑟)=sup_{{𝑥1,…,𝑥𝑛𝑥∣𝑟=𝑓1,…,𝑥𝑛}∩Supp(}𝐴⁾min_{1≤𝑖≤𝑛}𝜇𝐴_𝑖𝑥_𝑖, (5) where sup_𝑥∈∅𝑔(𝑥)=0.

Remark 12. It is necessary to point out that there is a term Supp(𝐴) in (5), which does not exist in [23–25]. Since sup_𝑥∈∅𝑔(𝑥)=0, the term Supp(𝐴) does not affect the proposition. However, it plays an important role in our results.

Proposition 13 (see [22]). Let 𝑆 be a nonempty compact set in ℝ^𝑛. If 𝑓 is upper semicontinuous on 𝑆, then 𝑓 attains maximum over 𝑆 and if 𝑓 is lower semicontinuous on 𝑆, then 𝑓 attains minimum over 𝑆.

Proposition 14. (i) [26, Theorem 4.25] Let 𝑓∶𝑆→𝑇 be a function from one metric space (𝑆,𝑑_𝑆) to another (𝑇,𝑑_𝑇). If 𝑓 is continuous on a compact subset 𝑋 of 𝑆, then the image 𝑓(𝑋) is a compact subset of 𝑇; in particular, 𝑓(𝑋) is a closed and bounded in 𝑇.
(ii) [26, Theorem 4.37] Let 𝑓∶𝑆→𝑇 be a function from one metric space (𝑆,𝑑_𝑆) to another (𝑇,𝑑_𝑇). Let 𝑋 be a connected subset of 𝑆. If 𝑓 is continuous on 𝑋, then the 𝑓(𝑋) is a connected subset of 𝑇.

We also review the results in [16, 18] about the formula for currency option under uncertain environment as follows: 𝐶_𝑡^𝐿_𝛼=𝑄_𝑡^𝐿_𝛼⋅𝑒⁻⁽𝑟̃_𝑓)^𝑈_{𝛼×(𝑇−𝑡)}⋅𝑁𝑑₁^𝐿_𝛼−𝐾⋅𝑒⁻⁽𝑟̃_𝑑)^𝐿_{𝛼×(𝑇−𝑡)}⋅𝑁𝑑₂^𝑈_𝛼,𝐶_𝑡^𝑈_𝛼=𝑄_𝑡^𝑈_𝛼⋅𝑒⁻⁽𝑟̃_𝑓)^𝐿_{𝛼×(𝑇−𝑡)}⋅𝑁𝑑₁^𝑈_𝛼−𝐾⋅𝑒⁻⁽𝑟̃_𝑑)^𝑈_{𝛼×(𝑇−𝑡)}⋅𝑁𝑑₂^𝐿_𝛼, (6) where 𝑑₁^𝐿_𝛼=𝑄ln^𝑡𝐿𝛼𝐾+𝑟̃_𝑑^𝐿_𝛼−𝑟̃_𝑓^𝑈_𝛼+12𝜎_𝑄^𝐿_𝛼²×(𝑇−𝑡)𝜎_𝑄^𝑈_𝛼√𝑇−𝑡⁻¹,𝑑₁^𝑈_𝛼=𝑄ln^𝑡𝑈𝛼𝐾+𝑟̃_𝑑^𝑈_𝛼−𝑟̃_𝑓^𝐿_𝛼+12𝜎_𝑄^𝑈_𝛼²×(𝑇−𝑡)𝜎_𝑄^𝐿_𝛼√𝑇−𝑡⁻¹,𝑑₂^𝐿_𝛼=𝑑₁^𝐿_𝛼−𝜎_𝑄^𝑈_𝛼√𝑑𝑇−𝑡,_2𝑈𝛼=𝑑₁^𝑈_𝛼−𝜎_𝑄^𝐿_𝛼√𝑇−𝑡. (7)

3. Expressing of 𝛼-Level Set of the Fuzzy-Valued Functions of Fuzzy Numbers

In this section, we will establish the fundamental proposition for this paper, which based on the decomposition theorem, the extension principle, and some properties about continuous function. The following proposition is a result about the fuzzy-valued functions of fuzzy subsets.

Proposition 15. Let 𝑓∶ℝ^𝑛→ℝ be a continuous, surjective, real-valued function and let 𝐴₁,𝐴₂,…,𝐴_𝑛 be fuzzy subsets of ℝ. Let 𝑓∶ℱ^𝑛→ℱ be a fuzzy-valued function induced by 𝑓(𝑥₁,𝑥₂,…,𝑥_𝑛) via the extension principle defined in (5). Suppose that(i)𝜇𝐴_𝑖(𝑥_𝑖)  is upper semicontinuous on ℝ, 𝑖=1,2,…,𝑛,(ii)Supp(𝐴)  is a bounded set of ℝ^𝑛,then (a)𝑓𝐴₁,𝐴₂,…,𝐴_𝑛𝛼=𝑓𝑥₁,𝑥₂,…,𝑥_𝑛∣𝑥₁∈𝐴_1𝛼,𝑥₂∈𝐴_2𝛼,…,𝑥_𝑛∈𝐴_𝑛𝛼,[],(b)𝑓𝐴∀𝛼∈0,1Supp₁,𝐴₂,…,𝐴_𝑛=𝑓𝑥₁,𝑥₂,…,𝑥_𝑛∣𝑥₁∈Supp𝐴₁,𝑥₂∈Supp𝐴₂,…,𝑥_𝑛∈Supp𝐴_𝑛. (8)(9)

Proof. (a) For 𝛼=0, (8) holds from Definition 4 and the property of surjection of 𝑓(𝑥₁,𝑥₂,…,𝑥_𝑛).
For 𝛼∈(0,1], let 𝑟∈{𝑓(𝑥₁,𝑥₂,…,𝑥_𝑛)∣𝑥₁∈(𝐴₁)_𝛼,𝑥₂∈(𝐴₂)_𝛼,…,𝑥_𝑛∈(𝐴_𝑛)_𝛼}. Then there exists (𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑛)∈(𝐴₁)_𝛼×(𝐴₂)_𝛼×⋯×(𝐴_𝑛)_𝛼 such that 𝑟=𝑓(𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑛). Thus min_{1≤𝑖≤𝑛}𝐴{𝜇_𝑖(𝑥^∗_𝑖)}≥𝛼. It follows that 𝜇_𝑓(𝐴₁^,𝐴₂^,…,𝐴_𝑛⁾(𝑟)=sup_{{𝑥1,…,𝑥𝑛𝑥∣𝑟=𝑓1,…,𝑥𝑛}∩Supp(}𝐴⁾min_{1≤𝑖≤𝑛}𝜇𝐴_𝑖𝑥_𝑖≥𝛼, (10) which means that 𝑓𝑥₁,𝑥₂,…,𝑥_𝑛∣𝑥₁∈𝐴_1𝛼,𝑥₂∈𝐴_2𝛼,…,𝑥_𝑛∈𝐴_𝑛𝛼⊂𝑓𝐴₁,𝐴₂,…,𝐴_𝑛𝛼. (11) On the other hand, 𝜇𝐴_𝑖(𝑥_𝑖) is upper semicontinuous on ℝ, then 𝑈_𝑖={𝑥_𝑖𝐴∣𝜇_𝑖(𝑥_𝑖)≥𝛼} is a closed subset of ℝ, 𝑖=1,2,…,𝑛. Therefore, 𝐶_𝑖=ℝ×⋯×ℝ|{z}_𝑖−1×𝑈_𝑖×ℝ×⋯×ℝ|{z}_𝑛−𝑖 is a closed subset of ℝ^𝑛𝑖=1,2,…,𝑛. Since 𝑥₁,𝑥₂,…,𝑥_𝑛∣min_{1≤𝑖≤𝑛}𝜇𝐴_𝑖𝑥_𝑖=𝑥≥𝛼₁,𝑥₂,…,𝑥_𝑛𝐴∣𝜇_𝑖𝑥_𝑖=≥𝛼,𝑖=1,2,…,𝑛_𝑛𝑖=1𝐶_𝑖 (12) is a closed subset of ℝ^𝑛, we have min_{1≤𝑖≤𝑛}𝐴{𝜇_𝑖(𝑥_𝑖)} is upper semicontinuous on ℝ^𝑛. Let 𝑟∈(𝑓(𝐴₁,𝐴₂,…,𝐴_𝑛))_𝛼, that is, sup_{{𝑥1,…,𝑥𝑛𝑥∣𝑟=𝑓1,…,𝑥𝑛}∩Supp(}𝐴₎min_{1≤𝑖≤𝑛}𝜇𝐴_𝑖𝑥_𝑖≥𝛼. (13) Since {(𝑥₁,…,𝑥_𝑛)∣𝑟=𝑓(𝑥₁,…,𝑥_𝑛)} is a closed subset of ℝ^𝑛and Supp(𝐴) is a compact set of ℝ^𝑛 and min_{1≤𝑖≤𝑛}𝐴{𝜇_𝑖(𝑥_𝑖)} is upper semicontinuous on ℝ^𝑛, we conclude from Proposition 13 that there exists (𝑥₁,𝑥₂,…,𝑥_𝑛) such that 𝑟=𝑓(𝑥₁,𝑥₂,…,𝑥_𝑛) and min_{1≤𝑖≤𝑛}𝐴{𝜇_𝑖(𝑥_𝑖)}≥𝛼. Therefore 𝜇𝐴_𝑖(𝑥_𝑖)≥𝛼, that is, 𝑥_𝑖∈(𝐴_𝑖)_𝛼,  𝑖=1,2,…,𝑛. This means 𝑟∈𝑓𝑥₁,𝑥₂,…,𝑥_𝑛∣𝑥₁∈𝐴_1𝛼,𝑥₂∈𝐴_2𝛼,…,𝑥_𝑛∈𝐴_𝑛𝛼. (14) Therefore, 𝑓𝐴₁,𝐴₂,…,𝐴_𝑛𝛼⊂𝑓𝑥₁,𝑥₂,…,𝑥_𝑛∣𝑥₁∈𝐴_1𝛼,𝑥₂∈𝐴_2𝛼,…,𝑥_𝑛∈𝐴_𝑛𝛼 (15) which completes the proof of (a).
(b) Let 𝑟∈Supp(𝑓(𝐴₁,𝐴₂,…,𝐴_𝑛)), we conclude from Definition 3 that there exists a sequence {𝑟_𝑚}^∞_𝑚=1 which satisfies 𝑟_𝑚→𝑟 and 𝜇_𝑓(𝐴_1,𝐴_2,…,𝐴_𝑛)(𝑟_𝑚)>0, that is, sup_{{𝑥1,…,𝑥𝑛∣𝑟𝑚𝑥=𝑓1,…,𝑥𝑛}∩Supp(}𝐴₎min_{1≤𝑖≤𝑛}𝜇𝐴_𝑖𝑥_𝑖>0. (16) Since {(𝑥₁,…,𝑥_𝑛)∣𝑟_𝑚=𝑓(𝑥₁,…,𝑥_𝑛)} is closed and Supp(𝐴) is bounded and closed, then {(𝑥₁,…,𝑥_𝑛)∣𝑟_𝑚=𝑓(𝑥₁,…,𝑥_𝑛)}∩Supp(𝐴) is compact. Moreover, we deduce that min_{1≤𝑖≤𝑛}𝐴{𝜇_𝑖(𝑥_𝑖)} is upper semicontinuous on ℝ^𝑛 according to the proof of (a). Consequently, we conclude from Proposition 13 that there exist {𝑥^𝑚₁}^∞_𝑚=1,{𝑥^𝑚₂}^∞_𝑚=1,…,{𝑥^𝑚_𝑛}^∞_𝑚=1 such that 𝜇_𝑓(𝐴_1,𝐴_2,…,A_𝑛)𝑟_𝑚=min_{1≤𝑖≤𝑛}𝜇𝐴_𝑖𝑥^𝑚_𝑖>0for𝑚∈N,𝑖=1,2,…,𝑛, (17) where 𝑟_𝑚=𝑓(𝑥^𝑚₁,…,𝑥^𝑚_𝑛) and 𝑥^𝑚_𝑖∈Supp(𝐴_𝑖). Since Supp(𝐴)∩{(𝑥₁,…,𝑥_𝑛)∣𝑟_𝑚=𝑓(𝑥₁,…,𝑥_𝑛)} is compact, for each 𝑖∈{1,2,…,𝑛}, there exists a subsequence {𝑥^𝑚𝑘_𝑖}^∞_𝑘=1 of {𝑥^𝑚_𝑖}^∞_𝑚=1 such that 𝑥^𝑚𝑘_𝑖→𝑥_𝑖∈Supp(𝐴_𝑖) as 𝑘→∞. Notice that 𝑟_𝑚𝑘=𝑓𝑥^𝑚𝑘₁,…,𝑥^𝑚𝑘_𝑛𝑓⟶𝑟,𝑥^𝑚𝑘₁,…,𝑥^𝑚𝑘_𝑛⟶𝑓𝑥₁,…,𝑥_𝑛as𝑘⟶∞. (18) We have 𝑟=𝑓(𝑥₁,…,𝑥_𝑛), where 𝑥_𝑖∈Supp(𝐴_𝑖). Then Supp𝑓𝐴₁,𝐴₂,…,𝐴_𝑛⊂𝑓𝑥₁,𝑥₂,…,𝑥_𝑛∣𝑥₁∈Supp𝐴₁𝑥,…,_𝑛∈Supp𝐴_𝑛. (19) For the converse, let 𝑧∈{𝑓(𝑥₁,𝑥₂,…,𝑥_𝑛)∣𝑥₁∈Supp(𝐴₁),…,𝑥_𝑛∈Supp(𝐴_𝑛)}. Then there exist sequences {𝑥^𝑚_𝑖}^∞_𝑚=1, 𝑖=1,2,…,𝑛, which satisfy 𝜇𝐴_𝑖(𝑥^𝑚_𝑖)>0 and 𝑥^𝑚_𝑖→𝑥_𝑖 as 𝑚→∞, where 𝑥_𝑖∈Supp(𝐴_𝑖) and 𝑧=𝑓(𝑥₁,𝑥₂,…,𝑥_𝑛). Recall the continuity of 𝑓(𝑥₁,𝑥₂,…,𝑥_𝑛), we have 𝑓𝑥₁,𝑥₂,…,𝑥_𝑛∣𝑥₁∈Supp𝐴₁,…,𝑥_𝑛∈Supp𝐴_𝑛⊂Supp𝑓𝐴₁,𝐴₂,…,𝐴_𝑛. (20) This completes the proof.

Remark 16. One of the conditions of Proposition in [13] is that {(𝑥₁,…,𝑥_𝑛)∣𝑟=𝑓(𝑥₁,…,𝑥_𝑛)} is compact forall𝑟∈Range(𝑓). This one is not verified there. Actually, it is difficult to be verified. Meanwhile, (8) only holds for 𝛼∈(0,1]. We change the above conditions to (i) 𝜇𝐴_𝑖(𝑥_𝑖) is upper semicontinuous on ℝ, 𝑖=1,2,…,𝑛; (ii) Supp(𝐴) is bounded of ℝ^𝑛 and; (iii) 𝑓∶ℝ^𝑛→ℝ is a continuous, surjective, real-valued function. Conditions (i)–(iii) can be easily satisfied. At the same time, (8) holds for 𝛼∈[0,1] and (9) is also obtained.

The following is the fundamental proposition for discussing the Garman-Kohlhagen formula for European currency option via the extension principle.

Proposition 17. Let 𝑓∶ℝ^𝑛→ℝ be a continuous, surjective, real-valued function and let 𝑎₁,𝑎₂,…,𝑎_𝑛 be fuzzy numbers. Let 𝑓∶ℱ^𝑛→ℱ be a fuzzy-valued function induced by 𝑓(𝑥₁,𝑥₂,…,𝑥_𝑛) via the extension principle defined in (5). Then 𝑓(𝑎₁,𝑎₂,…,𝑎_𝑛) is a fuzzy number and its 𝛼-level set is 𝑓𝑎₁,𝑎₂,…,𝑎_𝑛𝛼=𝑓𝑥₁,𝑥₂,…,𝑥_𝑛∣𝑥₁∈𝑎_1𝛼,𝑥₂∈𝑎_2𝛼,…,𝑥_𝑛∈𝑎_𝑛𝛼=𝑓𝑥₁,𝑥₂,…,𝑥_𝑛∣𝑎₁^𝐿_𝛼≤𝑥₁≤𝑎₁^𝑈_𝛼,…,𝑎_𝑛^𝐿_𝛼≤𝑥_𝑛≤𝑎_𝑛^𝑈_𝛼. (21)

Proof. Let 𝑋_𝛼≜{(𝑥₁,𝑥₂,…,𝑥_𝑛)∣𝑥₁∈(𝑎₁)_𝛼,𝑥₂∈(𝑎₂)_𝛼,…,𝑥_𝑛∈(𝑎_𝑛)_𝛼} for all 𝛼∈(0,1]. Then it follows from Definition 7 that 𝑋_𝛼 is an 𝑛-dimensional bounded interval; that is, 𝑋_𝛼 is a compact and connected subset of ℝ^𝑛. Next, we conclude from Propositions 15 and 14 that(i){𝜇_𝑓(𝑎_1,𝑎_2,…,𝑎_𝑛)≥𝛼}=(𝑓(𝑎₁,𝑎₂,…,𝑎_𝑛))_𝛼={𝑓(𝑥₁,𝑥₂,…,𝑥_𝑛)∣(𝑥₁,𝑥₂,…,𝑥_𝑛)∈𝑋_𝛼} is a compact and connected subset of ℝ for all 𝛼∈(0,1]; (ii)(𝑓(𝑎₁,𝑎₂,…,𝑎_𝑛))₀=ℝ is closed and convex set; (iii)Supp(𝑓(𝑎₁,𝑎₂,…,𝑎_𝑛))={𝑓(𝑥₁,𝑥₂,…,𝑥_𝑛)∣𝑥₁∈Supp(𝑎₁),𝑥₂∈Supp(𝑎₂),…,𝑥_𝑛∈Supp(𝑎_𝑛)}.Furthermore,(i′) the membership function 𝜇_𝑓(𝑎_1,𝑎_2,…,𝑎_𝑛) of 𝑓(𝑎₁,𝑎₂,…,𝑎_𝑛) is upper semicontinuous; (ii′)𝑓(𝑎₁,𝑎₂,…,𝑎_𝑛) is a convex fuzzy subset; (iii′)Supp(𝑓(𝑎₁,𝑎₂,…,𝑎_𝑛)) is bounded.
Obviously, 𝑓(𝑎₁,𝑎₂,…,𝑎_𝑛) is a normal fuzzy subset, as 𝑋₁ is nonempty. Therefore, from Definition 7 we conclude that 𝑓(𝑎₁,𝑎₂,…,𝑎_𝑛) is a fuzzy number and its 𝛼-level set is (21). This completes the proof.

4. Fuzzy Currency Options Pricing Formula via the Extension Principle

4.1. The G-K Formula for European Currency Option

The closed-form solution for European currency option pricing formula has been derived by Garman and Kohlhagen [1]. The G-K formula for European call currency option with the expiry date 𝑇 and the strike price 𝐾 is described as follows. Let 𝑄_𝑡, 𝐶_𝑡 denote the spot exchange rate, and the price of European call currency option at time 𝑡∈[0,𝑇], respectively, then 𝐶_𝑡=𝑄_𝑡⋅𝑒^{−𝑟𝑓𝜏}⋅𝑁𝑑₁−𝐾⋅𝑒^{−𝑟𝑑𝜏}⋅𝑁𝑑₂, (22) where 𝜏=𝑇−𝑡, 𝑑₁=[ln(𝑄_𝑡/𝐾)+(𝑟_𝑑−𝑟_𝑓+(1/2)𝜎²_𝑄)𝜏]/(𝜎_𝑄√𝜏), 𝑑₂=𝑑₁−𝜎_𝑄√𝜏, 𝑟_𝑓, 𝑟_𝑑, 𝜎_𝑄, 𝑁(⋅) denote the foreign interest rate, the domestic interest rate, the volatility of spot exchange rate and the standard Gaussian cumulative distribution function, respectively. Let 𝑓𝑞,𝑡,𝐾,𝑟_𝑑,𝑟_𝑓,𝜎=𝑞⋅𝑒^{−𝑟𝑓𝑡}⋅𝑁𝑑₁−𝐾⋅𝑒^{−𝑟𝑑𝑡}⋅𝑁𝑑₂, (23) then 𝐶_𝑡=𝑓𝑄_𝑡,𝑇−𝑡,𝐾,𝑟_𝑑,𝑟_𝑓,𝜎_𝑄, (24) for all 𝑡∈[0,𝑇]. Furthermore, the price 𝑃_𝑡 of European put option at time 𝑡∈[0,𝑇] with same expiry date 𝑇 and strike price 𝐾 can be obtained by the following put-call parity relationship (Musiela and Rutkowski [27]): 𝐶_𝑡−𝑃_𝑡=𝑒^{−𝑟𝑓(𝑇−𝑡)}⋅𝑄_𝑡−𝑒^{−𝑟𝑑(𝑇−𝑡)}⋅𝐾. (25) Let 𝑔𝑞,𝑡,𝐾,𝑟_𝑑,𝑟_𝑓,𝜎=𝑓𝑞,𝑡,𝐾,𝑟_𝑑,𝑟_𝑓,𝜎−𝑒^{−𝑟𝑓𝑡}⋅𝑞+𝑒^{−𝑟𝑑𝑡}⋅𝐾 (26) for all 𝑡∈[0,𝑇], then 𝑃_𝑡=𝑔𝑄_𝑡,𝑇−𝑡,𝐾,𝑟_𝑑,𝑟_𝑓,𝜎_𝑄. (27)

4.2. The Fuzzy Version of the G-K Formula for European Currency Option

As the discussion in Section 1, it is suitable to represent the four input variables in the G-K formula by fuzzy numbers. Thus, we give the following assumptions.

Assumption 1. (i)The spot exchange rate at time 𝑡∈[0,𝑇] is fuzzy number 𝑄_𝑡; (ii)the domestic interest rate is fuzzy number 𝑟_𝑑̃; (iii)the foreign interest rate is fuzzy number 𝑟_𝑓̃; (iv)the volatility of spot exchange rate is fuzzy number 𝜎_𝑄.

Based on Assumption 1, (24) and the extension principle (Proposition 11), the original price for European call currency option 𝐶_𝑡 at time 𝑡∈[0,𝑇] turns into fuzzy subset 𝐶_𝑡∈ℱ with the membership function 𝜇𝐶_𝑡(𝑐)=sup_{{𝑞,𝑟𝑑,𝑟𝑓,𝜎𝑄∣𝑐=𝑓𝑞,𝑇−𝑡,𝐾,𝑟𝑑,𝑟𝑓,𝜎𝑄}∩Supp(}𝐴⁾𝜇𝑄×min_𝑡𝑞,𝜇𝑟_𝑑̃𝑟_𝑑,𝜇𝑟_𝑓̃𝑟_𝑓,𝜇𝜎_𝑄𝜎_𝑄, (28) where Supp(𝐴)=Supp(𝑄_𝑡)×Supp(𝑟_𝑑̃)×Supp(𝑟_𝑓̃)×Supp(𝜎_𝑄). Since the function 𝑓(𝑄_𝑡,𝑇−𝑡,𝐾,𝑟_𝑑,𝑟_𝑓,𝜎_𝑄) satisfies conditions of Proposition 17, we conclude that the fuzzy price 𝐶_𝑡 at time 𝑡∈[0,𝑇] is a fuzzy number. Furthermore, its 𝛼-level set (𝐶_𝑡)_𝛼 is 𝐶_𝑡𝛼=𝑓𝑞,𝑇−𝑡,𝐾,𝑟_𝑑,𝑟_𝑓,𝜎_𝑄∣𝑞∈𝑄_𝑡𝛼,𝑟_𝑑∈𝑟̃_𝑑𝛼,𝑟_𝑓∈𝑟̃_𝑓𝛼,𝜎_𝑄∈𝜎_𝑄𝛼, (29) for all 𝛼∈[0,1]. On the other hand, from Definition 7, the 𝛼-level set (𝐶_𝑡)_𝛼 of 𝐶_𝑡 is a closed interval 𝐶_𝑡𝛼=𝐶_𝑡^𝐿_𝛼,𝐶_𝑡^𝑈_𝛼. (30) Notice that the 𝛼-level set of 𝑄_𝑡, 𝑟_𝑑̃, 𝑟_𝑓̃, and 𝜎_𝑄 are (𝑄_𝑡)_𝛼=[(𝑄_𝑡)^𝐿_𝛼,(𝑄_𝑡)^𝑈_𝛼], (𝑟_𝑑̃)_𝛼=[(𝑟_𝑑̃)^𝐿_𝛼,(𝑟_𝑑̃)^𝑈_𝛼], (𝑟_𝑓̃)_𝛼=[(𝑟_𝑓̃)^𝐿_𝛼,(𝑟_𝑓̃)^𝑈_𝛼], and (𝜎_𝑄)_𝛼=[(𝜎_𝑄)^𝐿_𝛼,(𝜎_𝑄)^𝑈_𝛼], respectively. Therefore, from (29) and (30), the left endpoint (𝐶_𝑡)^𝐿_𝛼 and the right-end point (𝐶_𝑡)^𝑈_𝛼 can be displayed as 𝐶_𝑡^𝐿_𝛼=min𝑄_𝑡^𝐿_{𝛼≤𝑞≤}𝑄_𝑡^𝑈_𝛼,𝑟̃_𝑑^𝐿_{𝛼≤𝑟𝑑≤}𝑟̃_𝑑^𝑈_𝛼,𝑟̃_𝑓^𝐿_{𝛼≤𝑟𝑓≤}𝑟̃_𝑓^𝑈_𝛼,𝜎_𝑄^𝐿_{𝛼≤𝜎𝑄≤}𝜎_𝑄^𝑈_𝛼𝑓𝑞,𝑇−𝑡,𝐾,𝑟_𝑑,𝑟_𝑓,𝜎_𝑄,𝐶_𝑡^𝑈_𝛼=max𝑄_𝑡^𝐿_{𝛼≤𝑞≤}𝑄_𝑡^𝑈_𝛼,𝑟̃_𝑑^𝐿_{𝛼≤𝑟𝑑≤}𝑟̃_𝑑^𝑈_𝛼,𝑟̃_𝑓^𝐿_{𝛼≤𝑟𝑓≤}𝑟̃_𝑓^𝑈_𝛼,𝜎_𝑄^𝐿_{𝛼≤𝜎𝑄≤}𝜎_𝑄^𝑈_𝛼𝑓𝑞,𝑇−𝑡,𝐾,𝑟_𝑑,𝑟_𝑓,𝜎_𝑄. (31) Moreover, according to the sensitivity analysis (Greek letters) of the G-K formula [18], we have 𝜕𝑓𝜕𝑞=𝑒^{−𝑟𝑓𝑡}𝑁𝑑₁>0,𝜕𝑓𝜕𝑟_𝑑=𝑡⋅𝐾⋅𝑒^{−𝑟𝑑𝑡}𝑁𝑑₂>0,𝜕𝑓𝜕𝑟_𝑓=−𝑡⋅𝑞⋅𝑒^{−𝑟𝑓𝑡}𝑁𝑑₁<0,𝜕𝑓𝜕𝜎_𝑄=𝑞⋅𝑒^{−𝑟𝑓𝑡}𝑁^′𝑑₁⋅√𝑡>0. (32) This mean 𝑓 is increasing with respect to 𝑞, 𝑟_𝑑, and 𝜎_𝑄 and is decreasing with respect to 𝑟_𝑓. Therefore, from (31), we conclude that 𝐶_𝑡^𝐿_𝛼=𝑓𝑄_𝑡^𝐿_𝛼,𝑇−𝑡,𝐾,𝑟̃_𝑑^𝐿_𝛼,𝑟̃_𝑓^𝑈_𝛼,𝜎_𝑄^𝐿_𝛼,𝐶_𝑡^𝑈_𝛼=𝑓𝑄_𝑡^𝑈_𝛼,𝑇−𝑡,𝐾,𝑟̃_𝑑^𝑈_𝛼,𝑟̃_𝑓^𝐿_𝛼,𝜎_𝑄^𝑈_𝛼. (33) On the other hand, from Proposition 10, the membership function of 𝐶_𝑡 can also be written as 𝜇𝐶_𝑡(𝑐)=sup_0≤𝛼≤1𝛼⋅1₍𝐶_𝑡)𝛼(𝑐). (34) Finally, (28), (33), and (34) constitute the fuzzy version of the G-K formula for European call currency option.

Similarly, the conclusions about European put currency option are as follows, through the above discussions.

The original price of European put currency option 𝑃_𝑡 at time 𝑡∈[0,𝑇] turns into fuzzy subset 𝑃_𝑡∈ℱ with the membership function under Assumption 1𝜇𝑃_𝑡𝑝=sup_{{𝑞,𝑟𝑑,𝑟𝑓,𝜎𝑄∣𝑝=𝑔𝑞,𝑇−𝑡,𝐾,𝑟𝑑,𝑟𝑓,𝜎𝑄}∩Supp(}𝐴₎𝜇𝑄×min_𝑡𝑞,𝜇𝑟_𝑑̃𝑟_𝑑,𝜇𝑟_𝑓̃𝑟_𝑓,𝜇𝜎_𝑄𝜎_𝑄, (35) where Supp(𝐴) is as above. Furthermore, the fuzzy price 𝑃_𝑡 at time 𝑡∈[0,𝑇] is a fuzzy number and the 𝛼-level set (𝑃_𝑡)_𝛼 of 𝑃_𝑡 is a closed interval 𝑃_𝑡𝛼=𝑃_𝑡^𝐿_𝛼,𝑃_𝑡^𝑈_𝛼. (36) Moreover, the left endpoint (𝑃_𝑡)^𝐿_𝛼 and the right endpoint (𝑃_𝑡)^𝑈_𝛼 can be displayed as 𝑃_𝑡^𝐿_𝛼=𝑔𝑄_𝑡^𝑈_𝛼,𝑇−𝑡,𝐾,𝑟̃_𝑑^𝑈_𝛼,𝑟̃_𝑓^𝐿_𝛼,𝜎_𝑄^𝐿_𝛼,𝑃_𝑡^𝑈_𝛼=𝑔𝑄_𝑡^𝐿_𝛼,𝑇−𝑡,𝐾,𝑟̃_𝑑^𝐿_𝛼,𝑟̃_𝑓^𝑈_𝛼,𝜎_𝑄^𝑈_𝛼. (37) On the other hand, the membership function of 𝑃_𝑡 can also be written as 𝜇𝑃_𝑡𝑝=sup_0≤𝛼≤1𝛼⋅1₍𝑃_𝑡)𝛼𝑝. (38)

Finally, (35), (37), and (38) constitute the fuzzy version of the G-K formula for European put currency option.

5. Defuzzification via Weighting Parameter Identification and Numerical Analysis

In this section, we defuzzify the fuzzy price for European currency option 𝐶_𝑡(𝑃_𝑡). Then an numerical example is presented. Up to now, it is proved that the fuzzy price for European currency option is a fuzzy number under Assumption 1. For practical purposes, it may be convenient to find a crisp number that synthesizes the fuzzy number 𝐶_𝑡(𝑃_𝑡). This type of problem is known as the “defuzzification procedure" in the literature. Defuzzification is the process to select an appropriate crisp value based on a fuzzy set such that the selected crisp value may represent the fuzzy set in some sense.

Of all the methods of defuzzification, two most popular methods are the center of area (COA) and the mean of maxima (MOM) which require explicit membership function. Now, the membership function of 𝐶_𝑡(𝑃_𝑡) cannot be expressed explicitly; it can be deduced by the decomposition theorem (see (34) or (38)) or the extension principle (see (28) or (35)) only. Since its 𝛼-level sets of 𝐶_𝑡(𝑃_𝑡) are available (see (33) or (37)), the weighted possibilistic mean value of Fullér and Majlender [20] about fuzzy number is fited to be the crisp number that we needed (see Definition 9) in the following.

Next, the proposed fuzzy version of the G-K formula (28) is tested with the daily market price data of EUR/USD currency option. To compare with the results in [16, 18] conveniently, the market data from [16] are used in Example 18. In the following example, we assumed that the weighting function is 𝑓_𝑛(𝑥)≜(1+𝑛)𝛼^𝑛, where 0≤𝛼≤1 is the belief degree and 𝑛 is the belief degree of weighting parameter.

Example 18. A European EUR/USD currency call option is studied on march 16, 2006. For the EUR/USD currency option, the spot exchange rate is 1.215, the 3-month volatility is around 9.0%, the domestic 3-month interest rate is around 4.93%, the foreign 3-month interest rate is around 2.71%, and the strike price is 1.21 with 3-month to expiry. The market price of this currency option is 0.0274 USD. The fuzzy foreign and domestic interest rates, the fuzzy spot exchange rate, and the fuzzy volatility are assumed to triangular fuzzy numbers: 𝑄_𝑡=(1.2138,1.2150,1.2162), 𝜎_𝑄=(7.2%,9.0%,10.8%), 𝑟_𝑑̃=(4.91%,4.93%,4.95%), and 𝑟_𝑓̃=(2.69%,2.71%,2.72%), respectively.

The fuzzy price 𝐶_𝑡 of European call currency option can be obtained by (28). Table 1 gives the 𝛼-level set of the fuzzy price 𝐶_𝑡 for different belief degrees 𝛼 and the 𝑓-weighted possibilistic mean value 𝑀_𝑓𝑛(𝐶_𝑡) for different belief degrees of weighting parameters 𝑛, respectively. In fact, if 𝑄_𝑡=1.215, 𝑟_𝑑=4.93%, 𝑟_𝑓=2.71%, and 𝜎_𝑄=9.0%, then European call currency option price is 0.027872$ by the G-K formula (24). This situation matches the observation 𝐶_𝑡=0.027872$ with belief degree 1.

Table 1 

Fuzzy EUR/USD currency option price.

Remark 19. There are some differences between the 𝛼-level sets of fuzzy call option price in this paper and those in [16, 18], which are shown in Table 1. The reason is that the division operation of fuzzy numbers in [16, 18] is invalid occasionally. Table 1 also shows that 𝑀_𝑓𝑛(𝐶_𝑡) is the best approximation to the real price when 𝑛=2.

Next, assuming the currency option prices 𝐶_𝑡1,𝐶_𝑡2,…,𝐶_𝑡𝑁 at 𝑡₁,𝑡₂,…,𝑡_𝑁 dates are available, how to acquire a more accurate 𝐶_𝑡𝑁+1 at 𝑡_𝑁+1 date is an important question. Based on minimizing the mean square error between the market prices and WPMs, a method of weighting parameter identification is presented as follows.

Frame:(i)collect the market price data: 𝐶_𝑡1,𝐶_𝑡2,…,𝐶_𝑡𝑁; (ii)calculate the performance index 𝐽(𝑛)=∑^𝑁_𝑖=1(∫¹₀(((𝐶_𝑡𝑖)^𝐿_𝛼+(𝐶_𝑡𝑖)^𝑈_𝛼)/2)(1+𝑛)𝛼^𝑛d𝛼−𝐶_𝑡𝑖)²; (iii)solve min_0≤𝑛≤1𝐽(𝑛) and optimal solution 𝑛^∗ is obtained and; (iv)calculate 𝑀_𝑓𝑛(𝐶_𝑡𝑁+1)=∫¹₀(((𝐶_𝑡𝑁+1)^𝐿_𝛼+(𝐶_𝑡𝑁+1)^𝑈_𝛼)/2)(1+𝑛^∗)𝛼^𝑛∗d𝛼, and let 𝐶_𝑡𝑁+1=𝑀_𝑓𝑛(𝐶_𝑡𝑁+1).

6. Conclusions

In this paper, the Fuzzy version of the Garman-Kohlhagen (FG-K) formula for pricing European currency option via the extension principle is proposed. In the FG-K formula, the interest rate, the spot exchange rate, and the volatility in conventional G-K formula are replaced by fuzzy numbers. So the FG-K formula is able to keep consistent with the real market. We prove that the fuzzy price for European currency option is a fuzzy number. And then the fuzzy price 𝐶_𝑡(𝑃_𝑡) is defuzzified by the weighted possibilistic mean value. The example shows that the fuzzy price in this paper is more accurate than the ones in [16, 18] and the fuzzy prices in [16, 18] are just approximations to our fuzzy price for European currency option. In addition, the example indicates that the WPM value has different approximation effects to real market price by taking different values of weighting parameter in the weighting function. So an identification method is presented. And the optimal parameter is obtained if we have already known some market prices for European currency option. The method in this paper can be extended to other option pricing. How to find out different kinds of weighting function in defuzzification requires a further study.

Acknowledgments

This work was supported by 863 Program (Grant no. 2007AA02Z208), 973 Program (Grant no. 2007CB714304), and the National Natural Science Foundation of China (Grants nos. 10871033 and 11171050).