1、1A model for foreign exchange rate dynamicsAbstract. The model of target zones and exchange rate dynamics plays an strikingly important role in discussing and mastering the behavior of the spot exchange rate. As we all known, a key feature of the target-zone model is that it has has two pre-specifie
2、d barriers. In this paper, we will use a reflected diffusion process with two boundaries to model the foreign exchange rate dynamics. Furthermore, we offer an approximation for its transient moment and obtain a perturbation expression. Key words: target-zone exchange rate; the central parity; reflec
3、ted diffusion processes; Kolmogorov backward equation; transient moments; Girsanov formula. MSC(2000): 60H15; 60J35; 60J60; 60J70; 60K25 1. Introduction As we all know, reflected diffusion processes with one or two reflected boundary conditions, especially reflected Brownian motion and reflected Orn
4、stein-Uhlenbeck process, have played an important role in many applications such as economics, finance, queueing, bioinformatics, and electrical engineering. 2Since the seminal book by Harrison (1986), quite a few research on reflected diffusion processes has been reported in queueing and economic l
5、iterature. For a specific review, the reader can refer Ward and Glynn (2003a, 2003b), or Linetsky (2005). Among economic and mathematical finance applications, we mainly discuss the target zones and exchange rate dynamics models which is originated from the classic paper by Krugman (1991). In exchan
6、ge rate target-zone regime, the spot exchange rate has two pre-specified lower and upper barriers and is allowed to freely float inside the two boundaries without any interventions. But the monetary authority or the central banks should enforce some interventions in the foreign exchange market as so
7、on as the spot log-exchange rate hits the boundaries in order to prevent the exchange rate from touching the upper boundary, or to prohibit the exchange rate from fall bellowing the minimum boundary. This characteristic stimulates us to employ reflecting diffusion process with two barriers to charac
8、terize the dynamics of the log-exchange rate in a target zone. We also can see Krugman (1991), or De Jong, Drost,and Werker (2001) in details. A well documented data of target zones can be derived from the European Monetary System. It is well known that EMS has 3been in effect since 1979. However, t
9、he EMS target zone system gradually disappeared with the introduction of the common currency (the Euro) on January 1, 1999. Understanding the role of the reflected diffusion processes in financial fields, the study of property of the diffusion processes with barriers is very of importance. The purpo
10、se of this paper is to propose and investigate a relatively simple and analytically tractable model for exchange rate in a target zone: that is a reflected diffusion model with two barriers which the central banks have promised to keep the exchange rate floating within the fixed bounds and to mainta
11、in a unchangeable central parity, which corresponds to a completely credible target zone, in the sense that there are no realignments. The existing theoretical and methods have been successfully exploited in dealing with the steady-state distribution of the reflected Brownian motion with one or two
12、boundaries, but do not apply to analyze other reflected diffusion processes until the appearance of the article by Ward and Glynn (2003b). In addition, Ward and Glynn (2003b) only discussed the reflected Ornstein-Uhlenbeck process with one reflected barrier conditions at 0 without considering the ca
13、se with two boundary conditions. Therefore, in this paper, our destination is to sponsor the discussion of the stationary 4distribution of the reflected diffusion model for exchange rate in a target zone. 2. Exchange rate dynamics In our paper, we try to construct a model to meet some typical featur
14、es of the exchange rate in a target zone which can be found in details in Vlaar and Palm (1993). A case in point is that EMS exchange rates have mean reversion phenomenon within the two barriers. That is to say that exchange rates tend towards their central parity in the long term. De Jong, Drost, a
15、nd Werker (2001) proposed a target zone model by applying a two-limit version of the square root process, popularized in the financial literature by Cox, Ingersoll and Ross (1985) for the term interest rate. However, we will employ a reflected diffusionprocess with two barriers to describe the excha
16、nge rate dynamics under the target zone based on some typical facts of the target-zone exchange rate. Let St denote the spot price of foreign exchange at time t, and assume that the monetary authority has promised to prohibit the exchange rate dynamics St deviating more than Z, which is a certain pe
17、rcentage, from a central parity U, then we have the following restriction Furthermore, the log process Xt = log(St) of the spot exchange rate dynamics St also satisfies the condition 5It is well known that the target-zone exchange rate theory pioneered by Krugman (1991) has two barriers. When the lo
18、g-exchange rate process X fluctuates near the boundaries - z and + z, the monetary authority may formulate monetary policy to intervene the exchange rate behavior. Specifically,the central bank is obliged to decrease the domestic money supply so as to prohibit the log-exchange rate X from exceeding
19、the upper boundary + z. On the contrary, the central bank is prepared to increase the domestic money supply to protect X from falling below the lower boundary - z. From the point of the stochastic process we can view the boundaries z of the target zone as two reflecting barriers. Inspired by Harriso
20、n (1986), Ramasubramanian (1996), and Ward and Glynn (2003a,2003b) and our front analysis, we can utilize the following reflected diffusion process with two-sided boundaries to characterize the spot exchange rate behavior in the target zone is called the barrier policy among economic and financial a
21、reas. In this paper, can be interpreted as the monetary policy of the monetary authority. In other words, the process is the 1. and have the continuous sample paths, and are both non-decrease processes with initial positions ; 62. and increase only on the sets A = t : Xt = - z and B = t : Xt = + z,
22、respectively,so that 3. Kolmogorov backward equation and transient moments Definition 3.1. Let stochastic process X = Xt, t 0 be a where the function satisfies f : R R and f C2(R). In this section, we mainly discuss the computation of transient expectations of Exf(Xt) . In the sequel, we shall first
23、 give the Kolmogorov backward equation of the transient expectations. Then, we exploit the Girsanov formula to offer an approach to calculate the expectation Exf(Xt) . Proposition 3.1. For any f(x) C2(-z, +z), we denote u(x, t) = Exf(Xt). Then u(x, t) satisfies the following Kolmogorov backward part
24、ial differential equation Now take expectation Ex() of both sides and resort to the Fubini theorem, we can attain the following equality On one hand, from the definition (2) of the infinitesimal generator A, it follows that On the other hand, for fixed t, by taking g(x) = u(x,t) = Ex(f(Xt) and using
25、 the strong Markov property of the log-exchange rate process, we yields Combining with the equations (4) and (5) permit us to 7arrive at The process of the log of the exchange rate is reflected once it attaches to the maximum value +z and the minimum value -z, because the central bank or the monetar
26、y authority is obliged to maintain the target zone. Consequently, the function u(x,t) must meet the additional boundary conditions Jointing (6), (7)and(8), we can immediately find that the Proposition 3.1 is valid. Lemma 3.1. Assume that W = Wt;t 0 is a standard Brownian motion on the probability sp
27、ace (,F,P) with the natural filtration generated by W and the log-exchange rate process X = Xt;t 0 with initial point X0 = x satisfies the following stochastic differential equation where L and U are the regulators of lower barrier -z and upper barrier +z ,respectively. Let 1. For every 0, the proce
28、ss M = Mt;t 0 is a martingale with respect to the filtration under the probability P. Furthermore, define a new measure as so, for each ,t 0, is a probability measure. 2.The log-exchange rate process X also follows the equation: 8X is reflected Brownian motion with two barriers which has drift Since
29、 the log-exchange rate process X has two reflecting boundaries, the Novikov condition is satisfied. Based on these facts, we can easily attain the conclusion that the process M = Mt;t 0 is a martingale with respect to the filtration 2. we can receive a similar demonstration from the literature Karat
30、zas and Shreve (1991). Theorem 3.1. Suppose that the log-exchange rate process X = Xt;t 0 on the probability space (,F,P) satisfies the stochastic differential equation (2) and the function f is non-negative and measur-able, then there exist a reflected Brownian motion Y = Yt;t 0 with two barriers o
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