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International Journal of Stochastic Analysis
Volume 2011 (2011), Article ID 942478, 19 pages
Optimal Harvesting When the Exchange Rate Is a Semimartingale
Department of Mathematics, Faculty of Science, University of Botswana, Private Bag UB 0022, Gaborone, Botswana
Received 8 August 2011; Revised 31 October 2011; Accepted 1 November 2011
Academic Editor: Onesimo Hernandez Lerma
Copyright © 2011 E. R. Offen and E. M. Lungu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider harvesting in the Black-Scholes Quanto Market when the exchange rate is being modeled by the process , where is a semimartingale, and we ask the following question: What harvesting strategy and the value function maximize the expected total income of an investment? We formulate a singular stochastic control problem and give sufficient conditions for the existence of an optimal strategy. We found that, if the value function is not too sensitive to changes in the prices of the investments, the problem reduces to that of Lungu and Øksendal. However, the general solution of this problem still remains elusive.
This paper is concerned with an optimal harvesting strategy in the Black-Scholes Quanto Market when the exchange rate is being driven by a general semimartingale. Specifically, it is proposed that the optimal harvesting strategy can be found under certain conditions. The paper aims to make a contribution by deriving the general formula for an optimal harvesting strategy when the exchange rate is a semimartingale. This study could shed light on the application of general semimartingales in optimization of harvests from investments. Optimal harvesting is one of the crucial areas in finance because investment into stocks and bonds can be used as a source of revenue to expand business. Therefore, making investors happy through an optimal harvesting strategy could lead to more investments and consequently to further expansion of business. This study will make reference to dividend policy to illustrate an optimal harvesting strategy. Indeed a suboptimal dividend policy can result in destruction of shareholder confidence. How much to payout and still maintain growth of investments has been a challenge. For example, Miller and Modigilliani  claimed that a dividend policy was irrelevant in perfect markets because it had no impact on firm value. However, research on dividend policy that followed Miller and Modigilliani  has further examined various market imperfections and have identified the relevance of dividend policy. A number of stochastic models for optimal dividend policy can also be found in Taksar  and the references therein.
A number of these models ([1, 3], etc.) have developed an optimal harvesting strategy as optimal stochastic control problems, and that is our approach in this paper. For example, Asmussen and Taksar  applied the theory of singular control in their study of a company’s optimal dividend policy that tries to maximize expected value of the total (discounted) payments to the shareholders. Asmussen and Taksar  made an assumption that no fixed costs are incurred during payment of dividends and that the liquid assets are modeled by Brownian Motion with drift. Others who also contributed to this problem are Højgaard and Taksar [5, 6], Radner and Shepp , Asmussen et al. , and Choulli et al. . The studies [5–7] treated the problem as a classical singular stochastic control problem but allowed a control to affect both potential profits and the risks of the financial corporation. Jeanblanc-Picqué and Shiryaev  investigated the problem of a company that tries to maximize the expected total (discounted) amount of dividend payments by modeling dividends as a stochastic impulse control problem. They  looked at a situation whereby the company faced a fixed cost each time a dividend was paid out by choosing optimally the timing and the size of the payments. They  assumed that, when there is no intervention, the liquid asset follows a Brownian Motion process with drift. Lungu and Øksendal  have considered the problem of optimizing flow of dividends for a market situation with two investments and determined an optimal harvesting strategy. They concluded that the optimal strategy was to do nothing as long as the investments were in the nonintervention region but to harvest when the investment reached a certain calculated value (see Lungu and Øksendal ). Motivated by Lungu and Øksendal , this study considers a similar problem but now in the Black-Scholes Quanto market with the exchange rate modeled by a general semimartingale. The paper is structured as follows. Section 2 states the model and the necessary theory. Section 3 applies the theory, while Section 4 gives the conclusions.
2. The Model
In the absence of interventions, the dynamics of , the value of the sterling risky investment, can be modeled by the equation where is the riskless interest rate, the constant is the volatility, and is Brownian Motion. Let the sterling to dollar exchange be modeled by the equation and suppose that these processes are on a filtered probability space , where and is the natural filtration generated by the stock price process while is the natural filtration generated by the exchange rate process. describes information about prices and the exchange rate revealed to investors. We assume that the probability space satisfies the usual conditions, that is, the -field is -complete and every contains all -null sets of . , that is, is a càdlàg process that admits the decomposition , where (a process of bounded variation), , and (a local martingale), .
We consider harvesting from the investment process given (2.1) previously, and we ask the following question: what value function and harvesting strategy maximise the total expected discounted utility harvested from a given time interval.
Since our asset is in sterling but our currency of businesses is the dollar, we need first of all to find the dollar equivalence of this asset. To do this, we let be the dollar value of the sterling asset price given by Using (2.1) and (2.2), we obtain where is a semimartingale since it is the sum of two semimartingales and . This in turn implies that is a semimartingale. Our approach will be probabilistic rather than statistical; hence, it becomes reasonable to express in stochastic exponential form.
Theorem 2.1 (Ito’s theorem for semimartingales ). Let be a semimartingale, and let be a real function. Then, is again a semimartingale and where and is the quadratic characteristic of the continuous martingale part of , that is, a predictable process such that .
In this study, we use Theorem 2.1 to rewrite in stochastic exponential form.
Let then, using Theorem 2.1, we have and, in differential form, this can be expressed as where We associate with this semimartingale an integer-valued random measure defined as where is an indicator function, that is, (see Shiryaev , for details). We define the integral-valued random measures of jumps , where and and by the compensator of , that is, the predictable measure with the property that is a local martingale measure. This means that, for each is a local martingale with value 0 for .
Let be a truncation function, for example, . Then . We now denote the jump part of corresponding to big jumps by Using random measure of jumps, (2.9) can be written in canonical form as where is a predictable process and is the continuous martingale part of .
Adopting the definition of harvesting strategy from Lungu and Øksendal , we define a harvesting strategy as a stochastic process with the following properties: ,(1) is measurable with respect to -algebra generated by (i.e., is adapted);(2) is nondecreasing with respect to for almost all (a.a.) ;(3) is right continuous as a function of for a.a. ;(4) for a.a. .
represents the total amount harvested from an initial time up to time . We let represent a set of all harvesting strategies. If we apply the harvesting strategy , then the corresponding process satisfies the equation It is important to note that the difference between and is the state before harvesting starts at time while is the state immediately after. If consists of an immediate harvest of size at , then Let the prices/utilities per unit investment when harvested at time be given by a constant nonnegative function . The expected total discounted payoff in this case is given by where denotes expectation with respect to probability law of for , assuming that , are the time of bankruptcy and is the solvency region, respectively. The optimal harvesting problem is then to find the value function and an optimal dividend strategy such that We let denote the jumping times of the given strategy , and we let be the jump of . We also let be the continuous part of . We formulate the sufficient conditions for the given function to be the value function of (2.19) and for a given strategy to be optimal in the following theorem.
Theorem 2.2 (extended Lungu and Øksendal ). Suppose that is twice continuously differentiable on with the following properties:(1)One has (2)One has Then (3)Define the nonintervention region as Suppose and there exists a harvesting strategy such that the following hold: where is the closure of , that is, , where is the boundary of .(4)One has(5)One has at all jumping times of and where Then and is an optimal harvest strategy.
Proof. Consider the following: Choose , and assume that satisfies (2.21)-(2.22). Then, by Ito’s formula for semimartingales and then computing the expectation throughout the equation From the theory of martingales for integrals, Substituting (2.28), (2.32), and (2.35) in (2.34), we have Now from (2.22), Using (2.37) in (2.36) gives Using the fact that and that (refer to (2.27)) yields the inequality From the relation we deduce that Using (2.41), the right-hand side of (2.32) becomes Furthermore, using (2.42) we can achieve the simplification of (2.34) as follows: taking the last three terms in (2.39) and using the fact that we have Substituting (2.44) into (2.39), inequality (2.39) becomes This will then lead to the inequality By the Mean-Value property, we have for some point on the line connecting the points and , and, using (2.41), we have which leads to From (2.21), we obtain Equation (2.21) can also be expressed in discrete form as Combining (2.50) and (2.51) gives the inequality Taking the expectation of (2.52) yields the inequality from which we obtain Since were arbitrary and , this proves that Let us now assume that is given by (2.24) and that (2.25)–(2.28) hold. If we replace in the above calculation by , then equality holds everywhere and we end up with the relation Letting and using (2.29), we get
3. Application of the Theory
From the form of our discounted utility function (2.17), it becomes reasonable to look for the function of the form Let Equations (2.21), (2.22) become where Inequality (3.7) is similar to inequality (3.11) in Lungu and Øksendal . The major difference is that in Lungu and Øksendal  the coefficients are constants whereas in our case the coefficient is not a constant. Since is a process with jumps, the general solution of is elusive. We still try to explore the behavior of the solution after fixing the value of ; that is, we want to see the general behavior of the solution with respect to .
3.1.1. Example 1 ( Constant with respect to )
The case is a constant that reduces to the problem in Lungu and Øksendal . The auxiliary equation for (3.9) is The solutions are Let us suppose the nonintervention region to be of the form for some . From (2.21) and (2.22), we try a solution of the form for . We want to determine parameters , and such that becomes a at . Using the continuity and differentiability of at and taking , we obtain With this choice of parameters, all the conditions of Theorem 2.2 are satisfied, and we have Similarly, as discussed in Lungu and Øksendal , the optimal strategy is obtained by doing nothing as long as (i.e., ) and to harvest a total amount of the reflected process in the direction of .
3.1.2. Example 2 (When Is Not a Constant)
We now look at a more general case corresponding to not a constant. Suppose that we can write as then (3.9) can be transformed into its canonical form given by where and the jumps are embedded in .
To solve (3.17), we use the jump transfer matrix method . The auxiliary equation of (3.17) is with solutions The general solution for can be written as where and are functions to be determined. The solution (3.21) can be expressed as where and the superscript denotes transpose of a matrix. is the solution of the equation where (cf. ).
Clearly, for the case corresponding to , the matrices (3.26) are given by
Using power series expansion for exponential square matrices and truncating the expansion at second-order terms, we have Similarly, From (3.25) and (3.28), we obtain an approximation for as From (3.19) and (3.20), we obtain The jump transfer matrix from region 1 to region 2 across the interface in our case takes the form This jump transfer matrix over singularities given by is simplified as in  to The reader is referred to  and the references therein for details regarding classification of singularities.
3.1.3. Analysis of the Results
Let and our solutions (3.21) have jumps at points , and we define for some , and we define For , we define
The optimal strategy is achieved by doing nothing during jumps and as long as but to harvest according to local time at the boundary .
Remarks. In each nonintervention region , is a reflected process at , since, as hits the boundary, a certain amount is harvested thereby forcing the process to go below .
Most of the conditions in Theorem 2.2 are extensions from Lungu and Øksendal  with exception of condition (2.5), that is, We note that this condition is achieved if is small. It is found under additional condition (2.27) that our problem can be reduced to a second-order differential equation similar to that in Lungu and Øksendal  though with some jumps. What is observed is that if the investment process is being modeled by a semimartingale in general, optimal value function and the optimal dividend strategy can be found if the rate of change of the value function with respect to the investment process itself, that is, , is small enough. In other words, should not be too sensitive to variations in investments. Our results further show that the general solution to this problem is still elusive.
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