|
Article Excerpt
We consider the problem of dynamically hedging the profits of a corporation when these profits are correlated with returns in the financial markets. In particular, we consider the general problem of simultaneously optimizing over both the operating policy and the hedging strategy of the corporation. We discuss how different informational assumptions give rise to different types of hedging and solution techniques. Finally, we solve some problems commonly encountered in operations management to demonstrate the methodology.
Key words: operations management; portfolio optimization; stochastic control; incomplete markets
MSC2000 subject classification: Primary: 90B05, 91B28; secondary: 90B30, 91B70
OR/MS subject classification: Primary: inventory/production: policies; secondary: finance: portfolio
**********
1. Introduction. In this paper, we propose a framework for modelling the operations of a nonfinancial corporation that also trades in the financial markets. The corporation must simultaneously choose an optimal operating policy and an optimal trading strategy in the financial markets. In practice, it has long been observed that nonfinancial corporations do, in fact, hedge using financial markets. The goal of this paper then is to describe a method by which operations and hedging might be conducted.
One immediate difficulty that arises when modelling this problem is that most of the operations literature assumes that corporations are risk neutral. Indeed, this is supported by the famous work of Modigliani and Miller [24] who argue that in a frictionless world there is no need for corporations to hedge as shareholders can do so themselves. While this argument has some merit, we do, of course, live in a world with many frictions. These frictions include the costs of financial distress, taxes, and agency costs, as well as frictions in the capital markets. As a result, it is often the case that corporations should and do hedge. Once this is recognized, it is no longer plausible to assume that corporations are always risk neutral.
In this paper, we therefore consider the problem of dynamically hedging the profits of a risk-averse corporation when these profits are correlated with returns in the financial markets. The central modelling insight is to view the operations and facilities of the corporation as an asset in the corporation's portfolio. This view enables us to pose the problem as one of financial hedging in incomplete markets, a problem that has been studied extensively in the recent literature in mathematical finance, e.g., Schweizer [31]. Though we pose the problem as one of hedging in incomplete markets, we also have the added complexity of simultaneously seeking to choose an optimal operating policy. As a result, we also have some control over the type of asset to be hedged. This is a distinguishing feature of this paper that is generally not found in the mathematical finance literature. We also discuss how different informational assumptions give rise to different types of hedging and solution techniques. In particular, the class of feasible hedging strategies that are available to the corporation will depend on whether or not the corporation can observe the evolution of all relevant state variables.
To maintain tractability, we will assume that the corporation has a mean-variance objective function. While this of course is somewhat restrictive, it is often used in practice and can serve as a useful first approximation. The techniques that we use are based on the mean-variance analysis of Schweizer [30], and the martingale approach of Cox and Huang [8] and Karatzas et al. [19]. While we are aware of the very recent progress that has been made towards solving hedging problems for more general utility functions and general price processes (e.g., Bertsimas et al. [2], Delbaen et al. [10], Gourieroux et al. [16], Laurent and Pham [22], Lim [23], Pham et al. [28], Schweizer [31]), we have not attempted to apply this work here. Indeed, much of this literature is concerned with issues regarding existence and uniqueness of solutions and does not lend itself easily to the computation of such solutions. Moreover, the purpose of this paper is simply to highlight the modelling framework and demonstrate that it can be used to solve some interesting problems in operations management.
We will see that our framework appears to be most useful when the operational control is a scalar or vector of scalars as opposed to a dynamic control policy. In the latter case, we can still use the framework to solve the problem numerically (see Appendix B), but it is not at all clear that this provides any improvement beyond the standard Hamilton-Jacobi-Bellman (HJB) approach.
The remainder of this paper is organized as follows. In [section] 2, we formulate the problem under two different informational assumptions and show how to solve the problem in each case. In [subsection] 3 and 4, we demonstrate the methodology by solving two problems from operations management, including the so-called newsboy problem. We finally conclude and discuss future research directions in [section] 5.
2. Model and problem formulation. Let ([OMEGA], [??], [??]) be a probability space endowed with two independent standard Brownian motions, [B.sub.1,t] and [B.sub.2,t]. We denote by [??] = [([[??].sub.t]).sub.0[less than or equal to] t [less than or equal to] T] the usual filtration generated by ([B.sub.1], [B.sub.2]) where T is a fixed-time horizon. We also define a subfiltration [??] of [??] that will represent the evolution of the observable information in the model. When all relevant information is observable, then we will have [??] = [??].
The financial market that we consider consists of a risk-free cash account and a risky stock. Without lost of generality, we will assume throughout that the risk-free interest rate, r, is identically zero. The time t stock price, [X.sub.t], satisfies the stochastic differential equation
d[X.sub.t] = [[mu].sub.t][X.sub.t] dt + [[sigma].sub.t][X.sub.t] d[B.sub.1,t], (1)
where [[mu].sub.t] and [[sigma].sub.t] are assumed to be bounded adapted processes. We define the set [THETA] of self-financing trading strategies to be the collection of [??]-predictable processes [([[theta].sub.t]).sub.0 [less than or equal to] t [less than or equal to] T] satisfying
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]
We interpret [[theta].sub.t] as the number of shares in the stock held at time t. Given [([[theta].sub.t]).sub.0[less than or equal to]t [less than or equal to] T], the self-financing condition then implicitly defines the position in the cash account for all t [member of] [0, T]. Recalling that r [equivalent to] 0, the gain process, [G.sub.t]([theta]), associated with a trading strategy, [theta] [member of] [THETA], is defined to be
[G.sub.t]([theta]) := [[integral].sup.t.sub.0] [[theta].sub.t] d[X.sub.t] for all t [member of] [0, T].
We consider the problem of a risk-averse nonfinancial corporation that earns a terminal payoff, [H.sub.T], that may be interpreted as the profits that are earned from operating in [0, T]. Of course [H.sub.T] will depend on the operating policy, [gamma], that is adopted in this interval. In this context, [gamma] represents a generic operational policy that may in principle be anything from a scalar control to a complex state-dependent control policy. We will usually write [H.sup.([gamma]).sub.T] to signify the dependence of the operating profits on [gamma]. The set of [??]-predictable admissible policies is denoted by F and we will assume that for [gamma] [member of] [GAMMA], the payoff [H.sup.([gamma]).sub.T] is an [[??].sub.T]-measurable random variable that satisfies [H.sup.([gamma]).sub.T] E [[??].sup.P]([??]) for some p > 2.
In addition to the nonfinancial operations, the corporation is able to trade in the financial market by employing a self-financing trading strategy, [theta] [member of] [THETA]. Therefore, for a given initial wealth, [W.sub.0], and for a given strategy, ([gamma], [theta]) [member of] [GAMMA] x [THETA], the corporation's time T wealth is given by [W.sub.0] + [H.sup.([gamma]).sub.T] + [G.sub.T]([theta]). (Due to the possibility of unlimited borrowing, we may assume that [THETA] is independent of the operating policy, [gamma].)
We assume that the corporation has a quadratic utility function, u(*), defined over terminal wealth so that u(w) = w - [lw.sup.2] where l is a positive constant. Then, the corporation's problem is to solve
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2)
An important potential weakness in our problem formulation can be seen in (2) where operational profits and trading profits are treated identically in the corporation's utility function. In practice, this is certainly not the case. For example, suppose the corporation has a choice over two possible random variables, [Y.sub.1] and [Y.sub.2], which represent terminal profits. If [Y.sub.1] and [Y.sub.2] are identically distributed, then in our problem formulation, the corporation should be indifferent between the two. Suppose now, however, that [Y.sub.1] is strongly positively correlated with the financial market and that [Y.sub.2] is independent of the financial market. Then, for diversification reasons, shareholders (and therefore the corporation) would prefer [Y.sub.2] to [Y.sub.1]. A simple argument based on the capital asset pricing model (CAPM) makes this clear, but the reasoning applies more generally.
A solution ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [G.sup.*.sub.T]) to (2), where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is highly correlated with the financial markets due to a significant reduction in operational activity, might therefore be unappealing. This would be true in particular if the solution was intended to be implemented on a permanent basis, i.e., if after each period of length T the corporation intended to implement the same optimal operating and hedging strategies. However, there are many reasons for why such a significant reduction in operations might make sense on a temporary basis. We also mention that, in many contexts, we would not expect the optimal solution to (2) to necessarily result in a reduction in operations. We mention finally that any problems associated with a solution resulting in a significant reduction in operational activity could be overcome by working instead with an equivalent martingale measure (EMM), [??], rather than the physical measure, [??]. The cumulative trading gains process is a martingale under any such measure, [??], and so the only motivation for trading would be to hedge operational cash flows. On the other hand, it is more difficult to interpret the quadratic objective function when we use an EMM, [??].
Let us now comment on the relationships between [X.sub.t], [B.sub.2], and [gamma]. We have assumed that X represents some financial market, e.g., an equity index, an exchange rate, or possibly an economic index. As such, we would not expect X to depend in any way on the operating policy, [gamma], of an individual corporation. It is possible that an argument could be made for introducing some dependency if the corporation was extremely large and the hedging security, X, was, for example, an index reflecting the performance of the industry in which the corporation operates. However, we do not have that situation in mind in this paper, and so we assume that [gamma] has no impact upon X. Similarly, [B.sub.2] represents nonfinancial or idiosyncratic noise. It is firm specific and might represent that part of the market demand for a particular good that is not explained by the current state of the economy or financial market. Or it might simply represent the uncertain quality of a product to be purchased or manufactured at some future date. Because of this interpretation, we make the natural assumption that [[mu].sub.t] and [[sigma].sub.t] are adapted to the filtration generated by [B.sub.1] only. Finally, the manner in which [H.sup.([gamma]).sub.T] then depends on X and [B.sub.2] will depend on the application in question. We will give some examples in [subsection] 3 and 4.
We now conclude this section with two observations. First, we mention that the models we consider in this paper could be extended to the case where [B.sub.1] and [B.sub.2] are correlated...
|
