Abstract

We study the question what value an agent in a generalized Black-Scholes model with partial information attributes to the complementary information. To do this, we study the utility maximization problems from terminal wealth for the two cases partial information and full information. We assume that the drift term of the risky asset is a dynamic process of general linear type and that the two levels of observation correspond to whether this drift term is observable or not. Applying methods from stochastic filtering theory we derive an analytical tractable formula for the value of information in the case of logarithmic utility. For the case of constant relative risk aversion (CRRA) we derive a semianalytical formula, which uses as an input the numerical solution of a system of ODEs. For both cases we present a comparative analysis.

1. Introduction

The economics of information and more precisely the way how information influences our strategic opportunities is a topic which is more and more discussed among economists, with applications in basically all behavioral sciences but particularly in finance. To an increase in strategic opportunities corresponds an increase in the level of maximal obtainable utility. This increase can be associated with a financial value and it is this value what is usually referred to as the value of information. This value depends in general on the whole model, assets, strategies, agents preferences, and so forth, and technically the same information can have different values for different agents and different underlying models. For more background on the general foundations of information economics we refer to Birchler and BΓΌttler [1] and Hirshleifer and Riley [2]. In the specific context of stock markets the aspect of additional information and its value has been studied by various authors; see, for example, Amendinger et al. [3], Imkeller [4], Ewald [5], and Kohatsu-Higa and Sulem [6] as only to mention a view. While the first three focus on the case of initially enlarged filtration in the sense of Jacod [7], Kohatsu-Higa provides a rather general framework, which appears however to be to technical as to obtain analytical or numerical expressions for the value of information. All of the four models mentioned above treat the case of a representative agent, which either does or does not have an increased level of information, but there is no interaction between differently informed agents and no active exchange of information from one agent to another. The aspect of agent interaction and its strategic consequences in the presence of information asymmetry as well as the effect of increased information on general welfare has been studied by Ewald and Xiao [8]. In this article we resume the classical representative agent framework, but instead of initially enlarged information are study the case of partial information, in which the drift rate of the risky asset is assumed to be an unobservable process. The framework of partial information has been studied before for various levels of generality; see, for example, Pham and Quenez [9], Sass and Hausmann [10], Genotte [11], Dothan and Feldman [12], Detemple [13] and Xia [14]. Pham and Quenez study the case of a stochastic volatility model with partial information and derive formulas for the optimal strategy and the optimal wealth. The model discussed in our article could be treated along the same lines of arguments. However the formulas obtained by Pham and Quenez are not very explicit. By this we mean that from these formulas alone, it is difficult to say how model parameters effect the value of full information, which is one focus of our article. Sass and Haussmann are far more explicit and in fact provide a numerical and empirical analysis. Their discussion however is limited to the case where the unobserved process is a finite state Markov chain. Genotte studies a model which includes our model as a particular case, the setup is in fact far more general than ours, but the focus is on consumption rather than terminal wealth as in our article and no explicit formulas are derived. The articles of Dothan and Feldman as well as Detemple address different economic models, bond markets and interest rates in the first case, production economies in the second. Xia provides a quite general framework in which he also deals with an optimal consumption/terminal wealth problem under CRRA, but relies strongly on the numerical solution of the corresponding HJB equation via finite difference methods. None of these articles however really addresses the issue of the value of information. This extension could be easily done by adding the standard valuation method provided by information economics, however as indicated before, with the exception of Sass and Haussmann the obtained formulas are not explicit enough as to allow a numerical analysis and comparative statics. This is partly due to the high level of generality which was assumed in the underlying models. More explicit results at the expense of a loss in model generality are obtained by Yang and Ma [15]. Here the aspect of valuation of information is discussed but as an objective the maximization of consumption rather than expected utility from terminal wealth is chosen. Furthermore the model studied in our article is more general than by Yang and Ma [15] where a rather static setup was chosen. Focusing at first on the potentially easiest case of logarithmic utility but allowing a general linear dynamic for the unobserved drift term, we present a direct computation which leads to an explicit and analytically tractable expression for the value of full information in the case of a nonobservable drift term which is assumed to follow a linear stochastic differential equation. We then perform a comparative statics analysis and study how individual model parameters influence the value of the information. Such an analysis in the framework of a continuous time model has to the best of our knowledge not been done before. For the case of CRRA we derive a semianalytical formula for the value function under partial information and use this to determine the value of information under CRRA. More precisely, the Hamilton-Jacobi-Bellman equation is reduced to a compact functional form, which as an input depends on the solution of a system of three ODEs. The numerical solution of this system of ODEs is far easier, than applying an implicit finite difference scheme in three dimensions, as done in Xia [14].

The remainder of the article is organized as follows. In Section 2, we introduce the investment model with partial information, while in Section 3, we compute the stochastic filtering estimate for the drift rate of the risky asset. We apply this result in Section 4 and give an explicit solution to the optimal investment problem with partial information under logarithmic utility. In Section 5, we provide a simple calculation formula for the information valuation while in Section 6 we study how individual parameters of the model influence the value of information. In Section 7, we repeat the analysis from the previous sections for the case of CRRA. Here we particularly focus on the effect of the risk-aversion parameter 𝛽 on the value of information. The main conclusions are summarized in Section 8.

2. The Partial Information Model

We assume that there are two types of assets an investor can invest in: a risky asset and a risk-free asset. We think of the risky asset as a stock and the risk free asset as a savings deposit which pays a deterministic interest rate. The investment problem the agent faces is how to choose the appropriate ratio between risky asset and risk-free asset. For the case of complete information, where the agent is able to observe the noise generating process, this problem has been studied by many authors, most famously by Merton [16]. However, in principal it is unrealistic that the agent can observe the noise generating process, neither can he directly observe the parameters which constitute the model, for example the drift rate of the risky asset. What the agent does instead is observing the price process of both the risky asset and the risk-free asset. In general these processes carry less information than the noise generating process. The investment problem under partial information is the problem of how to invest optimally, when information is generated by the asset price processes only and not by the noise generating process. In the following we give a mathematical precise formulation at hand of an explicit example, which we continue to study in this article. Our agent faces a finite time horizon [0,𝑇] and an economic environment whose uncertainty is modeled by a complete probability space (Ξ©,β„±,β„™). The prices of the two types of assets (risky and risk free) are denoted by (𝐡𝑑,𝑆𝑑) and are governed by the following SDEs:𝑑𝐡𝑑=π‘Ÿπ‘‘π΅π‘‘π‘‘π‘‘,𝑑𝑆𝑑=πœ‡π‘‘π‘†π‘‘π‘‘π‘‘+πœŽπ‘‘π‘†π‘‘π‘‘π‘Š1𝑑,0≀𝑑≀𝑇,(2.1) where the interest rate π‘Ÿπ‘‘ and the volatility πœŽπ‘‘ are deterministic processes, π‘Š1 is a Brownian motion, and the appreciation rate πœ‡π‘‘ is a stochastic process satisfying the following SDE:π‘‘πœ‡π‘‘=ξ€·π‘Ž1πœ‡π‘‘+π‘Ž0𝑑𝑑+𝑏1π‘‘π‘Š1𝑑+𝑏2π‘‘π‘Š2𝑑,0≀𝑑≀𝑇,(2.2) with π‘Ž1,π‘Ž0,𝑏1,𝑏2 constants and π‘Š2 a Brownian motion which is independent of π‘Š1. The filtrations β„±={ℱ𝑑}0≀𝑑≀𝑇 and 𝒒={𝒒𝑑}0≀𝑑≀𝑇 withβ„±π‘‘ξ€·π‘Š=𝜎1𝑠,π‘Š2π‘ ξ€ΈβˆΆ0≀𝑠≀𝑑,𝒒𝑑𝑆=πœŽπ‘ ξ€ΈβˆΆ0≀𝑠≀𝑑(2.3) represent the two different levels of information in our model. While the filtration β„± contains the information produced by the noise generating process the filtration 𝒒 does only contain the information produced by the asset price processes. Note that since the bond price follows a deterministic process, it does not really contribute toward the information flow. We refer to β„± as full information and to 𝒒 as partial information. Classically it is assumed that the investors have access to the full information flow {ℱ𝑑}0≀𝑑≀𝑇. This however means that the agent is able to observe the noise generating processes which in this case are the Brownian motions driving the stock price as well as the appreciation rate. This assumption is not very realistic. The case of partial information {𝒒𝑑}0≀𝑑≀𝑇 where the agent can only observe the asset prices is much more realistic. Let us now suppose that the investor has initial wealth 𝑋0 and that she will invest according to a self-financing trading strategy. Suppose that at time π‘‘βˆˆ[0,𝑇], she invests a proportion πœ‹π‘‘ of her wealth in the risky asset, and invests the rest in the risk-free asset. Then her wealth process 𝑋={𝑋𝑑}0≀𝑑≀𝑇 satisfies: for all 0≀𝑑≀𝑇,𝑑𝑋𝑑=πœ‹π‘‘π‘‹π‘‘π‘†π‘‘π‘‘π‘†π‘‘+ξ€·1βˆ’πœ‹π‘‘ξ€Έπ‘‹π‘‘π΅π‘‘π‘‘π΅π‘‘=ξ€·π‘Ÿπ‘‘+πœ‹π‘‘ξ€·πœ‡π‘‘βˆ’π‘Ÿπ‘‘π‘‹ξ€Έξ€Έπ‘‘π‘‘π‘‘+πœŽπ‘‘πœ‹π‘‘π‘‹π‘‘π‘‘π‘Š1𝑑,(2.4) where the process {πœ‹π‘‘}0≀𝑑≀𝑇 must be 𝒒-adapted. In summary, the optimal portfolio problem is described as follows:max{πœ‹π‘‘}0β‰€π‘‘β‰€π‘‡π”Όξ€Ίπ‘ˆξ€·π‘‹π‘‡ξ€Έξ€»(2.5) subject to the constraints:π‘‘πœ‡π‘‘=ξ€·π‘Ž1πœ‡π‘‘+π‘Ž0𝑑𝑑+𝑏1π‘‘π‘Š1𝑑+𝑏2π‘‘π‘Š2𝑑,𝑑𝑋𝑑=ξ€·π‘Ÿπ‘‘+πœ‹π‘‘ξ€·πœ‡π‘‘βˆ’π‘Ÿπ‘‘π‘‹ξ€Έξ€Έπ‘‘π‘‘π‘‘+πœŽπ‘‘πœ‹π‘‘π‘‹π‘‘π‘‘π‘Š1𝑑,0≀𝑑≀𝑇,(2.6) and {πœ‹π‘‘}𝒒-adapted. {𝑆𝑑}0≀𝑑≀𝑇 and therefore also {𝑋𝑑}0≀𝑑≀𝑇 can be observed, but {πœ‡π‘‘}0≀𝑑≀𝑇 is the unobservable state process. Therefore, (2.5) and (2.6) are an optimization problem for a partially observable stochastic dynamic system. In the first part of this article the utility function π‘ˆ is chosen to be logarithmic, that is, π‘ˆ(π‘₯)=log(π‘₯); later we consider the case of CRRA utility, that is, π‘ˆ(π‘₯)=π‘₯𝛽/𝛽.

3. Filtering Estimation for the Average Appreciation Rate

In this section, we apply the filtering technique to estimate the appreciation rate {πœ‡π‘‘}0≀𝑑≀𝑇 given the information flow 𝒒. Denoteπ‘šπ‘‘ξ€·πœ‡=π”Όπ‘‘βˆ£π’’π‘‘ξ€Έ,π›Ύπ‘‘ξ‚ƒξ€·πœ‡=π”Όπ‘‘βˆ’π‘šπ‘‘ξ€Έ2βˆ£π’’π‘‘ξ‚„.(3.1)

Note that π‘š0=𝔼(πœ‡0) and 𝛾0=Var(πœ‡0). The following lemma follows from Theorem 11.1 by Liptser and Shiryayev [17].

Lemma 3.1. If the conditional distribution 𝐹𝒒0ξ€·πœ‡(π‘₯)=β„™0≀π‘₯βˆ£π’’0ξ€Έ(3.2) is normal with mean π‘š0 and variance 𝛾0, a.s., then the conditional distribution πΉπ’’π‘‘ξ€·πœ‡(π‘₯)=ℙ𝑑≀π‘₯βˆ£π’’π‘‘ξ€Έ(3.3) is normal a.s. with mean with π‘šπ‘‘ and variance with 𝛾𝑑.

As a normally distributed random variable is uniquely determined by its expectation and variance we conclude from this lemma that the knowledge of π‘šπ‘‘ and 𝛾𝑑 reflects all the information on πœ‡π‘‘ the agent is able to obtain under the partial information 𝒒𝑑.

Lemma 3.2. Let {πœ‡π‘‘,𝑆𝑑}0≀𝑑≀𝑇 be the stochastic processes with differentials given by (2.2) and (2.1). Suppose that β„™(πœ‡0≀π‘₯βˆ£π’’0) is Gaussian with mean π‘š0 and variance 𝛾0. Further, one assume that there exist two constants 0<𝑐1<𝑐2<∞ such that for all π‘‘βˆˆ[0,𝑇], one has 𝑐1β‰€πœŽπ‘‘β‰€π‘2. Then π‘šπ‘‘ and 𝛾𝑑 satisfy the following equations: π‘‘π‘šπ‘‘=ξ€·π‘Ž1π‘šπ‘‘+π‘Ž0𝑏𝑑𝑑+1πœŽπ‘‘+π›Ύπ‘‘πœŽ2π‘‘ξ‚΅π‘‘π‘†π‘‘π‘†π‘‘βˆ’π‘šπ‘‘ξ‚Ά,𝑑𝑑̇𝛾𝑑=2π‘Ž1𝛾𝑑+𝑏21+𝑏22βˆ’ξ‚΅π‘1πœŽπ‘‘+π›Ύπ‘‘πœŽπ‘‘ξ‚Ά2,0≀𝑑≀𝑇,(3.4) where ̇𝛾𝑑 denotes the deterministic time-derivative of 𝛾𝑑.

Proof. Applying ItΓ΄'s formula to (2.1), we have 𝑑log𝑆𝑑=ξ‚€πœ‡π‘‘βˆ’12𝜎2𝑑𝑑𝑑+πœŽπ‘‘π‘‘π‘Š1𝑑.(3.5) It is then easy to see that the conditions of Theorem 12.1 by Liptser and Shiryayev [17] hold for (πœ‡,log𝑆). The conclusion of this lemma then follows from that theorem.
In the case of the classic Black-Scholes model where πœŽπ‘‘=𝜎 is constant, it follows by using the technique of separation of variables, that 𝛾𝑑 is explicitly given by the following expression:𝛾𝑑=⎧βŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽ©π›Ύβˆ’ξ”+2Ξ›2+𝑏22𝜎2βŽ›βŽœβŽœβŽœβŽπ›Ύ1βˆ’0βˆ’π›Ύ+𝛾0βˆ’π›Ύβˆ’βŽ›βŽœβŽœβŽœβŽβˆ’2expΞ›2+𝑏22𝜎2𝜎2π‘‘βŽžβŽŸβŽŸβŽŸβŽ βŽžβŽŸβŽŸβŽŸβŽ βˆ’1ifΞ›2+𝑏22ξ‚΅1β‰ 0,𝛾0+π‘‘πœŽ2ξ‚Άβˆ’1ifΞ›=𝑏2=0,(3.6) where Ξ›=π‘Ž1𝜎2βˆ’π‘1𝜎,𝛾±=Λ±Λ2+𝑏22𝜎2.(3.7)

Remark 3.3. (1) The magnitude of 𝛾𝑑 is a characterization of the accuracy of the estimate π‘šπ‘‘ of πœ‡π‘‘. If 𝛾𝑑 converges to zero as π‘‘β†’βˆž, then π‘šπ‘‘ is called a consistent estimate of πœ‡π‘‘.
(2) It is obvious that 𝛾𝑑→𝛾+ as π‘‘β†’βˆž. Note that 𝛾+=0 if and only if π‘Ž1=𝑏1=𝑏2=0. Therefore, π‘šπ‘‘ is consistent if and only if the appreciation rate is constant in time, that is, πœ‡π‘‘β‰‘πœ‡0.
(3) The points 𝛾+ and π›Ύβˆ’ are stationary points of the dynamics of 𝛾𝑑. As indicated in 2. 𝛾+ is globally stable while π›Ύβˆ’ is nonstable. In fact π›Ύβˆ’β‰€0 and as 𝛾0>0 it is easy to see that the trajectory of 𝛾𝑑 is confined to the interval [min(𝛾0,𝛾+),max(𝛾0,𝛾+)] and furthermore that it is strictly monotonic.

Let us now define the innovation process by the following equation:π‘‘π‘Šπ‘‘=1πœŽπ‘‘ξ‚΅π‘‘π‘†π‘‘π‘†π‘‘βˆ’π‘šπ‘‘ξ‚Άπ‘‘π‘‘.(3.8) It is well-known from the theory of filtering that π‘Š={π‘Šπ‘‘}0≀𝑑≀𝑇 is a Brownian motion with respect to the stochastic basis (Ξ©,𝒒𝑇,β„™,{𝒒𝑑}0≀𝑑≀𝑇). Therefore, the self-financing condition (2.4) can be rewritten as𝑑𝑋𝑑=ξ€·π‘Ÿπ‘‘π‘‹π‘‘+ξ€·π‘šπ‘‘βˆ’π‘Ÿπ‘‘ξ€Έπœ‹π‘‘π‘‹π‘‘ξ€Έπ‘‘π‘‘+πœŽπ‘‘πœ‹π‘‘π‘‹π‘‘π‘‘π‘Šπ‘‘.(3.9) Similarly, combining (3.8) and (3.4), we obtain thatπ‘‘π‘šπ‘‘=ξ€·π‘Ž1π‘šπ‘‘+π‘Ž0𝑏𝑑𝑑+1πœŽπ‘‘+π›Ύπ‘‘πœŽπ‘‘π‘‘π‘Šπ‘‘.(3.10) The stochastic differential equations (3.9) and (3.10) describe the dynamics from the point of view of the partially informed agent.

4. Optimal Investment under Partial Information for Logarithmic Utility

In this section we compute the optimal investment strategy under partial information for the case where the agents objective is maximization of expected logarithmic utility from terminal wealth. For this case we are able to obtain a reasonably tractable formula which allows us to study the effects of the model parameters on the value of the information. It follows from the ItΓ΄ formula and (3.9) that log𝑋𝑇=log𝑋0+ξ€œπ‘‡0ξ‚€π‘Ÿπ‘‘+ξ€·π‘šπ‘‘βˆ’π‘Ÿπ‘‘ξ€Έπœ‹π‘‘βˆ’12𝜎2π‘‘πœ‹2π‘‘ξ‚ξ€œπ‘‘π‘‘+𝑇0πœŽπ‘‘πœ‹π‘‘π‘‘π‘Šπ‘‘.(4.1) Hence𝔼log𝑋𝑇=log𝑋0+ξ€œπ‘‡0π”Όξ‚€π‘Ÿπ‘‘+ξ€·π‘šπ‘‘βˆ’π‘Ÿπ‘‘ξ€Έπœ‹π‘‘βˆ’12𝜎2π‘‘πœ‹2𝑑𝑑𝑑,(4.2) whose maximum is attained atπœ‹π‘‘=π‘šπ‘‘βˆ’π‘Ÿπ‘‘πœŽ2𝑑[].,βˆ€π‘‘βˆˆ0,𝑇(4.3) The optimal value under partial information is then given by𝑉𝑃=log𝑋0+ξ€œπ‘‡0ξƒ©π‘Ÿπ‘‘+π”Όξ€·π‘šπ‘‘βˆ’π‘Ÿπ‘‘ξ€Έ22𝜎2𝑑ξƒͺ𝑑𝑑.(4.4) From (3.10) we obtain thatπ‘šπ‘‘=π‘š0π‘’π‘Ž1𝑑+π‘Ž0π‘Ž1ξ€·π‘’π‘Ž1𝑑+ξ€œβˆ’1𝑑0π‘’π‘Ž1(π‘‘βˆ’π‘ )𝑏1+π›Ύπ‘ πœŽπ‘ ξ‚Άπ‘‘π‘Šπ‘ ,(4.5) where in the case that π‘Ž1=0 the expression (π‘Ž0/π‘Ž1)(π‘’π‘Ž1π‘‘βˆ’1) needs to be replaced by π‘Ž0𝑑. We then obtain thatπ”Όξ€·π‘šπ‘‘βˆ’π‘Ÿπ‘‘ξ€Έ2=ξ‚΅π‘š0π‘’π‘Ž1𝑑+π‘Ž0π‘Ž1ξ€·π‘’π‘Ž1π‘‘ξ€Έβˆ’1βˆ’π‘Ÿπ‘‘ξ‚Ά2+ξ€œπ‘‘0𝑒2π‘Ž1(π‘‘βˆ’π‘ )𝑏1+π›Ύπ‘ πœŽπ‘ ξ‚Ά2𝑑𝑠.(4.6) Combining (4.4) and (4.6), we have𝑉𝑃=log𝑋0+ξ€œπ‘‡0π‘Ÿπ‘‘ξ€œπ‘‘π‘‘+𝑇0ξ€·π‘š0π‘’π‘Ž1t+ξ€·π‘Ž0/π‘Ž1π‘’ξ€Έξ€·π‘Ž1π‘‘ξ€Έβˆ’1βˆ’π‘Ÿπ‘‘ξ€Έ22𝜎2𝑑+ξ€œπ‘‘π‘‘π‘‡0ξ€œπ‘‘0𝑒2π‘Ž1(π‘‘βˆ’π‘ )2𝜎2𝑑𝑏1+π›Ύπ‘ πœŽπ‘ ξ‚Ά2𝑑𝑠𝑑𝑑.(4.7) This expression only involves a deterministic integral, which can be computed using standard deterministic numerical integration methods. To the best of our knowledge no compact analytical expression for this integral exists in the general case. To summarize, we have the following theorem.

Theorem 4.1. At time π‘‘βˆˆ[0,𝑇], the optimal investment problem under partial information (2.5) with logarithmic utility subject to constraints (3.9) and (3.10) has the following solution: πœ‹π‘‘=π‘šπ‘‘βˆ’π‘Ÿπ‘‘πœŽ2𝑑.(4.8) where π‘šπ‘‘ is determined by (3.4). The optimal value function is given by (4.7).

It is important to identify (3.4) as the equation from which π‘šπ‘‘ is obtained. Even though we derived a more explicit expression for π‘šπ‘‘ in (4.6), the latter expression is not directly useful for the agent, as he does not observe the Brownian π‘Š but the stock 𝑆 instead.

Finally, we give an elementary example where the above quantities can be calculated in explicit form.

Example 4.2. Let us consider the special case that πœ‡π‘‘β‰‘πœ‡0 is a normal random variable with mean π‘š0 and variance 𝛾0. Namely, π‘Ž1=π‘Ž0=𝑏1=𝑏2=0. Further, we take π‘Ÿπ‘‘=π‘Ÿ and πœŽπ‘‘=𝜎 to be constants. In this case 𝛾𝑑=ξ‚΅1𝛾0+π‘‘πœŽ2ξ‚Άβˆ’1,(4.9) and hence 𝑉𝑃=log𝑋0ξ€·π‘š+π‘Ÿπ‘‡+0ξ€Έβˆ’π‘Ÿ2𝑇2𝜎2βˆ’12𝛾log1+0π‘‡πœŽ2ξ‚Ά+𝛾0𝑇2𝜎2.(4.10)

This example, which has also been studied by other authors; however is useless, when the question is "How do model parameters affect the value of information?", because in the setup of the example, there are to many coefficient restrictions and in fact the only remaining interesting parameters are 𝑇 and 𝜎.

5. Valuation of the Information under Logarithmic Utility

In this section we determine the value of full information for an agent who has only partial information. As we said in the introduction, the value of a piece of information depends on how much information the individual agent already has. The value of the information is then obtained in the way, that the optimal expected utility the agent can achieve with his current level of information is subtracted from the optimal expected utility the agent could achieve with the increased level of information. The computed value of information is then in terms of additional utility. Alternatively a monetary value can be obtained by comparing certainty equivalences instead. However, here we choose utility as a scale. In order to fulfill this agenda in our particular model we need to compute the optimal expected logarithmic utility from terminal wealth under full information. This problem can in fact be solved along the same line of arguments as in the previous section for the case of partial information. There are however some minor subtleties and for completeness we include the argument. By ItΓ΄'s formula, we have log𝑋𝑇=log𝑋0+ξ€œπ‘‡0ξ‚€π‘Ÿπ‘‘+ξ€·πœ‡π‘‘βˆ’π‘Ÿπ‘‘ξ€Έπœ‹π‘‘βˆ’12𝜎2π‘‘πœ‹2π‘‘ξ‚ξ€œπ‘‘π‘‘+𝑇0πœŽπ‘‘πœ‹π‘‘π‘‘π‘Š1𝑑.(5.1) Hence𝔼log𝑋𝑇=log𝑋0+ξ€œπ‘‡0π”Όξ‚€π‘Ÿπ‘‘+ξ€·πœ‡π‘‘βˆ’π‘Ÿπ‘‘ξ€Έπœ‹π‘‘βˆ’12𝜎2π‘‘πœ‹2𝑑𝑑𝑑,(5.2) whose maximum is attained atπœ‹βˆ—π‘‘=πœ‡π‘‘βˆ’π‘Ÿπ‘‘πœŽ2𝑑.(5.3) The value function is then given by𝑉𝐹=log𝑋0+ξ€œπ‘‡0ξƒ©π‘Ÿπ‘‘+π”Όξ€·πœ‡π‘‘βˆ’π‘Ÿπ‘‘ξ€Έ22𝜎2𝑑ξƒͺ𝑑𝑑.(5.4) By (2.6), we obtain thatπœ‡π‘‘=πœ‡0π‘’π‘Ž1𝑑+π‘Ž0π‘Ž1ξ€·π‘’π‘Ž1𝑑+ξ€œβˆ’1𝑑0π‘’π‘Ž1(π‘‘βˆ’π‘ )𝑏1π‘‘π‘Š1𝑠+𝑏2π‘‘π‘Š2𝑠.(5.5) It then follows from 𝔼(πœ‡0)=π‘š0 thatπ”Όξ€·πœ‡π‘‘βˆ’π‘Ÿπ‘‘ξ€Έ2=𝑒2π‘Ž1𝑑𝛾0+ξ‚΅π‘š0π‘’π‘Ž1𝑑+π‘Ž0π‘Ž1ξ€·π‘’π‘Ž1π‘‘ξ€Έβˆ’1βˆ’π‘Ÿπ‘‘ξ‚Ά2+𝑏21+𝑏22𝑒2π‘Ž2π‘Ž1𝑑.βˆ’1(5.6) Combining (5.4) and (5.6), we see thatπœ‹βˆ—π‘‘=πœ‡π‘‘βˆ’π‘Ÿπ‘‘πœŽ2𝑑,(5.7) and the optimal obtainable expected utility is given by𝑉𝐹=log𝑋0+ξ€œπ‘‡0π‘Ÿπ‘‘ξ€œπ‘‘π‘‘+𝑇012𝜎2𝑑𝑒2π‘Ž1𝑑𝛾0+ξ‚΅π‘š0π‘’π‘Žπ‘‘+π‘Ž0π‘Ž1ξ€·eπ‘Ž1π‘‘ξ€Έβˆ’1βˆ’π‘Ÿπ‘‘ξ‚Ά2+𝑏21+𝑏22𝑒2π‘Ž2π‘Ž1𝑑ξƒͺβˆ’1𝑑𝑑(5.8) with (π‘Ž0/π‘Ž1)(π‘’π‘Ž1π‘‘βˆ’1) replaced by π‘Ž0𝑑 in the case that π‘Ž1=0. Following the program described above we then obtain the following theorem which associates a numerical value to full information relative to partial information.

Theorem 5.1. The value of full information relative to partial information in the model described in Section 2 is given by ξ€œπ’±=𝑇012𝜎2𝑑𝑒2π‘Ž1𝑑𝛾0+𝑏21+𝑏222π‘Ž1𝑒2π‘Ž1π‘‘ξ€Έβˆ’ξ€œβˆ’1𝑑0𝑒2π‘Ž1(π‘‘βˆ’π‘ )𝑏1βˆ’π›Ύπ‘ πœŽπ‘ ξ‚Ά2ξƒͺ𝑑𝑠𝑑𝑑.(5.9)

Proof. As indicated above we simply have to compute the value π‘‰πΉβˆ’π‘‰π‘ƒ. The valuation formula is then obtained by substitution of (5.8) and (4.7).

Perhaps the most striking thing about Theorem 5.1. is that the parameter π‘Ž0 from the non observable drift term does not have any influence on the value of information. We will include a comparative static analysis on how the individual model parameters influence this value in the next section. For the moment let us come back to Example 4.2. In this case we obtain the following.

Example 5.2. For the same choice of coefficients as in Example 4.2. we obtain that the value of full information relative to partial information is given by 1𝒱=2𝛾log1+0π‘‡πœŽ2ξ‚Ά.(5.10)

6. Comparative Statics and Numerical Analysis: Log-Utility

In this section we restrict our analysis to the case where πœŽπ‘‘β‰‘πœŽ is constant and investigate the effect changes of the model parameters π‘Ž1, π‘Ž0, 𝑏1, 𝑏2 and 𝜎 have on the value of full information relative to partial information. For this case we have an explicit representation of the dynamic of the conditional variance 𝛾𝑑=var(πœ‡π‘‘βˆ£π’’π‘‘). We have already indicated in the previous section that the constant term π‘Ž0 in the unobservable drift term does not have any effect on the value of the information. The reason for this is that this parameter is not multiplied with a variable that produces uncertainty, that is, πœ‡π‘‘, respectively, π‘šπ‘‘ in the case of π‘Ž1, π‘‘π‘Š1 in the case of 𝑏1 and 𝜎1 and π‘‘π‘Š2 in the case of 𝑏2. The second observation which is easy to make is that an increase in the absolute value |𝑏2| of the parameter 𝑏2 always leads to an increase in the value of full information. Mathematically this can be seen as follows. Assume first that π‘Ž1β‰ 0. It then follows directly from the valuation formula in Theorem 5.1. thatπœ•π’±πœ•π‘2=𝑏2π‘Ž1ξ€œπ‘‡0𝑒2π‘Ž1π‘‘ξ€Έβˆ’1𝑑𝑑.(6.1) We have (1/π‘Ž1)(𝑒2π‘Ž1π‘‘βˆ’1)>0 for all choices of π‘Ž1 and positive 𝑑 and therefore our assertion holds. In the case that π‘Ž1=0 we obtain by taking the limit for π‘Ž1β†’0 in (6.1) thatπœ•π’±πœ•π‘2=𝑏2ξ€œπ‘‡02𝑑𝑑𝑑=𝑏2𝑇2,(6.2) and we see that our assertion holds in this case as well. The economical interpretation of this result is more or less clear. The parameter 𝑏2 increases the uncertainty in the unobserved process πœ‡π‘‘ and decreases the covariation between asset and unobserved noise 𝑏2π‘‘π‘Š2. Information about this process therefore becomes more valuable the higher |𝑏2| is. The effect of 𝑏1 on the value of full information is more diversified and difficult to identify. Intuitively we can find two effects here. The first one is that an increase in 𝑏1 (as does an increase in 𝑏2) increase the uncertainty in the unobserved process πœ‡π‘‘. However, an increase in 𝑏1 also leads to an increase rather than a decrease of the covariation between πœ‡π‘‘ and the risky asset 𝑆𝑑. While the first effect gives an inertia to an increase of the value of full information, the second effect gives an inertia to decrease its value. Formally by taking derivatives we see thatπœ•π’±πœ•π‘1=2𝑏1ξ€œπ‘‡0ξ€œπ‘‘0𝑒2π‘Ž1(π‘‘βˆ’π‘ )𝑏1βˆ’1βˆ’π›Ύπ‘ πœŽξ‚2𝑑𝑠𝑑𝑑+2𝑏21ξ€œπ‘‡0ξ€œπ‘‘0𝑒2π‘Ž1(π‘‘βˆ’π‘ )𝑏1βˆ’π›Ύπ‘ πœŽξ‚π‘‘π‘ π‘‘π‘‘.(6.3) Since π›Ύπ‘ βˆˆ[0,𝛾+] for all π‘‘βˆˆ[0,𝑇], we see that this is positive, whenever𝛾+𝜎<𝑏1<1.(6.4) Economically it is reasonable to assume that the volatilities 𝑏1 and 𝑏2 of the drift term are less than 1. Remembering that 𝛾+=limπ‘‘β†’βˆžπ›Ύπ‘‘ represents the ultimate level of informational uncertainty and 𝜎 the asset volatility, which in a way represents unresolvable model uncertainty, (6.4) can be interpreted as saying that informational uncertainty on the drift rate has to be comparably small relative to overall model uncertainty in order for 𝑏1 to have a positive affect on the value of full information. Let us now consider the individual effect of 𝜎. We haveπœ•π’±π›Ύπœ•πœŽ=βˆ’02π‘Ž1𝜎30𝑒2π‘Ž1π‘‡ξ€Έξ€·π‘βˆ’1+221+𝑏22ξ€Έξ€œπ‘‡0ξ€œπ‘‘0𝑒2π‘Ž1(π‘‘βˆ’π‘ )𝑏1βˆ’π›Ύπ‘ πœŽξ‚π›Ύπ‘ πœŽ2𝑑𝑠𝑑𝑑,(6.5) where the first summand needs to be replaced by βˆ’π›Ύ0𝑇/𝜎30 in the case that π‘Ž1=0. We are not able to provide a compact and conclusive analytical answer to the question whether this expression is positive or not at least for the general case. Numerically however it is easy to evaluate the integral and compute this expression. Obviously as can be seen from expression (6.5) high values of 𝑏1 and 𝑏2 increase the chance that πœ•π’±/πœ•πœŽ is positive. In the case that 𝑏1≀min(𝛾0,𝛾+)/𝜎 we see that both summands in (6.5) are negative and hence in this case πœ•π’±/πœ•πœŽ<0. Similarly it can be seen that if condition (6.4) holds we have πœ•π’±/πœ•π‘‡>0 which means that the value of information increases with the length of the time horizon the agent is facing. For the special case studied in Examples 4.2 and 5.2 things are more straightforward, however as indicated before, interesting effects are lost. It is easy to see from formula (5.10) that in this special case the larger the variability 𝛾0 of the drift rate πœ‡0 or the time 𝑇, the larger the value of the information while the larger the asset volatility 𝜎 of the risky asset, the smaller the value of the information. Returning back to the general case, Figures 1–6 in Appendices A and B illustrate our theoretical findings above. To produce these figures and in order to evaluate the double integral in Theorem 5.1 we used the following approach based on the classical Simpson rule. Consider a double integral ξ€œπ‘π‘Žξ€œπ‘‘(π‘₯)𝑐(π‘₯)𝑓(π‘₯,𝑦)𝑑𝑦𝑑π‘₯.(6.6) Let the step size for x be β„Ž=(π‘βˆ’π‘Ž)/2 and let the step size for 𝑦 be π‘˜(π‘₯)=(𝑑(π‘₯)βˆ’π‘(π‘₯))/2. Adapting the classical Simpson method iteratively, we obtainξ€œπ‘π‘Žξ€œπ‘‘(π‘₯)𝑐(π‘₯)β‰ˆξ€œπ‘“(π‘₯,𝑦)𝑑𝑦𝑑π‘₯π‘π‘Žπ‘˜(π‘₯)3[]β‰ˆβ„Žπ‘“(π‘₯,𝑐(π‘₯))+4𝑓(π‘₯,𝑐(π‘₯)+π‘˜(π‘₯))+𝑓(π‘₯,𝑑(π‘₯))𝑑π‘₯3ξ‚»π‘˜(π‘Ž)3[]+𝑓(π‘Ž,𝑐(π‘Ž))+4𝑓(π‘Ž,𝑐(π‘Ž)+π‘˜(π‘Ž))+𝑓(π‘Ž,𝑑(π‘Ž))4π‘˜(π‘Ž+β„Ž)3[]+𝑓(π‘Ž+β„Ž,𝑐(π‘Ž+β„Ž))+4𝑓(π‘Ž+β„Ž,𝑐(π‘Ž+β„Ž)+π‘˜(π‘Ž+β„Ž))+𝑓(π‘Ž+β„Ž,𝑑(π‘Ž+β„Ž))π‘˜(𝑏)3[]ξ‚Ό.𝑓(𝑏,𝑐(𝑏))+4𝑓(𝑏,𝑐(𝑏)+π‘˜(𝑏))+𝑓(𝑏,𝑑(𝑏))(6.7) In Theorem 5.1, we need to evaluate ξ€œπ’±=𝑇012𝜎2𝑑𝑒2π‘Ž1𝑑𝛾0+𝑏21+𝑏222π‘Ž1𝑒2π‘Ž1π‘‘ξ€Έβˆ’ξ€œβˆ’1𝑑0𝑒2π‘Ž1(π‘‘βˆ’π‘ )𝑏1βˆ’π›Ύπ‘ πœŽξ‚2𝑑𝑠𝑑𝑑.(6.8) We divide 𝒱 into two parts. The first one is a single integral and the other is a double integral. Let 𝑛 be the number of discretized steps in time and Δ𝑑=𝑇/𝑛. Then𝑑𝑖=π‘‘π‘–βˆ’1+Δ𝑑,for𝑖=1,…,𝑛,𝑑0=0.(6.9) We evaluate the first part by applying the standard Simpson method. Let β„Ž=Δ𝑑/2,πœŽπ‘‘=𝜎 be a constant, and 1𝑓(𝑑)=2𝜎2𝑒2π‘Ž1𝑑𝛾0+𝑏21+𝑏222π‘Ž1𝑒2π‘Ž1π‘‘ξ€Έξƒ­βˆ’1.(6.10) We obtainπ‘›βˆ’1𝑖=0ξ€œπ‘‘π‘–+1𝑑𝑖12𝜎2𝑒2π‘Ž1𝑑𝛾0+𝑏21+𝑏222π‘Ž1𝑒2π‘Ž1π‘‘ξ€Έξƒ­βˆ’1π‘‘π‘‘β‰ˆπ‘›βˆ’1𝑖=0β„Ž3𝑓𝑑𝑖𝑑+4𝑓𝑖𝑑+β„Ž+𝑓𝑖+1.ξ€Έξ€»(6.11) To evaluate the double integral, we let𝑒𝑔(𝑠,𝑑)=2π‘Ž1(π‘‘βˆ’π‘ )𝑏1βˆ’π›Ύπ‘ ξ€Έ/𝜎22𝜎2.(6.12) By applying our adapted Simpson method (6.7) and choosing π‘˜(𝑑)=π‘‘βˆ’02=𝑑2,(6.13) we obtain the following approximate integration formula: π‘›βˆ’1𝑖=0ξ€œπ‘‘π‘–+1π‘‘π‘–ξ€œπ‘‘0β‰ˆπ‘”(𝑠,𝑑)π‘‘π‘ π‘‘π‘‘π‘›βˆ’1𝑖=0ξ€œπ‘‘π‘–+1𝑑𝑖𝑑6𝑑𝑔(0,𝑑)+4𝑔2ξ‚ξ‚„β‰ˆ,𝑑+𝑔(𝑑,𝑑)π‘‘π‘‘π‘›βˆ’1𝑖=0β„Ž3𝑑𝑖6𝑔0,𝑑𝑖𝑑+4𝑔𝑖2,𝑑𝑖𝑑+𝑔𝑖,𝑑𝑖+2𝑑𝑖+β„Ž3𝑔0,𝑑𝑖𝑑+β„Ž+4𝑔𝑖+β„Ž2,𝑑𝑖𝑑+β„Ž+𝑔𝑖+β„Ž,𝑑𝑖+𝑑+β„Žπ‘–+16𝑔0,𝑑𝑖+1𝑑+4𝑔𝑖+12,𝑑𝑖+1𝑑+𝑔𝑖+1,𝑑𝑖+1ξ€Έ.ξ‚Ήξ‚Ό(6.14)


Figure 1 
𝒱 as a function of 𝜎 and π‘Ž 1 .

Figure 2 
𝒱 as a function of 𝜎 and 𝑏 1 .

Figure 3 
𝒱 as a function of 𝜎 and 𝑏 2 .

Figure 4 
𝒱 𝛽 as a function of π‘Ž 1 and 𝛽 .

Figure 5 
𝒱 𝛽 as a function of 𝑏 1 and 𝑏 2 .

Figure 6 
𝒱 𝛽 as a function of 𝜎 and 𝛾 0 .

7. The Value of Full Information under CRRA

In this section, we consider the case where the agent's utility function in (2.5) is given by π‘ˆ(π‘₯)=π‘₯𝛽/𝛽. This utility function is known as CRRA-utility. The parameter 𝛽 measures how risk averse the agent is. The case 𝛽=1 corresponds to risk neutrality, 𝛽<1 represents a risk averse agent while 𝛽>1 would represent a risk-seeking agent. The case 𝛽=0 formally corresponds to the case of logarithmic utility that we discussed in the previous sections.

It follows from classical stochastic optimal control theory that the value function for the problem of maximizing (2.5) subject to constraints (3.9), (3.10), that is, under partial information,𝐽𝑝(𝑑,π‘š,π‘₯)=max{πœ‹π‘’βˆˆπ’’π‘’}π‘‘β‰€π‘’β‰€π‘‡π”Όξ€Ίπ‘ˆξ€·π‘‹π‘‡ξ€Έβˆ£,π‘šπ‘‘=π‘š,𝑋𝑑=π‘₯(7.1) satisfies the following Hamilton-Jacobi-Bellman equation:0=maxπœ‹ξ‚΅π½π‘π‘‘+π‘Žπ‘šπ½π‘π‘š+ξ€·π‘Ÿπ‘‘+ξ€·π‘šβˆ’π‘Ÿπ‘‘ξ€Έπœ‹ξ€Έπ‘₯𝐽𝑝π‘₯+12𝑏1πœŽπ‘‘+π›Ύπ‘‘πœŽπ‘‘ξ‚Ά2π½π‘π‘šπ‘š+12𝜎2π‘‘πœ‹2π‘₯2𝐽𝑝π‘₯π‘₯+𝑏1πœŽπ‘‘+π›Ύπ‘‘ξ€Έπœ‹π‘₯π½π‘π‘šπ‘₯ξƒͺ,(7.2) where 𝐽𝑝𝑑,π½π‘π‘š,𝐽𝑝π‘₯,π½π‘π‘šπ‘š, and so forth. represent the corresponding first- and second-order derivatives of 𝐽𝑝 and the upper index 𝑝 indicates that this is the value function under partial information. We will later also consider the value function under full information, which we will denote with 𝐽𝑓. If the optimal solution exists, then 𝐽𝑝π‘₯π‘₯<0 and the maximum on the right-hand side of (7.2) is attained atπœ‹βˆ—π‘ξ€·=βˆ’π‘šβˆ’π‘Ÿπ‘‘ξ€Έπ½π‘π‘₯+𝑏1πœŽπ‘‘+π›Ύπ‘‘ξ€Έπ½π‘π‘šπ‘₯𝜎2𝑑π‘₯𝐽𝑝π‘₯π‘₯.(7.3) Substituting (7.3) back into the HJB equation (7.2), we find that the value function under partial information satisfies the following partial differential equation:𝐽𝑝𝑑+π‘Žπ‘šπ½π‘π‘š+π‘Ÿπ‘‘π‘₯𝐽𝑝π‘₯+12𝑏1πœŽπ‘‘+π›Ύπ‘‘πœŽπ‘‘ξ‚Ά2π½π‘π‘šπ‘šβˆ’ξ€·(π‘šβˆ’π‘Ÿπ‘‘)𝐽𝑝π‘₯+(𝑏1πœŽπ‘‘+𝛾𝑑)π½π‘π‘šπ‘₯ξ€Έ22𝜎2𝑑𝐽𝑝π‘₯π‘₯=0,(7.4) with the terminal condition:𝐽𝑝π‘₯(𝑇,π‘š,π‘₯)=𝛽𝛽.(7.5) In the following we will show how to reduce the PDE (7.4) to a system of three ODEs and in this way obtain a semianalytic form for the value function under partial information. In order to do this, let us first consider the reduction𝐽𝑝π‘₯(𝑑,π‘š,π‘₯)=𝐻(𝑑,π‘š)⋅𝛽𝛽.(7.6) Substitution into (7.4) gives𝐻𝑑π‘₯𝛽𝛽+π‘Žπ‘šπ»π‘šπ‘₯𝛽𝛽+π‘Ÿπ‘₯𝛽1𝐻+2𝑏1+π›Ύπ‘‘πœŽξ‚Ά2π»π‘šπ‘šπ‘₯π›½π›½βˆ’ξ€·(π‘šβˆ’π‘Ÿ)𝐻π‘₯π›½βˆ’1+(𝑏1𝜎+𝛽𝑑)π»π‘šπ‘₯π›½βˆ’1ξ€Έ22𝜎2𝐻(π›½βˆ’1)π‘₯π›½βˆ’2=0.(7.7) We see that the common factor π‘₯𝛽/𝛽 cancels out and that (7.7) can be rewritten asπ‘₯𝛽𝛽𝐻𝑑+π‘Žπ‘šπ»π‘š1+π‘Ÿπ›½π»+2𝑏1+π›Ύπ‘‘πœŽξ‚Ά2π»π‘šπ‘šπ‘₯π›½π›½βˆ’ξ€·π›½(π‘šβˆ’π‘Ÿ)𝐻+(𝑏1𝜎+𝛾𝑑)π»π‘šξ€Έ22𝜎2𝐻(π›½βˆ’1)=0.(7.8) Using the notation 𝛼(𝑑) for the deterministic function𝑏𝛼(𝑑)∢=1𝜎+π›Ύπ‘‘πœŽ(7.9) as well as discounting the function 𝐻(𝑑,π‘š) with an appropriate discount factor, that is,𝐻(𝑑,π‘š)∢=π‘’π‘Ÿπ›½π‘‘π»(𝑑,π‘š),(7.10) we obtain from (7.8), (7.6), and (7.5) that𝐻𝑑𝐻+π‘Žπ‘šπ‘š+12𝛼(𝑑)2ξ‚π»π‘šπ‘šξ‚ƒξ‚ξ‚π»βˆ’π›½((π‘šβˆ’π‘Ÿ)/𝜎)𝐻+𝛼(𝑑)π‘šξ‚„2𝐻2(π›½βˆ’1)=0,(7.11)𝐻(𝑇,π‘š)=π‘’π‘Ÿπ›½π‘‡.(7.12) Looking at the PDE (7.11) with boundary condition (7.12) we make the following sophisticated guess: 𝐻(𝑑,π‘š)=𝑒𝐴𝑝(𝑑)π‘š2+𝐡𝑝(𝑑)π‘š+𝐢𝑝(𝑑),(7.13) with 𝐴𝑝(𝑑), 𝐡𝑝(𝑑), and 𝐢𝑝(𝑑) function of 𝑑 only satisfying 𝐴𝑝(𝑇)=𝐡𝑝(𝑇)=0, and 𝐢𝑝(𝑇)=π‘Ÿπ›½π‘‡. The subindex 𝑝 indicates that we deal with the case of partial information. We conclude that 𝐻𝑑𝐻𝐴(𝑑,π‘š)=(𝑑,π‘š)ξ…žπ‘(𝑑)π‘š2+π΅ξ…žπ‘(𝑑)π‘š+πΆξ…žπ‘ξ€Έ,𝐻(𝑑)π‘šξ‚ξ€·(𝑑,π‘š)=𝐻(𝑑,π‘š)2𝐴𝑝(𝑑)π‘š+𝐡𝑝,𝐻(𝑑)π‘šπ‘šξ‚ξ‚ƒξ€·(𝑑,π‘š)=𝐻(𝑑,π‘š)2𝐴𝑝(t)π‘š+𝐡𝑝(𝑑)2ξ‚„,+2𝐴(𝑑)(7.14) and substitution of these expressions into (7.11) gives 𝐻𝐴(𝑑,π‘š)ξ…žπ‘(𝑑)π‘š2+π΅ξ…žπ‘(𝑑)π‘š+πΆξ…žπ‘ξ€Έ+ξ€·π‘Ž(𝑑)1π‘š+π‘Ž0𝐻(𝑑,π‘š)2𝐴𝑝(𝑑)π‘š+𝐡𝑝+𝛼(𝑑)2(𝑑)2𝐻(𝑑,π‘š)2𝐴𝑝(𝑑)π‘š+𝐡𝑝(𝑑)2ξ‚„βˆ’π›½ξ‚ƒξ‚ξ‚ξ€·+2𝐴(𝑑)((π‘šβˆ’π‘Ÿ)/𝜎)𝐻(𝑑,π‘š)+𝛼(𝑑)𝐻(𝑑,π‘š)2𝐴𝑝(𝑑)π‘š+𝐡𝑝(𝑑)22(π›½βˆ’1)𝐻(𝑑,π‘š)=0.(7.15) Rearranging terms according to the order of π‘š gives ξƒ¬π΄ξ…žπ‘(𝑑)+2π‘Ž1𝐴𝑝(𝑑)+2𝛼2(𝑑)𝐴2𝑝𝛽(𝑑)βˆ’2𝛼(𝑑)𝐴𝑝(𝑑)+1/πœŽξƒ­π‘š2(π›½βˆ’1)2+ξƒ¬π΅ξ…žπ‘(𝑑)+π‘Ž1𝐡𝑝(𝑑)+2𝛼2(𝑑)𝐴𝑝(𝑑)𝐡𝑝(𝑑)+2π‘Ž0π΄π‘βˆ’π›½ξ€·(𝑑)2𝛼(𝑑)𝐴𝑝(𝑑)+1/πœŽξ€Έξ€·π›Ό(𝑑)𝐡𝑝(𝑑)βˆ’π‘Ÿ/πœŽξƒ­π‘š+𝐢(π›½βˆ’1)ξ…žπ‘(𝛼𝑑)+2(𝑑)2ξ€·2𝐴𝑝(𝑑)+π΅ξ…žπ‘(𝑑)+π‘Ž0𝐡𝑝(𝛽𝑑)βˆ’π›Ό(𝑑)𝐡𝑝(𝑑)βˆ’π‘Ÿ/𝜎2ξƒ­2(π›½βˆ’1)=0.(7.16) The latter can only be identical zero, if the three brackets are identical zero. This provides us with three first-order ODEs for the functions 𝐴𝑝(𝑑), 𝐡𝑝(𝑑) and 𝐢𝑝(𝑑): π΄ξ…žπ‘π›½(𝑑)=ξ‚€2(π›½βˆ’1)2𝛼(𝑑)𝐴𝑝1(𝑑)+πœŽξ‚2ξ€·π‘Žβˆ’21+𝛼2(𝑑)𝐴𝑝𝐴(𝑑)𝑝𝐡(𝑑),ξ…žπ‘π›½(𝑑)=ξ‚€π›½βˆ’12𝛼(𝑑)𝐴𝑝1(𝑑)+πœŽξ‚ξ‚€π›Ό(𝑑)π΅π‘π‘Ÿ(𝑑)βˆ’πœŽξ‚βˆ’ξ€·π‘Ž1+2𝛼2(𝑑)𝐴𝑝𝐡(𝑑)𝑝(𝑑)βˆ’2π‘Ž0𝐴𝑝𝐢(𝑑),ξ…žπ‘π›½(𝑑)=ξ‚€2(π›½βˆ’1)𝛼(𝑑)π΅π‘π‘Ÿ(𝑑)βˆ’πœŽξ‚2βˆ’π›Ό2(𝑑)2ξ€·2𝐴𝑝(𝑑)+𝐡2𝑝(𝑑)βˆ’π‘Ž0𝐡𝑝(𝑑).(7.17) The equation for 𝐴𝑝(𝑑) is a Riccati ODE, however due to the presence of the function 𝛼(𝑑) it cannot be solved explicitly. The ODEs for 𝐡𝑝(𝑑) and 𝐢𝑝(𝑑) are linear. It would be possible to write down an analytic expression for their solutions, however they would depend on an integral of 𝐴𝑝(𝑑), which one would still need to compute numerically. The errors due to the discretization of the integral and the numerical error in 𝐴𝑝(𝑑) when computing this integral are of about the same order as when applying a standard Runge-Kutta scheme right from the beginning, which is what we did in our numerical analysis.

Let us now move to the case of full information. In this case the agent maximizes 𝔼{π‘₯𝛽(𝑇)/𝛽} under the constraint (2.6). We denote the value function of the corresponding stochastic optimal control problem with 𝐽𝑓(𝑑,πœ‡,π‘₯)=max{πœ‹π‘’βˆˆβ„±π‘’}π‘‘β‰€π‘’β‰€π‘‡π”Όξ€Ίπ‘ˆξ€·π‘‹π‘‡ξ€Έβˆ£,πœ‡π‘‘=πœ‡,𝑋𝑑=π‘₯.(7.18) Note that for the case 𝑑=0 this is the value function of the informed agent, once he knows the realization of πœ‡0, which by Lemma 3.2. is normal distributed. We pick up on this point later.

The HJB equation for this problem is given by0=maxπœ‹ξƒ―π½π‘“π‘‘+𝜎2π‘₯2πœ‹22𝐽𝑓π‘₯π‘₯+[]π‘Ÿ+(πœ‡βˆ’π‘Ÿ)πœ‹π‘₯𝐽𝑓π‘₯+ξ€·π‘Ž1πœ‡+π‘Ž0ξ€Έπ½π‘“πœ‡+𝑏21+𝑏222π½π‘“πœ‡πœ‡+𝑏1πœŽπœ‹π‘₯π½π‘“πœ‡π‘₯ξƒ°.(7.19) Assuming that 𝐽𝑓π‘₯π‘₯ is negative, the maximizer is given by πœ‹βˆ—π‘“=βˆ’(πœ‡βˆ’π‘Ÿ)𝐽𝑓π‘₯(𝑑,πœ‡,π‘₯)+𝑏1πœŽπ½π‘“πœ‡π‘₯(𝑑,πœ‡,π‘₯)𝜎2π‘₯𝐽𝑓π‘₯π‘₯.(𝑑,πœ‡,π‘₯)(7.20) Note that as one would expect, the maximizer formally coincides with the maximizer under partial information (7.3) in the case that 𝛾(𝑑)=0, replacing π‘š(𝑑) by πœ‡(𝑑). Substituting πœ‹βˆ—π‘“ in (7.19) implies𝐽𝑓𝑑+π‘Ÿπ‘₯𝐽𝑓π‘₯+ξ€·π‘Ž1πœ‡+π‘Ž0ξ€Έπ½π‘“πœ‡+𝑏21+𝑏222π½π‘“πœ‡πœ‡βˆ’ξ‚ƒ(πœ‡βˆ’π‘Ÿ)𝐽𝑓π‘₯+𝑏1πœŽπ½π‘“πœ‡π‘₯ξ‚„22𝜎2𝐽𝑓π‘₯π‘₯=0(7.21) with terminal condition𝐽𝑓π‘₯(𝑇,πœ‡,π‘₯)=𝛽𝛽.(7.22) Inspired by the analysis of the case 𝐽𝑝(𝑑,π‘š,π‘₯), we make the sophisticated guess:𝐽𝑓(𝑑,πœ‡,π‘₯)=𝑒𝐴𝑓(𝑑)πœ‡2+𝐡𝑓(𝑑)πœ‡+𝐢𝑓(𝑑)π‘₯𝛽𝛽.(7.23) Substitution into (7.22) leads in analogy to the analysis before to the following system of three ODEs: π΄ξ…žπ‘“(𝑑)=βˆ’2π‘Ž1𝐴𝑓̃(𝑑)βˆ’4𝑏𝐴2𝑓𝛽(𝑑)+2𝜎2(ξ€·π›½βˆ’1)2𝑏1πœŽπ΄π‘“ξ€Έ(𝑑)+12,π΅ξ…žπ‘“(𝑑)=βˆ’π‘Ž1𝐡𝑓(𝑑)βˆ’2π‘Ž0𝐴𝑓̃(𝑑)βˆ’4𝑏𝐴𝑓(𝑑)𝐡𝑓𝛽(𝑑)+𝜎2ξ€·(π›½βˆ’1)2𝑏1πœŽπ΄π‘“π‘(𝑑)+1ξ€Έξ€·1πœŽπ΅π‘“ξ€Έ,𝐢(𝑑)βˆ’π‘Ÿξ…žπ‘“(𝑑)=βˆ’π‘Ÿπ›½βˆ’π‘Ž0𝐡𝑓̃𝑏(𝑑)βˆ’2𝐴𝑓(𝑑)+𝐡2𝑓+𝛽(𝑑)2𝜎2𝑏(π›½βˆ’1)1πœŽπ΅π‘“ξ€Έ(𝑑)βˆ’π‘Ÿ2,(7.24) where ̃𝑏=(𝑏21+𝑏22)/2 and 𝐴𝑓(𝑇)=𝐡𝑓(𝑇)=𝐢𝑓(𝑇)=0. In summary, the maximizers for the case of full and partial information are therefore given byπœ‹βˆ—π‘“(𝑑,πœ‡)=βˆ’πœ‡βˆ’π‘Ÿ+𝑏1𝜎2ξ€·2𝐴𝑓(𝑑)πœ‡+𝐡𝑓(𝑑)𝜎2(,πœ‹π›½βˆ’1)βˆ—π‘ξ€·π‘(𝑑,π‘š)=βˆ’π‘šβˆ’π‘Ÿ+1𝜎+𝛾(𝑑)ξ€Έξ€·2𝐴𝑝(𝑑)π‘š+𝐡𝑝(𝑑)𝜎2,(π›½βˆ’1)(7.25) and the corresponding value functions are given by𝐽𝑓(𝑑,πœ‡,π‘₯)=𝑒𝐴𝑓(𝑑)πœ‡2+𝐡𝑓(𝑑)πœ‡+𝐢𝑓(𝑑)ξ‚΅π‘₯𝛽𝛽𝐽,(7.26)𝑝(𝑑,π‘š,π‘₯)=𝑒𝐴𝑝(𝑑)π‘š2+𝐡𝑝(𝑑)π‘š+𝐢𝑝(𝑑)βˆ’π‘Ÿπ›½π‘‘ξ‚΅π‘₯𝛽𝛽.(7.27) We can clearly see from (7.25) how the newly derived portfolio rules adjust the Merton [16] rule for the stochastic drift term under full and partial information. Note that like in the classical Merton problem, the optimal investment strategies do not depend on the level of wealth π‘₯, which is why this variable has not been included in the notation in (7.25).

In order to determine the value of information we now need to compute the difference between the two value functions of full information (7.26) and partial information (7.27). In order to do this, we need to realize one thing. While the initial condition π‘š0 for the partial information case is deterministic, the initial condition πœ‡0 for the full information case is by assumption a normal distributed random variable, whose realization the fully informed agent observes, and after that behaves optimally conditional on πœ‡0. According to Lemma 3.2, we know that πœ‡0βˆΌπ’©(π‘š0,𝛾0). In average the informed agents optimal utility is therefore given by 𝔼{𝐽𝑓(0,πœ‡0,π‘₯)}, where the expectation is taken over πœ‡0. Knowing the semiexplicit form of the value function 𝐽𝑓 in (7.26) as well as the density function of the normal distribution, it is not to difficult to carry out this integration. The result can be computed as follows: 𝔼𝑒𝐴𝑓(0)πœ‡2+𝐡𝑓(0)πœ‡+𝐢𝑓(0)=ξ€œβˆžβˆ’βˆž1√2πœ‹π›Ύ0𝑒𝐴𝑓(0)π‘₯2+𝐡𝑓(0)π‘₯+𝐢𝑓(0)π‘’βˆ’((π‘₯βˆ’π‘š0)2/2𝛾0)=1𝑑π‘₯√1βˆ’2𝐴𝑓(0)𝛾0𝑒(4𝐴𝑓(0)𝐢𝑓(0)𝛾0βˆ’2𝐢𝑓(0)βˆ’2𝐴𝑓(0)π‘š20βˆ’π΅2𝑓(0)𝛾0βˆ’2𝐡𝑓(0)π‘š0)/2(2𝐴𝑓(0)𝛾0βˆ’1),(7.28) if 1>2𝐴𝑓(0)𝛾0 and ∞ otherwise. In this way, the two value functions are now comparable and we obtain the value of information for the CRRA case: 𝒱𝛽𝐽=𝔼𝑓0,πœ‡0,π‘₯ξ€Έξ€Ύβˆ’π½π‘ξ€·π‘‘,π‘š0ξ€Έ.,π‘₯(7.29) Note that the value of information is infinite, if 1≀2𝐴𝑓(0)𝛾0. The latter however could not be observed for realistic parameters.

In the numerical examples presented in the appendix we assume that π‘Ÿ=0.08, π‘Ž1∈[βˆ’1,1], π‘Ž0=0.1, 𝑏1∈[βˆ’0.5,1.2], 𝑏2∈[βˆ’0.3,0.3], π›½βˆˆ[0.1,0.4], 𝜎∈[0.2,0.4], 𝛾0∈[0,0.2], 𝑇=0.25, π‘š0=0.2 and π‘₯=0.4. We observe that the value of information is increasing in the volatility parameter 𝜎 as well as in the risk aversion parameter 𝛽. This means that more risk averse agents value information higher than less risk averse agents, which makes sense from an intuitive point of view. The value of information is further increasing in 𝛾0 which is the initial uncertainty in πœ‡0.

8. Conclusions

We have studied the value of full information in a financial market model with partial information, where the drift rate of the risky asset is assumed to be an unobservable dynamic process, which we model as the solution of a linear stochastic differential equation with constant coefficients. We derived an analytic formula for the value of information taking logarithmic utility from terminal wealth as an objective, and a semianalytical formula for the case of CRRA-utility from terminal wealth. We performed a detailed comparative statics and singled out the various effects the model parameters have on the value of information.

Appendices

A. Logarithmic Utility

(See Figures 1, 2, and 3.)

B. CRRA Utility

(See Figures 4, 5, and 6.)

Acknowledgments

This research is supported by the national Natural Science Foundation of China (70971037) and the Rheinland Pfalz Excellence cluster DASMOD. Z. Yang would like to thank Professor Jie Xiong from the University of Tennessee for many helpful comments. All authors would like to thank two anonymous referees for their valuable feedback and suggestions.