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Journal of Probability and Statistics
Volume 2011, Article ID 238623, 23 pages
http://dx.doi.org/10.1155/2011/238623
Research Article

A Comparative Analysis of the Value of Information in a Continuous Time Market Model with Partial Information: The Cases of Log-Utility and CRRA

1School of Finance and Statistics, Hunan University, Changsha 410079, China
2School of Mathematics and Statistics, University of Sydney, Sydney NSW 2006, Australia

Received 16 March 2010; Accepted 12 May 2010

Academic Editor: Tak Kuen Siu

Copyright © 2011 Zhaojun Yang et al. 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.

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 Büttler [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 (Ω,,). The prices of the two types of assets (risky and risk free) are denoted by (𝐵𝑡,𝑆𝑡) and are governed by the following SDEs:𝑑𝐵𝑡=𝑟𝑡𝐵𝑡𝑑𝑡,𝑑𝑆𝑡=𝜇𝑡𝑆𝑡𝑑𝑡+𝜎𝑡𝑆𝑡𝑑𝑊1𝑡,0𝑡𝑇,(2.1) where the interest rate 𝑟𝑡 and the volatility 𝜎𝑡 are deterministic processes, 𝑊1 is a Brownian motion, and the appreciation rate 𝜇𝑡 is a stochastic process satisfying the following SDE:𝑑𝜇𝑡=𝑎1𝜇𝑡+𝑎0𝑑𝑡+𝑏1𝑑𝑊1𝑡+𝑏2𝑑𝑊2𝑡,0𝑡𝑇,(2.2) with 𝑎1,𝑎0,𝑏1,𝑏2 constants and 𝑊2 a Brownian motion which is independent of 𝑊1. The filtrations ={𝑡}0𝑡𝑇 and 𝒢={𝒢𝑡}0𝑡𝑇 with𝑡𝑊=𝜎1𝑠,𝑊2𝑠0𝑠𝑡,𝒢𝑡𝑆=𝜎𝑠0𝑠𝑡(2.3) represent the two different levels of information in our model. While the filtration contains the information produced by the noise generating process the filtration 𝒢 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 as full information and to 𝒢 as partial information. Classically it is assumed that the investors have access to the full information flow {𝑡}0𝑡𝑇. 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 {𝒢𝑡}0𝑡𝑇 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 𝑡[0,𝑇], she invests a proportion 𝜋𝑡 of her wealth in the risky asset, and invests the rest in the risk-free asset. Then her wealth process 𝑋={𝑋𝑡}0𝑡𝑇 satisfies: for all 0𝑡𝑇,𝑑𝑋𝑡=𝜋𝑡𝑋𝑡𝑆𝑡𝑑𝑆𝑡+1𝜋𝑡𝑋𝑡𝐵𝑡𝑑𝐵𝑡=𝑟𝑡+𝜋𝑡𝜇𝑡𝑟𝑡𝑋𝑡𝑑𝑡+𝜎𝑡𝜋𝑡𝑋𝑡𝑑𝑊1𝑡,(2.4) where the process {𝜋𝑡}0𝑡𝑇 must be 𝒢-adapted. In summary, the optimal portfolio problem is described as follows:max{𝜋𝑡}0𝑡𝑇𝔼𝑈𝑋𝑇(2.5) subject to the constraints:𝑑𝜇𝑡=𝑎1𝜇𝑡+𝑎0𝑑𝑡+𝑏1𝑑𝑊1𝑡+𝑏2𝑑𝑊2𝑡,𝑑𝑋𝑡=𝑟𝑡+𝜋𝑡𝜇𝑡𝑟𝑡𝑋𝑡𝑑𝑡+𝜎𝑡𝜋𝑡𝑋𝑡𝑑𝑊1𝑡,0𝑡𝑇,(2.6) and {𝜋𝑡}𝒢-adapted. {𝑆𝑡}0𝑡𝑇 and therefore also {𝑋𝑡}0𝑡𝑇 can be observed, but {𝜇𝑡}0𝑡𝑇 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 𝑈 is chosen to be logarithmic, that is, 𝑈(𝑥)=log(𝑥); later we consider the case of CRRA utility, that is, 𝑈(𝑥)=𝑥𝛽/𝛽.

3. Filtering Estimation for the Average Appreciation Rate

In this section, we apply the filtering technique to estimate the appreciation rate {𝜇𝑡}0𝑡𝑇 given the information flow 𝒢. Denote𝑚𝑡𝜇=𝔼𝑡𝒢𝑡,𝛾𝑡𝜇=𝔼𝑡𝑚𝑡2𝒢𝑡.(3.1)

Note that 𝑚0=𝔼(𝜇0) and 𝛾0=Var(𝜇0). The following lemma follows from Theorem 11.1 by Liptser and Shiryayev [17].

Lemma 3.1. If the conditional distribution 𝐹𝒢0𝜇(𝑥)=0𝑥𝒢0(3.2) is normal with mean 𝑚0 and variance 𝛾0, a.s., then the conditional distribution 𝐹𝒢𝑡𝜇(𝑥)=𝑡𝑥𝒢𝑡(3.3) is normal a.s. with mean with 𝑚𝑡 and variance with 𝛾𝑡.

As a normally distributed random variable is uniquely determined by its expectation and variance we conclude from this lemma that the knowledge of 𝑚𝑡 and 𝛾𝑡 reflects all the information on 𝜇𝑡 the agent is able to obtain under the partial information 𝒢𝑡.

Lemma 3.2. Let {𝜇𝑡,𝑆𝑡}0𝑡𝑇 be the stochastic processes with differentials given by (2.2) and (2.1). Suppose that (𝜇0𝑥𝒢0) is Gaussian with mean 𝑚0 and variance 𝛾0. Further, one assume that there exist two constants 0<𝑐1<𝑐2< such that for all 𝑡[0,𝑇], one has 𝑐1𝜎𝑡𝑐2. Then 𝑚𝑡 and 𝛾𝑡 satisfy the following equations: 𝑑𝑚𝑡=𝑎1𝑚𝑡+𝑎0𝑏𝑑𝑡+1𝜎𝑡+𝛾𝑡𝜎2𝑡𝑑𝑆𝑡𝑆𝑡𝑚𝑡,𝑑𝑡̇𝛾𝑡=2𝑎1𝛾𝑡+𝑏21+𝑏22𝑏1𝜎𝑡+𝛾𝑡𝜎𝑡2,0𝑡𝑇,(3.4) where ̇𝛾𝑡 denotes the deterministic time-derivative of 𝛾𝑡.

Proof. Applying Itô's formula to (2.1), we have 𝑑log𝑆𝑡=𝜇𝑡12𝜎2𝑡𝑑𝑡+𝜎𝑡𝑑𝑊1𝑡.(3.5) It is then easy to see that the conditions of Theorem 12.1 by Liptser and Shiryayev [17] hold for (𝜇,log𝑆). The conclusion of this lemma then follows from that theorem.
In the case of the classic Black-Scholes model where 𝜎𝑡=𝜎 is constant, it follows by using the technique of separation of variables, that 𝛾𝑡 is explicitly given by the following expression:𝛾𝑡=𝛾+2Λ2+𝑏22𝜎2𝛾10𝛾+𝛾0𝛾2expΛ2+𝑏22𝜎2𝜎2𝑡1ifΛ2+𝑏2210,𝛾0+𝑡𝜎21ifΛ=𝑏2=0,(3.6) where Λ=𝑎1𝜎2𝑏1𝜎,𝛾±=Λ±Λ2+𝑏22𝜎2.(3.7)

Remark 3.3. (1) The magnitude of 𝛾𝑡 is a characterization of the accuracy of the estimate 𝑚𝑡 of 𝜇𝑡. If 𝛾𝑡 converges to zero as 𝑡, then 𝑚𝑡 is called a consistent estimate of 𝜇𝑡.
(2) It is obvious that 𝛾𝑡𝛾+ as 𝑡. Note that 𝛾+=0 if and only if 𝑎1=𝑏1=𝑏2=0. Therefore, 𝑚𝑡 is consistent if and only if the appreciation rate is constant in time, that is, 𝜇𝑡𝜇0.
(3) The points 𝛾+ and 𝛾 are stationary points of the dynamics of 𝛾𝑡. As indicated in 2. 𝛾+ is globally stable while 𝛾 is nonstable. In fact 𝛾0 and as 𝛾0>0 it is easy to see that the trajectory of 𝛾𝑡 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:𝑑𝑊𝑡=1𝜎𝑡𝑑𝑆𝑡𝑆𝑡𝑚𝑡𝑑𝑡.(3.8) It is well-known from the theory of filtering that 𝑊={𝑊𝑡}0𝑡𝑇 is a Brownian motion with respect to the stochastic basis (Ω,𝒢𝑇,,{𝒢𝑡}0𝑡𝑇). Therefore, the self-financing condition (2.4) can be rewritten as𝑑𝑋𝑡=𝑟𝑡𝑋𝑡+𝑚𝑡𝑟𝑡𝜋𝑡𝑋𝑡𝑑𝑡+𝜎𝑡𝜋𝑡𝑋𝑡𝑑𝑊𝑡.(3.9) Similarly, combining (3.8) and (3.4), we obtain that𝑑𝑚𝑡=𝑎1𝑚𝑡+𝑎0𝑏𝑑𝑡+1𝜎𝑡+𝛾𝑡𝜎𝑡𝑑𝑊𝑡.(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 Itô formula and (3.9) that log𝑋𝑇=log𝑋0+𝑇0𝑟𝑡+𝑚𝑡𝑟𝑡𝜋𝑡12𝜎2𝑡𝜋2𝑡𝑑𝑡+𝑇0𝜎𝑡𝜋𝑡𝑑𝑊𝑡.(4.1) Hence𝔼log𝑋𝑇=log𝑋0+𝑇0𝔼𝑟𝑡+𝑚𝑡𝑟𝑡𝜋𝑡12𝜎2𝑡𝜋2𝑡𝑑𝑡,(4.2) whose maximum is attained at𝜋𝑡=𝑚𝑡𝑟𝑡𝜎2𝑡[].,𝑡0,𝑇(4.3) The optimal value under partial information is then given by𝑉𝑃=log𝑋0+𝑇0𝑟𝑡+𝔼𝑚𝑡𝑟𝑡22𝜎2𝑡𝑑𝑡.(4.4) From (3.10) we obtain that𝑚𝑡=𝑚0𝑒𝑎1𝑡+𝑎0𝑎1𝑒𝑎1𝑡+1𝑡0𝑒𝑎1(𝑡𝑠)𝑏1+𝛾𝑠𝜎𝑠𝑑𝑊𝑠,(4.5) where in the case that 𝑎1=0 the expression (𝑎0/𝑎1)(𝑒𝑎1𝑡1) needs to be replaced by 𝑎0𝑡. We then obtain that𝔼𝑚𝑡𝑟𝑡2=𝑚0𝑒𝑎1𝑡+𝑎0𝑎1𝑒𝑎1𝑡1𝑟𝑡2+𝑡0𝑒2𝑎1(𝑡𝑠)𝑏1+𝛾𝑠𝜎𝑠2𝑑𝑠.(4.6) Combining (4.4) and (4.6), we have𝑉𝑃=log𝑋0+𝑇0𝑟𝑡𝑑𝑡+𝑇0𝑚0𝑒𝑎1t+𝑎0/𝑎1𝑒𝑎1𝑡1𝑟𝑡22𝜎2𝑡+𝑑𝑡𝑇0𝑡0𝑒2𝑎1(𝑡𝑠)2𝜎2𝑡𝑏1+𝛾𝑠𝜎𝑠2𝑑𝑠𝑑𝑡.(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 𝑡[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: 𝜋𝑡=𝑚𝑡𝑟𝑡𝜎2𝑡.(4.8) where 𝑚𝑡 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 𝑚𝑡 is obtained. Even though we derived a more explicit expression for 𝑚𝑡 in (4.6), the latter expression is not directly useful for the agent, as he does not observe the Brownian 𝑊 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 𝜇𝑡𝜇0 is a normal random variable with mean 𝑚0 and variance 𝛾0. Namely, 𝑎1=𝑎0=𝑏1=𝑏2=0. Further, we take 𝑟𝑡=𝑟 and 𝜎𝑡=𝜎 to be constants. In this case 𝛾𝑡=1𝛾0+𝑡𝜎21,(4.9) and hence 𝑉𝑃=log𝑋0𝑚+𝑟𝑇+0𝑟2𝑇2𝜎212𝛾log1+0𝑇𝜎2+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 Itô's formula, we have log𝑋𝑇=log𝑋0+𝑇0𝑟𝑡+𝜇𝑡𝑟𝑡𝜋𝑡12𝜎2𝑡𝜋2𝑡𝑑𝑡+𝑇0𝜎𝑡𝜋𝑡𝑑𝑊1𝑡.(5.1) Hence𝔼log𝑋𝑇=log𝑋0+𝑇0𝔼𝑟𝑡+𝜇𝑡𝑟𝑡𝜋𝑡12𝜎2𝑡𝜋2𝑡𝑑𝑡,(5.2) whose maximum is attained at𝜋𝑡=𝜇𝑡𝑟𝑡𝜎2𝑡.(5.3) The value function is then given by𝑉𝐹=log𝑋0+𝑇0𝑟𝑡+𝔼𝜇𝑡𝑟𝑡22𝜎2𝑡𝑑𝑡.(5.4) By (2.6), we obtain that𝜇𝑡=𝜇0𝑒𝑎1𝑡+𝑎0𝑎1𝑒𝑎1𝑡+1𝑡0𝑒𝑎1(𝑡𝑠)𝑏1𝑑𝑊1𝑠+𝑏2𝑑𝑊2𝑠.(5.5) It then follows from 𝔼(𝜇0)=𝑚0 that𝔼𝜇𝑡𝑟𝑡2=𝑒2𝑎1𝑡𝛾0+𝑚0𝑒𝑎1𝑡+𝑎0𝑎1𝑒𝑎1𝑡1𝑟𝑡2+𝑏21+𝑏22𝑒2𝑎2𝑎1𝑡.1(5.6) Combining (5.4) and (5.6), we see that𝜋𝑡=𝜇𝑡𝑟𝑡𝜎2𝑡,(5.7) and the optimal obtainable expected utility is given by𝑉𝐹=log𝑋0+𝑇0𝑟𝑡𝑑𝑡+𝑇012𝜎2𝑡𝑒2𝑎1𝑡𝛾0+𝑚0𝑒𝑎𝑡+𝑎0𝑎1e𝑎1𝑡1𝑟𝑡2+𝑏21+𝑏22𝑒2𝑎2𝑎1𝑡1𝑑𝑡(5.8) with (𝑎0/𝑎1)(𝑒𝑎1𝑡1) replaced by 𝑎0𝑡 in the case that 𝑎1=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 𝒱=𝑇012𝜎2𝑡𝑒2𝑎1𝑡𝛾0+𝑏21+𝑏222𝑎1𝑒2𝑎1𝑡1𝑡0𝑒2𝑎1(𝑡𝑠)𝑏1𝛾𝑠𝜎𝑠2𝑑𝑠𝑑𝑡.(5.9)

Proof. As indicated above we simply have to compute the value 𝑉𝐹𝑉𝑃. 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 𝑎0 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=2𝛾log1+0𝑇𝜎2.(5.10)

6. Comparative Statics and Numerical Analysis: Log-Utility

In this section we restrict our analysis to the case where 𝜎𝑡𝜎 is constant and investigate the effect changes of the model parameters 𝑎1, 𝑎0, 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 𝛾𝑡=var(𝜇𝑡𝒢𝑡). We have already indicated in the previous section that the constant term 𝑎0 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, 𝜇𝑡, respectively, 𝑚𝑡 in the case of 𝑎1, 𝑑𝑊1 in the case of 𝑏1 and 𝜎1 and 𝑑𝑊2 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 𝑎10. It then follows directly from the valuation formula in Theorem 5.1. that𝜕𝒱𝜕𝑏2=𝑏2𝑎1𝑇0𝑒2𝑎1𝑡1𝑑𝑡.(6.1) We have (1/𝑎1)(𝑒2𝑎1𝑡1)>0 for all choices of 𝑎1 and positive 𝑡 and therefore our assertion holds. In the case that 𝑎1=0 we obtain by taking the limit for 𝑎10 in (6.1) that𝜕𝒱𝜕𝑏2=𝑏2𝑇02𝑡𝑑𝑡=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 𝜇𝑡 and decreases the covariation between asset and unobserved noise 𝑏2𝑑𝑊2. 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 𝜇𝑡. However, an increase in 𝑏1 also leads to an increase rather than a decrease of the covariation between 𝜇𝑡 and the risky asset 𝑆𝑡. 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 that𝜕𝒱𝜕𝑏1=2𝑏1𝑇0𝑡0𝑒2𝑎1(𝑡𝑠)𝑏11𝛾𝑠𝜎2𝑑𝑠𝑑𝑡+2𝑏21𝑇0𝑡0𝑒2𝑎1(𝑡𝑠)𝑏1𝛾𝑠𝜎𝑑𝑠𝑑𝑡.(6.3) Since 𝛾𝑠[0,𝛾+] for all 𝑡[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+=lim𝑡𝛾𝑡 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 have𝜕𝒱𝛾𝜕𝜎=02𝑎1𝜎30𝑒2𝑎1𝑇𝑏1+221+𝑏22𝑇0𝑡0𝑒2𝑎1(𝑡𝑠)𝑏1𝛾𝑠𝜎𝛾𝑠𝜎2𝑑𝑠𝑑𝑡,(6.5) where the first summand needs to be replaced by 𝛾0𝑇/𝜎30 in the case that 𝑎1=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 𝜕𝒱/𝜕𝜎 is positive. In the case that 𝑏1min(𝛾0,𝛾+)/𝜎 we see that both summands in (6.5) are negative and hence in this case 𝜕𝒱/𝜕𝜎<0. Similarly it can be seen that if condition (6.4) holds we have 𝜕𝒱/𝜕𝑇>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 𝜇0 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 16 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 𝑏𝑎𝑑(𝑥)𝑐(𝑥)𝑓(𝑥,𝑦)𝑑𝑦𝑑𝑥.(6.6) Let the step size for x be =(𝑏𝑎)/2 and let the step size for 𝑦 be 𝑘(𝑥)=(𝑑(𝑥)𝑐(𝑥))/2. Adapting the classical Simpson method iteratively, we obtain𝑏𝑎𝑑(𝑥)𝑐(𝑥)𝑓(𝑥,𝑦)𝑑𝑦𝑑𝑥𝑏𝑎𝑘(𝑥)3[]𝑓(𝑥,𝑐(𝑥))+4𝑓(𝑥,𝑐(𝑥)+𝑘(𝑥))+𝑓(𝑥,𝑑(𝑥))𝑑𝑥3𝑘(𝑎)3[]+𝑓(𝑎,𝑐(𝑎))+4𝑓(𝑎,𝑐(𝑎)+𝑘(𝑎))+𝑓(𝑎,𝑑(𝑎))4𝑘(𝑎+)3[]+𝑓(𝑎+,𝑐(𝑎+))+4𝑓(𝑎+,𝑐(𝑎+)+𝑘(𝑎+))+𝑓(𝑎+,𝑑(𝑎+))𝑘(𝑏)3[].𝑓(𝑏,𝑐(𝑏))+4𝑓(𝑏,𝑐(𝑏)+𝑘(𝑏))+𝑓(𝑏,𝑑(𝑏))(6.7) In Theorem 5.1, we need to evaluate 𝒱=𝑇012𝜎2𝑡𝑒2𝑎1𝑡𝛾0+𝑏21+𝑏222𝑎1𝑒2𝑎1𝑡1𝑡0𝑒2𝑎1(𝑡𝑠)𝑏1𝛾𝑠𝜎2𝑑𝑠𝑑𝑡.(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 Δ𝑡=𝑇/𝑛. Then𝑡𝑖=𝑡𝑖1+Δ𝑡,for𝑖=1,,𝑛,𝑡0=0.(6.9) We evaluate the first part by applying the standard Simpson method. Let =Δ𝑡/2,𝜎𝑡=𝜎 be a constant, and 1𝑓(𝑡)=2𝜎2𝑒2𝑎1𝑡𝛾0+𝑏21+𝑏222𝑎1𝑒2𝑎1𝑡1.(6.10) We obtain𝑛1𝑖=0𝑡𝑖+1𝑡𝑖12𝜎2𝑒2𝑎1𝑡𝛾0+𝑏21+𝑏222𝑎1𝑒2𝑎1𝑡1𝑑𝑡𝑛1𝑖=03𝑓𝑡𝑖𝑡+4𝑓𝑖𝑡++𝑓𝑖+1.(6.11) To evaluate the double integral, we let𝑒𝑔(𝑠,𝑡)=2𝑎1(𝑡𝑠)𝑏1𝛾𝑠/𝜎22𝜎2.(6.12) By applying our adapted Simpson method (6.7) and choosing 𝑘(𝑡)=𝑡02=𝑡2,(6.13) we obtain the following approximate integration formula: 𝑛1𝑖=0𝑡𝑖+1𝑡𝑖𝑡0𝑔(𝑠,𝑡)𝑑𝑠𝑑𝑡𝑛1𝑖=0𝑡𝑖+1𝑡𝑖𝑡6𝑡𝑔(0,𝑡)+4𝑔2,𝑡+𝑔(𝑡,𝑡)𝑑𝑡𝑛1𝑖=03𝑡𝑖6𝑔0,𝑡𝑖𝑡+4𝑔𝑖2,𝑡𝑖𝑡+𝑔𝑖,𝑡𝑖+2𝑡𝑖+3𝑔0,𝑡𝑖𝑡++4𝑔𝑖+2,𝑡𝑖𝑡++𝑔𝑖+,𝑡𝑖+𝑡+𝑖+16𝑔0,𝑡𝑖+1𝑡+4𝑔𝑖+12,𝑡𝑖+1𝑡+𝑔𝑖+1,𝑡𝑖+1.(6.14)

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Figure 1: 𝒱 as a function of 𝜎 and 𝑎1.
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Figure 2: 𝒱 as a function of 𝜎 and 𝑏1.
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Figure 3: 𝒱 as a function of 𝜎 and 𝑏2.
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Figure 4: 𝒱𝛽 as a function of 𝑎1 and 𝛽.
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Figure 5: 𝒱𝛽 as a function of 𝑏1 and 𝑏2.
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Figure 6: 𝒱𝛽 as a function of 𝜎 and 𝛾0.

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 𝑈(𝑥)=𝑥𝛽/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(𝑡,𝑚,𝑥)=max{𝜋𝑢𝒢𝑢}𝑡𝑢𝑇𝔼𝑈𝑋𝑇,𝑚𝑡=𝑚,𝑋𝑡=𝑥(7.1) satisfies the following Hamilton-Jacobi-Bellman equation:0=max𝜋𝐽𝑝𝑡+𝑎𝑚𝐽𝑝𝑚+𝑟𝑡+𝑚𝑟𝑡𝜋𝑥𝐽𝑝𝑥+12𝑏1𝜎𝑡+𝛾𝑡𝜎𝑡2𝐽𝑝𝑚𝑚+12𝜎2𝑡𝜋2𝑥2𝐽𝑝𝑥𝑥+𝑏1𝜎𝑡+𝛾𝑡𝜋𝑥𝐽𝑝𝑚𝑥,(7.2) where 𝐽𝑝𝑡,𝐽𝑝𝑚,𝐽𝑝𝑥,𝐽𝑝𝑚𝑚, 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 𝐽𝑝𝑥𝑥<0 and the maximum on the right-hand side of (7.2) is attained at𝜋𝑝=𝑚𝑟𝑡𝐽𝑝𝑥+𝑏1𝜎𝑡+𝛾𝑡𝐽𝑝𝑚𝑥𝜎2𝑡𝑥𝐽𝑝𝑥𝑥.(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:𝐽𝑝𝑡+𝑎𝑚𝐽𝑝𝑚+𝑟𝑡𝑥𝐽𝑝𝑥+12𝑏1𝜎𝑡+𝛾𝑡𝜎𝑡2𝐽𝑝𝑚𝑚(𝑚𝑟𝑡)𝐽𝑝𝑥+(𝑏1𝜎𝑡+𝛾𝑡)𝐽𝑝𝑚𝑥22𝜎2𝑡𝐽𝑝𝑥𝑥=0,(7.4) with the terminal condition:𝐽𝑝𝑥(𝑇,𝑚,𝑥)=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 reduction𝐽𝑝𝑥(𝑡,𝑚,𝑥)=𝐻(𝑡,𝑚)𝛽𝛽.(7.6) Substitution into (7.4) gives𝐻𝑡𝑥𝛽𝛽+𝑎𝑚𝐻𝑚𝑥𝛽𝛽+𝑟𝑥𝛽1𝐻+2𝑏1+𝛾𝑡𝜎2𝐻𝑚𝑚𝑥𝛽𝛽(𝑚𝑟)𝐻𝑥𝛽1+(𝑏1𝜎+𝛽𝑡)𝐻𝑚𝑥𝛽122𝜎2𝐻(𝛽1)𝑥𝛽2=0.(7.7) We see that the common factor 𝑥𝛽/𝛽 cancels out and that (7.7) can be rewritten as𝑥𝛽𝛽𝐻𝑡+𝑎𝑚𝐻𝑚1+𝑟𝛽𝐻+2𝑏1+𝛾𝑡𝜎2𝐻𝑚𝑚𝑥𝛽𝛽𝛽(𝑚𝑟)𝐻+(𝑏1𝜎+𝛾𝑡)𝐻𝑚22𝜎2𝐻(𝛽1)=0.(7.8) Using the notation 𝛼(𝑡) for the deterministic function𝑏𝛼(𝑡)=1𝜎+𝛾𝑡𝜎(7.9) as well as discounting the function 𝐻(𝑡,𝑚) with an appropriate discount factor, that is,𝐻(𝑡,𝑚)=𝑒𝑟𝛽𝑡𝐻(𝑡,𝑚),(7.10) we obtain from (7.8), (7.6), and (7.5) that𝐻𝑡𝐻+𝑎𝑚𝑚+12𝛼(𝑡)2𝐻𝑚𝑚𝐻𝛽((𝑚𝑟)/𝜎)𝐻+𝛼(𝑡)𝑚2𝐻2(𝛽1)=0,(7.11)𝐻(𝑇,𝑚)=𝑒𝑟𝛽𝑇.(7.12) Looking at the PDE (7.11) with boundary condition (7.12) we make the following sophisticated guess: 𝐻(𝑡,𝑚)=𝑒𝐴𝑝(𝑡)𝑚2+𝐵𝑝(𝑡)𝑚+𝐶𝑝(𝑡),(7.13) with 𝐴𝑝(𝑡), 𝐵𝑝(𝑡), and 𝐶𝑝(𝑡) function of 𝑡 only satisfying 𝐴𝑝(𝑇)=𝐵𝑝(𝑇)=0, and 𝐶𝑝(𝑇)=𝑟𝛽𝑇. The subindex 𝑝 indicates that we deal with the case of partial information. We conclude that 𝐻𝑡𝐻𝐴(𝑡,𝑚)=(𝑡,𝑚)𝑝(𝑡)𝑚2+𝐵𝑝(𝑡)𝑚+𝐶𝑝,𝐻(𝑡)𝑚(𝑡,𝑚)=𝐻(𝑡,𝑚)2𝐴𝑝(𝑡)𝑚+𝐵𝑝,𝐻(𝑡)𝑚𝑚(𝑡,𝑚)=𝐻(𝑡,𝑚)2𝐴𝑝(t)𝑚+𝐵𝑝(𝑡)2,+2𝐴(𝑡)(7.14) and substitution of these expressions into (7.11) gives 𝐻𝐴(𝑡,𝑚)𝑝(𝑡)𝑚2+𝐵𝑝(𝑡)𝑚+𝐶𝑝+𝑎(𝑡)1𝑚+𝑎0𝐻(𝑡,𝑚)2𝐴𝑝(𝑡)𝑚+𝐵𝑝+𝛼(𝑡)2(𝑡)2𝐻(𝑡,𝑚)2𝐴𝑝(𝑡)𝑚+𝐵𝑝(𝑡)2𝛽+2𝐴(𝑡)((𝑚𝑟)/𝜎)𝐻(𝑡,𝑚)+𝛼(𝑡)𝐻(𝑡,𝑚)2𝐴𝑝(𝑡)𝑚+𝐵𝑝(𝑡)22(𝛽1)𝐻(𝑡,𝑚)=0.(7.15) Rearranging terms according to the order of 𝑚 gives 𝐴𝑝(𝑡)+2𝑎1𝐴𝑝(𝑡)+2𝛼2(𝑡)𝐴2𝑝𝛽(𝑡)2𝛼(𝑡)𝐴𝑝(𝑡)+1/𝜎𝑚2(𝛽1)2+𝐵𝑝(𝑡)+𝑎1𝐵𝑝(𝑡)+2𝛼2(𝑡)𝐴𝑝(𝑡)𝐵𝑝(𝑡)+2𝑎0𝐴𝑝𝛽(𝑡)2𝛼(𝑡)𝐴𝑝(𝑡)+1/𝜎𝛼(𝑡)𝐵𝑝(𝑡)𝑟/𝜎𝑚+𝐶(𝛽1)𝑝(𝛼𝑡)+2(𝑡)22𝐴𝑝(𝑡)+𝐵𝑝(𝑡)+𝑎0𝐵𝑝(𝛽𝑡)𝛼(𝑡)𝐵𝑝(𝑡)𝑟/𝜎22(𝛽1)=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(𝑡), 𝐵𝑝(𝑡) and 𝐶𝑝(𝑡): 𝐴𝑝𝛽(𝑡)=2(𝛽1)2𝛼(𝑡)𝐴𝑝1(𝑡)+𝜎2𝑎21+𝛼2(𝑡)𝐴𝑝𝐴(𝑡)𝑝𝐵(𝑡),𝑝𝛽(𝑡)=𝛽12𝛼(𝑡)𝐴𝑝1(𝑡)+𝜎𝛼(𝑡)𝐵𝑝𝑟(𝑡)𝜎𝑎1+2𝛼2(𝑡)𝐴𝑝𝐵(𝑡)𝑝(𝑡)2𝑎0𝐴𝑝𝐶(𝑡),𝑝𝛽(𝑡)=2(𝛽1)𝛼(𝑡)𝐵𝑝𝑟(𝑡)𝜎2𝛼2(𝑡)22𝐴𝑝(𝑡)+𝐵2𝑝(𝑡)𝑎0𝐵𝑝(𝑡).(7.17) The equation for 𝐴𝑝(𝑡) is a Riccati ODE, however due to the presence of the function 𝛼(𝑡) it cannot be solved explicitly. The ODEs for 𝐵𝑝(𝑡) and 𝐶𝑝(𝑡) are linear. It would be possible to write down an analytic expression for their solutions, however they would depend on an integral of 𝐴𝑝(𝑡), which one would still need to compute numerically. The errors due to the discretization of the integral and the numerical error in 𝐴𝑝(𝑡) 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{𝑥𝛽(𝑇)/𝛽} under the constraint (2.6). We denote the value function of the corresponding stochastic optimal control problem with 𝐽𝑓(𝑡,𝜇,𝑥)=max{𝜋𝑢𝑢}𝑡𝑢𝑇𝔼𝑈𝑋𝑇,𝜇𝑡=𝜇,𝑋𝑡=𝑥.(7.18) Note that for the case 𝑡=0 this is the value function of the informed agent, once he knows the realization of 𝜇0, which by Lemma 3.2. is normal distributed. We pick up on this point later.

The HJB equation for this problem is given by0=max𝜋𝐽𝑓𝑡+𝜎2𝑥2𝜋22𝐽𝑓𝑥𝑥+[]𝑟+(𝜇𝑟)𝜋𝑥𝐽𝑓𝑥+𝑎1𝜇+𝑎0𝐽𝑓𝜇+𝑏21+𝑏222𝐽𝑓𝜇𝜇+𝑏1𝜎𝜋𝑥𝐽𝑓𝜇𝑥.(7.19) Assuming that 𝐽𝑓𝑥𝑥 is negative, the maximizer is given by 𝜋𝑓=(𝜇𝑟)𝐽𝑓𝑥(𝑡,𝜇,𝑥)+𝑏1𝜎𝐽𝑓𝜇𝑥(𝑡,𝜇,𝑥)𝜎2𝑥𝐽𝑓𝑥𝑥.(𝑡,𝜇,𝑥)(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(𝑡)=0, replacing 𝑚(𝑡) by 𝜇(𝑡). Substituting 𝜋𝑓 in (7.19) implies𝐽𝑓𝑡+𝑟𝑥𝐽𝑓𝑥+𝑎1𝜇+𝑎0𝐽𝑓𝜇+𝑏21+𝑏222𝐽𝑓𝜇𝜇(𝜇𝑟)𝐽𝑓𝑥+𝑏1𝜎𝐽𝑓𝜇𝑥22𝜎2𝐽𝑓𝑥𝑥=0(7.21) with terminal condition𝐽𝑓𝑥(𝑇,𝜇,𝑥)=𝛽𝛽.(7.22) Inspired by the analysis of the case 𝐽𝑝(𝑡,𝑚,𝑥), we make the sophisticated guess:𝐽𝑓(𝑡,𝜇,𝑥)=𝑒𝐴𝑓(𝑡)𝜇2+𝐵𝑓(𝑡)𝜇+𝐶𝑓(𝑡)𝑥𝛽𝛽.(7.23) Substitution into (7.22) leads in analogy to the analysis before to the following system of three ODEs: 𝐴𝑓(𝑡)=2𝑎1𝐴𝑓̃(𝑡)4𝑏𝐴2𝑓𝛽(𝑡)+2𝜎2(𝛽1)2𝑏1𝜎𝐴𝑓(𝑡)+12,𝐵𝑓(𝑡)=𝑎1𝐵𝑓(𝑡)2𝑎0𝐴𝑓̃(𝑡)4𝑏𝐴𝑓(𝑡)𝐵𝑓𝛽(𝑡)+𝜎2(𝛽1)2𝑏1𝜎𝐴𝑓𝑏(𝑡)+11𝜎𝐵𝑓,𝐶(𝑡)𝑟𝑓(𝑡)=𝑟𝛽𝑎0𝐵𝑓̃𝑏(𝑡)2𝐴𝑓(𝑡)+𝐵2𝑓+𝛽(𝑡)2𝜎2𝑏(𝛽1)1𝜎𝐵𝑓(𝑡)𝑟2,(7.24) where ̃𝑏=(𝑏21+𝑏22)/2 and 𝐴𝑓(𝑇)=𝐵𝑓(𝑇)=𝐶𝑓(𝑇)=0. In summary, the maximizers for the case of full and partial information are therefore given by𝜋𝑓(𝑡,𝜇)=𝜇𝑟+𝑏1𝜎22𝐴𝑓(𝑡)𝜇+𝐵𝑓(𝑡)𝜎2(,𝜋𝛽1)𝑝𝑏(𝑡,𝑚)=𝑚𝑟+1𝜎+𝛾(𝑡)2𝐴𝑝(𝑡)𝑚+𝐵𝑝(𝑡)𝜎2,(𝛽1)(7.25) and the corresponding value functions are given by𝐽𝑓(𝑡,𝜇,𝑥)=𝑒𝐴𝑓(𝑡)𝜇2+𝐵𝑓(𝑡)𝜇+𝐶𝑓(𝑡)𝑥𝛽𝛽𝐽,(7.26)𝑝(𝑡,𝑚,𝑥)=𝑒𝐴𝑝(𝑡)𝑚2+𝐵𝑝(𝑡)𝑚+𝐶𝑝(𝑡)𝑟𝛽𝑡𝑥𝛽𝛽.(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 𝑥, 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 𝑚0 for the partial information case is deterministic, the initial condition 𝜇0 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 𝜇0. According to Lemma 3.2, we know that 𝜇0𝒩(𝑚0,𝛾0). In average the informed agents optimal utility is therefore given by 𝔼{𝐽𝑓(0,𝜇0,𝑥)}, where the expectation is taken over 𝜇0. 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: 𝔼𝑒𝐴𝑓(0)𝜇2+𝐵𝑓(0)𝜇+𝐶𝑓(0)=12𝜋𝛾0𝑒𝐴𝑓(0)𝑥2+𝐵𝑓(0)𝑥+𝐶𝑓(0)𝑒((𝑥𝑚0)2/2𝛾0)=1𝑑𝑥12𝐴𝑓(0)𝛾0𝑒(4𝐴𝑓(0)𝐶𝑓(0)𝛾02𝐶𝑓(0)2𝐴𝑓(0)𝑚20𝐵2𝑓(0)𝛾02𝐵𝑓(0)𝑚0)/2(2𝐴𝑓(0)𝛾01),(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: 𝒱𝛽𝐽=𝔼𝑓0,𝜇0,𝑥𝐽𝑝𝑡,𝑚0.,𝑥(7.29) Note that the value of information is infinite, if 12𝐴𝑓(0)𝛾0. The latter however could not be observed for realistic parameters.

In the numerical examples presented in the appendix we assume that 𝑟=0.08, 𝑎1[1,1], 𝑎0=0.1, 𝑏1[0.5,1.2], 𝑏2[0.3,0.3], 𝛽[0.1,0.4], 𝜎[0.2,0.4], 𝛾0[0,0.2], 𝑇=0.25, 𝑚0=0.2 and 𝑥=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 𝜇0.

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.

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