The value of knowing the market price of risk
TTHE VALUE OF KNOWING THE MARKET PRICE OF RISK
KATIA COLANERI, STEFANO HERZEL, AND MARCO NICOLOSI
Abstract.
This paper presents an optimal allocation problem in a financial marketwith one risk-free and one risky asset, when the market is driven by a stochastic marketprice of risk. We solve the problem in continuous time, for an investor with a ConstantRelative Risk Aversion (CRRA) utility, under two scenarios: when the market price ofrisk is observable (the full information case ), and when it is not (the partial informationcase ). The corresponding market models are complete in the partial information case andincomplete in the other case, hence the two scenarios exhibit rather different features.We study how the access to more accurate information on the market price of risk affectsthe optimal strategies and we determine the maximal price that the investor would bewilling to pay to get such information. In particular, we examine two cases of additionalinformation, when an exact observation of the market price of risk is available either attime only (the initial information case ), or during the whole investment period (the dynamic information case ). Keywords : Portfolio optimization, Power utility, Martingale Method, Partial Infor-mation. 1.
Introduction
Ours is a classical expected utility maximization problem in continuous time, firststudied by Merton (1969) [28]. We solve it via the martingale method, proposed byKaratzas el al. (1987) [22] and by Cox and Huang (1989) [9]. The martingale method relieson duality theory and transforms the original dynamic problem, usually solved throughthe Hamilton-Jacobi-Bellman (HJB) partial differential equation, into an equivalent staticoptimization problem. It has two main advantages over the more direct HJB approach:it leads to a quasi-linear partial differential equation that is usually easier to solve and itprovides an expression of the optimal wealth as a function of the state price density, thatcan be used to relate the optimal strategy to the current state of the market.A necessary assumption to apply the standard martingale method is that the stateprice density is unique, that is the market is complete. However for our problem, in the
Katia Colaneri, Department of Economics and Finance, University of Rome Tor Ver-gata, Via Columbia 2, 00133 Roma, IT.Stefano Herzel, Department of Economics and Finance, University of Rome Tor Ver-gata, Via Columbia 2, 00133 Roma, IT.Marco Nicolosi, Department of Economics, University of Perugia, Via A. Pascoli 20,06123 Perugia, IT.
E-mail addresses : [email protected], [email protected],[email protected] . a r X i v : . [ q -f i n . P M ] S e p K. COLANERI, S. HERZEL, AND M. NICOLOSI full information case, the trading strategies may also depend on the market price of risk,that is a not traded asset, and this makes the corresponding market model incomplete.Therefore we must rely on a modification of the standard approach, the so called minimaxmartingale method, introduced by He and Pearson (1991) [20]. The minimax methodexploits the fact that, in an incomplete market, there are infinitely many state pricedensities but they all assign the same values to the marketable claims, i.e. those claimsattainable by admissible trading strategies involving the market securities. Hence, theoptimal final wealth is determined by selecting the state price density which minimizesthe maximal expected utility of the final wealth, the minimax state price density .We model the stock as a geometric brownian motion with a market price of risk givenby an Ornstein-Uhlenbeck process, which is a Gaussian, mean reverting process. This isa convenient assumption, adopted by several studies which will be mentioned below, andthat may be justified by empirical evidence. To solve the investment problem under partialinformation it is necessary to identify the filter , that is the conditional distribution of theunobservable process given the available information. The assumptions on the model ofthe market allows us to apply the linear finite dimensional Kalman filter to identify thedynamics of the filter and characterize its conditional distribution, see, e.g. Lipster andShiryaev (2001) [27]. Then, following Fleming and Pardoux (1982) [14], we transformthe original optimization problem into an equivalent one where all the state variables areadapted to the same filtration. Under this transformation the market model is completeand the classical martingale method can be used, see e.g. Björk, Davis and Landen (2010)[6].Another important consequence of the assumptions on the dynamics of the assets andon the utility of the investor is that the state variables of the market model, representedby the (logarithm of) minimax state price density and the market price of risk in the fullinformation case, and by the unique state price density and the filter in the partial infor-mation case, are jointly affine. This fact allows us to compute the corresponding optimalwealths (and strategies) in closed form, after solving a system of Riccati equations thatis homogeneous under full information and non-homogeneous under partial information.We apply the results for the full and partial information problem to compute the valueof initial and dynamic information. Of course, by increasing the information set, theinvestor gets a higher expected optimal utility. To measure the subjective value of suchenlargements we compute the corresponding reservation prices. Again, because of thestructure of the model, their expressions are simple. The last part of the paper is devotedto numerical examples that illustrate a few applications of our results. Some of the resultsare rather unexpected: to mention at least one of them, we will see that to maximize theSharpe ratio of an investment, one should follow the strategy of a partially informedportfolio manager with CRRA utility rather than that of a fully informed one!Our study relies on a long list of previous contributions, which we will mention below,but its closest references are Kim and Omberg (1996) [23] and Brendle (2006) [7], who
HE VALUE OF KNOWING THE MARKET PRICE OF RISK 3 study optimal investment problems similar to ours by using the HJB approach, showingthat the HJB equations can be reduced to a system of Riccati equations. In particular,while Kim and Omberg (1996) [23] are interested in the full information case for HARAutility functions, Brendle (2006) [7] also focuses on a partially informed investor endowedwith bounded CRRA preferences, extending his analysis to a multi-dimensional marketmodel. We rely on many of their results, especially those related to the solutions tothe Riccati equations. However, both these papers fail to provide a verification theoremfor their results, that is, they only show necessary, but not sufficient, conditions for theoptimality of their proposed solutions. In particular, Kim and Omberg (1996) [23] writeâĂIJ (...) we are not acquainted with any verification theorem that fits the model above,despite its relative simplicity (...) âĂİ. They mention the fact that the classical verificationtheorems cannot be applied because the indirect utility that solves the Bellman equation isa function of the investor wealth that is not restricted to a closed set. Hence, they can onlysolve numerically their HJB equation to suggest that, for a given choice of parameters,there should be no signs of multiple solutions. Instead, by using the martingale approach,we prove verification theorems for both the cases of our interest (Theorem 3.2 and 4.1),we apply them to our solutions to show that they are effectively optimal (Theorem 3.3and 4.3), and we provide sufficient conditions that are easy to verify directly on any setof parameters (Proposition 3.4 and 4.4) .Having mentioned what we believe are the most important theoretical contributionsof this paper, we summarize the other ones. We derive the distribution, conditional andunconditional, of the optimal wealth under full and under partial information at any timewithin the investment horizon (Proposition 3.5 and 4.5). To the best of our knowledge suchresults are new and have never been addressed before, despite the fact that the knowledgeof the distribution of the optimal wealth may be useful for applications in portfolio andrisk management. Another novelty inspired the title of our paper, that is we assign aprice to the information that an investor may buy from an expert who is able to providethe value of the market price of risk either at the beginning of the investment period orcontinuously in time. We hope that this result may provide a new tool to measure thelevel of uncertainty on the returns of an asset, to be used along with the standard onesbased on volatility or implied volatility. To support our theoretical findings we providea rich numerical analysis that also shows interesting and sometimes unexpected results.Last, but not least, by solving the optimization problem under full information, we showa new application of the powerful minimax martingale approach where it is possible toexplicitly identify the minimax state price density and the associated penalization process.This is a nice example that may be useful for didactical purposes.Before completing this introduction with a literature review, we provide a short de-scription of the rest of the paper. In Section 2 we introduce the modeling framework.Section 3 solves the optimization problem under full information, while Section 4 underpartial information. In both sections we characterize the optimal investment strategy and
K. COLANERI, S. HERZEL, AND M. NICOLOSI provide a closed form representation for the optimal wealth as a function of the relevantstate variables as well as for its distribution. In Section 5 we define and compute the valueof initial and dynamic information. A numerical study to illustrate the effects of the pa-rameters on the distribution of the wealth and on the value of information is presentedin Section 6. Section 7 concludes.1.1.
Literature review.
Optimal investment problem in continuous time started fromthe work of Merton (1969) [28], and extended since then in many directions with thescope of including more realistic situations. The extension considered by us, when thedrift of asset prices is not directly observable by the investor, leads to problems of partialinformation. Problems of this type have been addressed considering several modelizationsof the unobservable risk factors. Contributions in the case where prices are modeledas diffusions can be found for instance in Lackner (1995) [24], Lackner (1998) [25] andBrendle (2006) [7] under different approaches. In Brennan (1998) [8] and Xia (2001) [34]the authors discuss the effect of learning on the portfolio choices. The setting whereprices depend on an unobservable Markov chain is considered, for instance in Bäuerleand Rieder (2005) [5] and Haussmann and Sass (2004) [19] and Barucci and Marazzina(2015) [4]. Considering investors endowed with different levels of information motivatesthe work of Fouque, Papanicolaou, and Sircar (2015) [15] who analyze the loss of utilitydue to partial information. In Frey, Gabih and Wunderlich (2012) [17] and Gabih etal. (2014) [18] expert opinions in the form of signals at random discrete time points areinvestigated. This idea is extended to the case where expert opinions arrive continuouslyin time by Fouque, Papanicolaou and Sircar (2017) [16] and by Davis and Lleo (2013)[11] who implement the BlackâĂŞLitterman model in a continuous time setting and useseparability of the filtering problem and the stochastic control problem to incorporateanalyst views and non-tradeable assets as additional source of observation to estimate thefilter. A similar setting has been studied by Danilova, Monoyios and Ng (2010) [10] whoassumed that the investor has partial information about the drift of the stock price butalso some privileged information about the future of stock price. In Putschögl and Sass(2008) [32] an optimal investment and consumption problem under partial observation isanalysed using Malliavin calculus. Investment problems in a market with two cointegratedassets under partial information are studied in some recent works as for instance Lee andPapanicolaou (2016) [26] and Altay et al. (2018, 2019) [1], [2].The issue of assessing the value of information is a classical one in economics andfinance. Pikovsky and Karatzas (1996) [30] presented the problem of computing thevalue of initial information for an investor, defined as the informational gain in terms ofincremental utility provided by the access to an enlarged filtration. Amendinger, Bechererand Schweizer (2003) [3] addressed the problem of computing the value of informationfor a trader who has the opportunity to buy some extra information. The problem isformulated for a complete market in the mathematical framework of an initially enlarged
HE VALUE OF KNOWING THE MARKET PRICE OF RISK 5 filtration, and the value of information is derived via a comparison of the expected utilityfrom terminal wealth. Chau, Cosso and Fontana (2018) [13] extended their approach toestimate the value of an insider information that may allow for an arbitrage opportunity,assuming that unbounded profits cannot be reached with bounded risk.We model the market price of risk as an Ornstein-Uhlenbeck mean reverting process.We refer to Wachter (2002) [33] for a review of the most important empirical contribu-tions justifying such assumption. Wachter (2002) [33] solved the optimal investment andconsumption problem in a complete model where the market price of risk and the stockreturn are perfectly negatively correlated. A setting close to ours under full informationwhere returns of the risky asset are driven by an Ornstein-Uhlenbeck process was pro-posed by Kim and Omberg (1996) [23], who studied the portfolio optimization problemfor a HARA investor and discussed the existence of the so called nirvana solutions , whichhappen when an infinite expected utility is reached in finite time. An application of thissetting to the problem of a fund manager whose compensation depends on the relativeperformance with respect to a benchmark can be found in Nicolosi, Angelini and Herzel(2018) [29], and Herzel and Nicolosi (2019) [21].2.
The general setting
Let (Ω , F , P ) be a fixed probability space endowed with a complete and right continuousfiltration F = ( F t ) { ≤ t ≤ T } representing the global information flow where T is a fixed timehorizon. All processes defined below are assumed to be F -adapted. We consider a marketmodel with one risky asset S , the stock , and one risk-free asset B with dynamics dS t S t = µ t dt + σdZ St S = s ∈ R + ,dB t B t = rdt, B = 1 where σ > and r ≥ are constant, Z S is a standard, one dimensional, F -Brownianmotion and the drift process µ is of the form µ t = r + σX t . The process X represents the market price of risk X and is assumed to follow an Ornstein-Uhlenbeck process with dynamics dX t = − λ ( X t − ¯ X ) dt + σ X dZ Xt , where X is a normally distributed random variable with mean π and variance R , λ > is a constant representing the strength of attraction toward the long term expected mean ¯ X ≥ , σ X > is the volatility of the market price of risk and Z X is a standard one-dimensional F -Brownian motion correlated with Z S with d (cid:104) Z X , Z S (cid:105) t = ρ dt, K. COLANERI, S. HERZEL, AND M. NICOLOSI for a constant correlation coefficient ρ ∈ [ − , . Let Z ⊥ be a Brownian motion indepen-dent of Z S such that Z X = ρZ S + (cid:112) − ρ Z ⊥ . Then without loss of generality we canassume that F is the complete and right continuous filtration generated by ( Z S , Z ⊥ ) .An investor trades the risky asset and the risk-free asset continuously in time, startingfrom an initial capital w , to maximize the expected utility of her final wealth at time T . Her trading strategy is self-financing and given by the process θ = { θ t , t ∈ [0 , T ] } representing the proportion of portfolio value invested in the risky asset at time t ∈ [0 , T ] .We assume the standard integrability condition on θ E (cid:20)(cid:90) T (cid:0) | θ t X t | + θ t (cid:1) dt (cid:21) < ∞ . (2.1)to ensure that the associated wealth process dW t W t = ( r + θ t σX t ) dt + θ t σdZ St , W = w > . (2.2)is well defined (note that we also assume that no dividends are paid by the stock beforetime T ) and to exclude arbitrage opportunities. Further restrictions on the measurabilityof the process θ depending on the information set of the investor will be given in the nextsections.The investor has a power utility function u ( x ) = 11 − γ x − γ , (2.3)for every x > and for a positive risk aversion parameter γ (cid:54) = 1 . By setting γ = 1 we getthe logarithmic utility u ( x ) = log x . Note that for γ > the function u ( x ) is boundedabove while it becomes unbounded when γ < .We will solve optimization problems corresponding to two different assumptions on theinformation flow available to the investor. In the first case, we assume that the investorobserves both the stock price and the market price of risk; therefore her information flowis given by the global filtration F = ( F t ) t ∈ [0 ,T ] . In particular the initial information F isgiven by the sigma algebra generated by X enlarged with the collection of P -null sets.In the second case, we assume that the investor directly observes stock prices but not themarket price of risk. At any time t ∈ [0 , T ] , the value of X has to be inferred from theavailable information, represented by the natural filtration generated by the stock priceprocess completed by the P -null sets and denoted by F S , where the initial information is F S = {∅ , Ω } . 3. Optimal investment under full information
In this section we consider a fully informed investor who observes the path of the stockand of the market price of risk. The investor wants to maximize the expected utility ofher wealth at a time
T > , hence her problem is max θ ∈A ( w ) E [ u ( W T )] (3.1) HE VALUE OF KNOWING THE MARKET PRICE OF RISK 7 where A ( w ) is the set of F -predictable self-financing strategies satisfying the integrabilitycondition (2.1) starting from an initial wealth w . Problem (3.1) is equivalent to max θ ∈A ( w ) E (cid:20) − γ W − γT |F (cid:21) . (3.2)In fact, a strategy θ ∗ ∈ A ( w ) is optimal for (3.1) if and only if it is optimal for (3.2) almostsurely, because θ ∗ is F -measurable. In the sequel we will denote by E t the conditionalexpectation given F t .In this setting there are two risk factors and only one asset that can be used as ahedging instrument, therefore the market is not complete. The state price densities ξ ν satisfy the equation dξ νt ξ νt = − rdt − X t dZ St − ν t (cid:112) − ρ dZ ⊥ t , ξ ν = 1 (3.3)where ν = { ν t , t ∈ [0 , T ] } is such that E (cid:104)(cid:82) T ν t dt (cid:105) < ∞ and E (cid:104)(cid:82) T | ξ νt | dt (cid:105) < ∞ .The process Z ⊥ t is orthogonal to the space of attainable payoffs (i.e. payoffs that canbe reached by feasible self-financing strategies). If the stock price process and the marketprice of risk are perfectly correlated (positively or negatively), (cid:112) − ρ vanishes, themarket becomes complete and the state price density is unique.To solve the optimal investment problem in an incomplete market we follow He andPearson (1991) [20] and apply the martingale approach to transform the dynamic problem(3.2) into the equivalent static one min ν max W T E [ u ( W T )] , (3.4)subject to the constraint w = E [ ξ νT W T ] . (3.5)The optimal ν ∗ for the problem (3.4)-(3.5) determines the minimax state price density process ξ ∗ . The role of the process ν ∗ is to penalize contingent claims that cannot bereplicated by feasible portfolio strategies. For example, ν ∗ t = 0 P -a.s. for t ∈ [0 , T ] implies that no penalization is necessary and a feasible optimal strategy is naturallychosen by the investor. For power utility functions of the form (2.3), a sufficient conditionfor the existence of ξ ∗ is that γ > , see He and Pearson (1991) [20, Theorem 4 andTheorem 6].The Lagrangian function associated to problem (3.4)-(3.5) is L ( W T , λ ) = E [ u ( W T )] − λ ( E [ ξ ∗ T W T ] − w ) , where λ is the multiplier from the constraint (3.5). From standard results (e.g. Karatzaset al. (1987) [22]), the optimal final wealth satisfies W ∗ T = g ( λ ξ ∗ T ) (3.6) K. COLANERI, S. HERZEL, AND M. NICOLOSI where function g ( · ) is the inverse of the marginal utility u (cid:48) . For the power utility (2.3) g ( y ) = y − γ . Since ξ ∗ is a state-price density process, the optimal wealth at time t is W ∗ t = ξ ∗ t − E t [ ξ ∗ T g ( λ ξ ∗ T )]= ξ ∗ t − λ − γ E t (cid:104) ξ ∗ T − γ (cid:105) . Remark . This approach can also be applied to logarithmic utility functions. In thiscase g ( y ) = y , the optimal wealth at time t is W ∗ t = ξ ∗ t − E t [ ξ ∗ T g ( λ ξ ∗ T )] = ( λ ξ ∗ t ) − . By applying Itô formula to (3.8) we get dW ∗ t W ∗ t =( r + X t + (1 − ρ )( ν ∗ t ) ) dt + X t dZ St + (cid:112) − ρ ν ∗ t dZ ⊥ t . (3.9)By equating the predictable quadratic covariations of W ∗ and Z S computed from (2.2)and (3.9) we get the optimal strategy θ ∗ t = X t σ . This strategy is called myopic becauseit does not depend on the investment horizon. Note that in this case the minimax stateprice density is associated to the penalty process ν ∗ t = 0 .The following verification theorem states sufficient conditions to solve the optimizationproblem. Theorem 3.2 (Verification Theorem under full information) . Let the function F ( Y, X, t ) be the solution to the partial differential equation (where subscripts denote partial deriva-tives) F Y Y Y (cid:0) X + ν ∗ ( Y, X, t ) (1 − ρ ) (cid:1) + F XY Y σ X (cid:0) ρX + ν ∗ ( Y, X, t )(1 − ρ ) (cid:1) + 12 F XX σ X − F X (cid:0) σ X (cid:0) ρX + ν ∗ ( Y, X, t )(1 − ρ ) (cid:1) + λ ( X − ¯ X ) (cid:1) + F t − rF + rF Y Y = 0 where ν ∗ ( Y, X, t ) = − σ X F X ( Y, X, t ) Y F Y ( Y, X, t ) with the boundary conditions F ( Y, X, T ) = Y γ F ( Y , X ,
0) = w for some constant Y > , F Y ( Y, X, t ) (cid:54) = 0 , and F ( Y, X, t ) → F ( Y, X, T ) as t → T .Assume that the following conditions hold: (i) the function ν ∗ ( Y, X, t ) is sublinear and locally Lipschitz for ( Y, X, t ) ∈ R + × R × [0 , T ] (this condition implies that the process ξ ∗ t satisfying (3.3) with ν ∗ t := ν ∗ ( Y ( ξ ∗ t ) − , X t , t ) is a well defined local martingale), HE VALUE OF KNOWING THE MARKET PRICE OF RISK 9 (ii) E (cid:20)(cid:90) T (cid:0) ( Y − t F ( Y t , X t , t )) + ( F Y ( Y t , X t , t )) (cid:1) ( X t + ( ν ∗ ( Y t , X t , t )) ) dt (cid:21) < ∞ , (3.14) where Y t := Y ( ξ ∗ t ) − , for t ∈ [0 , T ] .Then the process ξ ∗ t is the minimax state price density, the optimal wealth is W ∗ t = F ( Y t , X t , t ) and the optimal investment strategy is θ ∗ t = F Y ( Y t , X t , t ) Y t X t + ρσ X F X ( Y t , X t , t ) σF ( Y t , X t , t ) , (3.15) for every t ∈ [0 , T ] .Proof. To show that F ( Y t , X t , t ) is the optimal wealth process, we need to verify that theinitial wealth satisfies the budget constraint and that the final wealth satisfies the firstorder condition (3.6) and is attainable by a self-financing strategy.The budget constraint (3.5) follows from (3.13), and the first order condition from(3.12), since g ( y ) = y − γ . Since ξ ∗ t is well defined by condition (i) and Y is given by(3.13), we can define the process Y t := Y ( ξ ∗ t ) − . From Itô formula the dynamics of Y t is dY t Y t =( r + X t + (1 − ρ )( ν ∗ ( Y t , X t , t )) ) dt + X t dZ St + (cid:112) − ρ ν ∗ ( Y t , X t , t ) dZ ⊥ t , and, still applying Itô, F ( Y t , X t , t ) = F ( Y , X ,
0) + (cid:90) t L F ( Y s , X s , s ) ds + (cid:90) t ( ρσ X F X ( Y s , X s , s ) + Y s X s F Y ( Y s , X s , s )) dZ Ss + (cid:90) t (cid:112) − ρ ( σ X F X ( Y s , X s , s ) + Y s ν ∗ ( Y s , X s , s ) F Y ( Y s , X s , s )) dZ ⊥ s , (3.16)where L is the differential operator L F = F t + 12 F Y Y Y (cid:0) X + ν ∗ ( Y, X, t ) (1 − ρ ) (cid:1) + σ X F XY Y (cid:0) ρX + ν ∗ ( Y, X, t )(1 − ρ ) (cid:1) + 12 F XX σ X − F X λ ( X − ¯ X ) + F Y Y ( r + X + ( ν ∗ ( Y, X, t )) (1 − ρ )) . By (3.11), the integral with respect to Z ⊥ in (3.16) vanishes, therefore the final wealth F ( Y T , X T , T ) belongs to the space of attainable payoffs. To show that it can be obtained bya self-financing strategy starting from w it remains to show that the process ξ ∗ t F ( Y t , X t , t ) is a true martingale. From Itô formula and assumption (3.10), we get ξ ∗ t F ( Y t , X t , t ) = F ( Y , X , (cid:90) t ξ ∗ s ( ρσ X F X ( Y s , X s , s ) + Y s X s F Y ( Y s , X s , s ) − F ( Y s , X s , s ) X s ) dZ Ss − (cid:112) − ρ (cid:90) t ξ ∗ s F ( Y s , X s , s ) ν ∗ ( Y s , X s , s ) dZ ⊥ s , which is a true martingale because E (cid:20)(cid:90) T (cid:0) ( ξ ∗ s ) ( ρσ X F X ( Y s , X s , s ) + Y s X s F Y ( Y s , X s , s ) − F ( Y s , X s , s ) X s ) +(1 − ρ )( ξ ∗ s ) ( F ( Y s , X s , s )) ( ν ∗ ( Y s , X s , s )) (cid:1) dt (cid:3) = E (cid:20)(cid:90) T (cid:0) ( ξ ∗ s ) ( Y s F Y ( Y s , X s , s )( X s − ρν ∗ ( Y s , X s , s )) − F ( Y s , X s , s ) X s ) +(1 − ρ )( ξ ∗ s ) ( F ( Y s , X s , s )) ( ν ∗ ( Y s , X s , s )) (cid:1) dt (cid:3) ≤ c E (cid:20)(cid:90) T (cid:0) ( X s + ( ν ∗ ( Y s , X s , s )) )( F Y ( Y s , X s , s )) +( ξ ∗ s ) ( F ( Y s , X s , s )) ( X s + ( ν ∗ ( Y s , X s , s )) ) (cid:1) dt (cid:3) , that is bounded by (3.14) ( c is a positive constant). Note that the first equality comesfrom (3.11), and in the inequality we have used ( a + b ) ≤ a + b ) , ρ < , − ρ < ,and the definition of Y t .Therefore, W ∗ t = F ( Y t , X t , t ) is the optimal wealth process and ξ ∗ t is the minimax stateprice density (see He and Pearson (1991) [20, Theorem 8]). Finally, by equating thepredictable quadratic covariations of W ∗ and Z S computed from (2.2) and (3.16) we getthe optimal strategy (3.15). (cid:3) To determine a closed form expression for W ∗ t we guess that the joint process (log( ξ ∗ ) , X, X ) is affine. From this guess it follows that the conditional expectation in (3.7) is E t (cid:104) ξ ∗ T − γ (cid:105) = ξ ∗ t − γ e A ( t )+ B ( t ) X t + C ( t ) X t where the functions A ( t ) , B ( t ) and C ( t ) satisfy the system of Riccati equations dCdt = − a − bC ( t ) − cC ( t ) ,dBdt = − C ( t ) λ ¯ X − (cid:18) b cC ( t ) (cid:19) B ( t ) ,dAdt = γ − γ r − B ( t ) λ ¯ X − C ( t ) σ X − cB ( t ) (3.17)with boundary conditions A ( T ) = B ( T ) = C ( T ) = 0 , (3.18) HE VALUE OF KNOWING THE MARKET PRICE OF RISK 11 for constants a = 1 − γγ , b = 2 (cid:18) − λ + 1 − γγ ρσ X (cid:19) , c = σ X (cid:0) ρ + γ (1 − ρ ) (cid:1) . To prove that our guess is correct we must show that it satisfies Theorem 3.2. Beforethat, we discuss the behavior of the solutions to the problem (3.17)-(3.18) without re-porting them explicitly, as they can be found, for instance, in Kim and Omberg (1996)[23].Let us define ∆ := b − ac = 4 (cid:18) p − qγ (cid:19) , where p := λ + 2 λρσ X + σ X , and q :=2 λρσ X + σ X . It is easily seen that p > q and p ≥ . In particular, if ρ (cid:54) = − and σ X (cid:54) = λ ,then p (cid:54) = 0 . We define the critical correlation value ρ ∗ = − σ X λ ∨ − , (3.19)and, for ρ ≥ ρ ∗ , the critical risk aversion parameter γ ∗ = qp . (3.20)Note that ≤ γ ∗ < , where the lower bound follows from the assumption on ρ . Accordingto the classification by Kim and Omberg (1996) [23], there are four possible cases:i. If ρ ≥ ρ ∗ and γ ∗ < γ < ∪ γ > , then ∆ > and the solution, called “well-behavednormal", exists for every t in [0,T].ii. If ρ > ρ ∗ and < γ < γ ∗ , then ∆ < . The solution is called “tangent" and isdefined on [0 , T ∗ ) , where T ∗ = πη − η arctan bη (3.21)with η = √− ∆ . In such a case the investment horizon T has to be lower than T ∗ in order that solution exists over the entire interval [0 , T ] .iii. If ρ > ρ ∗ and γ = γ ∗ , we get the “well-behaved hyperbolic" solution which existsfor every t in [0,T]. This case corresponds to ∆ = 0 and b < .iv. If ρ < ρ ∗ and γ > , γ (cid:54) = 1 , then q < and hence ∆ > . The solution is“well-behaved normal" and exists for every t in [0 , T ] .Wachter (2002) [33] noted that, for real market data, the correlation between returns andthe market price of risk is usually negative and close to -1, hence case (iv) should be themost relevant for financial applications.After providing the conditions under which the system of Riccati equations has a solu-tion, we can state some sufficient conditions for our guess to provide the optimal wealthand the optimal policy for the full information case. Theorem 3.3.
Let the functions A ( t ) , B ( t ) and C ( t ) satisfy (3.17) - (3.18) on [0 , T ] , let ν ∗ t := − γ ( B ( t ) + C ( t ) X t ) σ X ,λ := (cid:34) e A (0)+ B (0) X + C (0) X w (cid:35) γ , and let ξ ∗ t be the state price density process associated to ν ∗ t .If E (cid:20)(cid:90) T (cid:16) ( ξ ∗ t ) − γ e A ( t )+ B ( t ) X t + C ( t ) X t (cid:17) (1 + X t ) dt (cid:21) < ∞ , (3.24) then ξ ∗ t is the minimax state price density, λ is the Lagrange multiplier for problem (3.4) - (3.5) , and W ∗ t = ( λ ξ ∗ t ) − γ e A ( t )+ B ( t ) X t + C ( t ) X t , (3.25) θ ∗ t = 1 γ X t σ + ρ σ X σ ( B ( t ) + C ( t ) X t ) are the optimal wealth and the optimal strategy.Proof. Let us define F ( Y, X, t ) := Y γ e A ( t )+ B ( t ) X + C ( t ) X , (3.26)for ( Y, X, t ) ∈ R + × R × [0 , T ] . We need to check that the function F satisfies theassumptions of Theorem 3.2, and hence the optimal wealth process is equal to F ( Y t , X t , t ) ,where Y t := Y ( ξ ∗ t ) − .From assumption (3.22) we see that ν ∗ verifies Equation (3.11); moreover, it is sublinearand locally Lipschitz for ( Y, X, t ) ∈ R + × R × [0 , T ] , hence condition (i) of Theorem 3.2 issatisfied. Since the functions A ( t ) , B ( t ) and C ( t ) satisfy (3.17), function F ( Y, X, t ) solves(3.10). Moreover F Y ( Y, X, t ) (cid:54) = 0 . The boundary conditions (3.18) imply that F ( Y, X, T ) = Y γ , and therefore condition (3.12) is also true. Moreover, imposing the budget constraint Y γ e A (0)+ B (0) X + C (0) X = w we get that the value of Y > that satisfies condition (3.13) is Y = λ − , where λ is given by (3.23). From (3.6) it follows that λ is the Lagrange multiplier for problem(3.4)-(3.5). HE VALUE OF KNOWING THE MARKET PRICE OF RISK 13
To check that condition (3.14) in Theorem 3.2 is satisfied we use the fact that Y t =( λ ξ ∗ t ) − , and the definitions of F given in (3.26) and of ν ∗ in (3.22), to get E (cid:20)(cid:90) T (cid:0) ( Y − t F ( Y t , X t , t )) + ( F Y ( Y t , X t , t )) (cid:1) ( X t + ( ν ∗ ( Y t , X t , t )) ) dt (cid:21) = E (cid:20)(cid:90) T (cid:16) ( λ ξ ∗ t ) − /γ e A ( t )+ B ( t ) X + C ( t ) X ) (cid:17) ( X t + γ σ X ( B ( t ) + C ( t ) X t ) ) dt (cid:21) ≤ c E (cid:20)(cid:90) T (cid:16) ( ξ ∗ t ) − /γ e A ( t )+ B ( t ) X + C ( t ) X ) (cid:17) (1 + X t ) dt (cid:21) for some constant c > , because B ( t ) and C ( t ) are continuous functions on [0 , T ] . Thelast term is bounded by assumption (3.24). This completes the proof. (cid:3) The following result provides some conditions for Theorem 3.3 that are easier to checkthan (3.24).
Proposition 3.4.
If at least one of the following two holds: (i) γ > (ii) the functions A ( t ) , B ( t ) and C ( t ) satisfy (3.17) - (3.18) on [0 , T ] and − C (0) max (cid:18) R , R e − λT + σ X λ (1 − e − λT ) (cid:19) > . (3.27) Then all assumptions of Theorem 3.3 are verified.Proof.
Recall that for γ > the functions A, B, C are well defined on [0 , T ] .Then, we only need to show that condition (3.24) is satisfied. By Cauchy Schwartzinequality, using Fubini and Y t = ( λ ξ ∗ t ) − , E (cid:20)(cid:90) T e A ( t ) (cid:16) ξ ∗ t − γ (cid:17) e B ( t ) X t + C ( t ) X t (1 + X t ) dt (cid:21) ≤ κ (cid:90) T e A ( t ) E (cid:104) ξ ∗ t − γ ) (cid:105) E (cid:104) e B ( t ) X t +2 C ( t ) X t (cid:105) E [(1 + X t ) ] dt, where k is a positive constant. Considering each expectation separately, first we have E (cid:104) ξ ∗ t − γ ) (cid:105) < ∞ , for every t ∈ [0 , T ] , since X is an Ornstein-Uhlenbeck process (see, e.g. Revuz and Yor(2013) [31, Chapter 8, Ex. 3.14]). Second, E [(1 + X t ) ] < ∞ for every t ∈ [0 , T ] , since X t is a Gaussian random variable and hence has moments of allorders. Finally E (cid:104) e B ( t ) X t +2 C ( t ) X t (cid:105) < ∞4 K. COLANERI, S. HERZEL, AND M. NICOLOSI
Recall that for γ > the functions A, B, C are well defined on [0 , T ] .Then, we only need to show that condition (3.24) is satisfied. By Cauchy Schwartzinequality, using Fubini and Y t = ( λ ξ ∗ t ) − , E (cid:20)(cid:90) T e A ( t ) (cid:16) ξ ∗ t − γ (cid:17) e B ( t ) X t + C ( t ) X t (1 + X t ) dt (cid:21) ≤ κ (cid:90) T e A ( t ) E (cid:104) ξ ∗ t − γ ) (cid:105) E (cid:104) e B ( t ) X t +2 C ( t ) X t (cid:105) E [(1 + X t ) ] dt, where k is a positive constant. Considering each expectation separately, first we have E (cid:104) ξ ∗ t − γ ) (cid:105) < ∞ , for every t ∈ [0 , T ] , since X is an Ornstein-Uhlenbeck process (see, e.g. Revuz and Yor(2013) [31, Chapter 8, Ex. 3.14]). Second, E [(1 + X t ) ] < ∞ for every t ∈ [0 , T ] , since X t is a Gaussian random variable and hence has moments of allorders. Finally E (cid:104) e B ( t ) X t +2 C ( t ) X t (cid:105) < ∞4 K. COLANERI, S. HERZEL, AND M. NICOLOSI for every t ∈ [0 , T ] if and only if − C ( t ) v t > , where v t = R e − λt + σ X λ (1 − e − λt ) isthe variance of X t . To show that − C ( t ) v t > we use that C ( t ) is strictly negative and increasing on[0,T] if γ > , and is strictly positive and decreasing if γ < (see Kim and Omberg (1996)[23, Equation (23)]). Then, for γ > , C ( t ) < , therefore − C ( t ) v t > . When γ < , C ( t ) is positive and decreasing, hence C ( t ) < C (0) . Moreover, let v ∞ := σ X λ , then v t isincreasing and R ≤ v t ≤ v T if R < v ∞ and decreasing with v T ≤ v t ≤ R otherwise.This means that − C ( t ) v t > − C (0) max ( R , v T ) . The result then follows from(3.27). (cid:3) Condition (3.27) can be easily verified on any set of parameters, but it is more restrictivethan Condition (3.24), that is more difficult to check. In the section devoted to theapplications we show graphically, in Figure 2, how much restrictive Condition (3.27) iswith respect to the domain of existence of the corresponding system of Riccati equations.Kim and Omberg (1996) [23] and Brendle (2006) [7] solved a problem similar to ours byusing the Hamilton-Jacobi-Bellman (HJB) approach. To recover their results, we computethe expected optimal utility at time t E t [ u ( W ∗ T )] = 11 − γ λ − γ E t (cid:104) ( ξ ∗ T ) − γ (cid:105) = 11 − γ λ ξ ∗ t W ∗ t = 11 − γ W ∗ t − γ e γ ( A ( t )+ B ( t ) X t + C ( t ) X t ) , where (3.28) follows from (3.6), (3.29) from (3.7), and (3.30) from (3.25). Equation (3.30)corresponds to the formulas [23, Equation (16)] and [7, Equation (14)].By plugging (3.25) into (3.30), we can also compute the conditional expected optimalutility as E t [ u ( W ∗ T )] = 11 − γ ( λ ξ ∗ t ) − /γ e A ( t )+ B ( t ) X t + C ( t ) X t . (3.31)An advantage of the martingale approach over HJB is that it allows us to compute boththe optimal wealth (3.25) and the expected optimal utility (3.31) as functions of theminimax state price density ξ ∗ and of the market price of risk X . This may be useful tostudy the dependence on the current state of the market.From (3.30), we can also derive the (unconditional) expected optimal utility, that existswhen Q (0) := 1 − γC (0) R For a random variable ε ∼ N ( µ, σ ) , if − cσ > , E [ e a + bε + cε ] = √ − cσ exp (cid:16) a + b σ − cσ ) + bµ − cσ + cµ − cσ ) (cid:17) . HE VALUE OF KNOWING THE MARKET PRICE OF RISK 15 is strictly positive under the hypotheses of Proposition 3.4. Indeed, since X ∼ N ( π , R ) , and using the formula provided in Footnote 1, we get E [ u ( W ∗ T )] = EE [ u ( W ∗ T )] = w − γ − γ E (cid:104) e γ ( A (0)+ B (0) X + C (0) X ) (cid:105) = w − γ (1 − γ ) (cid:112) Q (0) e γA (0)+ γ Q (0) ( γB (0) R +2 π B (0)+ C (0) π ) . To study the conditional distribution of the optimal wealth we compute the conditionalmoment generating function of ln W ∗ t , φ s ( t, z ) := E s [( W ∗ t ) z ] , (3.33)on its domain of existence. Proposition 3.5.
Let φ s ( t, z ) < ∞ for ≤ s ≤ t ≤ T and z > . Then φ s ( t, z ) = ( λ ξ ∗ s ) − zγ e D ( s ; t,z )+ E ( s ; t,z ) X s + H ( s ; t,z ) X s (3.34) where the functions D : [0 , t ] → R , E : [0 , t ] → R and H : [0 , t ] → R satisfy the system ofdifferential equations dHds = d ( s ) + 2 e ( s ) H ( s ) − σ X H ( s ) dEds = f ( s ) + (cid:0) e ( s ) − σ X H ( s ) (cid:1) E ( s ) + g ( s ) H ( s ) dDds = h ( s ) + g ( s ) E ( s ) − σ X H ( s ) + E ( s ) ) (3.35) with boundary conditions D ( t ) = zA ( t ) , E ( t ) = zB ( t ) , H ( t ) = zC ( t ) , (3.36) and d ( s ) = − (cid:18) z γ + zγ (cid:19) (cid:0) γ σ X (1 − ρ ) C ( s ) (cid:1) e ( s ) = λ − zγ σ X ρ + zσ X (1 − ρ ) C ( s ) f ( s ) = − ( z + zγ ) σ X (1 − ρ ) B ( s ) C ( s ) g ( s ) = zσ X (1 − ρ ) B ( s ) − λ ¯ Xh ( s ) = −
12 ( z + zγ ) σ X (1 − ρ ) B ( s ) − zrγ where the functions A ( · ) , B ( · ) and C ( · ) solve (3.17) – (3.18) . In the next section we will provide further conditions for the positiveness of Q (0) (see Proposition4.2). The dependence on t and z for the functions D, E, H in system (3.35) is omitted for ease of notation.
Proof.
From (3.25) and the fact that process (ln ξ ∗ , X, X ) is affine, it follows that φ s ( t, z ) = λ − z/γ E s (cid:104) ξ ∗ t − z/γ e zA ( t )+ zB ( t ) X t + zC ( t ) X t (cid:105) = λ − z/γ G ( s, ξ ∗ s , X s ; t, z ) where G ( s, ξ ∗ s , X s ; t, z ) = ξ ∗ s − z/γ e D ( s ; t,z )+ E ( s ; t,z ) X s + H ( s ; t,z ) X s . (3.37)The boundary conditions (3.36) follow from φ t ( t, z ) = W ∗ t z . The function G ( s, ξ, x ; t, z ) isdifferentiable with respect to s , and twice differentiable with respect to ξ and x . Moreover,by definition, the process ( G ( s, ξ s , X s ; t, z )) { s ∈ [0 ,t ] } is a martingale. Hence, by applying Itô’sformula we get ∂G∂s − rξ ∂G∂ξ − λ ( x − ¯ X ) ∂G∂x + 12 (cid:18) ∂ G∂x σ X + ξ ∂ G∂ξ ( ν (1 − ρ ) + x ) + 2 ξσ X ∂ G∂ξ∂x ( − ν (1 − ρ ) − xρ ) (cid:19) . (3.38)By plugging (3.37) into Equation (3.38) and collecting the constant term and the factorsof X and X , we get that D, E, H are the unique solution to problem (3.35)-(3.36) (seeFilipović (2009) [12, Lemma 10.1]). (cid:3)
We note that the solution to problem (3.35)-(3.36) assumes a simple form in the specialcase corresponding to the computation of the conditional expectation of W ∗ T − γ . In fact,from (3.33) and (3.34), we get E t [ u ( W ∗ T )] = 11 − γ φ t ( T, − γ ) = 11 − γ ( λ ξ ∗ t ) − /γ e D ( t ; T, − γ )+ E ( t ; T, − γ ) X t + H ( t ; T, − γ ) X t . Hence, from (3.31), D ( t ; T, − γ ) = A ( t ) ,E ( t ; T, − γ ) = B ( t ) ,H ( t ; T, − γ ) = C ( t ) . Such relations can also be directly verified by substituting z = 1 − γ and t = T in(3.35)-(3.36). 4. Optimal investment under partial information
In this section we assume that the investor observes only the stock prices and notthe market price of risk. Hence, the available information is carried by the filtration F S generated by the process S and the investor can only adopt F S -adapted portfolio strate-gies. Here the standard procedure is to apply separability and transform the optimizationproblem under partial information into an equivalent one by means of filtering, see, e.g.Fleming and Pardoux (1982) [14]. The first step of this procedure consists of replacing theunobservable quantities by their filtered estimates. In this way, the dynamics of stock priceand of the “filtered" market price of risk turn out to be driven by a single, one-dimensionalBrownian motion, the so called Innovation process . Hence, after this transformation, we
HE VALUE OF KNOWING THE MARKET PRICE OF RISK 17 are in a complete market model and, in the second step of the procedure, we can solvethe optimization problem by following the standard martingale approach.Let us consider the information filtration F S := ( F St ) t ∈ [0 ,T ] , where, at any time t ∈ [0 , T ] , F St := σ { S u , ≤ u ≤ t } ∨ N and N is the collection of P -null sets. We recall that F S is the trivial σ -algebra. We denote by π the conditional expectation of X , given theinformation flow, that is π t = E (cid:2) X t |F St (cid:3) , for every t ∈ [0 , T ] and by R the conditionalvariance, R t := E (cid:104)(cid:0) X t − E [ X t |F St ] (cid:1) |F St (cid:105) for every t ∈ [0 , T ] . It is well known that theconditional distribution of X is Gaussian and hence completely identified by the dynamicsof expectation and variance.To characterize these dynamics we introduce the innovation process I = { I t , t ∈ [0 , T ] } , I t := Z St + (cid:90) t ( X u − π u ) du, for every t ∈ [0 , T ] . Following Lipster and Shiryaev (2001) [27, Chapter 10], it can beproved that I is an ( F S , P ) -Brownian motion and that the processes π and R are theunique solutions to the system dπ t = − λ ( π t − ¯ X ) dt + ( R t + ρσ X ) dI t , π ∈ R ,dR t = (cid:2) σ X − λR t − ( R t + ρσ X ) (cid:3) dt, R ∈ R + . (4.1)From equation (4.1), we see that R t is a deterministic function of time. Therefore toemphasise this fact, from now on we will write R ( t ) instead of R t .The semimartingale representations with respect to the information filtration F S of thestock price process and of the wealth produced by a strategy θ are dS t S t = ( r + σπ t ) dt + σdI t ,dW t W t = ( r + θ t σπ t ) dt + θ t σdI t . (4.2)The investor wants to solve the problem max θ ∈A S ( w ) E (cid:20) − γ W − γT (cid:21) where A S ( w ) is the set of F S -predictable self-financing strategies satisfying the integra-bility condition (2.1) with initial wealth w . The state price density process in this case isunique and is given by d ˜ ξ t ˜ ξ t = − rdt − π t dI t , ˜ ξ = 1 . By the martingale method we formulate the equivalent static problem max W T E [ u ( W T )] , (4.3)subject to the constraint w = E [ ˜ ξ T W T ] . (4.4) We note that, since F S -predictable strategies are also F -predictable, the optimal utilityunder partial information is always lower than that under full information, and hence ifproblem (3.4)-(3.5) is bounded, problem (4.3)-(4.4) is also bounded.By the usual Lagrangian approach, since ˜ ξ is the state price density process, the optimalwealth satisfies ˜ W ∗ t = ˜ λ − γ ˜ ξ − t E [ ˜ ξ − γ T |F St ] where ˜ λ is the Lagrangian multiplier from the budget constraint (4.4).We can now state a verification theorem for the partial information setting. Theorem 4.1 (Verification Theorem under partial information) . Let the function F ( Y, π, t ) solve the equation F Y Y Y π + F πY Y π ( R + ρσ X ) + 12 F ππ ( R + ρσ X ) + F t = rF − rF Y Y + F π (cid:0) ( R + ρσ X ) π + λ ( π − ¯ X ) (cid:1) with boundary conditions F ( Y, π, T ) = Y γ , and F ( Y , π ,
0) = w for some constant Y > and F ( Y, π, t ) → F ( Y, π, T ) as t → T .Let Y t := Y (cid:16) ˜ ξ t (cid:17) − and assume that E (cid:20)(cid:90) T (cid:0) ( F Y ( Y t , π t , t ) π t ) + ( Y − t F π ( Y t , π t , t )) + ( Y − t π t F ( Y t , π t , t )) (cid:1) dt (cid:21) < ∞ . (4.7) Then the optimal wealth is ˜ W ∗ t = F ( Y t , π t , t ) and the optimal investment strategy is ˜ θ ∗ t = F Y ( Y t , π t , t ) Y t π t + ( R ( t ) + ρσ X ) F π ( Y t , π t , t ) σF ( Y t , π t , t ) , (4.8) for all t ∈ [0 , T ] .Proof. Similarly to the proof of Theorem 3.2, we need to show that the initial wealthsatisfies the budget constraint and that the final wealth satisfies the first order conditionand is attainable by a self-financing strategy.The budget constraint and the first order condition follow from (4.6). By Itô formulawe get dY t Y t =( r + π t ) dt + π t dI t . Hence, F ( Y t , π t , t ) = F ( Y , π ,
0) + (cid:90) t (cid:101) L F ( Y s , π s , s ) ds + (cid:90) t (( R t + ρσ X ) F π ( Y s , π s , s ) + Y s π s F Y ( Y s , π s , s )) dI s (4.9) HE VALUE OF KNOWING THE MARKET PRICE OF RISK 19 where (cid:101) L is the differential operator (cid:101) L F = F t + 12 F Y Y Y π + F πY Y π ( R t + ρσ X )+ 12 F ππ ( R t + ρσ X ) − F π λ ( π − ¯ X ) + F Y Y ( r + π ) . To show that the optimal wealth can be obtained by a self-financing strategy starting from w it remains to prove that the process ˜ ξ t F ( Y t , π t , t ) is a true martingale. By applying theproduct rule and using (4.5), we get ˜ ξ t F ( Y t , π t , t ) = F ( Y , π , (cid:90) t ˜ ξ s (( R t + ρσ X ) F π ( Y s , π s , s ) + Y s π s F Y ( Y s , π s , s ) − F ( Y s , π s , s ) π s ) dI s . By (4.7) and the fact that R t is the solution to the Riccati equation (4.1) on [0 , T ] , weget that the integral with respect to I is a true martingale. Then ˜ W ∗ t = F ( Y t , π t , t ) isthe optimal wealth process and the optimal investment strategy in (4.8) is obtained byequating the predictable covariation processes with respect to I from (4.2) and (4.9). (cid:3) To obtain a closed form representation for the optimal wealth we guess that E [ ˜ ξ − γ T |F St ] = ˜ ξ − γ t e ˜ A ( t )+ ˜ B ( t ) π t + ˜ C ( t ) π t where the functions ˜ A ( t ) , ˜ B ( t ) and ˜ C ( t ) satisfy the system of Riccati Equations d ˜ Cdt = ˜ a + ˜ b ( t ) ˜ C ( t ) + ˜ c ( t ) ˜ C ( t ) ,d ˜ Bdt = − ˜ C ( t ) λ ¯ X + (cid:32) ˜ b ( t )2 + ˜ c ( t ) ˜ C ( t ) (cid:33) ˜ B ( t ) ,d ˜ Adt = γ − γ r − ˜ B ( t ) λ ¯ X + 12 ˜ c ( t ) (cid:16) ˜ B ( t ) + ˜ C ( t ) (cid:17) (4.10)with boundary conditions ˜ A ( T ) = ˜ B ( T ) = ˜ C ( T ) = 0 , (4.11)where ˜ a = γ − γ , ˜ b ( t ) = 2 (cid:18) λ + γ − γ ( R ( t ) + ρσ X ) (cid:19) , ˜ c ( t ) = − ( R ( t ) + ρσ X ) . The solutions to the non-homogeneous system of Riccati equations (4.10)-(4.11) arerelated to the solutions of the homogeneous system (3.17)-(3.18) arising in the full infor-mation case. This fact, shown in the next proposition, will be exploited to get simplerexpressions for many quantities of interest.
Proposition 4.2.
Let the pairs of functions B ( t ) , C ( t ) and ˜ B ( t ) , ˜ C ( t ) satisfy the problems (3.17) - (3.18) and (4.10) - (4.11) on [0 , T ] , respectively and let Q ( t ) := 1 − γC ( t ) R ( t ) . Then, for all t in [0 , T ] , Q ( t ) is strictly positive and ˜ C ( t ) = Q ( t ) − C ( t ) , ˜ B ( t ) = Q ( t ) − B ( t ) . Moreover, the functions C ( t ) and ˜ C ( t ) are strictly positive and decreasing on [0,T] if γ < and are strictly negative and increasing if γ > .Proof. The fact that the function C ( t ) is strictly positive and decreasing on [0,T] if γ < and it is negative and increasing for γ > has been proven by Kim and Omberg (1996)[23, Equation (23)].The function Q ( t ) is continuous, hence the set T := { t ∈ [0 , T ] | Q ( t ) = 0 } is closed;we want to show that it is empty. By contradiction, let us assume that it is not emptyand let ¯ t be its maximum. From the boundary condition (3.18) we see that Q ( T ) = 1 ,hence ¯ t < T . Relations (4.12) and (4.13) hold in the set T C ∩ [0 , T ] , where T C isthe complement of T . In fact they follow from the fact that Q ( t ) − C ( t ) and Q ( t ) − B ( t ) satisfy (4.10)-(4.11) when C ( t ) , B ( t ) satisfy (3.17) -(3.18), as it can be shown by followingBrendle (2006) [7, Equations (28)-(29)]. Therefore, for any (cid:15) > such that ¯ t + (cid:15) < T , Q (¯ t + (cid:15) ) ˜ C (¯ t + (cid:15) ) = C (¯ t + (cid:15) ) and, by continuity of all the functions involved in the equality, Q (¯ t ) ˜ C (¯ t ) = C (¯ t ) . Since C ( t ) is a monotone function (either increasing or decreasing,depending on the parameter γ ) and C ( T ) = 0 , then C (¯ t ) (cid:54) = 0 , hence ¯ t / ∈ T , that is acontradiction and T is the empty set.Since T is empty, Q ( t ) is continuous on [0 , T ] and C ( T ) = 1 , it follows that Q ( t ) isstrictly positive on [0 , T ] , hence the functions C ( t ) and ˜ C ( t ) must have the same sign(positive for γ < and negative for γ > ).Finally, we prove that for γ < , ˜ C ( t ) is strictly decreasing on [0 , T ] . Consider theequation d ˜ C ( t ) dt = f ( ˜ C ( t )) , where f ( ˜ C ( t )) is the right hand side of the first equation in (4.10)-(4.11). The boundarycondition implies that ˜ C ( T ) = 0 and that f (0) = γ − γ < . Then the function f ( t ) mustbe negative on [0 , T ] for the boundary condition to be satisfied and hence ˜ C ( t ) is strictlydecreasing. The same argument applies to the case γ > where the derivative of ˜ C ( t ) ispositive and hence ˜ C ( t ) is strictly increasing. (cid:3) We remark that from (4.12)-(4.13), we can get an explicit expression for ˜ B ( t ) and ˜ C ( t ) from those of B ( t ) and C ( t ) . Then ˜ A ( t ) can be obtained explicitly by integrating theright hand side of the third equation in system (4.10)-(4.11). HE VALUE OF KNOWING THE MARKET PRICE OF RISK 21
We are now ready to determine the optimal wealth and the optimal investment strategyfor the partial information problem.
Theorem 4.3.
Let the functions ˜ A ( t ) , ˜ B ( t ) and ˜ C ( t ) satisfy (4.10) – (4.11) on [0 , T ] andlet ˜ λ = (cid:34) e ˜ A (0)+ ˜ B (0) π + ˜ C (0) π w (cid:35) γ . Assume that E (cid:34)(cid:90) T (cid:18) ˜ ξ − γ t e ˜ A ( t )+ ˜ B ( t ) π t + ˜ C ( t ) π t (cid:19) (1 + π t ) (cid:35) < ∞ . (4.14) Then ˜ λ is the Lagrangian multiplier from the budget constraint (4.4) and the optimalwealth and the optimal investment strategy are given by ˜ W ∗ t = (˜ λ ˜ ξ t ) − γ e ˜ A ( t )+ ˜ B ( t ) π t + ˜ C ( t ) π t , (4.15) ˜ θ ∗ t = 1 γ π t σ + ( R ( t ) + ρσ X ) σ ( ˜ B ( t ) + ˜ C ( t ) π t ) , for every t ∈ [0 , T ] .Proof. The proof follows from the same argument of the analogous result under full in-formation, Theorem 3.3, and hence is omitted. (cid:3)
In the next proposition we provide sufficient conditions to apply Theorem 4.3 that areeasier to check for a given set of parameters.
Proposition 4.4.
Let the functions ˜ A ( t ) , ˜ B ( t ) and ˜ C ( t ) satisfy (4.10) – (4.11) on [0 , T ] and assume that at least one of the following two holds (i) γ > (ii) The functions A ( t ) , B ( t ) and C ( t ) satisfy (3.17) - (3.18) on [0 , T ] and − C (0) Q (0) max (cid:18) R , R e − λT + σ X λ (1 − e − λT ) (cid:19) > . (4.16) Then all assumptions of Theorem 4.3 are satisfied.Proof.
We only need to show that the integrability condition (4.14) in Theorem 4.3 issatisfied.Using Fubini and the Cauchy Schwartz inequality we get E (cid:34)(cid:90) T e A ( t ) (cid:18) ˜ ξ − γ t (cid:19) e B ( t ) π t + ˜ C ( t ) π t (1 + π t ) dt (cid:35) ≤ κ (cid:90) T e A ( t ) E (cid:20) ˜ ξ − γ ) t (cid:21) E (cid:104) e B ( t ) π t +2 ˜ C ( t ) π t (cid:105) E [(1 + π t ) ] dt. Since π t is Gaussian, E [(1 + π t ) ] < ∞ . The expectation E (cid:20) ˜ ξ − γ ) t (cid:21) is finite since π is Ornstein-Uhlenbeck (see, Revuz and Yor (2013)[31, Chaper 8, Ex. 3.14]). Finally, E (cid:104) e B ( t ) π t +2 ˜ C ( t ) π t (cid:105) is finite for all t ∈ [0 , T ] if and only if − C ( t )˜ v t > where ˜ v t = v t − R t is the variance of π t (and v t is the variance of X t ).If γ > , from Proposition 4.2, ˜ C ( t ) < . Hence − C ( t )˜ v t > and (4.14) is satisfied.If γ < , still from Proposition 4.2 ˜ C ( t ) is strictly positive and decreasing in [0 , T ] .Therefore − C ( t )˜ v t > − C (0) v t ≥ − C (0) Q (0) max (cid:18) R , R e − λT + σ X λ (1 − e − λT ) (cid:19) , where the first inequality follows from the monotonicity of ˜ C and from the fact that ˜ v t < v t . The second inequality follows from (4.12) and from the fact that v t is alwayslower than its maximum value on [0 , T ] that is equal to R or to v T depending on R ( t ) being decreasing or increasing. Then the result follows immediately from (4.16). (cid:3) Now we can compute the conditional moment generating function of the optimal wealthunder the partial information, ˜ φ s ( t, z ) := E (cid:104) ( ˜ W ∗ t ) z |F Ss (cid:105) . Proposition 4.5.
Let ˜ φ s ( t, z ) < ∞ for ≤ s ≤ t ≤ T and z > .Then ˜ φ s ( t, z ) = (˜ λ ˜ ξ s ) − zγ e ˜ D ( s ; t,z )+ ˜ E ( s ; t,z ) π s + ˜ H ( s ; t,z ) π s where ˜ D : [0 , t ] → R , ˜ E : [0 , t ] → R and ˜ H : [0 , t ] → R satisfy the system of differentialequations d ˜ Hds = ˜ d ( s ) + 2˜ e ( s ) ˜ F ( s ) + ˜ f ( s ) ˜ H ( s ) d ˜ Eds = (cid:16) ˜ e ( s ) + ˜ f ( s ) ˜ H ( s ) (cid:17) ˜ E ( s ) − λ ¯ X ˜ H ( s ) d ˜ Dds = − zrγ − λ ¯ X ˜ E ( s ) + 12 ˜ f ( s )( ˜ H ( s ) + ˜ E ( s ) ) with boundary conditions ˜ D ( t ) = z ˜ A ( t ) , ˜ E ( t ) = z ˜ B ( t ) , ˜ H ( t ) = z ˜ C ( t ) , (4.17) Note that the functions ˜ D ( s ) , ˜ E ( s ) and ˜ H ( s ) depend on t and z . We do not report such dependenceinto the formulas for a simpler notation. HE VALUE OF KNOWING THE MARKET PRICE OF RISK 23 and ˜ d ( s ) = − (cid:18) z γ + zγ (cid:19) , ˜ e ( s ) = λ − zγ ( R ( s ) + ρσ X ) , ˜ f ( s ) = − ( R ( s ) + ρσ X ) , for every s ≤ t , and where the functions ˜ A , ˜ B and ˜ C satisfy (4.10) - (4.11) .Proof. The proof replicates the steps of the proof of Proposition 3.5. Using that theprocess (ln ˜ ξ, π, π ) is affine we have ˜ φ s ( t, z ) = ˜ λ − z/γ E (cid:104) ˜ ξ t − z/γ e z ˜ A ( t )+ z ˜ B ( t ) π t + z ˜ C ( t ) π t |F Ss (cid:105) = ˜ λ − z/γ ˜ G ( s, ˜ ξ s , π s ; t, z ) where ˜ G ( s, ˜ ξ s , π s ; t, z ) = ˜ ξ s − z/γ e ˜ D ( s ; t,z )+ ˜ E ( s ; t,z ) π s + ˜ H ( s ; t,z ) π s . The boundary conditions (4.17) follow from ˜ φ t ( t, z ) = ( ˜ W ∗ t ) z . The function ˜ G ( s, ˜ ξ, π ; t, z ) is differentiable with respect to s , and twice differentiable with respect to ˜ ξ and π . More-over, by definition, the process ( ˜ G ( s, ˜ ξ s , π s ; t, z )) { s ∈ [0 ,t ] } is a martingale with respect tofiltration F S . Hence, by applying Itô’s formula we get that the function ˜ G satisfies theequation ∂ ˜ G∂s − r ˜ ξ s ∂ ˜ G∂ ˜ ξ − λ ( π s − ¯ X ) ∂ ˜ G∂π + 12 (cid:32) ∂ ˜ G∂π ( R s + ρσ X ) + ˜ ξ s π s ∂ ˜ G∂ ˜ ξ − ξ s π s ∂ ˜ G∂ ˜ ξ∂π ( R s + ρσ X ) (cid:33) . This completes the proof. (cid:3)
Note that, also under partial information, formulas simplify when t = T and z = 1 − γ ,in fact: ˜ D ( s ; T, − γ ) = ˜ A ( s ) , ˜ E ( s ; T, − γ ) = ˜ B ( s ) , ˜ H ( s ; T, − γ ) = ˜ C ( s ) . This allows to compute the optimal expected utility in closed form, since E (cid:104) u ( ˜ W ∗ T ) |F Ss (cid:105) = 11 − γ ˜ φ s ( T, − γ )= 11 − γ (˜ λ ˜ ξ s ) − /γ e ˜ A ( s )+ Q ( s ) − B ( s ) π s + Q ( s ) − C ( s ) π s = 11 − γ ( ˜ W ∗ s ) − γ e γ ( ˜ A ( s )+ Q ( s ) − B ( s ) π s + Q ( s ) − C ( s ) π s )4 K. COLANERI, S. HERZEL, AND M. NICOLOSI
Note that, also under partial information, formulas simplify when t = T and z = 1 − γ ,in fact: ˜ D ( s ; T, − γ ) = ˜ A ( s ) , ˜ E ( s ; T, − γ ) = ˜ B ( s ) , ˜ H ( s ; T, − γ ) = ˜ C ( s ) . This allows to compute the optimal expected utility in closed form, since E (cid:104) u ( ˜ W ∗ T ) |F Ss (cid:105) = 11 − γ ˜ φ s ( T, − γ )= 11 − γ (˜ λ ˜ ξ s ) − /γ e ˜ A ( s )+ Q ( s ) − B ( s ) π s + Q ( s ) − C ( s ) π s = 11 − γ ( ˜ W ∗ s ) − γ e γ ( ˜ A ( s )+ Q ( s ) − B ( s ) π s + Q ( s ) − C ( s ) π s )4 K. COLANERI, S. HERZEL, AND M. NICOLOSI where we have used the explicit expression of ˜ C and ˜ B in terms of C and B , given in(4.12)-(4.13) and where the last equality is obtained from (4.15).5. The value of information
We are now ready to define the value of information , that is to assign a monetary valueto the possibility of improving the knowledge of the market price of risk. We start bycomputing the reservation price, that is the maximal amount of money that a partiallyinformed investor would be willing to pay to get extra information. We will focus on twokinds of information, which we call initial and dynamic . While the initial informationgives the exact knowledge of X that is the value of the market price of risk at time ,the dynamic information provides the run-time values X t at all times t ∈ [0 , T ] .A partially informed investor, endowed with a starting wealth w , with a prior X ∼ N ( π , R ) obtains, at time T , the final (optimal) wealth ˜ W ∗ T ( w ) . Let us now assume thatthe value assumed by X is revealed to the investor at time . Then she will be able toimplement the optimal strategy, still under partial information because the following pathof X will remain unknown to her, but this time starting from the exact value X . Let usdenote by ˜ W IT ( w ) the optimal wealth obtained at time T , where the index I highlightsthe Initial information case. When dynamic information is provided to the investor, shewill reach the wealth produced at time T by the optimal strategy under full information,that is W ∗ T ( w ) . Since the sets of feasible strategies for the three scenarios are strictlyincreasing, the following inequalities hold E (cid:104) u ( ˜ W ∗ T ( w )) (cid:105) ≤ E (cid:104) u ( ˜ W IT ( w )) (cid:105) (5.1) ≤ E [ u ( W ∗ T ( w ))] . (5.2)The maximum amount that the investor is willing to pay to receive the initial informationis the quantity ∆ w < w that satisfies E [ u ( ˜ W ∗ T ( w ))] = E (cid:104) u ( ˜ W IT ( w − ∆ w )) (cid:105) . (5.3)Notice that ∆ w > because of (5.1) and the fact that the expected utility is increasingwith respect to the initial wealth. From (4.18) computed for s = 0 we get E (cid:104) u ( ˜ W ∗ T ( w )) (cid:105) = w − γ − γ e γ ( ˜ A ( R )+ Q − B π + Q − C π ) (5.4)where we use the notation Q := Q (0) , B := B (0) , C := C (0) and ˜ A ( R ) := ˜ A (0) tohighlight the dependence of ˜ A (0) on R . With an analogous computation, setting π = X and R = 0 , we get E (cid:104) u ( ˜ W IT ( w − ∆ w )) | X (cid:105) = ( w − ∆ w ) − γ − γ e γ ( ˜ A (0)+ B X + C X ) . HE VALUE OF KNOWING THE MARKET PRICE OF RISK 25
Hence, the right hand side of (5.3) is E (cid:104) E (cid:104) u ( ˜ W IT ( w − ∆ w )) | X (cid:105)(cid:105) = ( w − ∆ w ) − γ (1 − γ ) √ Q e γ ˜ A (0)+ γ Q ( γB R +2 B π + C π ) (5.5)which holds when Q = 1 − γC R > . Solving equation (5.3) using the explicit expres-sions in (5.4) and (5.5) we get ∆ w and we can define the Value of Initial Information V I as the ratio ∆ w/w , that is V I = 1 − (cid:18)(cid:112) Q e γ ( ˜ A ( R ) − ˜ A (0)) − γ B R Q (cid:19) − γ . (5.6)We remark that V I does not depend on the expected value of the market price of risk π but only on the variance of the initial estimate R .Let us now compute the reservation price for the dynamic information. Let the quantity ∆ w be the solution to the equation E (cid:104) u ( ˜ W ∗ T ( w )) (cid:105) = E [ u ( W ∗ T ( w − ∆ w ))] . (5.7)Inequality (5.2) implies that < ∆ w < w . To compute the right hand side of (5.7)we use equation(3.32). The left hand side of (5.7) is given in (4.18) for s = 0 . Againwe shorten the notation by using A = A (0) , B = B (0) , C = C (0) , Q = Q (0) and ˜ A ( R ) = ˜ A (0) . Hence, we can extract the reservation price ∆ w from (5.7) and definethe Value of Dynamic Information V D as the ratio ∆ w/w , that is V D = 1 − (cid:18)(cid:112) Q e γ ( ˜ A ( R ) − A ) − γ B R Q (cid:19) − γ . (5.8)From inequalities (5.1)- (5.2) we get < V I ≤ V D < . (5.9)We remark that the expression for V D can be obtained from that of V I (5.6) by replacing ˜ A (0) with A . We also note that V D does not depend on the expected value of the marketprice of risk. 6. Applications
In this section we discuss some applications of our results with the parameters of Table1 obtained from the estimates provided by Xia (2001) [34, Table I] on the U.S. stockmarket, from 1950 to 1997.Figure 1 shows the optimal penalization factor ν ∗ for the incomplete market under fullinformation derived in (3.22), as a function of the risk aversion parameter γ , for threedifferent values of the correlation between the stock price process and the market priceof risk and assuming X = π . We see that ν ∗ grows in absolute value as γ tends tozero. This is explained by the fact that investors with smaller risk-aversion need a higherpenalization factor (i.e. greater in absolute value) to be diverted from unattainable claims.The case γ = 1 corresponds to logarithmic utility. Here no penalty is necessary: investor r σ λ σ X ¯ X π R S W T γ
Table 1.
Parameter set adopted in this section (expressed on a yearlybasis and derived from [34, Table I])is myopic and selects only attainable claims (see Remark 1). For γ larger than the sizeof ν ∗ is first increasing and then decreasing towards zero. In fact, for values of γ slightlylarger than , investors are less myopic and attracted by not marketed claims. Wheninvestors are more risk averse they put a larger part of their wealth in the risk-free asset,and hence the penalization becomes again less necessary. -25-20-15-10-50510 = -0.80 = 0.00 = 0.80 Figure 1.
The optimal penalization factor ν ∗ , Equation (3.22), as a func-tion of the risk aversion parameter γ , when X = π , for correlations: ρ = 0 . , continuous line; ρ = 0 dotted line; ρ = − . dashed line.Figure 2 represents the critical time T ∗ , given by (3.21) , that is the maximal horizonof existence for the solution to (3.17)-(3.18), as a function of γ for two values of thecorrelation ρ . The analysis of existence of the system of Riccati equations states that T ∗ is finite when ρ is larger than ρ ∗ (cid:39) − . given by (3.19), and for values of γ smallerthan the value γ ∗ defined by (3.20). In this case γ ∗ is equal to . when ρ = 0 , andto . when ρ = 0 . . When ρ = 0 Figure 2 (left panel) shows that the critical timecorresponding to γ = 0 . is about 20 years and it becomes larger than 20 years for γ > . .In other words, for γ > . the solution to the system (3.17)-(3.18) is well defined up toan investment horizon of at least 20 years, and it is well defined for any horizon when γ > γ ∗ . In the same plot we also report T ∗∗ which is the maximal time such that (3.27)is satisfied. Remind that (3.27) is a sufficient condition for the optimal wealth under full HE VALUE OF KNOWING THE MARKET PRICE OF RISK 27 information W ∗ t to be expressed as in (3.25). When ρ > ρ ∗ and γ < γ ∗ there is a largeregion in the plane γ, T where the solution to the Riccati system if well defined but (3.27)does not hold, hence to state that formula (3.25) provides the optimal wealth, one shouldprove that the more general condition (3.24) of Theorem 3.3 holds. -1 = 0 T * T ** -1 = 0.8 T * T ** Figure 2.
Critical times T ∗ (continuous line), Equation (3.21), for thesystem of Riccati equations (3.17)-(3.18), as a function of γ , for correlations ρ = 0 (left panel) and ρ = 0 . (right panel). Superimposed the maximaltime T ∗∗ (dotted line) such that (3.27) is satisfied.Propositions 3.5 and 4.5 characterize the moment generating functions of the optimalwealth under full or partial information. Applying those results and Fast Fourier Trans-form we can compute the corresponding probability distributions very efficiently. Figure 3represents the probability density functions of the optimal wealth in T for a fully informedinvestor with γ = 4 . and for a partially informed one with γ = 2 . . We also plot theempirical distributions, obtained by simulations, for a visual check of the precision of ourcode. The values of γ have been chosen so that the expected returns of the two strategiesare equal to . Albeit with the same mean, the two distributions have very differentshapes, with the full information density being more skewed and with a heavier righttail. This has interesting consequence on the mean-variance curves corresponding to thefull and the partial information investment strategies for different level of risk aversions,represented in Figure 4. To connect Figure 4 and Figure 3 we also indicate the pointscorresponding to the two values of γ for which we computed the densities. We see that thecurve of expected returns under partial information dominates the full information one.Hence, if an investor following a mean-variance criterion (as, for instance, maximizing theSharpe ratio of her investment) had to choose between optimal strategies under full orunder partial information, she would always select the partial information one. This isa consequence of the heavier right tail of the wealth distribution under full information(clearly shown in Figure 3), a feature not much appreciated by a mean-variance kind ofinvestor. Partial InformationFull Information
Figure 3.
Probability distribution for the optimal final wealth under full(continuous line) and partial (dotted line) information starting from w = 1 .The two distributions have the same mean, and are obtained by setting γ = 2 . for the partially informed investor and γ = 4 . for the fullyinformed one. Standard deviation of returns E x pe c t ed r e t u r n s Full InformationPartial InformationS = 4.03 = 2.08
Figure 4.
Expected returns of optimal strategies under full (continuousline) or partial (dotted line) information as functions of their standard de-viations. The points obtained for γ = 2 . with partial information and γ = 4 . with full information are reported, for reference with Figure 3.The point S represents the risky asset.The cumulative probability distributions for the optimal final wealth under full andpartial information are represented in Figure 5. The plot shows that the optimal wealthunder full information (continuous line) stochastically dominates the optimal wealth inpartial information (dotted line). However such a dominance is lost if the partially in-formed investor adds to the initial budget w the reservation price for Dynamic Information ∆ w . In this case, by definition, the investor attains the same expected utility as the fully HE VALUE OF KNOWING THE MARKET PRICE OF RISK 29 informed investor and hence her optimal wealth is sometimes lower sometimes higher thanthe one obtained by the fully informed investor. W T* FI, w = 1PI, w = 1PI, w = 1+Value of Information Figure 5.
Cumulative probability distribution for the optimal final wealthunder full (continuous line) and partial (dotted line) information startingfrom w = 1 and for the optimal wealth under partial information startingfrom w = 1 + ∆ w (dashed line), where ∆ w is the reservation price ofDynamic Information .The certainty equivalent of the optimal utility under partial information with respect tothe initial conditional variance R , computed from (5.4), is represented in Figure 6. Theexpected utility does not always grow as the precision of the initial estimates increases.In particular, for different values of ρ , the certainty equivalent is either increasing or ittakes on the minimum value within the interval (0 . , . The intuitive explanation forthis fact is that, when the expected value of the market price of risk π is fixed, a greateruncertainty on its estimate may increase the likelihood of a better or a more favorableoutcome, consequently raising the expected utility of the optimal wealth.Figure 7 shows the value of the Initial Information V I (see Equation (5.6)) as a functionof R , for three values of the correlation ρ . As expected, the higher the uncertainty onthe initial estimate, the higher V I . It is perhaps less expected that the value is greater for ρ = − . than for the other two cases. Why is the investor willing to pay a larger shareof her initial wealth when the correlation of the changes in the market price of risk withthe stock returns is more negative? In our opinion, this is a combination of two effects:the first effect is related to the precision of the estimate of the market price of risk, thesecond effect to the expected return of the optimal strategy. To explain the first effect, wenote that, when ρ = 0 . , the variance of the estimate, R ( t ) , decreases faster to the steadystate R ∞ = 0 . , while for ρ = − . and ρ = 0 , it decreases, at a slower rate, towards R ∞ = 0 . and R ∞ = 0 . , respectively. Hence a more accurate information on X must be worth less when ρ = 0 . . As for the second effect, the certainty equivalent of theoptimal strategy under partial information when R = 0 obtained from (4.18), is . of the initial wealth for ρ = − . , . for ρ = 0 and . for ρ = 0 . . Therefore,when ρ = − . , the investor is expecting a higher return, and hence she is willing toinvest a larger share of her initial wealth to know the exact value of X . Initial conditional variance R C e r t a i n t y equ i v a l en t = -0.9 = 0.0 = 0.9 Figure 6.
The certainty equivalent under partial information computedfrom (5.4) as a function of the initial variance of the estimate R . Initial conditional variance R = -0.9 = 0.0 = 0.9 Figure 7.
The value of Initial Information (5.6) as a function of the initialvariance of the estimate R , for different correlation ρ .Figure 8 presents the ratio of the value of Dynamic Information V D , (5.8), over thevalue of Initial Information V I (5.6), as a function of the initial uncertainty R , and fordifferent values of ρ . The ratio is always positive and greater than because of (5.9).It is decreasing with R and converges to a constant as R increases. When R goes tozero, V I also goes to zero while V D converges to a positive value, hence the ratio growsunbounded. The ratio is larger for ρ = 0 and the difference between ρ = 0 . and ρ = − . is small. Intuitively, when the correlation is close to or − , the knowledge of the starting HE VALUE OF KNOWING THE MARKET PRICE OF RISK 31
Initial conditional variance R = -0.9 = 0.0 = 0.9 Figure 8.
The ratio of the values of information: Dynamic Information,Equation (5.8), over Initial Information, Equation (5.6), as a function ofthe initial uncertainty R .value for the process X is sufficient to estimate with good precision also its next values,and hence the value added by the full knowledge of X is low (for our set of parametersit is around of the value of knowing only X ). Instead, when there is no correlation( ρ = 0 ), knowing X alone is not sufficient to get a good future estimate for X , and hencethe value added by the dynamic information is more appreciated by the investor.Figure 9 provides the ratio V D / V I as a function of the investment horizon T , for a fixedvalue R = 0 . . The ratio increases almost linearly with T , but more steeply for ρ = 0 ,that is when having access to a dynamic information on the market price of risk adds asignificant improvement to the investment policy. T = -0.9 = 0.0 = 0.9 Figure 9.
The ratio of the values of information: dynamic information(5.8) over initial information (5.6), as a function of the length of the invest-ment period T , for fixed γ = 5 . Conclusions
We studied a portfolio optimization problem for an investor who aims to maximize herexpected utility from terminal wealth under two different hypotheses on the informationflows when the market price of risk is stochastic and mean-reverting. We solved theproblem via the martingale approach and found an explicit representation for the optimalwealth and its associated utility as function of the current state-price density process andof the market price of risk X in the full information case, or of its best estimate π underpartial information. We also provided verification theorems for our results.We introduced the notion of value of information as the maximum percentage of theinitial wealth that an investor would be willing to pay to access to more accurate informa-tion on the market price of risk X . In particular we considered the value of knowing thewhole path of X on-the-run and the value of knowing only its initial value X . Using thestructure of the solutions of the Riccati equations that characterize the optimal wealth,we determined closed form representations of such values. We provided applications toillustrate some consequences of our results. The empirical analysis of the distribution ofthe optimal wealth under full and partial information showed several features that couldnot be guessed a priori, like for instance the fact that, under our parameter setting, an in-vestor who cares for the Sharpe ratio of her investment would better allocate her wealth toa partially informed portfolio manager rather than to a fully informed one. Our measurefor the value of information can be applied to real market data, for example to determinein which periods of time the access to a better knowledge on the market price of risk ismore valuable. Our approach may also be used to assess the value of an improvement ofthe initial prior on the market parameters, and consequently to address issues related tothe evaluation of model error. Acknowledgements
The authors would like to thank the Referees for their useful suggestions. The work on thispaper initiated when Katia Colaneri was visiting the Department of Economics, Universityof Perugia, as a part of the ACRI Young Investigator Training Program (YITP): the As-sociation of Italian Banking Foundations and Savings Banks (ACRI) partially supportedthis work. Katia Colaneri also received partial financial support from INdAM-GNAMPAunder grant UFMBAZ- 2018/000349 and UFMBAZ-2019/000436. The research of Ste-fano Herzel and Marco Nicolosi was partially funded by the Swedish Research Councilgrant 2015-01713. Part of this article was written while Katia Colaneri was affiliated withthe School of Mathematics of the University of Leeds (UK).
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