Decomposition of Optimal Dynamic Portfolio Choice with Wealth-Dependent Utilities in Incomplete Markets
aa r X i v : . [ q -f i n . P M ] A p r Decomposition of Optimal Dynamic Portfolio Choicewith Wealth-Dependent Utilities in Incomplete Markets ∗ Chenxu Li † Guanghua School of Management
Peking University
Olivier Scaillet ‡ University of Geneva andSwiss Finance Institute
Yiwen Shen § Graduate School of BusinessColumbia University
April 22, 2020
Abstract
This paper establishes a new decomposition of optimal dynamic portfolio choice under generalincomplete-market diffusion models by disentangling the fundamental impacts on optimal policyfrom market incompleteness and flexible wealth-dependent utilities. We derive explicit dynamicsof the components for the optimal policy, and obtain an equation system for solving the shadowprice of market incompleteness, which is found to be dependent on both market state and wealthlevel. We identify a new important hedge component for non-myopic investors to hedge theuncertainty in shadow price due to variation in wealth level. As an application, we establish andcompare the decompositions of optimal policy under general models with the prevalent HARAand CRRA utilities. Under nonrandom but possibly time-varying interest rate, we solve in closed-form the HARA policy as a combination of a bond holding scheme and a corresponding CRRAstrategy. Finally, we develop a simulation method to implement the decomposition of optimalpolicy under the general incomplete market setting, whereas existing approaches remain elusive.
Keywords: optimal portfolio choice, decomposition, incomplete market, wealth-dependentutility, closed-form.
JEL Codes:
C61, C63, G11. ∗ We would like to thank T. Berrada, J. Detemple, I. Karatzas, H. Langlois, P. Glasserman, and M. Rindisbacher for valuablecomments and suggestions. We also benifited from the comments of participants at the 2018 3rd (resp. the 2019 4th) PKU-NUS Annual International Conference on Quantitative Finance and Economics, the 2019 Myron Scholes Financial Forum atNanjing University, and seminar at Columbia University. The research of Chenxu Li was supported by the Guanghua School ofManagement, the Center for Statistical Science, and the Key Laboratory of Mathematical Economics and Quantitative Finance(Ministry of Education) at Peking University, as well as the National Natural Science Foundation of China (Grant 71671003). † Address: Guanghua School of Management, Peking University, Beijing, 100871, P. R. China. E-mail address: [email protected]. ‡ Address: University of Geneva and Swiss Finance Institute, Bd du Pont d’Arve 40, CH - 1211 Geneve 4, Suisse. E-mailaddress: [email protected]. § Address: Graduate School of Business, Columbia University, New York, New York 10027, U.S.A. E-mail address:
[email protected]. Introduction
Optimal portfolio choice is a central topic in modern financial economics, drawing continual at-tention from both industry and academia. Hedge funds, asset management firms, and pensionfunds, which manage large positions of portfolios, as well as individual investors, are confrontedwith this type of decision frequently. Due to its own importance or as an indispensable tool am-ply applied in both theoretical and empirical literature, the optimal portfolio choice problem hasalso drawn long-standing interest in academia. The celebrated static mean-variance framework ofMarkowitz (1952) laid a foundation. Following the seminal work by Samuelson (1969) and Merton(1969, 1971), various studies have been developed for the optimal dynamic portfolio choice; see thecomprehensive surveys in,e.g., Detemple (2014), Brandt (2010), and Wachter (2010), as well as thereferences therein. In continuous-time setting, it is an optimal stochastic control problem, that com-bines stochastic modeling and optimization techniques. In early works, a large number of relevantcontributions relied on the dynamic programming approach, which employs the highly nonlinearHamilton-Jacobi-Bellman (HJB hereafter) equation to characterize the optimal policy. An alter-native notable approach is the martingale method pioneered and developed by, e.g., Pliska (1986),Karatzas et al. (1987), Cox and Huang (1989), Ocone and Karatzas (1991), Cvitanic and Karatzas(1992), and Detemple et al. (2003). Koijen (2014) explains how we can also use the martingaleapproach to estimate continuous-time optimization models.For the purpose of understanding and analyzing the behavior of optimal portfolios, existing workslargely focus on some specific affine models (see, e.g., Duffie et al. (2000)) and constant relative riskaversion (CRRA hereafter) utilities that yield closed-form optimal portfolio policies, though suchanalytically tractable cases are rare and limited. As an effective method for applying flexiblemodels without closed-form optimal portfolio policies, Detemple et al. (2003) further developed theaforementioned martingale approach and derived, at the theoretical level, an explicit decompositionof the optimal policy under general diffusion models as well as flexible utilities, and consequentlypioneered a flexible Monte Carlo simulation approach for implementation. See also Cvitanic et al.(2003) for an alternative simulation approach. However, this milestone of methods is by far limitedto the complete market setting. Besides simulation, other numerical methods were proposed; see, See, e.g., Kim and Omberg (1996) and Wachter (2002) for modeling stochastic market price of risk of the asset byusing an Ornstein-Uhlenbeck model, Lioui and Poncet (2001) for considering stochastic interest rates by employing aconstant-parameter instantaneous forward rate model, Liu and Pan (2003) for discussing dynamic derivative strategies,Liu et al. (2003) for studying impacts of event risk via affine stochastic volatility models with jumps, Liu (2007)for taking various stochastic environments (e.g., stochastic volatility) into account by modeling the asset returnsvia quadratic affine processes, Burraschi et al. (2010) for characterizing hedging components against both stochasticvolatility and correlation risk under Wishart processes, and Moreira and Muir (2019) allows for variation in volatilityand mean reversion in returns. We refer to the recentbook of Dumas and Luciano (2017) for a survey of different numerical methods available for optimalportfolio choice. It is a common consensus that the notable challenge lies in the incomplete marketsettings at not only the theoretical but also the numerical and even empirical levels.In this paper, we develop and implement a new decomposition for the optimal policy in generalincomplete market models, under which we cannot fully hedge the risk by investing in the riskyassets. Our contribution to the literature holds for general diffusion models of assets prices andstate variables, with flexible utilities (rather than limited to those of the CRRA type) over bothintermediate consumption and terminal wealth. The optimal policy is decomposed to the mean-variance component, the interest rate hedge component, and the price of risk hedge component,which are all functions of current market state variable and investor wealth level. This type ofdecomposition reconciles the seminal work in Merton (1971). In the decomposition, we explicitlyexpress each component of the optimal policy as conditional expectation of suitable random variablesunderlaid by sophisticated but explicit dynamics, with the necessary aid of Malliavin calculus (seean accessible survey of Malliavin calculus for finance in Appendix D of Detemple et al. (2003)).Our decomposition fundamentally and substantially extends the representation results under thecomplete market setting in Detemple et al. (2003) to general incomplete market models, and thuslays an important foundation for developing subsequent theoretical analysis, numerical methods, andempirical studies.To handle the market incompleteness, we apply and explore the “least favorable completion”principle developed by Karatzas et al. (1991) under general diffusion models. It completes the mar-ket by introducing suitable fictitious assets. Then, the equivalence between the optimal policy in thecompleted market and that in the original market is established via choosing the appropriate priceof risk associated with those fictitious assets. Such a price of risk is endogenously determined by theinvestor utility function and the investment horizon, and thus is referred to as the investor-specificprice of risk under market incompleteness. It is also known as the “shadow price” of market incom-pleteness in the literature; see, e.g., Detemple and Rindisbacher (2010). We begin by showing andapplying the following structure. Under general incomplete market models with wealth-dependentutilities, the appropriate investor-specific price of risk not only depends on current market state, butalso implicitly depends on investor’s wealth level. The latter dependence is completely absent in themarket price of risk associated with the real assets. Consequently, a new important hedge componentemerges in the optimal policy for hedging the uncertainty in investor-specific price of risk due to thevariation in investor wealth level. This reveals an additional type of hedging demand when investorsallocate their portfolio in incomplete markets, which essentially arises from the market incomplete- See, e.g., Fitzpatrick and Fleming (1991), Brennan et al. (1997), Hindy et al. (1997), Brennan (1998),Chacko and Viceira (2005), Brennan and Xia (2002), and Campbell et al. (2004). , our equation system offers more generality, explicitness, and analytical conveniencefor further analysis.Equipped with the decomposition results for general incomplete market models, we study how theoptimal policy is fundamentally impacted by market incompleteness and wealth-dependent utilities.Specifically, we derive the corresponding decomposition results for complete market models andincomplete market models with the wealth-independent CRRA utility, and compare them with theresult for the general incomplete market models. This new analytical contribution allows us toanalyse the impacts, not yet addressed in the literature, of market incompleteness and wealth-dependence utility. In two special cases, we show that the dynamics of components underlyingthe optimal policy are fundamentally different compared with the general case. Besides, the newcomponent in the optimal policy, which hedges the fluctuation in investor-specific price of risk due tovariation in wealth level, vanishes in these two special cases. These two comparative studies show thefundamental differences between our decomposition and the existing ones in, e.g., Detemple et al.(2003), Detemple and Rindisbacher (2005, 2010), and Detemple (2014). Obviously, the departurefrom the complete market setting and wealth-independent CRRA utility has both technical andeconomic implications on the resulting optimal policy, and we cannot simply use the results currentlyavailable in the literature for this investigation.Our decomposition of optimal policy under general incomplete market models with flexible utilityfunctions advances the frontier of related theoretical studies and offers a broader foundation for con-ducting relevant analysis. As the first application, among many others, of our representation results,we apply our new decomposition to disentangle the optimal policy under general incomplete marketmodels with hyperbolic absolute risk aversion (HARA) utility, which, compared with the CRRAutility, is more flexible and effective in reflecting investor preference due to its wealth-dependentproperty. While solving the optimal policy under HARA utility is commonly believed to be diffi-cult and is thus much less studied compared with the CRRA case in the literature, we apply ourdecomposition results to explicitly reveal how the optimal policy under HARA utility is impactedby investor’s wealth level and the minimum requirements for terminal wealth and intermediate con-sumption. Furthermore, under the special case with nonrandom but possibly time-varying interestrate, we apply our decomposition to solve and interpret the optimal policy under HARA utility. The See, e.g., the forward-backward stochastic differential equations in in Detemple and Rindisbacher (2005) andDetemple and Rindisbacher (2010) under a model featuring partially hedgeable Gaussian interest rate with CRRAutility, as well as the quasi-linear partial differential equation employed in He and Pearson (1991) for indirectly relatingto the optimal policy. We then use the closed-form formulae to explicitly analyze how the optimalpolicies under CRRA and HARA utilities are differently impacted by wealth level, interest rate,and investment horizon. We show that the wealth-dependent property of HARA utility shouldnot be taken only literally. The HARA utility impacts the optimal policy via not only the wealthlevel, but also the interest rate and investment horizon. This is because the latter two factorsimpact the prices of bonds held by HARA investors for the purpose of satisfying their minimumrequirements for terminal wealth, as revealed in the portfolio construction for HARA utility withnonrandom interest rate. Furthermore, we show that the optimal policies for HARA investors withdifferent (i.e., high and low) initial wealth levels become more (resp. less) resembling to each other,when the market experiences the bull (resp. bear) regime; such cycle impact is entirely absent forCRRA investors. These results illustrate the importance of our theoretical findings in analyzingthe behavior of optimal policies under market incompleteness and wealth-dependent utility. Theyalso corroborate empirical evidence in the literature that the investment in risky assets increasesconcavely in investors’ financial wealth; see, e.g., Roussanov (2010), Wachter and Yogo (2010), andCalvet and Sodini (2014). Besides, they imply that investment recommendations based on a CRRAutility are incorrect for a HARA investor except if her current wealth is sufficiently high. In thenumerical experiments of Section 3.2 (see Figure 2), we observe a relative (resp. absolute) increaseup to 24% (resp. 10%) in the optimal allocation in the risky asset when shifting from a low-wealthinvestor to a high-wealth investor.As demonstrated in the aforementioned application, our decomposition of optimal policy leads topotential success of solving optimal policy in closed-form under some analytically tractable models.Nevertheless, if no closed-form solutions are available in nature, as for most of the flexible models,its direct implementation obviously encounters significant challenges. We therefore propose andimplement a Monte Carlo simulation method for optimal policy in general incomplete market models,as the second application based on our decomposition results. Such a method provides a solutionto the open problem of extending the simulation approach proposed in Detemple et al. (2003) for While the optimal policy under Heston’s model has been solved in closed-form under CRRA utility as in Liu(2007), its counterpart under HARA utility, to our best knowledge, remains elusive in the existing literature. / − type investigated in, e.g.,Christoffersen et al. (2010). Due to its incomplete-market and nonaffine nature, the optimal policyunder this model is analytically intractable. Applying our simulation method, we explore and explainvarious impacts on optimal portfolio policies from model parameters that control the specification ofvolatility dynamics. This study substantially complements and extends the aforementioned closed-form impact analysis for the Heston model based on our decomposition results under HARA utility.Our findings bring comprehensive economic insights of the optimal policy under stochastic volatility.For instance, the price of risk hedge component increases with the degree of elasticity of the vari-ance process, but decreases as market becomes more incomplete, i.e., the leverage effect parameterapproaches zero. Besides, it exhibits a hump-shape with respect to the level of risk aversion.The rest of this paper is organized as follows. In Section 2, we establish the decompositionfor general incomplete market models with flexible utilities and analyze the impacts from marketincompleteness and/or wealth-dependent utilities. In Section 3, we apply our decomposition togeneral models under HARA utility, and reveal the fundamental connection between CRRA andHARA policies under nonrandom but possibly time-varying interest rate. In Section 4, we proposeand implement a Monte Carlo simulation method based on our decomposition. Section 5 concludesand provides discussions. We collect proofs for the decompositions of optimal policy in Appendix A.6 A new decomposition of optimal dynamic portfolio choice in in-complete markets
We begin by setting up the model, utility function, and optimal dynamic portfolio choice problemin Section 2.1. Then, we establish our new decomposition of the optimal policy in Section 2.2.In Sections 2.3 and 2.4, we analyze the impacts of market incompleteness and wealth-dependentutilities, respectively, which illustrate the fundamental differences between our decomposition andexisting results.
We assume that the market consists of m stocks and one savings account. The price of the stock S it , for i = 1 , , . . . , m , follows the generic SDE: dS it S it = ( µ i ( t, Y t ) − δ i ( t, Y t )) dt + σ i ( t, Y t ) dW t , (1)where Y t is an n -dimensional state variable driven by the following generic SDE: dY t = α ( t, Y t ) dt + β ( t, Y t ) dW t . (2)In (1), W t is a standard d − dimensional Brownian motion; µ i ( t, y ) is a scalar function for modelingthe mean rate of return; δ i ( t, y ) is a scalar function for modeling the dividend rate; σ i ( t, y ) is a d − dimensional vector-valued function for modeling the volatility. In (2), α ( t, y ) is an n − dimensionalvector-valued function for modeling the drift of the state variable Y t ; β ( t, y ) is an n × d matrix-valuedfunction for modeling the diffusion of the state variable Y t . Besides, we assume that the savingsaccount appreciates at the instantaneous interest rate r t = r ( t, Y t ) for some scalar-valued function r ( t, y ) . The state variable Y t governs all the investment opportunities in the market through the rateof return, the dividend rate, the volatility, and the instantaneous interest rate. We mainly focus onthe incomplete market case where the number of independent Brownian motions is strictly largerthan the number of tradable risky assets, i.e., d > m . In this case, we can not fully hedge therisk introduced by the Brownian motion. Owing to the market incompleteness, the model and thesubsequent portfolio choice problem enjoy their multidimensional nature, even for one-asset cases.Denote by X t the investor wealth process. Then, it satisfies the following wealth equation: dX t = ( r t X t − c t ) dt + X t π ⊤ t [( µ t − r t m ) dt + σ t dW t ] . (3)In (3), µ t and σ t represent the mean rate of return and volatility of the risky assets respectively,which satisfy µ t = µ ( t, Y t ) and σ t = σ ( t, Y t ) with the m –dimensional vector µ ( t, y ) and the m × d –dimensional matrix σ ( t, y ) defined by µ ( t, y ) := ( µ ( t, y ) , µ ( t, y ) , · · · , µ m ( t, y )) ⊤ and σ ( t, y ) :=( σ ( t, y ) , σ ( t, y ) , · · · , σ m ( t, y )) ⊤ . We assume the volatility function σ ( t, y ) has rank m, i.e., its rows7re linearly independent. Besides, c t is the instantaneous consumption rate; π t is an m − dimensionalvector representing the weights of the risky assets; 1 m denotes an m − dimensional column vectorwith all elements equal to 1 . The investor maximizes her expected utility over both intermediateconsumptions and terminal wealth by dynamically allocating her wealth among the risky assets andthe risk-free asset, subject to the non-bankruptcy condition. We can formulate this optimizationproblem as sup ( π t ,c t ) E (cid:20)Z T u ( t, c t ) dt + U ( T, X T ) (cid:21) , with X t ≥ t ∈ [0 , T ] , (4)where u ( t, · ) and U ( T, · ) are the utility functions of the intermediate consumptions and the terminalwealth, respectively.In (4), both utility functions u ( t, · ) and U ( T, · ) are time-varying, in order to reflect the time value,e.g., the discount effect. Furthermore, we assume them to be strictly increasing and concave withlim x →∞ ∂u ( t, x ) /∂x = 0 and lim x →∞ ∂U ( T, x ) /∂x = 0. One important specification is the constantrelative risk aversion (CRRA) utility. Following the convention (see, e.g., Pratt (1964)), the CRRAutility function is defined by u ( t, c ) = we − ρt c − γ − γ and U ( T, x ) = (1 − w ) e − ρT x − γ − γ , (5a)where γ ∈ (0 , + ∞ ) is the constant relative risk aversion coefficient, w ∈ [0 ,
1] is a weight for balancingthe intermediate consumption and the terminal wealth, and ρ is the discount rate. The CRRA utilityis wealth independent, as the investor relative risk aversion coefficient is a constant γ and does notvary with the wealth level. This property brings mathematical convenience that leads to closed-formformulae of optimal policy or simplifications of the optimization problem under some specific models;see, e.g., Wachter (2002), Kim and Omberg (1996), and Liu (2007) for closed-form optimal policies,as well as Detemple et al. (2003) for developing a Monte Carlo simulation approach.Another prevalent example is the hyperbolic absolute risk aversion (HARA) utility. Following theconvention (see, e.g., Pratt (1964)), other things being equal, the HARA utility function is definedby u ( t, c ) = we − ρt ( c − ¯ c ) − γ − γ and U ( T, x ) = (1 − w ) e − ρT ( x − ¯ x ) − γ − γ , (5b)for c > ¯ c and x > ¯ x, where ¯ c and ¯ x are set as the minimum allowable levels for the intermediateconsumption and terminal wealth, respectively. The HARA utility allows for imposing lower boundconstraints on the intermediate consumption and/or terminal wealth. This feature is particularlysuitable for incorporating, e.g., portfolio insurance, investment goal constraints, and subsistence levelconstraints. Unlike the CRRA utility in (5a), the HARA utility is wealth dependent. Besides, closed-form optimal policies under the HARA utility are generally rare; see Kim and Omberg (1996) forone such case with the stochastic market price of risk modelled by an Ornstein-Uhlenbeck process.The Monte Carlo simulation approaches of Detemple et al. (2003) and Cvitanic et al. (2003) offerremedies for such mathematical inconvenience. 8 .2 Decomposition of optimal policy To establish a decomposition of the optimal policy under the incomplete market setting ( d > m )for models with flexible dynamics (1) – (2) and general utility functions, we begin by applyingand further exploring the least favorable completion principle introduced in Karatzas et al. (1991)under the general diffusion model (1) – (2). First, we complete the market by introducing d − m candidate fictitious assets. Then, we solve the optimal portfolio choice problem in this completedmarket via the martingale method for complete market case. Finally, by letting the optimal weightsfor the fictitious assets be zero, we characterize the price of risk of these suitable fictitious assets andpin down the desired optimal policy for the real assets. In what follows, we develop the proceduresdescribed above to circumvent the challenge in explicitly decomposing and implementing the optimalportfolio policy for the general incomplete market model.We introduce d − m fictitious assets to complete the market as discussed in Karatzas et al. (1991).Their prices F it , for i = 1 , , . . . , d − m , satisfy the following SDE: dF it F it = µ Fit dt + σ Fi ( t, Y t ) dW t , (6)where the mean rates of returns µ Fit are stochastic processes adaptive to the filtration generated bythe Brownian motion W t . According to Karatzas et al. (1991), we can choose the volatility function σ F ( t, y ) := ( σ F ( t, y ) , · · · , σ Fd − m ( t, y )) ⊤ arbitrarily, as long as it has rank d − m and satisfies thefollowing orthogonal condition with respect to the volatility function σ ( t, y ) of the real risky assets S t : σ ( t, y ) σ F ( t, y ) ⊤ ≡ m × ( d − m ) . (7)This condition guarantees that the fictitious assets are driven by different Brownian shocks, and thusleads to the success of market completion. Moreover, we assume that the fictitious assets do not paydividend.Combining the m real risky assets with prices S t in (1) and the d − m fictitious risky assetswith prices F t in (6), we construct a completed market consisting of d risky assets and driven by d independent Brownian motions. In this completed market, we represent the prices of the risky assets,including both the real and the fictitious ones, by a d –dimensional column vector S S t = (cid:0) S ⊤ t , F ⊤ t (cid:1) ⊤ .According to (1) and (6), S S t is driven by the SDE: dS S t = diag( S S t ) (cid:2) µ S t dt + σ S ( t, Y t ) dW t (cid:3) , (8) As documented in the literature (see, e.g., Karatzas et al. (1991)), we can interpret the terminology “least fa-vorable completion” as follows: Consider all the possible fictitious completions and their associated optimal policies,we naturally say that a completion is more (resp. less) favorable if its corresponding optimal policy results in higher(resp. lower) expected utility. The completion (23) below, which leads to an optimal portfolio with zero weight on thefictitious assets, must be the least favorable one. Indeed, in any other fictitious completion, since this portfolio withoutthe fictitious assets is admissable (i.e., a candidate portfolio strategy), the optimal one must result in a higher expectedutility and thus becomes more favorable. S S t ) = diag( S t , S t , · · · , S mt , F t , F t , · · · , F ( d − m ) t ) , the d –dimensionalcolumn vector µ S t = (cid:0) ( µ ( t, Y t ) − δ ( t, Y t )) ⊤ , ( µ Ft ) ⊤ (cid:1) ⊤ , and the d × d dimensional matrix σ S ( t, Y t )= (cid:0) σ ( t, Y t ) ⊤ , σ F ( t, Y t ) ⊤ (cid:1) ⊤ . By linear algebra, the orthogonal condition (7) implies that σ S ( t, y ) mustbe nonsingular. Thus, we are now in a complete market, where we can fully hedge the uncertaintyfrom all Brownian motions by investing in the real and fictitious assets. The completed marketallows for investing in both the real assets S t and fictitious assets F t . We denote by π t and π Ft theircorresponding weights, which are m and ( d − m )–dimensional vectors, respectively. Similar to (4),we consider the utility maximization problem in this completed market, with the non-bankruptcyconstraint X t ≥ θ S t := σ S ( t, Y t ) − ( µ S t − r ( t, Y t )1 d ) . (9)By the orthogonal condition (7), it follows from matrix calculations that( σ S ( t, y )) − = (cid:0) σ ( t, y ) + σ F ( t, y ) + (cid:1) , (10)where the notation A + := A ⊤ ( AA ⊤ ) − (11)denotes the Moore–Penrose inverse (see, e.g., Penrose (1955)) of a general matrix A with linearlyindependent rows. Thus, σ ( t, y ) + (resp. σ F ( t, y ) + ) is a d × m dimensional (resp. d × ( d − m ) dimen-sional) matrix satisfying σ ( t, y ) σ ( t, y ) + = I m (resp. σ F ( t, y ) σ F ( t, y ) + = I d − m ), with I m being the m − dimensional identity matrix. By (10), the total price of risk θ S t allows the following decomposi-tion: θ S t = θ h ( t, Y t ) + θ ut , (12)where θ h ( t, Y t ) and θ ut are the prices of risk associated with the real and fictitious assets respectively.They are defined by d –dimensional column vectors θ h ( t, Y t ) := σ ( t, Y t ) + ( µ ( t, Y t ) − r ( t, Y t )1 m ) , (13a)and θ ut := σ F ( t, Y t ) + ( µ Ft − r ( t, Y t )1 d − m ) , (13b)respectively. The term θ h ( t, Y t ) in (13a) is referred to as the market price of risk, as it is fullydetermined by the real assets shared by all investors in the market. The term θ ut in (13b), however,is purely associated with the fictitious assets, which are introduced for solving the optimal portfoliochoice problem (18) in the incomplete market. As we will show momentarily, θ ut is endogenouslydetermined by the investor utility function and the investment horizon. Thus, we refer to θ ut as theinvestor-specific price of risk, since it varies from one investor to another.10y definitions (11), (13a), and (13b), we can translate the orthogonal condition (7) as σ F ( t, Y t ) θ h ( t, Y t ) ≡ d − m and σ ( t, Y t ) θ ut ≡ m . (14a)As we assume matrix σ ( t, Y t ) has linear independent rows, the second condition imposes m linearconstraints for the d − dimensional vector θ ut . Besides, definition (13a) and the second condition implythat θ h ( t, Y t ) ⊤ θ ut ≡ , (14b)i.e., the market and investor-specific price of risk are orthogonal. According to Karatzas et al. (1991),we can fully determine the optimal policy π t in the real market by the choice of θ us . More precisely,the fictitious mean return µ Fs and volatility σ F ( s, Y s ) impact the optimal policy π t only through theinvestor-specific price of risk θ us defined by (13b). In what follows, we investigate how to characterize θ us and apply it, among other building blocks, in decomposing the optimal portfolio.Next, we introduce the state price density as ξ S t := exp (cid:18) − Z t r ( v, Y v ) dv − Z t ( θ S v ) ⊤ dW v − Z t ( θ S v ) ⊤ θ S v dv (cid:19) . (15)For any s ≥ t ≥ , we define the relative state price density as ξ S t,s = ξ S s /ξ S t , (16)which satisfies dξ S t,s = − ξ S t,s [ r ( s, Y s ) ds + ( θ S s ) ⊤ dW s ] , (17)with initial value ξ S t,t = 1 , according to a straightforward application of Ito formula. The abovedynamics of ξ S t,s clearly hinges on the undetermined investor-specific price of risk θ us .The martingale approach pioneered by Karatzas et al. (1987) and Cox and Huang (1989) startsby formulating the dynamic problem (4) with information up to time t as an equivalent staticoptimization problem:sup ( c t ,X T ) E t (cid:20)Z Tt u ( s, c s ) ds + U ( T, X T ) (cid:21) subject to E t (cid:20)Z Tt ξ S t,s c s ds + ξ S t,T X T (cid:21) ≤ X t , (18)where, throughout the paper, E t denotes the expectation condition on the information up to time t and X t is the wealth level assuming that the investor always follows the optimal policy. Then, we cansolve this problem via the method of Lagrangian multiplier. The optimal intermediate consumptionand terminal wealth satisfy c t = I u ( t, λ ∗ t ) and X T = I U ( T, λ ∗ T ) , (19) To guarantee the martingale property of ξ S t exp( R t r ( v, Y v ) dv ), we assume that the total price of risk θ S v satisfiesthe Novikov condition: E (cid:20) exp (cid:18) Z T ( θ S v ) ⊤ θ S v dv (cid:19)(cid:21) < ∞ . I u ( t, · ) and I U ( t, · ) being the inverse marginal utility functions of u ( t, · ) and U ( t, · ),i.e., the functions satisfying ∂u/∂x ( t, I u ( t, y )) = y and ∂U/∂x ( t, I U ( t, y )) = y . In (19), we employ λ ∗ t to denote the Lagrangian multiplier for the wealth constraint in (18). It is uniquely characterizedby following wealth equation: X t = E t [ G t,T ( λ ∗ t )] , (20)where G t,T ( λ ∗ t ) is defined as G t,T ( λ ∗ t ) := Γ Ut,T ( λ ∗ t ) + Z Tt Γ ut,s ( λ ∗ t ) ds. (21)Here, Γ Ut,T ( λ ∗ t ) and Γ ut,s ( λ ∗ t ) are given byΓ Ut,T ( λ ∗ t ) = ξ S t,T I U (cid:0) T, λ ∗ t ξ S t,T (cid:1) and Γ ut,s ( λ ∗ t ) = ξ S t,s I u (cid:0) s, λ ∗ t ξ S t,s (cid:1) . (22a)By (20), we can determine the multiplier λ ∗ t with information up to time t . For representing theportfolio decomposition momentarily, we also introduce following quantities:Υ Ut,T ( λ ∗ t ) = λ ∗ t (cid:0) ξ S t,T (cid:1) ∂I U ∂y (cid:0) T, λ ∗ t ξ S t,T (cid:1) and Υ ut,s ( λ ∗ t ) = λ ∗ t (cid:0) ξ S t,s (cid:1) ∂I u ∂y (cid:0) s, λ ∗ t ξ S t,s (cid:1) . (22b)Consequently, we can represent the optimal policy ( π t , π Ft ) for the completed market via themartingale representation theorem (see, e.g., Section 3.4 in Karatzas and Shreve (1991)). Withthe Clark-Ocone formula (see, e.g., the survey provided in Detemple et al. (2003)), we can furtherrepresent the optimal policy in the form of conditional expectations of suitable random variables (seeOcone and Karatzas (1991)). Under a general and flexible diffusion model, Detemple et al. (2003)propose an explicit conditional expectation form for the optimal policy, and develop a Monte Carlosimulation method for its implementation; see also, e.g., Detemple and Rindisbacher (2010) along thisline of contributions and Detemple (2014) for a comprehensive survey of the related developments.We aim at explicitly developing such results for the incomplete market case.By the least favorable completion principle, the optimal policy π t for the real assets in the originalincomplete market coincides with its counterpart in the completed market, as long as we properlychoose the investor-specific price of risk θ uv such that the optimal weights for the fictitious assets arealways identically zero, i.e., π Fv ≡ d − m , for any 0 ≤ v ≤ T. (23)Given an arbitrary choice of the volatility function σ F ( v, y ) , the least favorable constraint (23)together with the second orthogonal condition in (14a) determines the desired θ uv for 0 ≤ v ≤ T .Then, the corresponding optimal policy π t of the real assets for the completed market is also optimalfor the original incomplete market. In particular, according to Karatzas et al. (1991), the desired θ uv π t are independent of the specific choice of σ F ( v, y ) . Thus, in what follows, we focus on characterizing θ uv .Now, we notice and apply the following useful representation of the unknown investor-specificprice of risk θ uv that satisfies the least favorable completion constraint (23): θ uv = θ u ( v, Y v , λ ∗ v ; T ) (24)for some function θ u ( v, y, λ ; T ) endogenously determined by investor’s utility function and invest-ment horizon. We can derive such a result by combining the fictitious completion approach inKaratzas et al. (1991) and the minimax local martingale approach in He and Pearson (1991); wedocument a brief verification in Appendix A.1. Representation (24) reveals the fundamental struc-ture of the investor-specific price of risk θ u ( v, Y v , λ ∗ v ; T ), which is strikingly different from that ofthe market price of risk θ h ( v, Y v ) defined in (13a), and thus leads to fundamental difference betweenincomplete and complete market cases. First, θ u ( v, Y v , λ ∗ v ; T ) depends on the constraint multiplier λ ∗ v , which solves X v = E v [ G v,T ( λ ∗ v )] by (20). Thus, θ u ( v, Y v , λ ∗ v ; T ) implicitly depends on the wealthlevel X v . Second, θ u ( v, Y v , λ ∗ v ; T ) also depends on the investment horizon T , which is economicallymeaningful and technically indispensable at the level of implementation. However, neither of thesetwo types of dependence exists in the market price of risk θ h ( v, Y v ) . Accordingly, we get the clearinsight that, when completing the market following the least favorable principle (23), the introducedfictitious assets ought to depend on both the current wealth level and the investment horizon. Inaddition to this economic message, representation (24) is key not only for establishing the decompo-sition of optimal policy in Theorems 1 and 2 below, but also for developing the simulation methodfor implementation purposes in Section 4.
Theorem 1.
Under the incomplete market model (1) and (2), the optimal policy π t for the realassets with prices S t admits the following decomposition π t = π mv ( t, X t , Y t ) + π r ( t, X t , Y t ) + π θ ( t, X t , Y t ) . (25) Here, the mean-variance component π mv , the interest rate hedge component π r , and the price of riskhedge component π θ satisfy π mv ( t, X t , Y t ) = − X t ( σ ( t, Y t ) + ) ⊤ θ h ( t, Y t ) E t [ Q t,T ( λ ∗ t )] , (26a) π r ( t, X t , Y t ) = − X t ( σ ( t, Y t ) + ) ⊤ E t [ H rt,T ( λ ∗ t )] , (26b) π θ ( t, X t , Y t ) = − X t ( σ ( t, Y t ) + ) ⊤ E t [ H θt,T ( λ ∗ t )] , (26c) where, throughout the paper, E t denotes the expectation condition on the information up to time t ; X t is the wealth level assuming that the investor always follows the optimal policy; the quantities t,T ( λ ∗ t ) , H rt,T ( λ ∗ t ) , and H θt,T ( λ ∗ t ) are given by Q t,T ( λ ∗ t ) := Υ Ut,T ( λ ∗ t ) + Z Tt Υ ut,s ( λ ∗ t ) ds, (27a) H rt,T ( λ ∗ t ) := (Γ Ut,T ( λ ∗ t ) + Υ Ut,T ( λ ∗ t )) H rt,T + Z Tt (Γ ut,s ( λ ∗ t ) + Υ ut,s ( λ ∗ t )) H rt,s ds, (27b) H θt,T ( λ ∗ t ) := (Γ Ut,T ( λ ∗ t ) + Υ Ut,T ( λ ∗ t )) H θt,T ( λ ∗ t ) + Z Tt (Γ ut,s ( λ ∗ t ) + Υ ut,s ( λ ∗ t )) H θt,s ( λ ∗ t ) ds. (27c) Hereof, λ ∗ t is the multiplier uniquely determined by (20), i.e., X t = E t [ G t,T ( λ ∗ t )] , and thus it dependson X t and satisfies the relation λ ∗ t = λ ∗ ξ S t ; (28)Γ Ut,T ( λ ∗ t ) , Γ ut,s ( λ ∗ t ) , Υ Ut,T ( λ ∗ t ) , and Υ ut,s ( λ ∗ t ) are defined in (22a) and (22b) except for replacing therelative state price density ξ S t,s by a λ ∗ t − dependent version ξ S t,s ( λ ∗ t ) , for t ≤ s ≤ T, which solves theSDE: dξ S t,s ( λ ∗ t ) = − ξ S t,s ( λ ∗ t )[ r ( s, Y s ) ds + θ S s ( λ ∗ t ) ⊤ dW s ] , (29) with θ S s ( λ ∗ t ) = θ h ( s, Y s ) + θ u ( s, Y s , λ ∗ t ξ S t,s ( λ ∗ t ) ; T ) (30) being the λ ∗ t − parameterized version of the total price of risk (12). Herein and thereafter, as anindispensable foundation, θ u ( v, y, λ ; T ) is the function for representing the desired investor-specificprice of risk θ uv , which satisfies the least favorable completion constraint (23), via θ uv = θ u ( v, Y v , λ ∗ v ; T ) as introduced in (24). Besides, H rt,s and H θt,s ( λ ∗ t ) in (27b) and (27c) are both d –dimensional vector-valued processes evolving according to SDEs: dH rt,s = ( D t Y s ) ∇ r ( s, Y s ) ds, (31) and dH θt,s ( λ ∗ t ) = [( D t Y s ) ( ∇ θ h ( s, Y s ) + ∇ θ u ( s, Y s , λ ∗ t ξ S t,s ( λ ∗ t ); T )) (32) − λ ∗ t ξ S t,s ( λ ∗ t )( θ S t ( λ ∗ t ) + H rt,s + H θt,s ( λ ∗ t )) ∂θ u /∂λ ( s, Y s , λ ∗ t ξ S t,s ( λ ∗ t ); T )][ θ S s ( λ ∗ t ) ds + dW s ] , for t ≤ s ≤ T, with initial values H rt,t = H θt,t ( λ ∗ t ) = 0 d and θ S s ( λ ∗ t ) given in (30). Here and throughoutthis paper, ∇ denotes the gradient of functions with respect to the arguments in the place of Y s .In (32), the random variable D t Y s is the time– t Malliavin derivative for the time– s state variable Y s , i.e., a d × n matrix with D t Y s = (( D t Y s ) ⊤ , ( D t Y s ) ⊤ , · · · , ( D dt Y s ) ⊤ ) ⊤ , where each D it Y s is an n –dimensional column vector satisfying SDE: d D it Y s = ( ∇ α ( s, Y s )) ⊤ D it Y s ds + d X j =1 ( ∇ β j ( s, Y s )) ⊤ D it Y s dW js , lim s → t D it Y s = β i ( t, Y t ) , (33) For an m − dimensional vector-valued function f ( t, y ) = ( f ( t, y ) , f ( t, y ) , · · · , f m ( t, y )), its gradient is an n × m matrix with each element given by [ ∇ f ( t, y )] ij = ∂f j /∂y i ( t, y ) , for i = 1 , , . . . , n and j = 1 , , . . . , m. or t ≤ s ≤ T, where β i ( s, y ) denotes the i th column of β ( s, y ) ; W js denotes the j th dimension ofBrownian motion W s . Finally, the optimal intermediate consumption c t and terminal wealth X T aregiven by (19), i.e., c t = I u ( t, λ ∗ t ) and X T = I U ( T, λ ∗ T ) . Proof.
See Appendix A.1.The contribution of Theorem 1, relative to the previous literature, lies in that it reveals thestructure of optimal policy under general incomplete market models with flexible utilities. The firstcomponent π mv ( t, X t , Y t ) is the mean-variance component, as reflected in (26a) through the marketprice of risk θ h ( t, Y t ) defined in (13a) – a mean-variance trade-off. The component π r ( t, X t , Y t ) (resp. π θ ( t, X t , Y t )) is for hedging the uncertainty in interest rate (resp. price of risk), as reflected by theinterest rate sensitive term ∇ r involved in (26b) via H rt,T ( λ ∗ t ) in (27b) and H rt,s in (31) (resp. the priceof risk sensitive terms ∇ θ h , ∇ θ u , and ∂θ u /∂λ involved in (26c) via H θt,T ( λ ∗ t ) in (26c) and H θt,s ( λ ∗ t ) in(32)) as well as the Malliavin derivative D t Y s appearing in (31) (resp. (32)). In particular, naturallyanalogous to classical derivatives, we can intuitively understand the Malliavin derivative D t Y s as thesensitivity of the state variable Y s to the underlying Brownian motion W t ; see an accessible surveyof Malliavin calculus for finance in Appendix D of Detemple et al. (2003). This type of decomposition of optimal policy was first proposed in Merton (1971) for a completemarket model. By (26a) – (26c), we represent each component as an explicit conditional expectation,which substantially extends the complete-market results in Detemple et al. (2003). In contrast tothe complete market counterpart, the hedge component π θ ( t, X t , Y t ) in (26c) is designed to hedge theuncertainty in both market and investor-specific price of risk. Indeed, this is because, as shown in(26c), π θ ( t, X t , Y t ) depends on the term H θt,T ( λ ∗ t ) defined in (27c) via H θt,s ( λ ∗ t ) , of which the dynamicsinvolves both θ h and θ u according to (32).In line with the time– t formulation of the optimization problem (18), we express these componentsby the time– t state variable Y t and the current wealth X t , rather than the time–0 wealth X asemployed in most of the existing literature, e.g., Detemple et al. (2003). We can easily see that, bysolving constraint (20), λ ∗ t is a function of t , X t , and Y t . In addition to the consistent expressionsbased on the time- t information, this setup leads to convenience for further developing numericalmethods for implementing our decomposition of optimal policy, e.g., the Monte Carlo simulationapproach to propose momentarily in Section 4. Technically, the expressions based on time– t andtime–0 variables are related according to relation (28) linking the time– t multiplier λ ∗ t and the time-0 multiplier λ ∗ via state price density ξ S t , which by (15) depends on the entire path of interest rate We can view Malliavin calculus as the stochastic calculus of variation in the space of sample paths. Malli-avin calculus has proven its important role in financial economics through its merit in solving portfolio choicesproblems, see, e.g., Ocone and Karatzas (1991), Detemple et al. (2003), Detemple and Rindisbacher (2005), andDetemple and Rindisbacher (2010). We can find a book-length discussion of the theory of Malliavin calculus in,for example, Nualart (2006). ( v, Y v ) and total price of risk θ S v for 0 ≤ v ≤ t .Moreover, we establish the explicit dynamics for ξ S t,s ( λ ∗ t ) and H θt,s ( λ ∗ t ) as in (29) and (32), respec-tively, which are key for both decomposition and implementation of the optimal policy under theincomplete market models. The representation of individual-specific price of risk θ uv in (24) plays acrucial role in deriving these dynamics and then further establishing the representations in Theorem1. In particular, to naturally separate the given information up to time t , we follow (24) to express θ us as the λ ∗ t − dependent version: θ us ( λ ∗ t ) = θ u ( s, Y s , λ ∗ s ; T ) = θ u (cid:0) s, Y s , λ ∗ t ξ S t,s ( λ ∗ t ); T (cid:1) , (34)where the second equality follows relations (16) and (28), i.e., λ ∗ s = λ ∗ ξ S s = λ ∗ t ξ S t,s . This leads to (30)– the parameterized version of the total price of risk (12). By (29) and (32), we see the separation of λ ∗ t and ξ S t,s ( λ ∗ t ) is fully reflected in the dynamics of ξ S t,s ( λ ∗ t ) and H θt,s ( λ ∗ t ). Besides, these λ ∗ t –dependentversions naturally isolate the information up to time t , by which we can fully determine the multiplier λ ∗ t . As we will show in Section 4, such a separation not only clearly reveals the structure regardinginformation, but also provides much convenience for our implementation of the decomposition bysimulation method. Besides, in Sections 2.3 and 2.4, we employ the dynamics of ξ S t,s ( λ ∗ t ) and H θt,s ( λ ∗ t )to analyze the impact on optimal policy from market incompleteness and wealth dependent propertiesof utility functions.As an extension and enhancement of Theorem 1, we can further decompose the price of risk hedgecomponent π θt into three parts according to the economic nature of the uncertainties embedded inthe market and investor-specific price of risk. Proposition 1 below shows that our decompositionnot only brings insights to the structure of price of risk hedge component, but also disentangles thefundamental impacts on optimal policy from market incompleteness and wealth-dependent utilities.In particular, we identify a new important hedge component for non-myopic investors to hedge theuncertainty in investor-specific price of risk due to variation in wealth level. Proposition 1.
The price of risk hedge component π θ ( t, X t , Y t ) in (26c) can be further decomposedas π θ ( t, X t , Y t ) = π h,Y ( t, X t , Y t ) + π u,Y ( t, X t , Y t ) + π u,λ ( t, X t , Y t ) , (35) where components π h,Y ( t, X t , Y t ) , π u,Y ( t, X t , Y t ) , and π u,λ ( t, X t , Y t ) are given by: π h,Y ( t, X t , Y t ) = − X t ( σ ( t, Y t ) + ) ⊤ E t [ H h,Yt,T ] , (36a) π u,Y ( t, X t , Y t ) = − X t ( σ ( t, Y t ) + ) ⊤ E t [ H u,Yt,T ( λ ∗ t )] , (36b) π u,λ ( t, X t , Y t ) = − X t ( σ ( t, Y t ) + ) ⊤ E t [ H u,λt,T ( λ ∗ t )] . (36c) Here, the terms H h,Yt,T , H u,Yt,T ( λ ∗ t ) , and H u,λt,T ( λ ∗ t ) in (36a)–(36c) are defined in the same way as thatfor H θt,T ( λ ∗ t ) in (27c) except for replacing H θt,s by H h,Yt,s , H u,Yt,s ( λ ∗ t ) , and H u,λt,s ( λ ∗ t ) for t ≤ s ≤ T , espectively, which follow the SDEs: dH h,Yt,s = ( D t Y s ) ∇ θ h ( s, Y s )( θ h ( s, Y s ) ds + dW s ) , (37a) dH u,Yt,s ( λ ∗ t ) = ( D t Y s ) ∇ θ u ( s, Y s , λ ∗ t ξ S t,s ( λ ∗ t ); T )( θ us ( λ ∗ t ) ds + dW s ) , (37b) and dH u,λt,s ( λ ∗ t ) = − λ ∗ t ξ S t,s ( λ ∗ t )( θ S t ( λ ∗ t ) + H rt,s + H θt,s ( λ ∗ t )) · ∂θ u /∂λ ( s, Y s , λ ∗ t ξ S t,s ( λ ∗ t ); T )( θ us ( λ ∗ t ) ds + dW s ) , (37c) with initial values H h,Yt,t = H u,Yt,t ( λ ∗ t ) = H u,λt,t ( λ ∗ t ) = 0 d . Here, the λ ∗ t − parameterized variables ξ S t,s ( λ ∗ t ) , θ S t ( λ ∗ t ) , H θt,s ( λ ∗ t ) and θ us ( λ ∗ t ) are given in (29), (30), (32), and (34), respectively.Proof. It follows by straightforward algebraic calculations based on the orthogonal condition (14b),representation (30), and the dynamics (32).Decomposition (35) of the price of risk hedge component π θ ( t, X t , Y t ) has the following interpreta-tions that reveal the fundamental structure of market incompleteness. The component π h,Y ( t, X t , Y t )hedges the uncertainty in the market price of risk θ h associated with the real assets. This because ithinges on (37a), in which the gradient ∇ θ h ( s, Y s ) reflects the sensitivity of market price of risk withrespect to the random state variable Y s . The other two components π u,Y ( t, X t , Y t ) and π u,λ ( t, X t , Y t )are both hedges for the investor-specific price of risk associated with the fictitious assets. However,their causes are fundamentally different. The first component π u,Y ( t, X t , Y t ) hedges the fluctuationin investor-specific price of risk that arises from the state variable, as explicitly reflected from thegradient ∇ θ u ( s, Y s , λ ∗ t ξ S t,s ( λ ∗ t ); T ) in (37b). Its structure resembles that of π h,Y ( t, X t , Y t ) , except forreplacing θ h by θ u in (37a). The second component π u,λ ( t, X t , Y t ), however, is introduced by thesensitivity with respect to the multiplier λ ∗ s = λ ∗ t ξ S t,s ( λ ∗ t ), i.e., ∂θ u /∂λ ( s, Y s , λ ∗ t ξ S t,s ( λ ∗ t ); T ) in (37c).Recall that, by (20), the multiplier λ ∗ s is directly related to wealth level X s via X s = E s [ G s,T ( λ ∗ s )].Thus, as a new important hedge component, π u,λ ( t, X t , Y t ) essentially hedges the fluctuation ininvestor-specific price of risk due to the variation in wealth level, which is completely absent fromthe market price of risk. This structure reconciles and further develops the discussion following(24): under general incomplete market models, the suitable investor-specific price of risk dependson current market state and wealth level, and thus the uncertainties from both channels need to behedged.With the decomposition in Theorem 1 and Proposition 1, we face only one remaining difficulty insolving the optimal policy – the investor-specific price of risk function θ u ( v, y, λ ; T ) is still undeter-mined. Indeed, by checking the function definitions for the four conditional expectations E t [ G t,T ( λ ∗ t )], E t [ Q t,T ( λ ∗ t )], E t [ H rt,T ( λ ∗ t )], and E t [ H θt,T ( λ ∗ t )] in (21) and (27a) – (27c) with ξ S t,s ( λ ∗ t ) and H θt,s ( λ ∗ t )given in (29) and (32), we can verify that we can fully determine all of them after we choose an17nvestor-specific price of risk θ us . This reconciles the previous claim, in Karatzas et al. (1991) andHe and Pearson (1991), that the characterization of θ us plays a key role in solving the desired op-timal policy for the real assets via fictitious completion. Meanwhile, by inspecting, e.g., (29) and(34) jointly, we find that the desired investor-specific price of risk function θ u ( s, y, λ ; T ) is entangledin a complex stochastic system. To circumvent this difficulty, our next task is to establish a novelequation system for solving this function in Theorem 2 below, which then suffices to explicitly ex-press the optimal policy. The equation technically follows representation (24), the second orthogonalconstraint in (14a), and the least favorable completion principle (23). Theorem 2.
The function θ u ( v, y, λ ; T ) for representing the investor-specific price of risk θ uv viarepresentation (24) satisfies the orthogonal condition σ ( v, y ) θ u ( v, y, λ ; T ) ≡ m (38) and the least favorable completion constraint (23). Or equivalently, θ u ( v, y, λ ; T ) satisfies the follow-ing d − dimensional equation θ u ( v, y, λ ; T ) = σ ( v, y ) + σ ( v, y ) − I d E [ Q v,T ( λ ) | Y v = y ] × ( E [ H rv,T ( λ ) | Y v = y ] + E [ H θv,T ( λ ) | Y v = y ]) , (39) for ≤ v ≤ T , where I d denotes the d –dimensional identity matrix; σ ( v, y ) + is given by (11). Thefunction θ u ( s, y, λ ; T ) is fully characterized by a multidimensional coupled equation system consistingof equation (39), as well as the SDEs of Y s , ξ S t,s ( λ ) , H rt,s , H θt,s ( λ ) , and D it Y s given in (2), (29),(31), (32), and (33), respectively, except for replacing λ ∗ t by λ. Consequently, given the function θ u ( v, y, λ ; T ) , the optimal portfolio is determined by decomposition (25) based on (26a) – (26c).Proof. See the derivation of equation (39) in Appendix A.2.We now analyze the structure of above-mentioned equation system that characterizes function θ u ( v, y, λ ; T ). It suffices to disentangle the structure of equation (39) by revealing how the unknownfunction θ u gets involved in the conditional expectations E [ Q v,T ( λ ) | Y v = y ] , E [ H rv,T ( λ ) | Y v = y ], and E [ H θv,T ( λ ) | Y v = y ] therein. Without loss of generality, we take E [ Q v,T ( λ ) | Y v = y ] as an example. Bydefinitions in (27a) and (22b), it can be expressed as: E [ Q v,T ( λ ) | Y v = y ] = E (cid:20) λξ S v,T ( λ ) ∂I U ∂y (cid:0) T, λξ S v,T ( λ ) (cid:1) + Z Tv λξ S v,s ( λ ) ∂I u ∂y (cid:0) s, λξ S v,s ( λ ) (cid:1) ds (cid:12)(cid:12)(cid:12)(cid:12) Y v = y (cid:21) . Here, we see that the conditional expectation is taken over the entire path of ξ S v,s ( λ ) for v ≤ s ≤ T . Meanwhile, the dynamics of ξ S v,s ( λ ) follows dξ S v,s ( λ ) = − ξ S v,s ( λ )[ r ( s, Y s ) ds + ( θ h ( s, Y s ) + θ u ( s, Y s , λξ S v,s ( λ ) ; T )) ⊤ dW s ] , by (29) and (30). It clearly depends on the unknown investor-specificprice of risk function θ u ( s, y ′ , λ ′ ; T ) for v ≤ s ≤ T and all possible values of y ′ and λ ′ . A similarstructure holds for the conditional expectations E [ H rv,T ( λ ) | Y v = y ] and E [ H θv,T ( λ ) | Y v = y ]. From18he above analysis, we verify that the function θ u ( s, y, λ ; T ) is indeed fully characterized by themultidimensional equation system consisting of equation (39), as well as SDEs of Y s , ξ S t,s ( λ ) , H rt,s ,H θt,s ( λ ) , and D it Y s given in (2), (29), (31), (32), and (33), respectively, except for replacing λ ∗ t by λ . Finally, by checking the definitions in (27a) – (27c), as well as the SDEs of Y s , ξ S t,s ( λ ) , H rt,s , H θt,s ( λ ) , and D it Y s , we verify that all the components involved in this equation system are independent ofthe functional form of σ F ( v, y ) . Thus, θ u ( v, y, λ ; T ), as the solution to the aforementioned equationsystem, is obviously also independent of the choice of σ F ( v, y ). Such an independence lines up withthe claim in Karatzas et al. (1991) that the desired price of risk θ uv for fictitious assets is independentof σ F ( v, y ) . Second, equation (39) obviously implies its terminal condition as θ u ( T, y, λ ; T ) = 0 d , (40)which corresponds to the investor-specific price of risk with the investment horizon shrinking to zero.An immediate consequence of this condition is that, as the investment horizon shrinks to zero, thereturn of the fictitious assets µ Ft employed in the least favorable completion converges to the risk-freereturn r ( t, Y t ) . This indeed directly follows from θ u ( v, Y v , λ ∗ v ; T ) = σ F ( v, Y v ) + ( µ Fv − r ( v, Y v )1 d − m ) , which combines (13b) and (24). The terminal condition (40) plays an important role in potentialnumerical methods for solving function θ u ( v, y, λ ; T ) . For example, when employing a discretizationand simulation approach, it serves as the terminal condition for numerical backward induction.This demonstrates the importance of highlighting the investment horizon T as an argument of theinvestor-specific price of risk function θ u ( v, y, λ ; T ) in (24).In the literature, other types of differential equations were employed for characterizing optimalportfolios in incomplete markets; see, e.g., Detemple and Rindisbacher (2005) for deriving and solv-ing a forward-backward SDE (FBSDE) governing the shadow price under a model featuring partiallyhedgeable Gaussian interest rate and CRRA utility, also see Detemple and Rindisbacher (2010) forgeneralizing such results, as well as He and Pearson (1991) for indirectly relating the optimal policyand the equivalent local martingale measure to the solution of a quasi-linear PDE. Thanks to theapplication of representation (24), when compared with, e.g., the forward-backward SDE, our equa-tion system with (39), as given in Theorem 2 for solving the deterministic function θ u ( v, y, λ ; T ) , appears more explicit in terms of revealing the fundamental structure of the investor-specific priceof risk for our general incomplete market model with flexible utilities, and it is relatively easier tohandle technically. Besides, compared with the quasi-linear PDE based approach in He and Pearson(1991), our Theorem 1 together with Theorem 2 jointly provide an explicit decomposition of the If seeking for a purely analytical characterization, we can obtain an integral-partial differential equation systemconsisting of an integral-form expression of equation (39) and a Kolmogorov forward or backward PDE for governingthe transition density of the Markov process ( Y s , ξ S t,s ( λ ) , D t Y s , H rt,s , H θt,s ( λ )) underlying the stochastic system. For thesake of space limitation, we omit such routine and technical details. θ u ( s, y, λ ; T ) , Y s , ξ S t,s ( λ ) , H rt,s , H θt,s ( λ ) , and D it Y s as established in Theorem 2. Owing to its intricatestructure and high-dimensional nature even for the one-asset case, it is a significant challenge tosolve the policy analytically or even numerically via existing methods. We will tackle this difficultyin Section 4 via a Monte Carlo simulation method, which provides a practical method for numericallysolving optimal policies under incomplete market models without closed-form solutions. In this section, we demonstrate the role of market incompleteness in the decomposition of optimalpolicy. For this purpose, we compare the optimal policy in general incomplete market models,as established in the previous section, with its complete market counterpart. More precisely, thecomplete market model for this comparison is set under the general model (1) and (2) by requiringthe number of risky assets and the number of driving Brownian motions are equal, i.e., m = d . Insuch a complete market, we can fully hedge the risk by investing in the risky assets, and we do notneed the above completion procedure with fictitious assets. We can decompose the optimal policyand implement it following the framework of Detemple et al. (2003).In this complete market model, we assume that the square matrix σ ( t, y ) := ( σ ( t, y ) , · · · , σ m ( t, y )) ⊤ is non-singular; see, for example, the related discussion in Ocone and Karatzas (1991). Then, wedefine the market price of risk as θ ( t, Y t ) := σ ( t, Y t ) − ( µ ( t, Y t ) − r ( t, Y t )1 m ) , which is the complete-market counterpart of the total price of risk θ St in the completed market given in (9). Next, the stateprice density follows ξ t := exp (cid:18) − Z t r ( v, Y v ) dv − Z t θ ( v, Y v ) ⊤ dW v − Z t θ ( v, Y v ) ⊤ θ ( v, Y v ) dv (cid:19) . (41)For any s ≥ t ≥ , the relative state price density ξ t,s = ξ s /ξ t satisfies dξ t,s = − ξ t,s [ r ( s, Y s ) ds + θ ( s, Y s ) ⊤ dW s ] , (42)with initial value ξ t,t = 1 , according to an application of Ito formula. Similarly, the dynamics of H θt,s under the complete market setting specifies to dH θt,s = ( D t Y s ) ∇ θ ( s, Y s ) [ θ ( s, Y s ) ds + dW s ] , (43)with initial value H θt,t = 0 d .Comparing the above dynamics with their incomplete market counterparts (29) and (32), weobserve that the dynamics (42) and (43) here are now explicitly given, as opposed to involving theundetermined investor-specific price of risk process θ us . Besides, we see an important feature thatdifferentiates the complete and incomplete market settings: the dynamics (42) and (43) for the20omplete market only depend on the time s , state variable Y s , and relative state price density ξ t,s ,while the dynamics of ξ S t,s ( λ ∗ t ) and H θt,s ( λ ∗ t ) for the incomplete market, as given by (29) and (32), alsodepend on the time– t multiplier λ ∗ t and the investment horizon T . Such an additional dependencein the incomplete market setting naturally stems from the representation of investor-specific price ofrisk θ us in (34), and thus quantifies the stylized fact, as discussed below (24), that the desired marketcompletion by fictitious assets nontrivially depends on both the current wealth level and investmenthorizon.The above structure of investor-specific price of risk is crucial for deriving the optimal policy. Ifwe fail to recognize such structure that originates from market incompleteness, we might incorrectlywrite the dynamics of ξ S t,s ( λ ∗ t ) and H θt,s ( λ ∗ t ) for incomplete market models via a mechanical imitationof the complete-market dynamics (42) and (43), i.e., by simply changing the notation θ to θ S . Thisleads to not only wrong dynamics of ξ S t,s ( λ ∗ t ) and H θt,s ( λ ∗ t ), e.g., missing the term related to ∂θ u /∂λ inthe second line of (32), but also the failure to capture the hedge term π u,λ ( t, X t , Y t ) in (35) that stemsfrom the uncertainty in investor-specific price of risk due to variation in wealth level. This highlightsagain the importance of applying the representation (24) and the equation system in Theorem 2 forthe investor-specific price of risk function θ u ( v, y, λ ; T ) . With the building blocks in (42) and (43), the decomposition of optimal policy in the completemarket follows as a special case of the incomplete market results in Theorem 1, via replacing thecomponents by their counterparts in the complete market. Specifically, the optimal policy π t stilladmits decomposition (25). The three components are given by (26a)–(26c), except for replacing( σ ( t, Y t ) + ) ⊤ and θ h ( t, Y t ) by ( σ ( t, Y t ) ⊤ ) − and θ ( t, Y t ), respectively. The time– t multiplier λ ∗ t is stillcharacterized by the wealth equation (20). The functions G t,T ( λ ∗ t ), Q t,T ( λ ∗ t ) , H rt,T ( λ ∗ t ) , and H θt,T ( λ ∗ t )appearing therein still satisfy (21), (27a), (27b), and (27c), respectively, except for replacing therelative state price density ξ S t,s ( λ ∗ t ) by ξ t,s . Obviously, unlike the incomplete market case, the optimalpolicy π t for complete market models does not involve the investor-specific price of risk θ uv or thefunction θ u ( v, y, λ ; T ) for representing it via (24). We omit the details of these adaptation andreductions from incomplete market to complete market cases; see, e.g., Detemple et al. (2003), forthe complete market policy decomposition. The previous literature on optimal portfolio choice in incomplete markets largely assumes that in-vestors have the wealth-independent CRRA utility (see, e.g., Detemple and Rindisbacher (2005),Liu (2007), and Detemple and Rindisbacher (2010)). Hence, little is known about the cases withwealth-dependent utilities, e.g., the HARA utility (5b). We hereby start from discussing explicitlyhow the wealth-dependent utility impacts the structure of optimal policy, by comparing the policydecomposition results in Theorems 1 and 2 under the general utility case and those given in Corollary21 below under the CRRA utility.
Corollary 1.
Under the incomplete market model (1) – (2) with the CRRA utility function givenin (5a), the investor-specific price of risk function θ u ( v, y, λ ; T ) introduced through (24) does notdepend on the parameter λ , and thus can be written as θ u ( v, y ; T ) . Consequently, the relative stateprice density ξ S t,s ( λ ∗ t ) and Malliavin term H θt,s ( λ ∗ t ) characterized in general by dynamics (29) and(32), respectively, become independent of λ ∗ t . So we write them simply as ξ S t,s and H θt,s , and spell theirdynamics as dξ S t,s = − ξ S t,s [ r ( s, Y s ) ds + ( θ h ( s, Y s ) + θ u ( s, Y s ; T )) ⊤ dW s ] , (44a) and dH θt,s = ( D t Y s ) ( ∇ θ h ( s, Y s ) + ∇ θ u ( s, Y s ; T ))[ θ S ( s, Y s ; T ) ds + dW s ] , (44b) with θ S ( s, Y s ; T ) = θ h ( s, Y s )+ θ u ( s, Y s ; T ) . Then, the optimal portfolio policy π t in (25) is independentof the current wealth level X t , and admits the following decomposition: π t = π mv ( t, Y t ) + π r ( t, Y t ) + π θ ( t, Y t ) , (45) where the mean-variance component π mv ( t, Y t ) follows π mv ( t, Y t ) = 1 γ ( σ ( t, Y t ) + ) ⊤ θ h ( t, Y t ) = 1 γ ( σ ( t, Y t ) σ ( t, Y t ) ⊤ ) − ( µ ( t, Y t ) − r ( t, Y t )1 m ) ; (46a) the interest rate hedge and price of risk hedge components are given by π r ( t, Y t ) = − ( σ ( t, Y t ) + ) ⊤ E t [ ˜ H rt,T ] E t [ ˜ G t,T ] and π θ ( t, Y t ) = − ( σ ( t, Y t ) + ) ⊤ E t [ ˜ H θt,T ] E t [ ˜ G t,T ] . (46b) The functions ˜ H rt,T , ˜ H θt,T , and ˜ G t,T in above are defined as ˜ H rt,T := (cid:18) − γ (cid:19) (cid:20) (1 − w ) γ e − ρTγ (cid:0) ξ S t,T (cid:1) − γ H rt,T + w γ Z Tt e − ρsγ (cid:0) ξ S t,s (cid:1) − γ H rt,s ds (cid:21) , (47a)˜ H θt,T := (cid:18) − γ (cid:19) (cid:20) (1 − w ) γ e − ρTγ (cid:0) ξ S t,T (cid:1) − γ H θt,T + w γ Z Tt e − ρsγ (cid:0) ξ S t,s (cid:1) − γ H θt,s ds (cid:21) , (47b) and ˜ G t,T := (1 − w ) γ e − ρTγ (cid:0) ξ S t,T (cid:1) − γ + w γ Z Tt e − ρsγ (cid:0) ξ S t,s (cid:1) − γ ds, (47c) with ξ S t,s , H rt,T , and H θt,s evolving according to (44a), (31), and (44b), respectively. The investor-specific price of risk function θ u ( v, y ; T ) satisfies the following d − dimensional equation θ u ( v, y ; T ) = σ ( v, y ) + σ ( v, y ) − I d E [ ˜ Q v,T | Y v = y ] × ( E [ ˜ H rv,T | Y v = y ] + E [ ˜ H θv,T | Y v = y ]) , (48) where ˜ Q v,T = − ˜ G v,T /γ , defined by ˜ Q v,T := − γ (cid:20) (1 − w ) γ e − ρTγ (cid:0) ξ S v,T (cid:1) − γ + w γ Z Tv e − ρsγ (cid:0) ξ S v,s (cid:1) − γ ds (cid:21) . (49)22 he function θ u ( s, y ; T ) is fully characterized by a multidimensional equation system consisting ofequation (48), as well as the SDEs of Y s , ξ S t,s , H rt,s , H θt,s , and D it Y s given in (2), (44a), (31), (44b),and (33), respectively, which are all independent of the parameter λ. Proof.
See Appendix A.3.Though as a special case, the explicit decomposition in Corollary 1 is new relative to the existinganalysis on the structure of optimal policy under CRRA utility by means of, e.g., HJB equations(see, e.g., Liu (2007).) By comparing the decomposition in Corollary 1 and that in Theorems 1 and2, we explicitly observe how the structure of optimal policy under wealth-independent CRRA utilitydiffers from that under the general wealth-dependent utilities, as well as how the specific structureof CRRA utility allows for significant simplification of the decomposition. This comparative studydemonstrates again the importance of our explicit decomposition of optimal policy in Theorems 1and 2.First, the building blocks employed in these two decompositions obviously have different struc-tures. Since the function θ u ( v, y, λ ; T ) does not depend on the parameter λ , the investor-specific priceof risk for the CRRA case takes the form θ us = θ u ( s, Y s ; T ), which is independent of the multiplier λ ∗ s .By the analysis similar to those immediately prior to Theorem 1, θ us here is also independent of thewealth level X s . Thus, the market completion under the CRRA utility enjoys a simpler mechanism.We now compare the dynamics of ξ S t,s ( λ ∗ t ) in (29) with that of ξ S t,s in (44a), as well as the dynamicsof H θt,s ( λ ∗ t ) in (32) with that of H θt,s in (44b). Owing to the absence of parameter λ from function θ u ( v, y, λ ; T ) , dynamics (44a) and (44b) are obviously simpler than (29) and (32). In particular, theterm related to ∂θ u /∂λ in the second line of (32) vanishes in the corresponding equation (44b) forthe CRRA utility case.Next, our decomposition results illustrate how current wealth level impacts the optimal policyunder general wealth-dependent utilities, but not under the CRRA utility. By Theorems 1 and 2for general wealth-dependent utilities, the current wealth level X t impacts the optimal policy throughtwo channels. First, it directly appears in the optimal policy as the denominator in (26a) – (26c).Second, due to the wealth equation X t = E t [ G t,T ( λ ∗ t )], X t is implicitly involved in the optimal policythrough the time– t multiplier λ ∗ t in the functions Q t,T ( λ ∗ t ), H rt,T ( λ ∗ t ), and H θt,T ( λ ∗ t ) with the buildingblocks ξ S t,s ( λ ∗ t ) and H θt,s ( λ ∗ t ). However, both channels are absent under the CRRA utility, thanksto its special structure. As shown in (46a) and (46b), both X t and λ ∗ t vanish in the componentsof the optimal policy. Furthermore, by (44a) and (44b), the building blocks ξ S t,s and H θt,s are alsoindependent of λ ∗ t . Such independence guarantees that X t is not implicitly involved in the optimalpolicy through λ ∗ t as in the case with wealth-dependent utility. In essence, this is again because theinvestor-specific price of risk θ u ( s, Y s ; T ) does not depend on λ ∗ s under the CRRA utility. This wealth-independent property reconciles the conclusions in Detemple et al. (2003) and Ocone and Karatzas(1991) for complete market models. π mv ( t, Y t ) under CRRA utility is independent of investment horizon T and only depends on currenttime t and state variable Y t . Besides, it reflects the classic mean-variance trade-off structure, as it isproportional to the excess return of the risky assets µ ( t, Y t ) − r ( t, Y t )1 m and is inversely proportionalto the covariance matrix σ ( t, Y t ) σ ( t, Y t ) ⊤ as well as the risk aversion level γ .As a direct implication of Proposition 1, the price of risk hedge component π θ ( t, Y t ) has thefollowing decomposition under CRRA utility π θ ( t, Y t ) = π h,Y ( t, Y t ) + π u,Y ( t, Y t ) , with π h,Y ( t, Y t ) = − ( σ ( t, Y t ) + ) ⊤ E t [ ˜ H h,Yt,T ] /E t [ ˜ G t,T ] and π u,Y ( t, Y t ) = − ( σ ( t, Y t ) + ) ⊤ E t [ ˜ H u,Yt,T ] /E t [ ˜ G t,T ] . Here, the func-tions ˜ H h,Yt,T and ˜ H u,Yt,T are defined in the same way as ˜ H θt,T in (47b), except for replacing H θt,T by H h,Yt,T and H u,Yt,T , respectively, which follow (37a) and (37b) with θ u ( s, Y s , λ ∗ t ξ S t,s ( λ ∗ t ); T ) replaced by theCRRA–utility counterpart θ u ( s, Y s ; T ). It is straightforward to verify that π h,Y ( t, Y t ) and π u,Y ( t, Y t )are both independent of the wealth level X t . The last component π u,λ ( t, X t , Y t ) in (35), however,vanishes under the CRRA utility, as the investor-specific price of risk θ us = θ u ( s, Y s ; T ) is independentof multiplier λ ∗ s . As discussed after Proposition 1, the term π u,λ ( t, X t , Y t ) hedges the uncertainty ininvestor-specific price of risk that arises from the variation in wealth level. However, as the investor-specific price of risk does not depend on wealth level under CRRA utility, this hedge componentnaturally disappears. With the wealth-dependent property and minimum requirements for terminal wealth and/or inter-mediate consumption, the HARA utility offers more flexibility in capturing investor risk preferencethan the CRRA utility. However, though desired, solving for optimal policy under HARA utilityis usually regarded as a notoriously difficult and even prohibitive task. To the our best knowledge,only a few analytical results on optimal portfolio under HARA utility exist in the literature; see, e.g.,Kim and Omberg (1996). Instead, previous literatures largely assume CRRA utility for simplicity,because of the technical inconvenience with HARA utility.In the first part of this section, we apply our decomposition results in Theorems 1 and 2 toexplicitly characterize optimal policy under HARA utility, illustrating the potential role of our de-composition for solving optimal policies in closed form, which facilitate subsequent economic analysis.We first develop the results under general incomplete market models with HARA utility. Then, moresurprisingly, we further show that under the special case with nonrandom but possibly time-varyinginterest rate, we can fundamentally connect the HARA optimal policy to its CRRA–utility counter-part. We can explicitly solve the HARA policy as a product of its counterpart under CRRA utilityand an explicit multiplier involving current wealth level and bond prices, further revealing how the24ptimal policy behaves. In the second part of this section, we specialize our decomposition to arepresentative example of incomplete market – the celebrated stochastic volatility model of Heston(1993), leading to closed-form optimal policy under the wealth-dependent HARA utility and the sub-sequent explicitly analysis of the behavior of optimal policies under the complex settings of marketincompleteness and wealth-dependent utility.
We start our discussions from the general case under the incomplete market model (1) and (2).
Corollary 2.
Under the HARA utility (5b) with w > , the investor-specific price of risk function θ u ( v, y, λ ; T ) for representing the investor-specific price of risk θ uv via representation (24) satisfiesthe following d − dimensional equation θ u ( v, y, λ ; T ) = σ ( v, y ) + σ ( v, y ) − I d E [ ˜ Q v,T ( λ ) | Y v = y ] × ( E [ ˜ H rv,T ( λ ) | Y v = y ]+ E [ ˜ H θv,T ( λ ) | Y v = y ]+ λ γ E [ ζ v,T ( λ ) | Y v = y ]) . (50) Here, ˜ H rv,T ( λ ) , ˜ H θv,T ( λ ) , and ˜ Q v,T ( λ ) are defined as ˜ H rv,T ( λ ) := (cid:18) − γ (cid:19) (cid:20) (1 − w ) γ e − ρTγ (cid:0) ξ S v,T ( λ ) (cid:1) − γ H rv,T + w γ Z Tv e − ρsγ (cid:0) ξ S v,s ( λ ) (cid:1) − γ H rv,s ds (cid:21) , (51a)˜ H θv,T ( λ ) := (cid:18) − γ (cid:19) (cid:20) (1 − w ) γ e − ρTγ (cid:0) ξ S v,T ( λ ) (cid:1) − γ H θv,T ( λ ) + w γ Z Tv e − ρsγ (cid:0) ξ S v,s ( λ ) (cid:1) − γ H θv,s ( λ ) ds (cid:21) , (51b) and ˜ Q v,T ( λ ) := − γ (cid:20) (1 − w ) γ e − ρTγ (cid:0) ξ S v,T ( λ ) (cid:1) − γ + w γ Z Tv e − ρsγ (cid:0) ξ S v,s ( λ ) (cid:1) − γ ds (cid:21) , (51c) with ξ S t,s ( λ ) , H rt,T , and H θt,T ( λ ) evolving according to SDEs (29), (31), and (32), respectively, exceptfor replacing λ ∗ t by λ. Besides, ζ v,T ( λ ) is a d –dimensional column vector given by ζ v,T ( λ ) = ζ rv,T ( λ ) + ζ θv,T ( λ ) , (52a) where ζ rv,T ( λ ) := ¯ xξ S v,T ( λ ) H rv,T + ¯ c Z Tv ξ S v,s ( λ ) H rv,s ds, (52b) ζ θv,T ( λ ) := ¯ xξ S v,T ( λ ) H θv,T ( λ ) + ¯ c Z Tv ξ S v,s ( λ ) H θv,s ( λ ) ds, (52c) with ¯ x and ¯ c being the minimum allowable levels for terminal wealth and intermediate consumptionunder the HARA utility (5b). The optimal policy under HARA utility follows by π t = π mv ( t, X t , Y t )+ π r ( t, X t , Y t ) + π θ ( t, X t , Y t ) , where π mv ( t, X t , Y t ) = − X t ( σ ( t, Y t ) + ) ⊤ θ h ( t, Y t ) ( λ ∗ t ) − γ E t [ ˜ Q t,T ( λ ∗ t )] , (53a)25 nd the hedging components given by: π r ( t, X t , Y t ) = − X t ( σ ( t, Y t ) + ) ⊤ (cid:16) ( λ ∗ t ) − γ E t [ ˜ H rt,T ( λ ∗ t )] + E t (cid:2) ζ rt,T ( λ ∗ t ) (cid:3)(cid:17) , (53b) π θ ( t, X t , Y t ) = − X t ( σ ( t, Y t ) + ) ⊤ (cid:16) ( λ ∗ t ) − γ E t [ ˜ H θt,T ( λ ∗ t )] + E t [ ζ θt,T ( λ ∗ t )] (cid:17) . (53c) The multiplier λ ∗ t is characterized as the unique solution for the wealth constraint: ( λ ∗ t ) − γ E t [ ˜ G t,T ( λ ∗ t )] + xE t (cid:2) ξ S t,T ( λ ∗ t ) (cid:3) + cE t (cid:20)Z Tt ξ S t,s ( λ ∗ t ) ds (cid:21) = X t , (54) with ˜ G t,T ( λ ) = − γ ˜ Q v,T ( λ ) , defined by ˜ G t,T ( λ ) := (1 − w ) γ e − ρTγ (cid:0) ξ S t,T ( λ ) (cid:1) − γ + w γ Z Tt e − ρsγ (cid:0) ξ S t,s ( λ ) (cid:1) − γ ds. (55) For case of w = 0 in utility (5b), the above representation still holds except for dropping the termsrelated to c in (52b), (52c), and (54). We now analyze the structure of equation (50) and the fundamental difference from its CRRAcounterpart. Comparing equation (50) under the HARA utility with its CRRA utility counterpart(48), a striking difference lies in that equation (50) contains the additional term λ γ E [ ζ v,T ( λ ) | Y v = y ]on its right-hand side. This explicitly introduces parameter λ to the equation and then to thesolution of θ u ( v, y, λ ; T ). Then, it follows from (51a), (51b), and (51c) that the components ˜ H rv,T ( λ ),˜ H θv,T ( λ ) , and ˜ G v,T ( λ ) also depend on λ. Besides, by (52a) – (52c), λ γ E [ ζ v,T ( λ ) | Y v = y ] appears asa linear combination of the minimum allowable levels ¯ x and ¯ c for terminal wealth and intermediateconsumption under the HARA utility, and is the only term explicitly involving these two constantsthat feature the HARA utility. From this aspect, equation (50) provides a further decompositionfor equation (39) under the HARA utility by isolating the parts involving ¯ x and ¯ c . As a result, themultiplier λ ∗ t and thus the current wealth level X t also get involved in the optimal policy (53a) –(53c) under HARA utility.Based on the general HARA results in Corollary 2, we further show in Proposition 2 below thatunder nonrandom but possibly time-varying interest rate, investor-specific price of risk under HARAutility is indeed identical to that under the corresponding CRRA utility, and furthermore, the optimalpolicies under HARA and CRRA utilities are connected to each other by a simple multiplier relatedto current wealth level and bond prices, which shed lights on the construction of the optimal policyunder HARA utility. Proposition 2.
With nonrandom but possibly time-varying interest rate, the investor-specific price ofrisk θ uv under HARA utility (5b) coincides with its counterpart under the CRRA utility (5a). That is,it does not depend on multiplier λ ∗ v and indeed allows the following representation θ uv = θ u ( v, Y v ; T ) , with the function θ u ( v, y ; T ) characterized by equation θ u ( v, y ; T ) = σ ( v, y ) + σ ( v, y ) − I d E [ ˜ Q v,T | Y v = y ] × E [ ˜ H θv,T | Y v = y ] , (56)26 ith ˜ H θv,T and ˜ Q v,T given in (47b) and (49), respectively. The optimal policy under HARA utilitysatisfies the following simple ratio relationship with its counterpart under CRRA utility: π mvH ( t, X t , Y t ) = π mvC ( t, Y t ) ¯ X t X t and π θH ( t, X t , Y t ) = π θC ( t, Y t ) ¯ X t X t , (57) as well as π rH ( t, X t , Y t ) = π rC ( t, Y t ) ¯ X t /X t ≡ d due to deterministic nature of interest rate. Here,the subscripts H and C represent for the HARA and CRRA utilities, respectively. The CRRAcomponents π mvC ( t, Y t ) and π θC ( t, Y t ) are given in (46a) and (46b). Besides, ¯ X t in (57) is given by ¯ X t = X t − xB t,T − c Z Tt B t,s ds, for w > , (58a) and ¯ X t = X t − xB t,T , for w = 0 , (58b) where B t,s := exp( − R st r v dv ) is the price at time t for a zero-coupon bond maturing at time s .Proof. See Appendix A.4.Let us further discuss the fundamental and explicit connection stated in Proposition 2. First,the investor-specific price of risk under the two utilities indeed agree with each other. That is, whenthere is no uncertainty in interest rate, the investor completes market in exactly the same way underthe two utilities, and the impact of current wealth level entirely vanishes in the investor-specificprice of risk, which is different from the situation under otherwise more general cases as analyzedin Corollary 2 and its follow-up discussions. Second, the ratio relationship (57) bridges the gapbetween HARA and CRRA policies, and thus renders a convenient way to compute or approximatethe optimal policy under HARA utility based on its CRRA counterpart, which is more advantageousto solve in closed-form or implement via, e.g., simulation methods due to the wealth independentnature.Although deterministic interest rate is assumed in Proposition 2, no assumptions are imposed onthe state variable. Thus, we can apply the relationship (57) to various models with sophisticatedstate variable evolving according to complex dynamics. As long as the investor-specific price ofrisk and optimal policy under CRRA utility can be solved, so do those under the HARA utility.For example, Liu (2007) assumes constant interest rate in his incomplete market with stochasticvolatility, and obtains the closed-form optimal policy under CRRA utility. With relationship (57),we can now solve the policy in closed form under HARA utility for this typical incomplete marketexample. Extension of Proposition 2 to the case with random interest rate can be regarded as anopen research topic, for which the change of numeraire techniques in Detemple and Rindisbacher(2010) may render a useful tool.We can view relationship (57) as a decomposition of the optimal policy under HARA utility bydisentangling how the state variable and current wealth level get involved in the optimal policy: the27tate variable Y t impacts the optimal policy only through the CRRA counterparts π mvC ( t, Y t ) and π θC ( t, Y t ), while the current wealth level X t impacts optimal policy only through the ratio ¯ X t /X t .Such a structure leads to an explicit explanation of the puzzle on how HARA investor optimallyallocates her/his wealth, while fulfilling the minimum terminal wealth and intermediate consumptionrequired by the utility specification. With deterministic interest rate, B t,s represents the time– t price of a zero-coupon bond with face value one maturing at time s . Thus, ¯ X t given in (58a), i.e.,¯ X t = X t − xB t,T − c R Tt B t,s ds is the remaining wealth, after the investor buys x zero-coupon bondsmaturing at T , and a continuum of c ds zero-coupon bond maturing at s for all s ∈ [ t, T ]. Thecontinuum payments from this bonds holding position exactly render the minimum terminal wealth x and intermediate consumption c required by the HARA utility (5b), i.e., x at time T and c ds at each s ∈ [ t, T ]. Immediately after the bonds purchasing, the investor allocates the remainingwealth following the optimal policy under CRRA utility, i.e., π mvC ( t, Y t ) ¯ X t and π θC ( t, Y t ) ¯ X t amountof wealth for the mean-variance and price of risk hedge components respectively, thus leading to theoptimal policy under HARA utility given in (57). We can summarize the insights out of relationship(57) as follows. With deterministic interest rate, the investor first buys a series of zero-couponbonds to satisfy his minimum requirements for terminal wealth and intermediate consumption in theentire future investment horizon, then allocates his remaining wealth just as a pure CRRA investor.The reason why the neat and seemingly simple relationship (57) was not discovered, to our bestknowledge, in the literature though much desired, is probably because the investment scheme (58a)and the corresponding mathematical formulation are not easy to conjecture and handle.Moreover, the decomposition in Proposition 2 leads to explicit evidences for the following eco-nomic common consensus. First, the HARA investor allocates more on risky assets as his wealthlevel increases, since the ratio ¯ X t /X t monotonically increases in current wealth level X t . Second,the optimal components π mvH ( t, X t , Y t ) and π θH ( t, X t , Y t ) converge to their CRRA counterparts as X t approaches to infinity, since it follows from (57) and (58a) thatlim X t →∞ π mvH ( t, X t , Y t ) = π mvC ( t, Y t ) lim X t →∞ X t (cid:18) X t − xB t,T − c Z Tt B t,s ds (cid:19) = π mvC ( t, Y t ),and lim X t →∞ π θH ( t, X t , Y t ) = π θC ( t, Y t ) lim X t →∞ X t (cid:18) X t − xB t,T − c Z Tt B t,s ds (cid:19) = π θC ( t, Y t ) . These findings reconcile the fact implied by the Arrow-Pratt relative risk aversion coefficient γ U ( x ) := − (cid:18) ∂U∂x ( t, x ) (cid:19) − x ∂U ∂x ( t, x ) . (59)Under the HARA utility for terminal wealth, it is given by γ U ( X t ) = γX t / ( X t − x ). It clearlydecreases as X t increases, and approaches its CRRA counterpart γ as X t goes to infinity. Thus, thedecrease in risk aversion degree motivates HARA investor to invest more in risky assets.28 .2 Illustrations by the incomplete-market stochastic volatility model of Heston(1993) In this section, we apply the decomposition developed in the previous section to a representativeexample of incomplete market – the celebrated stochastic volatility model of Heston (1993). Ourdecomposition allows for solving the optimal policy under the wealth-dependent HARA utility inclosed form. We then use the closed-form formulae to explicitly analyze the behavior of optimalpolicies by comparing it with the CRRA counterpart. In particular, we will investigate how they areimpacted by, e.g., wealth level, interest rate, and investment horizon, that play important roles in thewealth-dependent HARA utility. This example illustrates how we can further apply our theoreticalfindings on the decompositions established in the previous sections to explicitly analyze the behaviorof optimal policies under complex settings such as market incompleteness and wealth-dependentutility. We now begin by setting the model:
Example 1 (The incomplete-market stochastic volatility model of Heston (1993)) . The asset price S t follows dS t /S t = ( r + λV t ) dt + p (1 − ρ ) V t dW t + ρ p V t dW t , (60a)and the variance V t follows dV t = κ ( θ − V t ) dt + σ p V t dW t , (60b)where W t and W t are two independent standard one-dimensional Brownian motions. Here, theparameter r denotes the risk-free interest rate; the parameter λ controls the market price of risk; thepositive parameters κ, θ, and σ represent the speed of mean-reversion, the mean-reverting level, andthe proportional volatility of the variance process V t , respectively. We assume the Feller’s conditionholds: 2 κθ > σ . The leverage effect parameter ρ ∈ [ − ,
1] measures the instantaneous correlationbetween the asset return and the change in its variance.As shown in Example 1, the stochastic volatility model of Heston (1993) (Heston SV thereafter)is an incomplete market model featuring stochastic volatility, and it belongs to the class of affinemodels (Duffie et al. (2000)). We study the optimal policy under the Heston SV model for risk averseinvestors with HARA or CRRA utilities over the terminal wealth, i.e., the following maximizationproblem sup π t E ( U ( T, X T )) , with the HARA utility function U ( T, x ) = ( x − x ) − γ − γ , (61a)for some risk aversion coefficient γ > x , and for comparisonpurposes, the corresponding CRRA utility function U ( T, x ) = x − γ − γ . (61b)29y applying Proposition 2 on decomposing optimal policy for HARA investors, we immediatelyobtain the closed-form formulae for the mean-variance component, the price of risk hedge component,and the interest rate hedge component under the HARA utility as π mvH ( t, X t , V t ) = π mvC ( t, V t ) ¯ X t X t , π θH ( t, X t , V t ) = π θC ( t, V t ) ¯ X t X t , (62)and π rH ( t, X t , V t ) = π rC ( t, V t ) ¯ X t /X t , where π mvC ( t, V t ), π θC ( t, V t ) , and π rC ( t, V t ) are the counterpartsunder the corresponding CRRA utility, and ¯ X t follows¯ X t = X t − x exp( − r ( T − t )) , (63)according to (58b) . In particular, for the interest rate hedge components, we have π rH ( t, X t , V t ) = π rC ( t, V t ) ¯ X t /X t ≡ . (64)This follows from the representation of interest rate hedge component in (46b) and the constantnature of risk-free interest rate r . From an obvious economics perspective, it leads to no hedgingdemand for interest rate; from a technical perspective, it leads to a zero for the Malliavin derivativeterm H rt,s defined in (31), and, according to (47a), further leads to a zero of the component ˜ H rt,T , which determines π rC ( t, V t ) according to (46b). Thus, the optimal policy under HARA utility is givenby π H ( t, X t , V t ) = π mvH ( t, X t , V t ) + π θH ( t, X t , V t ) = ¯ X t X t π C ( t, V t ) , (65)where π C ( t, V t ) is the optimal policy under the CRRA utility, given by: π C ( t, V t ) = π mvC ( t, V t ) + π θC ( t, V t ). (66)Formulae (62), (65), and (66) are all in closed-form, since the corresponding CRRA components π mvC ( t, V t ) and π θC ( t, V t ) can be obtained in closed form based on our decomposition (45) in Corollary1 and the closed-form CRRA optimal policy solved in Liu (2007). While the optimal policy forCRRA investors under the Heston SV model is available in closed-form, its counterpart for HARAinvestors, to our best knowledge, was absent in the literature. This is probably due to the commonconsensus that HARA utility causes mathematical inconvenience. We now briefly state the closed-form results for the CRRA components π mvC ( t, V t ) and π θC ( t, V t ) for purposes of representing theirHARA counterparts and our comparative studies: The mean-variance component follows π mvC ( t, V t ) = λγ , (67a)which we can obtain by (46a). For an investor with risk aversion coefficient γ > π θC ( t, V t ) = − ρσδ ς ( T − t )) − κ + ς ) [exp( ς ( T − t )) −
1] + 2 ς , (67b)30ith ˜ κ = κ − (1 − γ ) λρσ/γ, δ = − (1 − γ ) λ / (2 γ ), and ς = p ˜ κ + 2 δσ ( ρ + γ (1 − ρ )).These closed-form formulae obviously illustrate again the potential role of our decomposition forsolving optimal policies in closed form. For our subsequent impact analysis, it is helpful to notice theobvious fact that both the mean-variance component (67a) and the price of risk hedge component(67b) are independent of the wealth level X t and the interest rate r ; so, the wealth level X t and theinterest rate r impact the HARA policy only through ¯ X t in (63). We now apply the above closed-form formulae to explicitly analyze the behavior of optimal policiesunder HARA utility by comparing with the corresponding CRRA case. We focus on the impactsfrom wealth level X t , interest rate r , and investment horizon T − t , that significantly distinguish theHARA and CRRA cases. In particular, the next comparative study provides us with an explanationfor empirical evidence that the investment in risky assets increases concavely in investors’ financialwealth; see, e.g., Roussanov (2010), Wachter and Yogo (2010), and Calvet and Sodini (2014). Forour numerical experiments, we set the model parameters at the following representative annualizedvalues κ = 5 . , ρ = − . , λ = 1 . , θ = 0 . , and σ = 0 .
48 according to the MaximumLikelihood estimation results in A¨ıt-Sahalia and Kimmel (2007), while choosing γ = 2 and V t =0 . Impact of wealth level X t : According to the closed-form formulae (62), the optimal policy π H ( t, X t , V t ) under HARA utility is impacted by current wealth level X t only via the ratio ¯ X t /X t ,which is exactly the ratio between the optimal policies (resp. the corresponding components) underHARA and CRRA utilities according to (65) (resp. (62)). By (63), we express it as¯ X t X t = 1 − (cid:18) X t x (cid:19) − exp( − r ( T − t )) , (68)where X t /x measures the current wealth X t relative to the minimum requirement x . This formulaexplicitly shows that ¯ X t /X t increases with X t /x , and it approaches 1 as X t /x approaches infinity. Inthe upper left panel of Figure 1, we follow formulae (67a), (67b), (66), (65) and (68) to show how theoptimal policies under CRRA and HARA utilities depend on the relative wealth X t /x. As exhibitedin the figure, the optimal policy under CRRA utility (red dotdash line) is independent of the wealthlevel. However, the optimal policy under HARA utility (blue solid line) increases concavely with In addition to the closed-form optimal policies, we can also solve equation (56) to obtain the investor-specific priceof risk in closed-form as θ u ( s, V s ; T ) = θ u ( s, V s ; T ) θ u ( s, V s ; T ) ! = ρ p (1 − ρ ) − (cid:0) − ρ (cid:1) ! σδ [exp( ς ( T − t )) − κ + ς ) [exp( ς ( T − t )) −
1] + 2 ς √ V s . We document this result here as a by-product an illustration for the solution of equation (56) without further analysis. t /x and approaches its asymptote at level of the optimal policy under CRRA utility, i.e., π C ( t, V t ) , showing the shape of a hyperbola, as revealed by formulae (65) and (68). Besides, we observe thatthe impact of wealth level can be substantial. As X t /x increases from 1 to 10, the allocation on riskyasset from HARA investors increases by approximately three times from less than 20% to more than50%. But such an impact decreases as the wealth level becomes higher. Impact of interest rate r : The optimal policy π H ( t, X t , V t ) under HARA utility is impacted by r only via ¯ X t /X t , according to (65) as well as the fact that the corresponding CRRA optimal policy π C ( t, V t ) given in (66) is independent of interest rate r. Formulae (68) explicitly shows that ¯ X t /X t increases with r. In the upper right panel of Figure 1, we follow formulae (67a), (67b), (66), (65) and(68) to show how the optimal policies under CRRA and HARA utilities depend on interest rate r. As exhibited in the figure, the optimal policy under CRRA utility (red dotdash line) is independentof r . However, the optimal policy under HARA utility (blue solid line) increases concavely with r, showing the shape of an exponential function, as revealed by formulae (65) and (68). An economicinterpretation based on the optimal investment strategy designed following Proposition 2 proceedsas follows. While higher risk-free rate does not impact the optimal policy under CRRA utility, itlowers the bond price B t,T . So, it costs less for HARA investors to satisfy their minimum allowablelevel x by investing in bonds, and thus increases their remaining wealth ¯ X t for investing in the riskyasset according to the corresponding CRRA policies. Impact of investment horizon T − t : The mean-variance component π mvH ( t, X t , V t ) under HARAutility is impacted by the investment horizon T − t only via ¯ X t /X t , according to (62) as well as thefact that the corresponding CRRA mean-variance component π mvC ( t, V t ) given in (67a) is independentof T − t. Formulae (68) explicitly shows that ¯ X t /X t increases with T − t, and it approaches 1 as T − t approaches infinity. On the other hand, according to (62), the price of risk hedge component π θH ( t, X t , V t ) under HARA utility depends on T − t via two channels: the ratio ¯ X t /X t given in (68)and the corresponding CRRA component π θC ( t, V t ) given in (67b). We now follow formulae (67a),(67b), (62), and (68) to show the different behaviors of the mean-variance and price of risk hedgecomponents with respect to T − t in the lower left and lower right panels in Figure 1 respectively. Asexhibited by the lower left panel, the mean-variance component under CRRA utility (red dotdashline) is independent of T − t . However, the mean-variance component under HARA utility (bluesolid line) increases concavely with T − t . Similar to our previous economic interpretation for theimpact of interest rate r, the observed behavior of π mvH ( t, X t , V t ) with respect to T − t is because,given the interest rate r , longer investment horizon lowers the bond price B t,T , and thus increasesthe remaining wealth ¯ X t for investing in risky asset according to the corresponding CRRA policies.Next, as shown in the lower right panel, both the price of risk hedge components under the CRRAand HARA utilities increase with investment horizon T − t . For the CRRA case, the sharp increase of π θC ( t, V t ) mainly occurs when investment horizon is short; however, under longer investment horizons,32 θC ( t, V t ) becomes almost insensitive to the increase in T − t . For the HARA case, besides thesimilar sharp increase for short horizons, π θH ( t, X t , V t ) increases faster relatively than its CRRAcounterpart π θC ( t, V t ) for longer horizons. An economic interpretation proceeds as follows: Theincrease in π θC ( t, V t ) under the CRRA utility is because longer investment horizon increases theuncertainty in the price of risk, thus leading to larger hedging demand. On the other hand, theincrease in the HARA component π θH ( t, X t , V t ) is generated by a combination of two, as analyzedabove based on the closed-form formula: first, the decrease of bond price and thus increase ofremaining wealth ¯ X t to be invested in the risky asset according to the corresponding CRRA policies,and second, the increase of the CRRA hedging demand π θC ( t, V t ) as mentioned above. In particular,the first effect is significant even under long investment horizons. The combination of these twoeffects leads to a faster and more lasting increase under the HARA utility compared with its CRRAcounterpart.The above comparative analysis illustrates how our theoretical decompositions allow for un-derstanding behaviors of optimal policy under incomplete market models with wealth-dependentutilities. In particular, it reveals that the wealth-dependent property of HARA utility should not betaken only literally. We observe that the HARA utility impacts the optimal policy via other channelsbeyond current wealth level, i.e., the interest rate and investment horizon, as shown in the upper rightand the lower two panels of Figure 1. This is supported by the fact that the remaining wealth ¯ X t , asa whole, plays the key role in determining how HARA policy fundamentally differs from its CRRAcounterpart, as established in Proposition 2 regarding the decomposition of optimal policy underHARA utility; in particular, under the Heston example illustrated above, the remaining wealth ¯ X t depends on current wealth level X t , minimum requirement x , interest rate r , and investment horizon T − t according to the closed-form formula (63). Dynamic impact of HARA utility:
In addition to the above static analysis of optimal policiesat any arbitrary fixed time t, we now examine the wealth impact of HARA utility from a dynamicperspective. Consider a market with the stock price S t and its variance V t following the Heston SVmodel. Without loss of generality, the initial price is set as S = 100 and the initial variance isset as V = θ = 0 . . We consider two investors with HARA utilities over terminal wealth for aninvestment horizon of T = 10 years. Their risk aversion coefficient and minimum allowable level forterminal wealth are set as γ = 2 and x = 10 , i.e., one million, respectively. The two investors onlydiffer in their initial wealth levels: the high-wealth investor has an initial wealth of X H = 10 , i.e.,ten million, while the low-wealth investor has an initial wealth of X L = 3 × , i.e., three million.Thus, their ratios of initial wealth over the minimum allowable level are given by X H /x = 10 and X L /x = 3, respectively.Denote by π Ht and π Lt the optimal policies of the two investors, respectively. Then, the ratio π Ht /π Lt measures how the optimal policy of the high-wealth investor differs from that of the low-33ealth investor. We simulate the market as well as the dynamics of optimal policies π Ht and π Lt during the entire investment horizon, and thus the dynamics of π Ht /π Lt simply by ratio calculation.The simulation is conducted based on a standard Euler scheme on the Heston model (60a) – (60b).Along the simulated path, the optimal policies π Ht and π Lt are evaluated via the closed-form solution,and the investors’ wealth evolve according to equation (3) with c t ≡ π t being the optimal policygiven in (65).Figure 2 shows a representative simulated path of ratio π Ht /π Lt (red solid line with the righty-axis) and the corresponding path of stock price S t (blue dotdash line with the left y-axis). Theoptimal policy under CRRA utility, as given in (67a) and (67b), only depends on t via the investmenthorizon T − t , and is independent of variance level V t , stock price S t , and wealth level X t . Thuswe have π Ht /π Lt ≡ t under the CRRA utility, i.e., the optimal policies from the high-wealthand low-wealth investors always coincide. However, under HARA utility, we observe an interestingpattern that the ratio π Ht /π Lt and the stock price S t are negatively correlated, i.e., π Ht /π Lt tends to behigh (resp. low) when S t is low (resp. high). Actually, this is not an incidental result out of a specificpath. With 1000 trials of simulations, we calculate the average correlation between π Ht /π Lt and S t as approximately − .
95. We can explain this negative correlation in the following way. As shownin the lower two panels of Figure 1 under the Heston SV model with the representative parameters,the HARA investors always hold positive positions in the stock. Thus, when the stock price S t increases, the wealth levels X t of the investors increase, making both of them wealthier. We nowrecall that, as shown in the upper left panel of Figure 1, under HARA utility, the impacts of wealthlevel on optimal policies decrease and the HARA optimal policy approaches its CRRA counterpart,as wealth level becomes higher. So, when the stock price increases and thus both investors becomewealthier, the difference in their optimal policies becomes smaller, in particular, the ratio π Ht /π Lt approaches 1 , as in the case under CRRA utility. We can also interpret this as follows. As investorsbecome wealthier, the minimum constraint x is less binding for them, and thus they behave morelike CRRA investors. This example demonstrates the wealth-dependent property of HARA utilityfrom a dynamic perspective. As opposed to being fully determined by their different initial wealthlevels, the investors’ dynamic optimal investment decisions depend on the historical performance ofthe market, e.g., the time spent in the bull or bear regimes. Obviously, cycles matter for HARAinvestors. This important path dependence is totally absent under the wealth-independent CRRAutility, as discussed above. Besides, Figure 2 shows that the policy ratio π Ht /π Lt can reach levels aslarge as 1 .
24, namely a 24% relative increase when shifting from the low-wealth type to the high-wealth type. The corresponding values for π Lt and π Ht at the peak of the red solid line in year 10 are41% and 51%, which yields an absolute increase of 10%. On the 1000 trials of simulations, we haveobserved a relative difference as large as 80% corresponding to an absolute difference of 19%. Thisimplies that delegated portfolio management assuming a CRRA utility can be completely erroneous34or a HARA investor and her investment profile can be suboptimal to a large extent except if hercurrent wealth X t is sufficiently high. The decomposition developed in Section 2 provides an indispensable foundation for revealing thestructure of optimal policy and conducting relevant analysis under incomplete market models. Asshown in Section 3, it facilitates the potential success in solving optimal policy in closed-form undersome settings. Now, a more important question is of course how to implement the decompositionunder the general setting as proposed in Section 2.1. Monte Carlo simulation is obviously oneof the most natural methods. For the complete market counterpart, Detemple et al. (2003) andCvitanic et al. (2003) proposed simulation approaches under general diffusion models. However,due to the essential challenge stemming from the market incompleteness, their extension or evenalternative approach, that is applicable to incomplete market model, remains an important openproblem so far.In this section, we propose and implement a Monte Carlo simulation method for optimal dynamicportfolio choice in the general incomplete market model, as an application of and based on thedecomposition developed in Section 2. It potentially allows for not only computing the optimalpolicies, but also conducting various comparative studies regarding their behavior and economicimplications. Our Monte Carlo method is an extension of the simulation approach for completemarket models developed in Detemple et al. (2003). By fully exploiting the explicit structure ofour decomposition, including, e.g., the equation system characterizing the investor-specific priceof risk θ u ( v, y, λ ; T ) , it successfully handles the essential difficulty due to market incompletenessand the generality of utility specification. It is thus of special importance for settings under whichoptimal policies are difficult or impossible to explicitly solve and/or existing numerical approachesdo not efficiently apply. As an illustrative example, we apply the simulation method in analyzingthe behavior of optimal policies under a flexible class of stochastic volatility models that in generallack analytical solutions. We now outline the major technical aspects for our simulation method. First, it is natural to followTheorem 1 to simulate for estimating the conditional expectations E t [ Q t,T ( λ ∗ t )], E t [ H rt,T ( λ ∗ t )], and E t [ H θt,T ( λ ∗ t )] of the optimal policy. Nevertheless, the difficulty stems from the unknown form ofthe investor-specific price of risk function θ u ( v, y, λ ; T ), which is involved in the dynamics (29)and (32) for simulation. Second, we numerically solve the unknown function θ u ( v, y, λ ; T ) basedon the equation system established in Theorem 2, in which the involved conditional expectations35 [ Q v,T ( λ ) | Y v = y ], E [ H rv,T ( λ ) | Y v = y ], and E [ H θv,T ( λ ) | Y v = y ] need to be simulated at the sametime. In other words, equations (2), (29), (31), (32), (33), and (39) in Theorems 1 and 2 form togetheran equation systems to solve simultaneously. For the expectations to be simulated, we resort to, e.g.,the standard discretization techniques for simulation, e.g., the Euler scheme (see, e.g., Chapter 6of Glasserman (2004)); for numerically solving function θ u , we resort to, e.g., the standard finitedifference techniques for numerical solution of initial value problems of differential and/or integralequations; see, e.g., Section 2.4 of Morton and Mayers (2005). Third, in principle, we need to solvethe wealth constraint multiplier λ ∗ t involved in the dynamics (29) and (32). It can be numericallysolved from equation (20) via standard root-finding methods, where we can simulate the conditionalexpectation E t [ G t,T ( λ ∗ t )] by the same procedure as proposed above for a given candidate value of λ ∗ t .This demonstrates the benefit of employing the time– t multiplier λ ∗ t in the decomposition instead ofthe initial version λ ∗ . Since we have clarified the major steps of our numerical approach, we omitthe routine details for the sake of space.We now examine the numerical performance of our simulation method for optimal portfoliochoice. By employing two representative incomplete-market examples with closed-form formulaefor optimal policy, we demonstrate the accuracy of simulated policies by comparing with the cor-responding explicitly known benchmarks. The first example is the stochastic volatility model ofHeston (1993) as given in Example 1. The second example is the mean-reverting return model ofKim and Omberg (1996), which is given in Example 2 below. These two examples cover two mainfeatures of the dynamics in real market: the stochastic volatility and stochastic drift, which featuremarket incompleteness. Example 2 (The incomplete-market mean-reverting return model of Kim and Omberg (1996)) . The asset price S t follows dS t /S t = ( r + σθ t ) dt + σdW t , where r, σ, and θ t represent the constantinterest rate, market price of risk, and volatility, respectively; W t is a standard one dimensionalBrownian motion. Assume that θ t is governed by the following Ornstein-Uhlenbeck process dθ t = λ ( θ − θ t ) dt − σ θ dW t . Assume that the instantaneous correlation between W t and W t is given by ρ ∈ [ − , dW t dW t = ρdt. Both Examples 1 and 2 are affine models (Duffie et al. (2000)), and closed-form formulae for theoptimal policies exist for both HARA utility (61a) and CRRA utility (61b) over terminal wealthwith risk aversion coefficient γ >
1. We can find the closed-form policies in Kim and Omberg (1996)for the mean-reverting return model under the HARA utility, in Liu (2007) for the Heston SV modelunder the CRRA utility, and (62) – (65) derived in this paper for the Heston SV model under theHARA utility. Therefore, we are allowed to use these closed-form formulae as benchmarks to examinethe accuracy of our simulation method, which is generally applicable but of course not confined toaffine models. Due to the relation between optimal policies under CRRA and HARA utilities withdeterministic interest rate, as established in Proposition 2, it suffices to consider the CRRA case in36ur numerical experiments. The two models can be nested into our general framework (1)–(2) bysetting m = 1, n = 1, and d = 2 . The state variable Y t is specialized as V t in Example 1, and θ t inExample 2, respectively.As shown in Corollary 1, the decomposition of the optimal policy enjoys the special structureunder the CRRA utility, while maintaining the impact from market incompleteness. The mean-variance component can be directly obtained in closed-form by (46a), which is given by π mv ( t, V t ) = λ/γ for Example 1 and π mv ( t, θ t ) = θ t /γ for Example 2. Besides, as both the two examples assumeconstant interest rate, we have zero interest rate hedge components for them by its representationin (46b), i.e., π r ( t, V t ) = π r ( t, θ t ) ≡
0. Thus, we only need to simulate the price of risk hedgecomponent π θ ( t, V t ) or π θ ( t, θ t ) and then compare results with the closed-form benchmark formulae.In particular, as shown in Corollary 1, the investor-specific price of risk function θ u ( v, y, λ ; T ) isindependent of parameter λ and we can represent the hedge components of the optimal policy asin (46b). These structures lead to slight simplification of our simulation method outlined in thebeginning of this section, but without reducing the essential challenges, e.g., we still need to followthe general method to deal with the equation (48) for the unknown investor-specific price of riskfunction θ u ( v, y ; T ) as well as the complex dynamics of variables ξ S t,s and H θt,s in (44a) and (44b),respectively.In our numerical experiments under Examples 1 and 2, we set for illustration purposes the riskaversion level at γ = 2 and consider three representative investment horizons with T = 0 . T = 1 . T = 3 . T , we compare the simulated optimal hedgecomponents π θ ( t, V t ) (resp. π θ ( t, θ t )) with the benchmark for five representative values of the statevariable V t (resp. θ t ). In the implementation, we simulate for M = 10 trials, and for each, we set thetime increment for discretization as ∆ = 0 . π θ and the true value π θ true calculated from benchmark formulae. The fourth column shows the relative error of simulation e Rel ,which is computed by e Rel = | ˆ π θ − π θ true | / | π θ true | . The fifth and sixth columns, i.e., Std and CI ,report the standard error of estimator ˆ π θ calculated from the asymptotic standard deviation of ratioestimator (the first method mentioned above) and the corresponding 95% level confidence intervalfor π θ true . The next two columns, i.e., Std B and CI , B , report bootstrap standard error of estimator37 π θ calculated based on J = 1 ,
000 bootstrap samples (the second method mentioned above) andthe corresponding 95% level confidence interval for π θ true . The last column reports the relative rootmean square error (RMSE) estimated based on the bootstrap estimators, i.e., RMSE B / | π θ true | , whereRMSE B is calculated according to RMSE B = qP Jj =1 (ˆ π θ, ( j ) − π θ true ) /J , with ˆ π θ, (1) , ˆ π θ, (2) , ..., ˆ π θ, ( J ) being the bootstrap estimators of π θ true based on J bootstrap samples.As shown in the results from Tables 1 and 2, our simulation method performs accurately acrossall the representative choices of investment horizons and state variables for both Examples 1 and2. For all entries, the relative errors and the relative RMSE are at the magnitude of 10 − or less;the standard errors are relatively small compared with the corresponding simulated values ˆ π θ andthe true values π θ true all lie in the corresponding confidence intervals. Moreover, by comparing thecolumns of Std and Std B (resp. CI and CI , B ), we observe that the standard errors (resp. confidenceintervals) calculated by the asymptotic standard deviation and bootstrap are closed to each other.This further demonstrates the numerical validity and accuracy of our simulation method. In this section, we illustrate by an example how to apply our decomposition and the subsequentsimulation method in analyzing the behavior of optimal policies under incomplete market models.Under a flexible class of incomplete market stochastic volatility models, which is more general andflexible than the Heston-SV model, we reveal and explain various impacts on optimal portfolio frommodel parameters that control the volatility dynamics, i.e., to analyze how the optimal policy behaveswith respect to different model specifications. This is as a substantial complement and extension tothe impact analysis conducted in Section 3.2 under the Heston SV model with our closed-form HARAoptimal policy (67a) and (67b). Nevertheless, as the closed-form solution for optimal policy is nolonger available for now, our simulation method plays an important role. The analysis in this sectionfurther demonstrates the importance and application potential of our decomposition developed inSection 2.2.We begin by introducing the model. Consider the following incomplete-market bivariate CEV-type stochastic volatility (CEV-SV hereafter) model of Jones (2003), which generalizes the volatilitydynamics of Heston SV model in Example 1.
Example 3 (The incomplete-market CEV-type stochastic volatility model of Jones (2003)) . Theasset price S t follows dS t /S t = ( r + λV t ) dt + p (1 − ρ ) V t dW t + ρ p V t dW t , (69a)and the variance V t follows dV t = κ ( θ − V t ) dt + σV νt dW t , (69b)38here W t and W t are two independent standard Brownian motions. Here, the parameters r, λ,κ, θ, σ, and ρ have the same interpretations as those for the stochastic volatility model of Heston(1993) given in Example 1. The constant elasticity parameter ν controls the elasticity of volatility.The CEV-SV model (69a) – (69b) belongs to the nonaffine class (see, e.g., Duffie et al. (2000)).It was proposed to offer great flexibility in modeling volatility dynamics; see, e.g., Jones (2003),A¨ıt-Sahalia and Kimmel (2007), Medvedev and Scaillet (2007), and Christoffersen et al. (2010) forempirical evidence in favour of this model. Owing to its generality, the CEV-SV model reduces,by letting ν = 1 / , , and 3 / , to the stochastic volatility model of Heston (1993) as in Example1, the GARCH diffusion type (GARCH-SV hereafter) proposed in Nelson (1990), and the 3 / − type (3 / − SV hereafter) investigated in, e.g., Christoffersen et al. (2010). While the literature haswitnessed a large amount of enthusiastic discussions on what is the proper dynamics of stochas-tic volatility for fitting prices of underlying asset or/and its derivative securities, little is knownabout the impact of such dynamics on optimal portfolios. As an attractive issue featuring marketincompleteness, this now becomes our task.For these flexible models, there is no general closed-form formulae for the optimal policies, exceptfor the affine case with ν = 1 / π r ( t, Y t ) = 0 as the interest rate is constant, and the mean-variance component is given by π mv ( t, V t ) = λγ (70)according to (46a). Both components are independent of the specification of volatility dynamics.In particular, the behaviors of π mv ( t, V t ) are clear: π mv ( t, V t ) increases (resp. decreases) with themarket price of risk parameter λ (resp. the risk aversion level γ ). The explanation is that investorsare more willing (resp. reluctant) to hold the risky asset as the market price of risk increases (resp.as their risk aversion increases.) See, e.g., Chacko and Viceira (2005), Liu (2007), and Moreira and Muir (2019) for investigations on optimalportfolio choice under stochastic volatility models.
39n what follows, we focus on understanding the price of risk hedge component π θ ( t, V t ) in the CEV-SV model. We simulate π θ ( t, V t ) by the method introduced in Section 4.1. For illustration purpose,we consider a representative medium investment horizon T − t = 1 year. The three panels of Figure3 illustrate how the optimal hedge component π θ ( t, V t ) depends on the current level of variance V t ,the correlation ρ , and the risk aversion γ , under the CEV-SV model in Example 3, complementingthe impact with respect to wealth level, interest rate, and investment horizon analyzed in Section 3.2under the Heston-SV model as a representative illustration. To take various volatility dynamics intoconsideration, instead of only an arbitrary one, we consider three choices of the elasticity parameter: ν = 1 /
2, 1, and 3 /
2, which correspond to the Heston-SV, GARCH-SV, and 3 / ν while controlling the effect of volatility,we keep σθ ν , the volatility of the variance process V t evaluated at the mean-reverting level θ, at aconstant and realistic level φ V := σθ ν by setting σ accordingly as σ = φ V /θ ν for each value of ν .Otherwise, if we merely change the value of ν , the volatility φ V = σθ ν would change accordingly, evento some unrealistic levels, and thus simultaneously impacts π θ ( t, V t ). By controlling the volatility φ V , we are able to expose and analyze the neat impact from the elasticity of the variance process,without changing the volatility φ V simultaneously. Impact of variance level V t : The left panel demonstrates how the optimal hedge component π θ ( t, V t ) varies with the current wealth level V t . The optimal hedge component π θ ( t, V t ) remainsconstant under the Heston-SV model. This observation reconciles the corresponding result in Liu(2007) that the optimal policy in the Heston model is independent of current variance level, as shownin (67b). However, this mathematical property does not necessarily hold under other specificationsof volatility dynamics, and instead optimal policies may depend on variance level. In the GARCH-SV model and 3 / π θ ( t, V t ) increases with the current variance V t . Moreover, the increment in the 3 / π θ ( t, V t ) with respect tovariance V t as follows. By (69a), the market price of risk in the CEV-SV model is calculated as λ √ V t , which represents the premium in expected return per a unit of volatility. Obviously, the uncertaintyin λ √ V t depends on that of V t . As λ √ V t is a concave increasing function in V t , the impact of V t on market price of risk λ √ V t decreases as V t increases . On the other hand, the volatility of the This can also be seen from the derivative of market price of risk with respect to V t , since this derivative is givenby λ/ (2 √ V t ) and clearly decreases in V t . σV νt in (69b), increases in V t given the elasticity parameter ν >
0, andit increases faster in V t when ν is larger. Thus, according these helpful calculus facts, the increase of V t has two effects that oppositely impact the hedge component π θ ( t, V t ). First, as the market priceof risk λ √ V t becomes less sensitive to V t , its uncertainty decreases and leads to a smaller hedgingdemand. Second, as the volatility of the variance process increases, increase in V t leads to moreuncertainty in the variance process, which translates to a larger hedging demand. For Heston-SVmodel with ν = 1 /
2, the independence of π θ ( t, V t ) with respect to V t can be possibly attributed tothe offset of these two opposite effects. For the GARCH-SV and 3/2-SV models however, as theyhave larger elasticity parameters ν , the volatility of the variance process σV νt increases faster in V t compared with the case under the Heston-SV model. Thus, the second effect of increasing V t (i.e.,more volatility in the variance process) outweighs the first one (i.e., less sensitivity of the marketprice of risk) in the GARCH-SV and 3/2-SV models. Furthermore, as the volatility in the varianceprocess σV νt increases faster in V t in the 3/2-SV model with ν = 3 / ν = 1, the hedging demand also increases faster in V t in the 3/2-SV model, as shown theleft panel. Impact of leverage effect parameter ρ : In the middle panel, we analyze how the optimal hedgecomponent π θ ( t, V t ) depends on the correlation parameter ρ, namely on the leverage effect when ρ <
0. As a by-product, this analysis provides an angle for understanding the impact of marketincompleteness. This is because the absolute value of ρ is related to the degree of market incomplete-ness: the smaller is | ρ | , the more incomplete is the market. In particular, the market is incomplete(resp. complete) for ρ ∈ ( − ,
1) (resp. ρ = − ρ = 1). For ρ = − ν = 1 / , the CEV-SVmodel reduces to the Heston-Nandi model in Heston and Nandi (2000). As exhibited in the middlepanel, the hedge component π θ ( t, V t ) interestingly changes its sign at ρ = 0 , and it decreases as ρ increases. This pattern is robust across all the three representative model specifications.An economic interpretation of the above relationship between the hedge component π θ ( t, V t )and correlation ρ proceeds as follows. Recall the market price of risk is calculated as λ √ V t by (69a).According to (70), investors take a long position in the mean-variance component given the condition λ >
0. Thus, they clearly benefit from a higher market price of risk, therefore regard its decrease, i.e.,the decrease of variance V t , as the downside risk. To hedge against this adverse situation, investorsought to take a position via π θ ( t, V t ) so that the decrease of V t is favorable, i.e., leading to potentiallylarger terminal wealth and thus the utility. To achieve this, we can buy (resp. short sell) the asset,as its price S t tends to increase (resp. decrease) as V t decreases when ρ < ρ > π θ ( t, V t ) when ρ < ρ > ρ = − ρ = 1, the instantaneous changes of price S t and its variance V t areperfectly correlated and thus result in a complete market. In this situation, the risky asset serves asa perfect hedge for the uncertainty in market price of risk λ √ V t . This naturally leads to the largest41ossible absolute value of the hedge component π θ ( t, V t ) across all values of ρ, as shown in the middlepanel. However, when ρ = 0, the instantaneous changes of price S t and its variance V t are totallyuncorrelated and accordingly lead to the “most incomplete” market. Thus, investors are not able touse the asset to hedge the uncertainty in the market price risk λ √ V t as under ρ = 0 . This naturallyresults in the zero hedge component π θ ( t, V t ) = 0 when ρ = 0. Impact of risk aversion level γ : The right panel demonstrates that the optimal hedge component π θ ( t, V t ) exhibits a hump shape with respect to the risk aversion γ for all the three representativemodel specifications. The aforementioned hump shape is also discussed in Detemple et al. (2003)under the complete-market non-linear mean-reverting elastic volatility (NMREV) model. We providethe following economic interpretation in our incomplete-market environment. When γ approaches1, the CRRA utility reduces to the logarithm utility. As discussed in Merton (1971), the investorswith the logarithm utility behave myopically and do not invest to hedge the risk in their investmentopportunities. Thus, their hedge component π θ ( t, V t ) is identically zero. As the risk aversion γ increases above 1, investors become more risk averse than myopic and start to hedge the risk byusing the risky asset. This explains the increase in the absolute value of the hedge component π θ ( t, V t ). However, as γ further increases significantly, the investors become much more risk averseso that they do not want to bear risk associated with the risky asset. This can be seen from themean-variance component (70), which obviously decreases with γ. As investors reduce their holdingsin the risky asset, the hedging demand for market price of risk also decreases. These two effectstogether yield the hump shape. The documentation in our incomplete-market CEV-SV model andin the complete-market NMREV model studied in Detemple et al. (2003) suggest that the humpshape effect for the optimal price of risk hedge component with respect to risk aversion may hold ina robust fashion insensitive to model specifications.
Impact of elasticity parameter ν : Finally, we analyze the impact from the elasticity parameter ν for different values of V t and ρ, as shown in the first two panels, respectively. As shown in the leftpanel, if V t > θ (resp. V t < θ ), as ν increases from 1 / / , the curves turn anti-clockwise and thus the positive value of the hedge component π θ ( t, V t ) increases (resp. decreases).We can interpret the aforementioned behavior of π θ ( t, V t ) with respect to elasticity ν as follows.Plugging σ = φ V /θ ν into (69b), i.e., fixing the volatility of V t evaluated at its mean-reverting level θ as the constant φ V , the dynamics of V t specifies to dV t = κ ( θ − V t ) dt + φ V ( V t /θ ) ν dW t . We see fromthis dynamics that the instantaneous volatility of V t , as given by φ V ( V t /θ ) ν , clearly increases (resp.decreases) with elasticity ν when V t > θ (resp. V t < θ ). So, when V t > θ , a higher ν yields morevolatility in the variance process V t , and thus more uncertainty in the market price of risk λ √ V t thatneeds to be hedged. This demand naturally leads to a larger absolute value of the hedge component π θ ( t, V t ). By similar argument, when V t < θ , a higher ν leads to a smaller absolute value of thehedge component π θ ( t, V t ) . Such interpretations reconcile the behavior of π θ ( t, V t ) with respect to ν This paper establishes and implements a new decomposition of the optimal dynamic portfolio choiceunder general incomplete-market diffusion models with flexible wealth-dependent utilities. By notic-ing and applying the functional form of the investor-specific price of risk in a suitable market com-pletion procedure, we derive explicit dynamics of the components underlying the optimal policy andobtain an equation system for characterizing the investor-specific price of risk, which is shown todepend on both market state and wealth level, and further for expressing the optimal policy. Our de-composition substantially extends the representation results under complete market setting in, e.g.,Detemple et al. (2003) to general incomplete market models. The decomposition reveals the impactson the optimal policy from market incompleteness and wealth-dependent utilities. In particular, wereport a new important hedge component for non-myopic investors with wealth-dependent utilities.This new component hedges the uncertainty in investor-specific price of risk due to variation inwealth level.As the first application, we establish and compare the decompositions of optimal policy undergeneral models with the prevalent HARA and CRRA utilities. Moreover, under nonrandom butpossibly time-varying interest rate, we explicitly solve the HARA policy as a combination of a bondholding scheme and the corresponding CRRA strategy. As a representative illustration, we apply thedecomposition results to solve the optimal policy for HARA investors under the incomplete marketstochastic volatility model of Heston (1993) in closed-form and then conduct in-depth comparativestudies for understanding the nature of wealth dependency of optimal policies.In addition to the theoretical findings and the potential applications in explicitly solving optimalpolicies, our decomposition renders an indispensable and flexible foundation for implementing theoptimal policy by appropriate numerical methods. As the second application, we propose a MonteCarlo simulation approach for computing the optimal policies in general incomplete market models,where existing numerical approaches do not efficiently apply. Such a simulation method is madepossible, owing to the indispensable characterization of market incompleteness through our decom-position. As a representative illustration, we apply this simulation approach in a novel analysis ofthe behavior of optimal portfolio policies under a flexible class of stochastic volatility models.We can adapt or generalize our decomposition for optimal portfolio policies to other settings, e.g.,the forward measure based representation of optimal portfolios considered in Detemple and Rindisbacher(2010). Moreover, it is interesting, among other possible extensions, to consider other (exotic) typesof market incompleteness, e.g., the short-selling constraint or the “rectangular” constraint consideredin Cvitanic and Karatzas (1992) and/or Detemple and Rindisbacher (2005), as well as the presence43f jumps considered in, e.g., A¨ıt-Sahalia et al. (2009) and Jin and Zhang (2012). Regarding imple-mentation by simulation, it is beneficial to investigate the convergence rate of our approach andenhance its efficiency via various techniques; see, e.g., the relevant analysis in Detemple et al. (2006)for the simulation method under complete market models. Besides, it is also interesting to developalternative numerical methods that can be efficiently applied to incomplete market models, basedon our theoretical decomposition results. We defer these investigations, among others, to futureprojects. 44 eferences
A¨ıt-Sahalia, Y., Cacho-Diaz, J., Hurd, T., 2009. Portfolio choice with a jumps: A closed formsolution. Annals of Applied Probability 19, 556–584.A¨ıt-Sahalia, Y., Kimmel, R., 2007. Maximum likelihood estimation of stochastic volatility models.Journal of Financial Economics 83, 413–452.Basak, S., Chabakauri, G., 2010. Dynamic mean-variance asset allocation. Review of Financial Stud-ies 23 (8), 2970–3016.Bazdresch, S., Kahn, R. J., Whited, T. M., 2017. Estimating and Testing Dynamic Corporate FinanceModels. The Review of Financial Studies 31 (1), 322–361.Brandt, M. W., 2010. Portfolio choice problems. In: A¨ıt-Sahalia, Y., Hansen, L. P. (Eds.), Handbookof Financial Econometrics, 1st Edition. Vol. 1. North-Holland, San Diego, pp. 269 – 336.Brennan, M., Schwartz, E., Lagnado, R., 1997. Strategic asset allocation. Journal of EconomicDynamics and Control 21, 1377–1403.Brennan, M. J., 1998. The role of learning in dynamic portfolio decisions. European Finance Review1 (3), 295–306.Brennan, M. J., Xia, Y., 2002. Dynamic asset allocation under inflation. Journal of Finance 57 (3),1201–1238.Burraschi, A., Porchia, P., Trojani, F., 2010. Correlation risk and optimal portfolio choice. Journalof Finance 65 (1), 393–420.Calvet, L. E., Sodini, P., 2014. Twin picks: Disentangling the determinants of risk-taking in householdportfolios. The Journal of Finance 69 (2), 867–906.Campbell, J. Y., Chacko, G., Rodriguez, J., Viceira, L. M., 2004. Strategic asset allocation in acontinuous-time VAR model. Journal of Economic Dynamics and Control 28 (11), 2195–2214.Chacko, G., Viceira, L. M., 2005. Dynamic consumption and portfolio choice with stochastic volatilityin incomplete markets. Review of Financial Studies 18 (4), 1369–1402.Christoffersen, P., Jacobs, K., Mimouni, K., 2010. Volatility dynamics for the S&P500: Evidence fromrealized volatility, daily returns, and option prices. Review of Financial Studies 23 (8), 3141–3189.Cox, J. C., Huang, C.-f., 1989. Optimal consumption and portfolio policies when asset prices followa diffusion process. Journal of Economic Theory 49 (1), 33–83.45vitanic, J., Goukasian, L., Zapatero, F., 2003. Monte Carlo computation of optimal portfolios incomplete markets. Journal of Economic Dynamics and Control 27, 971–986.Cvitanic, J., Karatzas, I., 1992. Convex duality in constrained portfolio optimization. Annals ofApplied Probability 2 (4), 767–818.Detemple, J., 2014. Portfolio selection: A review. Journal of Optimization Theory and Applications161 (1), 1–21.Detemple, J., Garcia, R., Rindisbacher, M., 2003. A Monte Carlo method for optimal portfolios.Journal of Finance 58 (1), 401–446.Detemple, J., Garcia, R., Rindisbacher, M., 2006. Asymptotic properties of monte carlo estimatorsof diffusion processes. Journal of Econometrics 134 (1), 1–68.Detemple, J., Rindisbacher, M., 2005. Closed-form solutions for optimal portfolio selection withstochastic interest rate and investment constraints. Mathematical Finance 15 (4), 539–568.Detemple, J., Rindisbacher, M., 2010. Dynamic asset allocation: Portfolio decomposition formulaand applications. Review of Financial Studies 23, 25–100.Duffie, D., Pan, J., Singleton, K. J., 2000. Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68, 1343–1376.Duffie, D., Singleton, K. J., 1993. Simulated moments estimation of markov models of asset prices.Econometrica 61 (4), 929–952.Dumas, B., Luciano, E., 2017. The Economics of Continuous-Time Finance. MIT Press, Cambridge.Efron, B., 1979. Bootstrap methods: Another look at the jackknife. The Annals of Statistics 7 (1),1–26.Fitzpatrick, B. G., Fleming, W. H., 1991. Numerical methods for an optimal investment-consumptionmodel. Mathematics of Operations Research 16 (4), 823–841.Glasserman, P., 2004. Monte Carlo methods in financial engineering. Springer, New York.He, H., Pearson, N. D., 1991. Consumption and portfolio policies with incomplete markets andshort-sale constraints: The infinite dimensional case. Journal of Economic Theory 54, 259–304.Heston, S., 1993. A closed-form solution for options with stochastic volatility with applications tobonds and currency options. Review of Financial Studies 6, 327–343.46eston, S., Nandi, S., 2000. A closed-form GARCH option pricing model. Review of Financial Studies13, 585–626.Hindy, A., Huang, C. F., Zhu, S. H., 1997. Numerical analysis of a free-boundary singular controlproblem in financial economics. Journal of Economic Dynamics and Control 21(2-3), 297–327.Jin, X., Zhang, A. X., 2012. Decomposition of optimal portfolio weight in a jump-diffusion modeland its applications. Review of Financial Studies 25, 2877–2919.Jones, C. S., 2003. The dynamics of stochastic volatility: Evidence from underlying and optionsmarkets. Journal of Econometrics 116, 181–224.Karatzas, I., Lehoczky, J. P., Shreve, S. E., 1987. Optimal portfolio and consumption decisions for asmall investor on a finite horizon. SIAM Journal of Control and Optimization 25, 1557–1586.Karatzas, I., Lehoczky, J. P., Shreve, S. E., Xu, G.-L., 1991. Martingale and duality methods forutility maximization in an incomplete market. SIAM Journal on Control and Optimization 29 (3),702–730.Karatzas, I., Shreve, S. E., 1991. Brownian Motion and Stochastic Calculus, 2nd Edition. Vol. 113of Graduate Texts in Mathematics. Springer-Verlag, New York.Kim, T., Omberg, E., 1996. Dynamic nonmyopic portfolio behavior. Review of Financial Studies 9,141–161.Koijen, R., 2014. The cross-section of managerial ability, incentives, and risk preferences. Journal ofFinance 69 (3), 1051–1098.Lioui, A., Poncet, P., 2001. On optimal portfolio choice under stochastic interest rates. Journal ofEconomic Dynamics and Control 25, 1141–1865.Liu, J., 2007. Portfolio selection in stochastic environments. Review of Financial Studies 20 (1), 1–39.Liu, J., Longstaff, F., Pan, J., 2003. Dynamic asset allocation with event risk. Journal of Finance58, 231–259.Liu, J., Pan, J., 2003. Dynamic derivative strategies. Journal of Financial Economics 69, 401–430.Markowitz, H., 1952. Portfolio selection. Journal of Finance 7 (1), 77–91.Medvedev, A., Scaillet, O., 2007. Approximation and calibration of short-term implied volatilitiesunder jump-diffusion stochastic volatility. Review of Financial Studies 20 (2), 427–459.47erton, R. C., 1969. Lifetime portfolio selection under uncertainty: The continuous-time case. Re-view of Economics and Statistics 51, 247–257.Merton, R. C., 1971. Optimum consumption and portfolio rules in a continuous-time model. Journalof Economic Theory 3, 373–413.Moreira, A., Muir, T., 2019. Should long-term investors time volatility? Journal of Financial Eco-nomics 131 (3), 507–527.Morton, K. W., Mayers, D. F., 2005. Numerical Solution of Partial Differential Equations: AnIntroduction, 2nd Edition. Cambridge University Press.Nelson, D. B., 1990. ARCH models as diffusion approximations. Journal of Econometrics 45, 7–38.Nualart, D., 2006. The Malliavin Calculus and Related Topics, 2nd Edition. Probability and ItsApplications. Springer, Berlin.Ocone, D., Karatzas, I., 1991. A generalized Clark representation formula, with application to optimalportfolios. Stochastics and Stochastics Reports 34, 187–220.Penrose, R., 1955. A generalized inverse for matrices. Mathematical Proceedings of the CambridgePhilosophical Society 51, 406–413.Pliska, S. R., 1986. A stochastic calculus model of continuous trading: Optimal portfolios. Mathe-matics of Operations Research 11, 239–246.Pratt, J. W., 1964. Risk aversion in the small and in the large. Econometrica 32, 122–136.Roussanov, N., 2010. Diversification and its discontents: Idiosyncratic and entrepreneurial risk inthe quest for social status. Journal of Finance 65 (5), 1755–1788.Samuelson, P., 1969. Lifetime portfolio selection by dynamic stochastic programming. Review ofEconomics and Statistics 51, 239–246.Touzi, N., 2012. Optimal stochastic control, stochastic target problems, and backward SDE. Vol. 29.Springer Science and Business Media, New York.Wachter, J. A., 2002. Portfolio and consumption decisions under mean-reverting returns: An exactsolution for complete markets. Journal of Financial and Quantitative Analysis 37, 63–91.Wachter, J. A., 2010. Asset allocation. Annual Review of Financial Economics 2 (1), 175–206.Wachter, J. A., Yogo, M., 2010. Why do household portfolio shares rise in wealth? The Review ofFinancial Studies 23 (11), 3929–3965. 48 ppendix A Proofs
In this appendix, we document the detailed proofs for Theorem 1, Theorem 2, and Corollary 1.
Appendix A.1 Proof of Theorem 1
We first provide the following lemma that represents the optimal policy under the completed marketwith both real and fictitious assets, assuming the investor-specific price of risk process θ us were known. Lemma 1.
In the completed market with dynamics (8) and (2), the optimal policy ( π t , π Ft ) ⊤ for boththe real and fictitious assets admits the following representation ( π t , π Ft ) ⊤ = − X t ( σ S ( t, Y t ) ⊤ ) − (cid:16) θ S t E t [ Q t,T ( λ ∗ ξ S t )] + E t [ H rt,T ( λ ∗ ξ S t )] + E t [ H θt,T ( λ ∗ ξ S t )] (cid:17) , (A.1) where θ S t is the total price of risk defined in (12); E t denotes the expectation conditional on theinformation up to time t ; ξ S t is the state price density defined in (15); λ ∗ is the multiplier uniquelydetermined by the wealth equation X = E [ G ,T ( λ ∗ )] , (A.2) where X is the initial wealth and the function G ,T ( · ) is defined in (21); the components Q t,T ( λ ∗ ξ S t ) , H rt,T ( λ ∗ ξ S t ) , and H θt,T ( λ ∗ ξ S t ) are given by Q t,T ( λ ∗ ξ S t ) = Υ Ut,T ( λ ∗ ξ S t ) + Z Tt Υ ut,s ( λ ∗ ξ S t ) ds, (A.3a) H rt,T ( λ ∗ ξ S t ) = (Γ Ut,T ( λ ∗ ξ S t ) + Υ Ut,T ( λ ∗ ξ S t )) H rt,T + Z Tt (Γ ut,s ( λ ∗ ξ S t ) + Υ ut,s ( λ ∗ ξ S t )) H rt,s ds, (A.3b) H θt,T ( λ ∗ ξ S t ) = (Γ Ut,T ( λ ∗ ξ S t ) + Υ Ut,T ( λ ∗ ξ S t )) H θt,T + Z Tt (Γ ut,s ( λ ∗ ξ S t )) + Υ ut,s ( λ ∗ ξ S t )) H θt,s ds, (A.3c) with functions Γ Ut,T ( · ) , Γ ut,s ( · ) , Υ Ut,T ( · ) , and Υ ut,s ( · ) defined in (22a) and (22b). Here, the terms H rt,s and H θt,s in (A.3b) and (A.3c) satisfy dH rt,s = D t r ( s, Y s ) ds and dH θt,s = D t θ S s [ θ S s ds + dW s ] , (A.4) with initial values H rt,t = H θt,t ( λ ∗ t ) = 0 d , where D t r ( s, Y s ) and D t θ S s denote the Malliavin derivativesof the interest rate r ( s, Y s ) and total price of risk θ S s , respectively.Proof. The statement follows from the martingale approach arguments that lead to Theorem 1 inDetemple et al. (2003) (see also, e.g., Karatzas et al. (1987) and Cox and Huang (1989)).In what follows, we prove Theorem 1. 49 roof.
This proof consists of three parts consecutively. In the first part, we prove the relationship(28) and apply it to verify the representation of the investor-specific price of risk in (24). In thesecond part, we start to apply Lemma 1 and focus on deriving the explicit dynamics of ξ S t,s ( λ ∗ t ), H rt,s ,and H θt,s ( λ ∗ t ) as (29), (31), and (32), respectively, based on the representation (24) of θ uv and thedynamics in (15) and (A.4). In the third part, we consequently establish the decomposition of theoptimal policy given in (26a) – (26c). Part 1:
We first briefly prove the relationship in (28). As a foundation, the existence and unique-ness of λ ∗ t , as the solution to equation (20), follow from standard calculus: the utilities u ( t, · ) and U ( t, · ) are strictly increasing and concave with lim x →∞ ∂u ( t, x ) /∂x = 0 and lim x →∞ ∂U ( T, x ) /∂x = 0(see similar discussions in Cox and Huang (1989)). We now proceed to show (28), i.e., λ ∗ t = λ ∗ ξ S t . Assuming the investor follows the optimal policy in the completed market, we follow Karatzas et al.(1987) and Cox and Huang (1989) to derive that the time– t optimal wealth satisfies ξ S t X t = E t (cid:2) ξ S T I U ( T, λ ∗ ξ S T )+ R Tt ξ S s I u ( s, λ ∗ ξ S s ) ds (cid:3) , where λ ∗ is characterized by (A.2). By dividing ξ S t on both sides of the aboveequation and using the relation ξ S s = ξ S t ξ S t,s for any s ≥ t , we obtain X t = E t (cid:2) ξ S t,T I U ( T, λ ∗ ξ S t ξ S t,T ) + R Tt ξ S t,s I u ( s, λ ∗ ξ S t ξ S t,s ) ds (cid:3) . By the definition of G t,T ( · ) in (21), the above equation is equivalent to X t = E t [ G t,T ( λ ∗ ξ S t )] . By the uniqueness of solution to equation (20), we establish the relationship(28).Then, we verify the representation of the investor-specific price of risk in (24), i.e., θ uv = θ u ( v, Y v , λ ∗ v ; T )for some function θ u ( v, y, λ ; T ). This verification hinges on linking the least favorable completionapproach of Karatzas et al. (1991) and the minimax local martingale approach of He and Pearson(1991), two independently developed martingale approaches for solving optimal portfolios underincomplete market settings.By Theorem 9.3 of Karatzas et al. (1991), the investor-specific price of risk θ uv satisfying (23)must lead to the smallest utility among all possible completions, i.e., the least favorable completion.More precisely, the desired θ uv satisfying (23) serves as the optimizer for the following dual probleminf θ u ∈ Ker( σ ) ( sup ( c t ,X T ) ∈A θu E (cid:20)Z T u ( t, c t ) dt + U ( T, X T ) (cid:21) ) , (A.5)where A θ u = { ( c t , X T ) : E [ R T ξ S t c t dt + ξ S T X T ] ≤ X and X t ≥ t ∈ [0 , T ] } . Here, correspondingto the second orthogonal condition in (14a), we use θ u ∈ Ker( σ ) to abbreviate θ uv ∈ Ker( σ ( v, Y v ))for any 0 ≤ v ≤ T , with Ker( σ ( v, Y v )) := { w ∈ R d : σ ( v, Y v ) w ≡ m } denoting the kernel of σ ( v, Y v ) . Problem (A.5) is also discussed in He and Pearson (1991) for the same goal of characterizing theoptimal portfolio in the incomplete market case, though the language of He and Pearson (1991)hinges on the class of arbitrage-free state prices, which indeed correspond to the state price density ξ S t of the completed market defined by (15). According to Theorem 2 and the discussion priorto Theorem 7 of He and Pearson (1991), the solution of problem (A.5) also solves the following50ptimization problem: inf θ u ∈ Ker( σ ) E (cid:20)Z T ˜ u ( v, λ ∗ v ) dv + ˜ U ( T, λ ∗ T ) (cid:21) , (A.6)where λ ∗ v is the time– v multiplier characterized by the equation X v = E v [ G v,T ( λ ∗ v )]. By (28), i.e., λ ∗ v = λ ∗ ξ S v , an application of Ito formula leads to dλ ∗ v = − λ ∗ v [ r ( v, Y v ) dv + ( θ hv ( v, Y v ) + θ uv ) ⊤ dW v ] , in which θ uv serves as a control process. Besides, ˜ u ( t, x ) and ˜ U ( t, x ) in (A.6) denote the conjugatesof utility functions u ( t, x ) and U ( t, x ), defined by ˜ u ( t, y ) := sup x ≥ ( u ( t, x ) − yx ) and ˜ U ( t, y ) :=sup x ≥ ( U ( t, x ) − yx ) , respectively. We can check that sup x ≥ ( u ( t, x ) − yx ) and sup x ≥ ( u ( t, x ) − yx )take their maximum at x = I u ( t, y ) and x = I U ( t, y ) respectively.To temporarily summarize, by linking the problems (A.5) and (A.6), we verify that the desiredinvestor-specific price of risk θ uv satisfying the least favorable completion (23) is also the solution ofthe optimization problem (A.6). Next, we proceed to obtain the functional representation of θ u bylooking into the optimization problem (A.6). Since ( Y v , λ ∗ v ) = (cid:0) Y v , λ ∗ ξ S v (cid:1) forms a Markov process, anapplication of the feedback law (see, e.g., Theorem 9.1 of Touzi (2012)) to the control problem (A.6)implies that the control process θ uv must be a measurable function of the time v , the state variable Y v , and the multiplier λ ∗ v . Besides, θ uv also depends on the investment horizon T, since it is involvedin the objective function of the optimization problem (A.6). Although T is a fixed parameter, itis economically and technically important for revealing the structural differences of optimal policiesbetween complete and incomplete market settings. Thus, we arrive to the representation (24) ofthe control process θ uv , i.e., θ uv = θ u ( v, Y v , λ ∗ v ; T ) for some investor-specific price of risk function θ u ( v, y, λ ; T ). Part 2:
In this part, we prove the dynamics of ξ S t,s ( λ ∗ t ), H rt,s , and H θt,s ( λ ∗ t ) given in (29), (27b),and (32), respectively. To begin, by applying the representation of θ uv in (24) and the relationship(28), we obtain the following λ ∗ t − dependent version of the total price of risk θ S s introduced in (12): θ S s ( λ ∗ t ) = θ h ( s, Y s ) + θ u ( s, Y s , λ ∗ t ξ S t,s ( λ ∗ t ); T ) , (A.7)which plays an important role in explicitly deriving the desired dynamics in what follows. We nowapply Lemma 1 to the completed market (8) with the total price of risk θ S s taking the specific formgiven in (A.7).Accordingly, by the generic dynamics of ξ S t,s in (17) and that of H θt,s in (A.4), it is easy to verifythat λ ∗ t gets involved in them through the term θ S s ( λ ∗ t ). Thus, similar to the spirit of creating the λ ∗ t –dependent notation θ S s ( λ ∗ t ), we express ξ S t,s and H θt,s by their λ ∗ t –dependent versions ξ S t,s ( λ ∗ t ) and H θt,s ( λ ∗ t ), respectively, for emphasizing their dependences on λ ∗ t . Applying the representation (A.7)to the generic dynamics of ξ S t,s ( λ ∗ t ) given in (17), we obtain the explicit dynamics of ξ S t,s ( λ ∗ t ) in (29),i.e., dξ S t,s ( λ ∗ t ) = − ξ S t,s ( λ ∗ t )[ r ( s, Y s ) ds + θ S s ( λ ∗ t ) ⊤ dW s ].For H rt,s from the first SDE in (A.4), it is straightforward to apply the chain rule of Malliavin51erivative to obtain (31), i.e., dH rt,s = ( D t Y s ) ∇ r ( s, Y s ) ds. Here, D t Y s denotes the Malliavin deriva-tive of the state variable, which satisfies SDE (33). Next, to derive the explicit dynamics of H θt,s ( λ ∗ t ),we plug the representation of θ S s ( λ ∗ t ) in (A.7) into the second SDE in (A.4) to obtain dH θt,s = D t θ S s ( λ ∗ t ) [ θ S s ( λ ∗ t ) ds + dW s ] . (A.8)By the chain rule of Malliavin derivative, we express the term D t θ S s ( λ ∗ t ) as D t θ S s ( λ ∗ t ) = ( D t Y s ) ( ∇ θ h ( s, Y s ) + ∇ θ u ( s, Y s , λ ∗ t ξ S t,s ( λ ∗ t ); T ))+ λ ∗ t ( D t ξ S t,s ( λ ∗ t )) ∂θ u /∂λ ( s, Y s , λ ∗ t ξ S t,s ( λ ∗ t ); T ) , (A.9)where we must take into account the dependence of θ S s ( λ ∗ t ) on both state variable Y s and relativestate price density ξ S t,s ( λ ∗ t ). Besides, by the property of Malliavin derivative on Ito integrals (see,e.g., the survey in Appendix D of Detemple et al. (2003)), we have D t ξ S t,s ( λ ∗ t ) = − ξ S t,s ( λ ∗ t ) ( θ S t ( λ ∗ t ) + H rt,s + H θt,s ( λ ∗ t )) . Further applying this to (A.9), we have D t θ S s ( λ ∗ t ) = ( D t Y s ) ( ∇ θ h ( s, Y s ) + ∇ θ u ( s, Y s , λ ∗ t ξ S t,s ( λ ∗ t ); T )) − λ ∗ t ξ S t,s ( λ ∗ t )( θ S t ( λ ∗ t ) + H rt,s + H θt,s ( λ ∗ t )) ∂θ u /∂λ ( s, Y s , λ ∗ t ξ S t,s ( λ ∗ t ); T ) . (A.10)Then, the explicit dynamics of H θt,s ( λ ∗ t ) in (32) follows by plugging (A.7) and (A.10) into SDE (A.8). Part 3:
We now proceed to prove the decomposition of optimal policy given in (26a) – (26c).Since we apply Lemma 1 to the completed market (8) with the total price of risk θ S s taking thespecific form given in (A.7), the components Q t,T ( λ ∗ ξ S t ), H rt,T ( λ ∗ ξ S t ), and H θt,T ( λ ∗ ξ S t ) in (A.3a) –(A.3c) of Lemma 1 exactly coincide with the components Q t,T ( λ ∗ t ), H rt,T ( λ ∗ t ), and H θt,T ( λ ∗ t ) in (27a)– (27c) of Theorem 1. Indeed, this correspondence hinges on the following two reasons. First, by therelationship (28), i.e., λ ∗ t = λ ∗ ξ S t , we can substitute λ ∗ ξ S t in Q t,T ( λ ∗ ξ S t ), H rt,T ( λ ∗ ξ S t ), and H θt,T ( λ ∗ ξ S t )by the time– t multiplier λ ∗ t . Second, as proved in Part 2 above, the building blocks ξ S t,s and H θt,s inLemma 1 are realized by their λ ∗ t –dependent versions ξ S t,s ( λ ∗ t ) and H θt,s ( λ ∗ t ), with explicit dynamics(29) and (32), while the dynamics of H rt,s is explicitly computed as (31).Following the above discussions, we can represent the optimal policy ( π t , π Ft ) in (A.1) for thecompleted market as( π t , π Ft ) ⊤ = − X t ( σ S ( t, Y t ) ⊤ ) − (cid:16) θ S t ( λ ∗ t ) E t [ Q t,T ( λ ∗ t )] + E t [ H rt,T ( λ ∗ t )] + E t [ H θt,T ( λ ∗ t )] (cid:17) , (A.11)where θ S t ( λ ∗ t ) = θ h ( t, Y t ) + θ u ( t, Y t , λ ∗ t ; T ) according to the λ ∗ t -dependent representation in (A.7) andthe fact that ξ S t,t ( λ ∗ t ) = 1. Here, in (A.11), the components Q t,T ( λ ∗ t ), H rt,T ( λ ∗ t ), and H θt,T ( λ ∗ t ) are This dynamics coincides with the complete market counterpart derived in Detemple et al. (2003). However, asshown in what follows, the dynamics of H θt,s ( λ ∗ t ) is further sophisticated and fundamentally different from its completemarket counterpart. ξ S t,s ( λ ∗ t ), H rt,s , and H θt,s ( λ ∗ t )now follow the dynamics in (29), (31), and (32), respectively.Next, combining (A.11) with the following algebraic fact:( σ S ( t, Y t ) ⊤ ) − = ( σ S ( t, Y t ) − ) ⊤ = (( σ ( t, Y t ) + ) ⊤ , ( σ F ( t, Y t ) + ) ⊤ ) ⊤ , (A.12)where the second equality follows (10), we explicitly represent the optimal policy for real assets as π t = − X t ( σ ( t, Y t ) + ) ⊤ (cid:16) θ S t ( λ ∗ t ) E t [ Q t,T ( λ ∗ t )] + E t [ H rt,T ( λ ∗ t )] + E t [ H θt,T ( λ ∗ t )] (cid:17) . (A.13)We can further simplify this expression using the following algebraic fact( σ ( t, Y t ) + ) ⊤ θ S t ( λ ∗ t ) = ( σ ( t, Y t ) + ) ⊤ θ h ( t, Y t ) (A.14)with θ h ( t, Y t ) defined in (13a). To verify this, we use definition (11) for ( σ ( t, Y t ) + ) ⊤ , the second or-thogonal condition in (14a), as well as representation (24) to deduce that ( σ ( t, Y t ) + ) ⊤ θ u ( t, Y t , λ ∗ t ; T ) =( σ ( t, Y t ) σ ( t, Y t ) ⊤ ) − σ ( t, Y t ) θ u ( t, Y t , λ ∗ t ; T ) = 0 m . By (12), we can compute the terms ( σ ( t, Y t ) + ) ⊤ θ S t ( λ ∗ t )in (A.13) as ( σ ( t, Y t ) + ) ⊤ θ S t ( λ ∗ t ) = ( σ ( t, Y t ) + ) ⊤ ( θ h ( t, Y t ) + θ u ( t, Y t , λ ∗ t ; T )) = ( σ ( t, Y t ) + ) ⊤ θ h ( t, Y t ) . Hence, by (A.14), we can further simplify the representations (A.13) as π t = − X t ( σ ( t, Y t ) + ) ⊤ (cid:16) θ h ( t, Y t ) E t [ Q t,T ( λ ∗ t )] + E t [ H rt,T ( λ ∗ t )] + E t [ H θt,T ( λ ∗ t )] (cid:17) . (A.15)Finally, the decomposition in (26a), (26b), and (26c) of the optimal policy π t for real assets directlyfollows the representation (A.15). Appendix A.2 Proof of Theorem 2
Proof.
We begin by verifying the simple fact that E v [ Q v,T ( λ ∗ v )] = E [ Q v,T ( λ ∗ v ) | Y v , λ ∗ v ] , E v [ H rv,T ( λ ∗ v )] = E [ H rv,T ( λ ∗ v ) | Y v , λ ∗ v ] , E v [ H θv,T ( λ ∗ v )] = E [ H θv,T ( λ ∗ v ) | Y v .λ ∗ v ] . (A.16)Without loss of generality, we take E v [ Q v,T ( λ ∗ v )] as an example to verify this fact. Indeed, it followsfrom (2), (29), and (27a) that the joint process ( Y s , ξ S v,s ( λ ∗ v ) , Q v,s ( λ ∗ v )) in the time variable s ≥ v isMarkovian with the starting point given by ( Y v , ξ S v,v ( λ ∗ v ) , Q v,v ( λ ∗ v )) ≡ ( Y v , , λ ∗ v ∂I U /∂y ( v, λ ∗ v )) . Thus,the conditioning in E v [ Q v,T ( λ ∗ v )] is reduced to Y v and λ ∗ v , i.e., E v [ Q v,T ( λ ∗ v )] = E [ Q v,T ( λ ∗ v ) | Y v , λ ∗ v ] . The orthogonal condition (38), i.e., σ ( v, y ) θ u ( v, y, λ ; T ) ≡ m , easily follows from the secondcondition in (14a). Next, we establish equation (39) for governing the d − dimensional column vector-valued function θ u ( v, y, λ ; T ) . For this purpose, we explicitly deduce the least favorable completionconstraint (23), i.e., π Fv ≡ d − m , for any 0 ≤ v ≤ T. To begin, we first explicitly represent the optimalpolicy for fictitious assets as π Fv = − X v ( σ F ( v, Y v ) + ) ⊤ (cid:16) θ u ( v, Y v , λ ∗ v ; T ) E v [ Q v,T ( λ ∗ v )] + E v [ H rv,T ( λ ∗ v )] + E v [ H θv,T ( λ ∗ v )] (cid:17) . (A.17)53e establish this representation following a similar argument for proving (A.15). By combining(A.11) with the algebraic fact (A.12), we can represent the optimal policy for fictitious assets as π Fv = − X v ( σ F ( v, Y v ) + ) ⊤ (cid:16) θ S v ( λ ∗ v ) E v [ Q v,T ( λ ∗ v )] + E v [ H rv,T ( λ ∗ v )] + E v [ H θv,T ( λ ∗ v )] (cid:17) . (A.18)We can further simplify this representation using the following algebraic fact( σ F ( v, Y v ) + ) ⊤ θ S v ( λ ∗ v ) = ( σ F ( v, Y v ) + ) ⊤ θ u ( v, Y v , λ ∗ v ; T ) , (A.19)with θ u ( v, Y v , λ ∗ v ; T ) introduced in (24) for representing θ uv . To verify this, we use definition (11) for( σ F ( v, Y v ) + ) ⊤ and the first orthogonal condition in (14a) to deduce that ( σ F ( v, Y v ) + ) ⊤ θ hv ( v, Y v ) =( σ F ( v, Y v ) σ F ( v, Y v ) ⊤ ) − σ F ( v, Y v ) θ hv ( v, Y v ) = 0 d − m . Then, by (12), we can compute the term( σ F ( v, Y v ) + ) ⊤ θ S v ( λ ∗ v ) = ( σ F ( v, Y v ) + ) ⊤ ( θ h ( v, Y v ) + θ u ( v, Y v , λ ∗ v ; T )) = ( σ F ( v, Y v ) + ) ⊤ θ u ( v, Y v , λ ∗ v ; T )in (A.18). By (A.19), we can further simplify representation (A.18) to obtain (A.17).Thus, by plugging (A.17) into the least favorable completion constraint (23), we have( σ F ( v, Y v ) + ) ⊤ (cid:16) θ u ( v, Y v , λ ∗ v ; T ) E v [ Q v,T ( λ ∗ v )] + E v [ H rv,T ( λ ∗ v )] + E v [ H θv,T ( λ ∗ v )] (cid:17) ≡ d − m , (A.20)for any 0 ≤ v ≤ T . By representation (A.16), equation (A.20) is equivalent to( σ F ( v, Y v ) + ) ⊤ (cid:16) θ u ( v, Y v , λ ∗ v ; T ) E [ Q v,T ( λ ∗ v ) | Y v , λ ∗ v ] + E [ H rv,T ( λ ∗ v ) | Y v , λ ∗ v ] + E [ H θv,T ( λ ∗ v ) | Y v , λ ∗ v ] (cid:17) ≡ d − m . Since this equation holds for any value of Y v and λ ∗ v , we replace them with arbitrary deterministicarguments y and λ to obtain( σ F ( v, y ) + ) ⊤ (cid:16) θ u ( v, y, λ ; T ) E [ Q v,T ( λ ) | Y v = y ] + E [ H rv,T ( λ ) | Y v = y ] + E [ H θv,T ( λ ) | Y v = y ] (cid:17) ≡ d − m . (A.21)It is straightforward to obtain from (A.21) that( σ F ( v, y ) + ) ⊤ θ u ( v, y, λ ; T ) = − ( σ F ( v, y ) + ) ⊤ E [ H rv,T ( λ ) | Y v = y ] + E [ H θv,T ( λ ) | Y v = y ] E [ Q v,T ( λ ) | Y v = y ] . (A.22a)Since ( σ F ( v, y ) + ) ⊤ is a ( d − m ) × d matrix, (A.22a) provides ( d − m ) equations governing the d − dimensional column vector θ u ( v, y, λ ; T ). We get the other m equations for governing θ u ( v, y, λ ; T )out of the second orthogonal condition in (38), i.e., σ ( v, y ) θ u ( v, y, λ ; T ) = 0 m . Thus, it follows that( σ ( v, y ) + ) ⊤ θ u ( v, y, λ ; T ) = ( σ ( v, y ) σ ( v, y ) ⊤ ) − σ ( v, y ) θ u ( v, y, λ ; T ) = 0 m . (A.22b)By combining (A.22a) and (A.22b), the function θ u ( v, y, λ ; T ) solves θ u ( v, y, λ ; T ) = − ( σ ( v, y ) + ) ⊤ (cid:0) σ F ( v, y ) + (cid:1) ⊤ ! − m × d (cid:0) σ F ( v, y ) + (cid:1) ⊤ ! E [ H rv,T ( λ ) | Y v = y ] + E [ H θv,T ( λ ) | Y v = y ] E [ Q v,T ( λ ) | Y v = y ] . (A.23)54e now further simplify the above equation. By (A.12), we have ( σ ( v, y ) + ) ⊤ (cid:0) σ F ( v, y ) + (cid:1) ⊤ ! − = σ S ( v, y ) ⊤ = ( σ ( v, y ) ⊤ σ F ( v, y ) ⊤ ) . (A.24)Thus, equation (A.23) can be further deduced as θ u ( v, y, λ ; T ) = − ( σ ( v, y ) ⊤ σ F ( v, y ) ⊤ ) m × d (cid:0) σ F ( v, y ) + (cid:1) ⊤ ! E [ H rv,T ( λ ) | Y v = y ] + E [ H θv,T ( λ ) | Y v = y ] E [ Q v,T ( λ ) | Y v = y ] ≡ − σ F ( v, y ) ⊤ ( σ F ( v, y ) + ) ⊤ E [ H rv,T ( λ ) | Y v = y ] + E [ H θv,T ( λ ) | Y v = y ] E [ Q v,T ( λ ) | Y v = y ] . (A.25)By (11), we can simplify the coefficient in the above equation as σ F ( v, y ) ⊤ ( σ F ( v, y ) + ) ⊤ = σ F ( v, y ) ⊤ ( σ F ( v, y ) σ F ( v, y ) ⊤ ) − σ F ( v, y ) = σ F ( v, y ) + σ F ( v, y ) . (A.26)Besides, by (10), we note that I d = ( σ ( v, y ) + σ F ( v, y ) + ) σ ( v, y ) σ F ( v, y ) ! ≡ σ ( v, y ) + σ ( v, y ) + σ F ( v, y ) + σ F ( v, y ) . (A.27)Combining (A.27) with (A.26), we get σ F ( v, y ) ⊤ ( σ F ( v, y ) + ) ⊤ = σ F ( v, y ) + σ F ( v, y ) = I d − σ ( v, y ) + σ ( v, y ) . (A.28)Then, (39) follows by plugging (A.28) into (A.25). Appendix A.3 Proof of Corollary 1
Proof.
For the incomplete market model with CRRA utility (5a), we follow the general decompositionestablished in Theorems 1 and 2, and then develop substantial structural simplifications of the resultsbased on the special properties of CRRA utility.First, we prove that the investor-specific price of risk function θ u ( v, y, λ ; T ) is independent ofthe parameter λ for any 0 ≤ v ≤ T under the CRRA utility. Equivalently, this leads to that theinvestor-specific price of risk θ uv , which ought to be θ u ( v, Y v , λ ∗ v ; T ) under general utilities accordingto representation (24), is independent of the multiplier λ ∗ v . To begin, with the explicit forms offunctions I u ( t, y ) and I U ( t, y ) under the CRRA utility (5a), we can specify the dual problem in(A.6) for characterizing the investor-specific price of risk function as:inf θ u ∈ Ker( σ ) E (cid:20) (1 − w ) γ e − ρTγ ( λ ∗ T ) − γ + w γ Z T e − ρsγ ( λ ∗ s ) − γ ds (cid:21) . (A.29)According to the principle of dynamic programming, we can solve the optimal θ u at arbitrary time v from the following time- v version of problem (A.29):inf θ u ∈ Ker( σ ) E v (cid:20) (1 − w ) γ e − ρTγ ( λ ∗ T ) − γ + w γ Z Tv e − ρsγ ( λ ∗ s ) − γ ds (cid:21) . (A.30)55sing the relationship λ ∗ s = λ ∗ ξ S s = λ ∗ v ξ S v,s as well as the fact that the multiplier λ ∗ v is known withinformation available up to time v , we can extract the factor ( λ ∗ v ) − γ from the conditional expectationin (A.30) to get inf θ u ∈ Ker( σ ) ( λ ∗ v ) − γ E v (cid:2) (1 − w ) γ e − ρTγ ( ξ S v,T ) − γ + w γ R Tv e − ρsγ ( ξ S v,s ) − γ ds (cid:3) . Accordingto He and Pearson (1991), as the Lagrangian multiplier of the static optimization problem (18), λ ∗ must be positive. This implies that λ ∗ v = λ ∗ ξ S v is also positive. Thus we can drop the factor ( λ ∗ v ) − γ inthis optimization problem. Besides, the process (cid:0) Y s , ξ S v,s (cid:1) for v ≤ s ≤ T is Markovian with the initialvalue ( Y v , θ uv admits the representation θ uv = θ u ( v, Y v ; T ) for some function θ u ( v, y ; T ) . In other words, the function θ u ( v, y, λ ; T ) introducedin (24) is independent of the parameter λ under the CRRA utility.Then, by (12), the total price of risk under the CRRA utility can be parameterized and rep-resented as θ S ( s, Y s ; T ) = θ h ( s, Y s ) + θ u ( s, Y s ; T ) . Plugging this representation into (29) and (32),we can prove that ξ S t,s and H θt,s , under the CRRA utility, satisfy the dynamics in (44a) and (44b),respectively. In particular, as both ξ S t,s and H θt,s are independent of the time– t multiplier λ ∗ t underthe CRRA utility, we drop λ ∗ t as opposed to writing their general expressions ξ S t,s ( λ ∗ t ) and H θt,s ( λ ∗ t ) . We now establish the representations of optimal policy in (46a) and (46b) as well as the equationgoverning function θ u ( v, y ; T ) in (48). First, we note the following algebraic fact: with the specifi-cation of the CRRA utility function given in (5a), the functions Q t,T ( λ ∗ t ), H rt,T ( λ ∗ t ), H θt,T ( λ ∗ t ) , and G t,T ( λ ∗ t ) defined in (27a) – (27c) and (21) are simplified to the following separable forms: Q t,T ( λ ∗ t ) = ( λ ∗ t ) − γ ˜ Q t,T , H rt,T ( λ ∗ t ) = ( λ ∗ t ) − γ ˜ H rt,T , H θt,T ( λ ∗ t ) = ( λ ∗ t ) − γ ˜ H θt,T , (A.31)and G t,T ( λ ∗ t ) = ( λ ∗ t ) − γ ˜ G t,T , where ˜ H rt,T , ˜ H θt,T , and ˜ G t,T are introduced in (47b), (47a), and (47c),respectively, and the function ˜ Q t,T is given by˜ Q t,T = − γ ˜ G t,T . (A.32)With the separable forms, the wealth equation in (20), i.e., X t = E t [ G t,T ( λ ∗ t )], is equivalent to X t = ( λ ∗ t ) − γ E t [ ˜ G t,T ] . (A.33)For the mean-variance component π mv ( t, X t , Y t ) = − ( σ ( t, Y t ) + ) ⊤ θ h ( t, Y t ) E t [ Q t,T ( λ ∗ t )] /X t in (26a),we use the relationships (A.31) and (A.32) to get E t [ Q t,T ( λ ∗ t )] = ( λ ∗ t ) − γ E t [ ˜ Q t,T ] = − ( λ ∗ t ) − γ E t [ ˜ G t,T ] /γ. Then, plugging it into (26a) yields π mv ( t, X t , Y t ) = ( λ ∗ t ) − γ E t [ ˜ G t,T ]( σ ( t, Y t ) + ) ⊤ θ h ( t, Y t ) / ( γX t ) =( σ ( t, Y t ) + ) ⊤ θ h ( t, Y t ) /γ, where the second equality follows by (A.33). Finally, by the definition of θ h ( t, Y t ) in (13a), we obtain the representation in (46a) as π mv ( t, X t , Y t ) = ( σ ( t, Y t ) + ) ⊤ θ h ( t, Y t ) /γ =( σ ( t, Y t ) σ ( t, Y t ) ⊤ ) − ( µ ( t, Y t ) − r ( t, Y t )1 m ) /γ. Similarly, for the interest rate and price of risk hedgecomponents given by (26b) and (26c), their representations under CRRA utility (46b) follow bycombining the separable forms in (A.31) and the constraint (A.33).Finally, we derive equation (48) for the investor-specific price of risk function θ u ( v, y ; T ) under theCRRA utility. With the separable forms given in (A.31) and the relationship (A.32), we can express56he conditional expectations in the general equation (39) as E [ H rv,T ( λ ) | Y v = y ] = λ − γ E [ ˜ H rv,T | Y v = y ] , E [ H θv,T ( λ ) | Y v = y ] = λ − γ E [ ˜ H θv,T | Y v = y ] , and E [ Q v,T ( λ ) | Y v = y ] = λ − γ E [ ˜ Q t,T | Y v = y ] = − λ − γ E [ ˜ G v,T | Y v = y ] /γ. Then, (48) follows directly by plugging them in (39). The term λ − γ cancelsout in both the nominator and denominator, which reconciles with the investor-specific price of riskfunction θ u ( v, y ; T ) being indeed independent of parameter λ . Appendix A.4 Proof of Proposition 2
We first prove the following lemma that plays a crucial role in proving Proposition 2.
Lemma 2.
With deterministic interest rate r s , the following relationship holds: E t [ ξ S t,s ( λ ) H θt,s ( λ )] ≡ d , for any s ≥ t. (A.34) Here, ξ S t,s ( λ ) is the relative state price density, and H θt,s ( λ ) is the Malliavin term related to the un-certainty in the total price of risk, with their dynamics given explicitly in (29) and (27c), respectively.Proof. By (15), we get ξ S t := exp( − R t r s ds − R t ( θ S s ) ⊤ dW s − R t ( θ S s ) ⊤ θ S s ds ) for the state price densityin incomplete markets. We can decompose it to two parts related to interest rate and total price ofrisk respectively, i.e., ξ S t = B t η t , where B t = exp (cid:18) − Z t r s ds (cid:19) and η t = exp (cid:18) − Z t (cid:0) θ S s (cid:1) ⊤ dW s − Z t (cid:0) θ S s (cid:1) ⊤ θ S s ds (cid:19) . (A.35)With deterministic interest rate r s , the discount term B t is also deterministic. A straightforwardapplication of Ito formula leads to the SDEs η t as dη t = − η t (cid:0) θ S t (cid:1) ⊤ dW t . (A.36)The martingale property of η t leads to E t [ η s ] = η t , for any s ≥ t. (A.37)Next, we prove that E t [ η s H θt,s ( λ )] ≡ d . (A.38)On one hand, by computing Malliavin derivative (see, e.g., the tutorial in Appendix D of Detemple et al.(2003)), we obtain the time– t Malliavin derivative of η s as D t η s = − η s ( θ S t + H θt,s ( λ )) . Taking condi-tional expectation on the both sides, we have E t [ D t η s ] = − E t (cid:2) η s θ S t (cid:3) − E t [ η s H θt,s ( λ )] = − E t [ η s ] θ S t − E t [ η s H θt,s ( λ )] = − η t θ S t − E t [ η s H θt,s ( λ )] , (A.39)where the last equality follows from the martingale property of η s in (A.37). On the other hand,we compute the Malliavin derivative D t η s again, using the SDE of η s . By (A.36), we have η s =57 R s η t (cid:0) θ S t (cid:1) ⊤ dW t + η , with η = 1 by definition. We then take Malliavin derivative on the both sidesof this equation, and D t η s = − R st (cid:2) D t ( η v θ S v ) (cid:3) ⊤ dW v − η t θ S t , by the property of Malliavin derivativeon Itˆo integrals. Taking time– t conditional expectations on the both sides, we obtain E t [ D t η s ] = − E t (cid:20)Z st (cid:2) D t (cid:0) η v θ S v (cid:1)(cid:3) ⊤ dW v (cid:21) − η t θ S t = − η t θ S t , (A.40)which follows the martingale property of Itˆo integrals. Thus, (A.38) follows by comparing (A.40)and (A.39).Finally, relationship (A.34) comes from E t [ ξ S t,s ( λ ) H θt,s ( λ )] = E t [ ξ S s H θt,s ] /ξ S t = B s E t [ η s H θt,s ( λ )] /ξ S t =0 d , where the second equality follows from the deterministic nature of the discount term B s as wellas (A.38).Now, we are ready to prove Proposition 2 for the optimal policy under HARA utility withdeterministic interest rate. Without loss of generality, we assume w > w = 0 follows in a similar fashion. Proof. Part 1:
First, we show that with deterministic interest rate, the investor-specific price of riskfunction θ u ( v, y, λ ; T ) under HARA utility coincides with its counterpart under CRRA utility, andthus is independent of parameter λ . To begin, like in other proofs, we employ the dual problem(A.6) as a tool. Using the explicit forms of functions I u ( t, y ) and I U ( t, y ) under HARA utility, wecan explicitly specify the dual problem in (A.6) asinf θ u ∈ Ker( σ ) E v (cid:20) (1 − w ) γ e − ρTγ ( λ ∗ T ) − γ + w γ Z Tv e − ρsγ ( λ ∗ s ) − γ ds + γ − γ A v,T (cid:21) , (A.41)where A v,T = ¯ xλ ∗ T + ¯ c R Tv λ ∗ s ds. Here, ¯ x and ¯ c are the minimum requirements for terminal wealth andintermediate consumption, respectively. Comparing (A.41) and (A.29), we see that the term A v,T distinguish the dual problem under HARA utility from that under CRRA utility.With deterministic interest rate, we then verify that E v [ A v,T ] does not depend on the controlprocess θ uv for v ∈ [ v, T ] and thus can be dropped from the dual problem (A.41) to simplify it asinf θ u ∈ Ker( σ ) E v (cid:20) (1 − w ) γ e − ρTγ ( λ ∗ T ) − γ + w γ Z Tv e − ρsγ ( λ ∗ s ) − γ ds (cid:21) . (A.42)To see this, we use the relationship λ ∗ s = λ ∗ ξ S s = λ ∗ v ξ S v,s to derive that E v [ A v,T ] = ¯ xE v [ λ ∗ T ] + ¯ c Z Tv E v [ λ ∗ s ] ds = λ ∗ v (cid:20) ¯ xE v [ ξ S v,T ] + ¯ c Z Tv E v [ ξ S v,s ] ds (cid:21) . (A.43)We express the conditional expectation E v [ ξ S v,T ] as E v [ ξ S v,s ] = E v [ B v,s η v,s ], where B v,s := B s /B v and η v,s := η s /η v following (A.35). A straightforward application of Ito formula leads to the SDE of η v,s For the proof under the case of w = 0 , we just need to drop all the terms related to ¯ c . dη v,s = − η v,s (cid:0) θ S s (cid:1) ⊤ dW s . As we assume a deterministic interest rate, B v,s is also deterministic.Thus, we have E v [ ξ S v,s ] = E v [ B v,s η v,s ] = B v,s E v [ η v,s ] = B v,s , (A.44)where the last equality follows from the martingale property of η v,s as a process in s and the factthat η v,v = 1. Plugging E v (cid:2) ξ S v,s (cid:3) into (A.43), we obtain that E v [ A v,T ] = λ ∗ v [¯ xB v,T + ¯ c R Tv B v,s ds ] , which obviously does not depend on the control process θ uv for v ∈ [ v, T ]. Thus, we can drop theterm A v,T from (A.41).By the above arguments, we show that with deterministic interest rate, the investor-specific priceof risk θ uv under HARA utility is uniquely characterized as the control process for the dual problem(A.42), with the underlying Markov process (cid:0) Y s , ξ S v,s (cid:1) for v ≤ s ≤ T . Comparing the dual problem(A.42) with its counterpart (A.30) under CRRA utility, we can verify that the two dual problems,as well as the underlying Markov process, are actually the same under the two utility specifications.Thus, the unique optimal control process θ uv is also the same for the two dual problems. This provesthat with deterministic interest rate, the investor-specific price of risk function θ u ( v, y, λ ; T ) underHARA utility coincides with its counterpart under CRRA utility, and thus is independent of theparameter λ . So, we can express it as θ uv = θ u ( v, Y v ; T ) with the same function θ u ( v, y ; T ) thatsatisfies the equation (50) under the CRRA utility. Consequently, with deterministic interest rate,the quantities ˜ H rv,T ( λ ), ˜ H θv,T ( λ ), ˜ Q v,T ( λ ), and ˜ G v,T ( λ ) in (51a), (51b), (51c), and (55) for HARAutility are also independent of parameter λ , and coincide with their counterparts under CRRA utility,which are given in (47b), (47b), (49), and (47c), respectively.Next, we establish equation (56) that governs the investor-specific price of risk function θ u ( v, y ; T ).It follows from equation (50) that, whether the interest rate is deterministic or not, θ u ( v, y, λ ; T )under HARA utility is characterized by θ u ( v, y, λ ; T ) = σ ( v, y ) + σ ( v, y ) − I d E [ ˜ Q v,T ( λ ) | Y v = y ] × ( E [ ˜ H θv,T ( λ ) | Y v = y ]+ E [ ˜ H rv,T ( λ ) | Y v = y ]+ λ γ E [ ζ v,T ( λ ) | Y v = y ]) , (A.45)where ζ v,T ( λ ) = ζ rv,T ( λ ) + ζ θv,T ( λ ) according to (52a). With deterministic interest rate, we have H rv,s ≡ d due to (31) and ∇ r ( s, Y s ) ≡ n . Thus, it follows from (51a) and (52b) that ˜ H rv,T ( λ ) ≡ d and ζ rv,T ( λ ) ≡ d . Also recall that θ u ( v, y, λ ; T ) is independent of λ and thus simplifies to θ u ( v, y ; T ) . Then, we simplify equation (A.45) to θ u ( v, y ; T ) = σ ( v, y ) + σ ( v, y ) − I d E [ ˜ Q v,T ( λ ) | Y v = y ] × ( E [ ˜ H θv,T ( λ ) | Y v = y ] + λ γ E [ ζ θv,T ( λ ) | Y v = y ]) . (A.46)where ζ θv,T ( λ ) is defined by (52c) as ζ θv,T ( λ ) = ¯ xξ S v,T ( λ ) H θv,T ( λ ) + ¯ c R Tv ξ S v,s ( λ ) H θv,s ( λ ) ds. By Lemma2, its expectation is always zero under deterministic interest rate, i.e., E v [ ζ θv,T ( λ )] = ¯ xE v [ ξ S v,T ( λ ) H θv,T ( λ )] + ¯ c Z Tv E v [ ξ S v,s ( λ ) H θv,s ( λ )] ds = 0 d . (A.47)59hus, the last term λ γ E [ ζ θv,T ( λ ) | Y v = y ] vanishes in (A.46), and the equation further simplifies to θ u ( v, y ; T ) = σ ( v, y ) + σ ( v, y ) − I d E [ ˜ Q v,T ( λ ) | Y v = y ] × E [ ˜ H θv,T ( λ ) | Y v = y ] . By examining the definitions of ˜ H θv,T ( λ ) and ˜ Q v,T ( λ ) in (51b) and (51c), as well as the SDEs of ξ S v,s ( λ ) and H θv,s ( λ ) in (29) and (32), we confirm that ˜ H θv,T ( λ ) and ˜ Q v,T ( λ ) reduce to ˜ H θv,T and ˜ Q v,T given in (47b) and (49), respectively. Hence, the parameter λ does not show up in either the aboveequation system or its solution θ u ( v, y ; T ). Part 2:
Next, we look into the optimal policy under HARA utility with deterministic interestrate. Under this circumstance, we have H rv,s ≡ d , and thus it follows from that (51a), (52b), and(53b) the interest hedge component π rH ( t, X t , Y t ) = 0 m , i.e., there is no need to hedge uncertainty ininterest rate. So, we only need to focus on the mean-variance and price of risk hedge components.First, we solve for the multiplier λ ∗ t from the wealth equation (54), i.e., ( λ ∗ t ) − γ E t [ ˜ G t,T ] + xE t [ ξ S t,T ] + cE t (cid:2) R Tt ξ S t,s ds (cid:3) = X t . Here, we drop the dependence on λ ∗ t from ˜ G t,T and ξ S t,s . This is because we haveshown in Part 1 that the investor-specific price of risk function θ u ( v, y ; T ) does not depend on λ, andneither do ˜ G t,T and ξ S t,s according to (55) and (29), respectively. By (A.44), we have E t [ ξ S t,s ] = B t,s .Plugging it to the above equation, we solve ( λ ∗ t ) − γ as( λ ∗ t ) − γ = ¯ X t E t [ ˜ G t,T ] , (A.48)where ¯ X t is defined in (58a), i.e., ¯ X t = X t − xB t,T − c R Tt B t,s ds. Plugging (A.48) into the mean-variance component in (53a) and invoking the relationship ˜ G t,T = − γ ˜ Q t,T , we can derive π mvH ( t, X t , Y t ) = − X t ( σ ( t, Y t ) + ) ⊤ θ h ( t, Y t ) ( λ ∗ t ) − γ E t [ ˜ Q t,T ]= − X t ( σ ( t, Y t ) + ) ⊤ θ h ( t, Y t ) ¯ X t E t [ ˜ G t,T ] E t [ ˜ Q t,T ]= ¯ X t γX t ( σ ( t, Y t ) + ) ⊤ θ h ( t, Y t ) . (A.49)Next, in (53c), we have π θH ( t, X t , Y t ) = − X t ( σ ( t, Y t ) + ) ⊤ (cid:16) ( λ ∗ t ) − γ E t [ ˜ H θt,T ] + E t [ ζ θt,T ] (cid:17) = − X t ( σ ( t, Y t ) + ) ⊤ ( λ ∗ t ) − γ E t [ ˜ H θt,T ]for the price of risk hedge component, where the second equality follows from (A.47). Plugging(A.48) into the right-hand side, we obtain π θH ( t, X t , Y t ) = − ( σ ( t, Y t ) + ) ⊤ ¯ X t X t E t [ ˜ H θt,T ] E t [ ˜ G t,T ] . (A.50)Finally, relationships (57) follow by comparing the optimal components in (A.49) and (A.50) withtheir counterparts (46a) and (46b) under the CRRA utility.60 -3 Figure 1: Behavior of optimal policy in the Heston-SV model
Note: These figures plot for optimal policies under both the HARA and CRRA utilities. The upper left (resp.upper right) panel plots the optimal policy π ( t, X t , V t ) for different wealth level X t /x (resp. interest rate r ).The lower left (resp. lower right) panel plots the mean-variance component π mv ( t, X t , V t ) (resp. price of riskhedge component π θ ( t, X t , V t )) at different investment horizon T − t . These figures are generated accordingto the closed-form formulae (62) – (67b). The parameters are set as follows in annualized form. In the upperleft panel, we set r = 0 .
04 and T − t = 10. In the upper right panel, we set X t /x = 3 and T − t = 10.In the two lower panels, we set r = 0 .
04 and X t /x = 3. Besides, we set the rest of the parameters at thefollowing representative values κ = 5 . , ρ = − . , λ = 1 . , θ = 0 . , σ = 0 .
48 according to the MaximumLikelihood estimation results of A¨ıt-Sahalia and Kimmel (2007), while choosing γ = 2 and V t = 0 . Figure 2: The stock price S t (blue line with left y-axis) and the optimal policy ratio π Ht /π Lt (red linewith right y-axis) in a simulated path of the Heston-SV model. Note: This figure plots for a simulated path of stock price S t and that of the corresponding ratio π Ht /π Lt between the optimal policies of the high– and low–wealth investors under HARA utility. The ratios arecalculated according to the closed-form formulae (65) and (66). The parameters for the Heston SV model arechosen as those representative ones employed for producing Figure 1. .1 0.15 0.2 0.25 0.3123456 10 -3 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-4-2024 10 -3 -3 Figure 3: Behavior of optimal hedging component π θ ( t, V t ) in the CEV-SV model Note: The three panels plot the optimal hedge component π θ ( t, V t ) with respect to different choices of thelevel of current variance V t , the leverage effect parameter ρ, and the risk aversion parameter γ, respectively.In each panel, three values are considered: ν = 1 / , ν = 1 (the GARCH diffusion model) , and ν = 3 / / γ = 2 and ρ = − .
8; in the middle and right panels, weset V t = 0 .
15. For all the results, we employ realistic annualized parameters. We set r = 0 . , λ = 0 . , κ = 1 . ,θ = 0 . , and T − t = 1 in common. In particular, to control the effect of volatility for different values of ν , we keep σθ ν , the volatility of the variance process V t evaluated at the mean-reverting level θ, at a constantlevel φ V = σθ ν by setting σ according to σ = φ V /θ ν for each value of ν . Without loss of generality, we choosethis constant level as φ V = 0 . × √ . ≈ .
11 according to a realistic parameter set of the Heston model.Then we set σ according to σ = φ V /θ ν for each value of ν , i.e., σ = 0 .
25 for Heston-SV model, σ ≈ .
56 forGARCH-SV model, and σ = 1 .
25 for 3/2-SV model. t ˆ π θ π θ true e Rel
Std CI Std B CI , B RMSE B / | π θ true | (in 10 − ) (in 10 − ) (in %) (in 10 − ) (in 10 − ) (in 10 − ) (in 10 − ) (in %) T − t = 0 . .
10 2 .
194 2 .
223 1 .
284 3 .
690 [2 . , . .
679 [2 . , . . .
15 2 .
199 2 .
223 1 .
051 3 .
203 [2 . , . .
197 [2 . , . . .
20 2 .
202 2 .
223 0 .
907 2 .
878 [2 . , . .
875 [2 . , . . .
25 2 .
205 2 .
223 0 .
807 2 .
640 [2 . , . .
637 [2 . , . . .
30 2 .
206 2 .
223 0 .
733 2 .
454 [2 . , . .
452 [2 . , . . T − t = 1 . .
10 3 .
281 3 .
299 0 .
535 4 .
007 [3 . , . .
039 [3 . , . . .
15 3 .
283 3 .
299 0 .
473 3 .
528 [3 . , . .
551 [3 . , . . .
20 3 .
285 3 .
299 0 .
420 3 .
206 [3 . , . .
222 [3 . , . . .
25 3 .
286 3 .
299 0 .
376 2 .
966 [3 . , . .
979 [3 . , . . .
30 3 .
287 3 .
299 0 .
340 2 .
778 [3 . , . .
788 [3 . , . . T − t = 3 . .
10 4 .
290 4 .
252 0 .
890 4 .
149 [4 . , . .
118 [4 . , . . .
15 4 .
281 4 .
252 0 .
683 3 .
673 [4 . , . .
645 [4 . , . . .
20 4 .
276 4 .
252 0 .
551 3 .
351 [4 . , . .
325 [4 . , . . .
25 4 .
272 4 .
252 0 .
459 3 .
113 [4 . , . .
089 [4 . , . . .
30 4 .
269 4 .
252 0 .
391 2 .
927 [4 . , . .
903 [4 . , . . Table 1: Simulation results of the incomplete-market stochastic volatility model of Heston given inExample 1.
Note: For the incomplete-market Heston SV model in Example 1, we choose the following representativeannualized parameter set: λ = 0 . , ρ = − . , κ = 1 . , σ = 0 . , and θ = 0 .
2. These values correspond tothe numbers that A¨ıt-Sahalia and Kimmel (2007) set for Monte Carlo simulations under Heston’s model toproduce their Table 2. t ˆ π θ π θ true e Rel
Std CI Std B CI , B RMSE B / | π θ true | (in 10 − ) (in 10 − ) (in %) (in 10 − ) (in 10 − ) (in 10 − ) (in 10 − ) (in %) T − t = 0 . .
10 0 .
230 0 .
233 1 .
459 5 .
041 [0 . , . .
038 [0 . , . . .
20 0 .
395 0 .
398 0 .
825 5 .
061 [0 . , . .
056 [0 . , . . .
30 0 .
559 0 .
562 0 .
559 5 .
091 [0 . , . .
083 [0 . , . . .
40 0 .
724 0 .
727 0 .
412 5 .
132 [0 . , . .
120 [0 . , . . .
50 0 .
888 0 .
891 0 .
316 5 .
184 [0 . , . .
166 [0 . , . . T − t = 1 . .
10 0 .
493 0 .
495 0 .
475 6 .
476 [0 . , . .
096 [0 . , . . .
20 0 .
765 0 .
767 0 .
245 6 .
529 [0 . , . .
134 [0 . , . . .
30 1 .
037 1 .
039 0 .
135 6 .
606 [1 . , . .
198 [1 . , . . .
40 1 .
309 1 .
310 0 .
070 6 .
706 [1 . , . .
287 [1 . , . . .
50 1 .
581 1 .
582 0 .
027 6 .
830 [1 . , . .
403 [1 . , . . T − t = 3 . .
10 1 .
311 1 .
327 1 .
219 8 .
697 [1 . , . .
685 [1 . , . . .
20 1 .
735 1 .
751 0 .
890 8 .
875 [1 . , . .
853 [1 . , . . .
30 2 .
159 2 .
174 0 .
695 9 .
103 [2 . , . .
071 [2 . , . . .
40 2 .
583 2 .
598 0 .
569 9 .
383 [2 . , . .
338 [2 . , . . .
50 3 .
006 3 .
021 0 .
482 9 .
717 [2 . , . .
658 [2 . , . . Table 2: Simulation results of the incomplete-market mean-reverting return model of Kim-Omberggiven in Example 2.
Note: For the incomplete-market mean-reverting return model in Example 2, we choose the following repre-sentative annualized parameter set: r = 0 . , σ = 0 . , λ = 0 . , σ θ = 0 . , and θ = 0 .
33 according to itscomplete-market counterpart in Wachter (2002), and choose ρ = − .
5, which is a typical value according toits economic interpretation.5, which is a typical value according toits economic interpretation.