Horizon-unbiased Investment with Ambiguity
HHorizon-unbiased Investment with Ambiguity ∗ Qian Lin † , Xianming Sun ‡ , and Chao Zhou § School of Economics and Management, Wuhan University, China School of Finance, Zhongnan University of Economics and Law, China Department of Mathematics, National University of Singapore, Singapore
April 23, 2019
Abstract
In the presence of ambiguity on the driving force of market randomness, we consider the dy-namic portfolio choice without any predetermined investment horizon. The investment criteria isformulated as a robust forward performance process, reflecting an investor’s dynamic preference.We show that the market risk premium and the utility risk premium jointly determine the investors’trading direction and the worst-case scenarios of the risky asset’s mean return and volatility. Theclosed-form formulas for the optimal investment strategies are given in the special settings of theCRRA preference.
Keywords : Ambiguity, Forward Performance, Robust Investment, Risk Premium ∗ We are grateful for the funding from the NSF of China (11501425 and 71801226). † Email: [email protected] ‡ Email: [email protected] § Email: [email protected] a r X i v : . [ q -f i n . M F ] A p r Introduction
Dynamic portfolio choice problems usually envisage an investment setting where an investor is ex-ogenously assigned an investment performance criteria and stochastic models for the price processesof risky assets. However, the investor may extemporaneously change the investment horizon, consis-tently update her preference with the market evolution, and conservatively invest due to ambiguity onthe driving force of market randomness or the dynamics of the risky assets. Motivated by these invest-ment realities, we study a robust horizon-unbiased portfolio problem in a continuous-time framework.In the seminal work of Merton (1969), continuous-time portfolio choice is formulated as a stochas-tic control problem to maximize the expected utility at a specific investment horizon by searching forthe optimal strategy in an admissible strategy space. Note that if the investor has two candidate in-vestment horizon T , T , ( T > T > , the resulting optimal strategies associated with these twohorizons are generally not consistent over the common time interval [0 , T ] , ( T ≤ T < T ) (Musiela &Zariphopoulou, 2007). Hence, Merton’s framework is neither suitable for the case where an investormay extend or shorten her initial investment horizon, nor the case where the investor may updateher preference in accordance to the accumulated market information. In these quite realistic settings,the investor needs an optimal strategy which is independent of the investment horizon and reflectsher dynamic preference in time and wealth. The horizon-unbiased utility or forward performancemeasure, independently proposed by Choulli et al. (2007), Henderson & Hobson (2007), Musiela &Zariphopoulou (2007), provides a portfolio framework satisfying the aforementioned requirements. Insuch framework, an investor specifies initial preferences (utility function), and then propagates them forward as the financial market evolves. This striking characteristic contrasts the portfolio choicebased on the forward performance measure from that in Merton’s framework, in which intertemporalpreference is derived from the terminal utility function in a backward way. Musiela & Zariphopoulou(2010b) specify the generic forward performance measure as a stochastic flow U = U ( t, x ) t ≥ , takingtime ( t ) and wealth ( x ) as arguments. The randomness of the forward performance measure is drivenby the Brownian motion which is the same as the driving force of the randomness of asset price. Itimplies that the driving force of market randomness is simultaneously embedded into the investor’s2reference and the risky asset price process. Such modeling approach implicitly assumes that theBrownian motion represents the essential source of risk behind the financial market and the riskyassets. Especially, the volatility of a forward performance measure reflects the investor’s uncertaintyabout her future preference due to the randomness of the financial market states. However, due to theepistemic limitation or limited information, an investor may have ambiguity about the driving forceof market randomness and her future preference. Focusing on such ambiguities, we will introduce arobust forward performance measure, and investigate the corresponding portfolio selection problems.The mean return rate and volatility are important factors characterizing the dynamics of risky as-sets. In the traditional portfolio theory, these two factors are usually modeled by stochastic processes,the distributions of which are known to the decision-maker at each time node before the specified in-vestment horizon. In this case, the investor is actually assumed to have full information on the drivingforce of market randomness, and can accurately assigns probabilities to the various possible outcomesof investment or factors associated with the investment. However, in so complicated financial mar-kets, it is unrealistic for investors to have accurate information on the dynamics or distributions ofthe risk factors, essentially due to the cognitive limitation on the driving force of market randomness.This situation is referred to as “ambiguity” in the sense of Knight, while “risk” in the former situation.Ambiguity has raised researchers’ attention in the area of asset pricing and portfolio management (seee.g. Maenhout, 2004, Garlappi et al., 2007, Wang, 2009, Bossaerts et al., 2010, Liu, 2011, Chen et al.,2014, Luo et al., 2014, Guidolin & Liu, 2016, Luo, 2017, Zeng et al., 2018, Escobar et al., 2018).We assume that an investor has ambiguous beliefs on the paths of the risky asset price. Ambiguousbeliefs are characterized by a set P of probability measures ( P ∈ P ) defined on the canonical space Ω , the set of continuous paths starting from the current price of the risky asset. We incorporate theinvestor’s ambiguity on the risky asset price into her preference, by defining the forward performancemeasure on the canonical space Ω .We first characterize ambiguity on the dynamics of risky asset in terms of ambiguity on its meanreturn and volatility. More specifically, we assume that the mean return and the volatility processes ofthe risky asset lie in a convex compact set Θ ⊂ R , which then leads to the set of probability measures3 . This formulation is different from the stochastic models with the known distributions at eachtime node, and generalizes the framework defined on a probability space with only one probabilitymeasure. Within in this general setting, we investigate an ambiguity-averse investor’s investmentstrategy, and her conservative beliefs on the mean return and the volatility of risky assets.We then define the robust forward performance measure, by taking the investor’s ambiguity onthe deriving force of market randomness. In turn, we propose a method to construct such robustforward performance measure for a given initial preference, and derive the corresponding investmentstrategy and conservative beliefs on the mean return and the volatility of risky assets. We show thatthe sum of the market risk premium and the utility risk premium determines the trading direction.We further specify the initial preference of the constant relative risk aversion (CRRA) type, andinvestigate the determinants of the conservative beliefs on the mean return and the volatility of riskyassets in three settings, i.e., ambiguity on the mean return rate, ambiguity on the volatility, and thestructured ambiguity. When we consider ambiguity on the mean return rate, we keep the volatility asa constant, and vise versa. Such ambiguities have been investigated in Merton’s framework (see e.g.Lin & Riedel, 2014, Luo, 2017). The third setting is motivated by the fact that there is no consensuson the relation between the mean return and the volatility of risky assets in the empirical literature(see e.g. Omori et al., 2007, Bandi & Ren `o, 2012, Yu, 2012), and investigated by Epstein & Ji (2013).We show that the sign of the total risk premium determines the conservative belief on the mean returnin the first setting, while the risk attitude and the relative value of the market risk premium over theutility risk premium jointly determine the conservative belief on the volatility in the second setting. Inthe third setting, we would not derive the closed-form formula for the conservative beliefs, but showthat the corresponding beliefs can take some intermediate value within the candidate value interval,as well as the upper and lower bounds. To our knowledge, such interesting results are new in theportfolio selection literature.This paper contributes to the existing literature in three folds. First , we propose a generic formula-tion of robust forward performance accommodating an investor’s ambiguity on the dynamics of riskyassets.
Second , we figure out the determinants of trading direction for an investor in a market with4ne risk-free asset and one risky asset. From the economic point of view, it is the sum of the marketrisk premium and the utility risk premium that determines an investor’s trading direction.
Third , weshow that the market risk premium, the utility risk premium, and the risk tolerance affect an investor’sconservative belief on the mean return and volatility. Especially, if the maximum of the total riskpremium is negative, an investor will take the maximum of the mean return as the worst-case value;if the minimum of the total risk premium is positive, an investor will take the minimum of the meanreturn as the value in the worst-case scenario; otherwise, the worst-case mean return lies between itsminimum and maximum. The market risk premium, the utility risk premium, and the risk tolerancejointly determine an investor’s conservative belief on the volatility of risky assets. We emphasizethat the conservative belief is related to the optimization associated with risk premiums, and theseconservative beliefs may be some intermediate values within their candidate value intervals, as wellas boundaries.
Related Literature . Most of the existing results on forward performance measures have so farfocused on its construction and portfolio problems in the setting of risk, rather than ambiguity (Za-riphopoulou & ˇZitkovi´c, 2010, Musiela & Zariphopoulou, 2010a, Alghalith, 2012, El Karoui & Mrad,2013, Kohlmann et al., 2013, Anthropelos, 2014, Nadtochiy & Tehranchi, 2017, Avanesyan & May,2018, Shkolnikov et al., 2016, Angoshtari et al., 2018, to name a few). As one of the few exceptions,K¨allblad et al. (2018) investigate the robust forward performance measure in the setting of ambiguitycharacterized by a set of equivalent probability measures. However, this approach fails to solve therobust “forward” investment problem under ambiguous volatility, since volatility ambiguity is char-acterized by a set of mutually singular probability measures (Epstein & Ji, 2013). We fill this gap bycharacterizing an investor’s ambiguity with a set of probability measures, which may not be equivalentwith each other. Similar to our work, Chong & Liang (2018) investigate robust forward investmentunder parameter uncertainty in the framework where a unique probability measure is aligned to thecanonical space (Ω) . Different from such model setup, we align a set of probability measures on thecanonical space (Ω) , accounting for an investor’s ambiguity on the future scenarios of the risky assetprice. This approach is not only technically more general than the approach with a set of dynamic5odels under a unique probability measure (as detailed in Remark 4 by Epstein & Ji (2013)), but alsoallows an investor to explicitly incorporate ambiguity on the risk source into her preference. Thatis the key difference between our framework and the framework of Chong & Liang (2018). On theother hand, Chong & Liang (2018) construct the forward performance measure based on the solu-tion of an infinite horizon backward stochastic differential equation (BSDE). Our approach associatesthe forward performance measure with a stochastic partial differential equation (SPDE), which pro-vides the analogue of the Hamilton-Jacobi-Bellman equation (HJB) in Merton’s framework. For thereason of tractability, we limit ourself to forward performance measures of some special forms, andinvestigate the corresponding robust investment. It is out of this paper’s scope to investigate the exis-tence, uniqueness, and regularity of the solution of the associated SPDE in the general setting. Suchsimplified model setup and the corresponding results shed light on how ambiguity-aversion investorsdynamically revise their preferences as the market involves.The remainder of this paper is organized as follows. Section 2 introduces the model setup forrobust forward investment. The construction of the robust forward performance measure is investi-gated in Section 3. In Section 4, we study the conservative belief of an ambiguity-averse investor withpreference of the constant relative risk aversion (CRRA) type. Section 5 concludes.
We consider a financial market with two tradable assets: the risk-free bond and the risky asset. Therisk-free bond has a constant return rate r , i.e., d P t = rP t d t , (2.1)where P is bond price with P = 1 .The risk asset price S = ( S t ) t ∈ [0 , ∞ ) is modelled by the canonical process of Ω , defined by Ω = (cid:8) ω = ( ω ( t )) t ∈ [0 , ∞ ) ∈ C ([0 , ∞ ) , R + ) : ω (0) = S (cid:9) , S is the current price of the risky asset and S t ( ω ) = ω ( t ) . We equip Ω with the uniformnorm and the corresponding Borel σ -field F . F = ( F t ) t ∈ [0 , ∞ ) denotes the canonical filtration, i.e., thenatural (raw) filtration generated by S . Due to the complication of financial market and the limitationof individual cognitive ability, an investor may have ambiguous belief on the risky asset price, i.e.,ambiguity on the mean return ( µ ) or volatility ( σ ) in our model setup. We assume that ( µ t , σ t ) cantake any value within a convex compact set Θ ⊂ R , but without additional information about theirdistributions for any time t ∈ [0 , ∞ ) . That is, Θ represents ambiguity on the return and volatility ofthe risky asset. More explicitly, we characterize ambiguity by Γ Θ , defined by Γ Θ = (cid:110) θ | θ = ( µ t , σ t ) t ≥ is an F -progressively measurable process and ( µ t , σ t ) ∈ Θ for any t > (cid:111) . (2.2)For θ = ( µ t , σ t ) t ≥ ∈ Γ Θ , let P θ be the probability measure on (Ω , F ) such that the following stochas-tic differential equation (SDE) d S t = S t ( µ t d t + σ t d W θt ) , (2.3)admits a unique strong solution S = ( S t ) t ≥ , where W θ = ( W θt ) t ≥ is a Brownian motion under P θ . Let P Θ denote the set of probabilities P θ on (Ω , F ) such that the SDE (2.3) has a unique strongsolution, corresponding to the ambiguity characteristic Θ ( θ ∈ Γ Θ ) . The Brownian motion W θ canbe interpreted as the driving force of randomness behind the risky asset under the probability measure P θ . Such model setup allows us to analyze how the investor’s belief on the risky asset affects herpreference and investment strategy, especially the effect of ambiguity on the risk source.An investor is endowed with some wealth x > at time t = 0 , and allocates her wealth dynam-ically between the risky asset and the risk-free bond. For t ≥ and s ≥ t , let π s be the proportionof her wealth invested in the stock at time s ≥ . Due to the self-financing property, the discountedwealth X π = ( X πs ) s ≥ t is given by d X πs = ( µ s − r ) π s X πs d s + π s X πs σ s d W θs , X πt = x , (2.4)where r is the risk-free interest rate, W θ is a Brownian motion under P θ ∈ P Θ . The set of admissible7trategies A ( x ) is defined by A ( x ) = (cid:26) π (cid:12)(cid:12)(cid:12)(cid:12) π is self-financing and F -adapted , (cid:90) st π r d r < ∞ , s ≥ t, P θ -a.s., for all P θ ∈ P Θ (cid:27) . The optimal investment strategy π ∗ and the corresponding wealth process X ∗ are usually as-sociated with an optimization problem, such as utility maximization or risk minimization. WithinMerton’s framework for portfolio theory (Merton, 1969), the value process U ( t, x ; ˜ T ) is formulatedas U ( t, x ; ˜ T ) := sup π ∈A ˜ T E [ u ( X π ˜ T ) | F t , X πt = x ] , (2.5)where the investment horizon ˜ T is predetermined, u is a utility function, A ˜ T is the set of admissiblestrategy, and X π ˜ T is the terminal wealth corresponding to an admissible strategy π ∈ A ˜ T . The expec-tation ( E ) in (2.5) is taken under some probability measure P , if there is no ambiguity on the derivingforce of market randomness. Then, the dynamic programming principle can be applied to solve theoptimal control problem (2.5), namely, U ( t, x ; ˜ T ) = sup π ∈A ˜ T E [ U ( s, X πs ; ˜ T ) | F t , X πt = x ] . (2.6)By verification arguments, U is the solution of the Hamilton-Jacobi-Bellman (HJB) equation (Yong &Zhou, 1999). The dynamic programming equation (2.6) essentially signifies that { U ( t, X πt ; ˜ T ) } t ∈ [0 , ˜ T ] is a martingale at the optimum, and a supermartingale otherwise, associated with some probabilitymeasure P . This property can be interpreted as follows: if the system is currently at an optimal state,one needs to seek for controls which preserve the same level of the average performance over all futuretimes before the predetermined investment horizon ˜ T . We refer to this property as the martingaleproperty of the value process. On the other hand, (2.6) hints that U ( ˜ T , X π ˜ T ; ˜ T ) coincides with u ( X π ˜ T ) ,where u represents the preference at t = ˜ T . Note that the future utility function u is specified at t = 0 . However, it is not intuitive to specify the future preference at the initial time with completeisolation from the evolution of the market. Musiela & Zariphopoulou (2007, 2008) propose the so-called forward performance measure U ( t, x ) which keeps the martingale property of { U ( t, X πt ) } t ∈ [0 ,T ] for any horizon T > , and coincides with the initial preference, namely U (0 , X π ) = u ( X π ) . In thisframework, the future preference dynamically changes in accordance with the market evolution.8n the similar spirit of Musiela & Zariphopoulou (2007, 2008), we will generalize the definition offorward performance measure by considering ambiguity on the risk source. For P θ ∈ P Θ , we define P Θ ( t, P θ ) P Θ ( t, P θ ) := { P (cid:48) ∈ P Θ | P (cid:48) = P θ on F t } (2.7)which facilitates the definition of the robust forward performance measure. Definition 2.1 (Robust forward performance) . An F t -progressively measurable process ( U ( t, x )) t ≥ is called a robust forward performance if for t ≥ and x ∈ R + , the following holds.(i) The mapping x → U ( t, x ) is strictly concave and increasing.(ii) For each π ∈ A ( x ) , ess inf P ∈P Θ E P [ U ( t, X πt )] + < ∞ , and ess inf P (cid:48) ∈P Θ ( t, P θ ) E P (cid:48) [ U ( s, X πs ) | F t ] ≤ U ( t, X πt ) , t ≤ s, P θ -a.s.(iii) There exists π ∗ ∈ A ( x ) for which ess inf P (cid:48) ∈P Θ ( t, P θ ) E P (cid:48) [ U ( s, X π ∗ s ) | F t ] = U ( t, X π ∗ t ) , t ≤ s, P θ -a.s. Given the dynamics of the forward performance measure ( U ( t, x )) t ≥ , we will solve the problemfor optimal investment strategy, which can be formulated as a similar problem as (2.6). Problem 2.1 (Robust Investment Problem) . Given the robust forward performance ( U ( t, x )) t ≥ , theinvestment problem is to solve U ( t, x ) = sup π ∈A ( x ) inf P (cid:48) ∈P Θ ( t, P θ ) E P (cid:48) [ U ( s, X πs ) | F t , X πt = x ] , P θ -a.s. , (2.8) where X π follows (2.4) and P Θ ( t, P θ ) is given in (2.7) . The solution of this problem provides the robust investment strategy π ∗ and the worst-case sce-nario of ( µ π ∗ , σ π ∗ ) under ambiguity. In turn, they will implicitly provide the corresponding proba-bility measure P θ ∗ . In the next section, we will introduce the construction methods for the forwardperformance under ambiguity, and then solve the robust investment problem.9 Robust Investment under the Forward Performance Measures
The specification of a forward performance measure ( ¯ U ( t, x )) t ≥ can take the market state and in-vestor’s wealth level into account at time t . Mathematically, ( ¯ U ( t, x )) t ≥ is called stochastic flow, astochastic process with space parameter. It can be characterized by its drift random field and diffusionrandom field. Under certain regularity hypotheses (El Karoui & Mrad, 2013), it can be written in theintegration form ¯ U ( t, x ) = u ( x ) + (cid:90) t β ( s, x )d s + (cid:90) t γ ( s, x )d ¯ B s , (3.9)where ¯ B is the standard Brownian motion defined on some probability space, u is the initial utility, and β and γ are the so-called drift random field and the diffusion random field, respectively. To guaranteea stochastic flow ( ¯ U ( t, x )) t ≥ satisfy the definition 2.1, its drift random filed β and diffusion randomfield γ should satisfy some structure. By exploring such structure, Musiela & Zariphopoulou (2010b)constructed some examples of forward performance measures. In this framework, the driving forceof market randomness is modelled by the standard Brownian motion ¯ B . We will generalize suchframework, and account for the ambiguity on the driving force of market randomness or the riskyasset price. Different from the dynamics of the risky asset price (2.3), we give even more freedom toan investor’s preference, and propose the robust forward performance measure of the following form, U ( t, x ) = u ( x ) + (cid:90) t (cid:2) β ( s, x ) + δ ( s, x ) µ s + γ ( s, x ) σ s (cid:3) d s + (cid:90) t η ( s, x ) σ s d W θs , (3.10)where W θ a Brownian motion under P θ ∈ P Θ and θ = ( µ t , σ t ) t ≥ ∈ Γ Θ . The random field ( β, δ, γ, η ) characterizes an investor’s attitudes toward wealth level, ambiguity, and market risk. Especially, thevolatility term η ( t, x ) σ t of the robust forward performance measure reflects the investor’s ambiguityabout her preferences in the future, and is subject to her choice. The BSDE-based approach proposedby Chong & Liang (2018) captures the investor’s concern on parameter uncertainty by the generatorof the associated BSDE. Different from this BSDE-based approach, we explicitly embed such concerninto the axiomatic formulation (3.10).For any given robust forward preference of the form (3.10), the investment problem 2.1 allowsan investor to maximize her utility under the worst-case scenario of ( µ t , σ t ) t ≥ ∈ Γ Θ . To make the10nvestment problem tractable, the forward performance measure is assumed to be regular enough. Forthis reason, we introduce the notation of L ( P ) -smooth stochastic flow. Definition 3.1 ( L ( P ) -Smooth Stochastic Flow) . Let F : Ω × [0 , ∞ ) × R → R be a stochastic flowwith spatial argument x and local characteristics ( β, γ ) , i.e., F ( t, x ) = F (0 , x ) + (cid:90) t β ( s, x )d s + (cid:90) t γ ( s, x )d B P s , where B P is a Brownian motion defined on a filtered probability space (Ω , ( F t ) t ≥ , P ) . F is said tobe L ( P ) -smooth or belong to L (Ω , ( F t ) t ≥ , P ) , if(i) for x ∈ R , F ( · , x ) is continuous; for each t > , F ( t, · ) is a C -map from R to R , P -a.s.,(ii) β : Ω × [0 , ∞ ) × R → R and γ : Ω × [0 , ∞ ) × R → R d are continuous process continuous in ( t, x ) such that(a) for each t > , β ( t, · ) , γ ( t, · ) belong to C ( R ) , P -a.s.;(b) for each x ∈ R , β ( · , x ) and γ ( · , x ) are F -adapted. For P θ ∈ P Θ , we are ready to formulate the robust forward performance as a L ( P θ ) -smoothstochastic flow U ( t, x ) = u ( x ) + (cid:90) t (cid:2) β ( s, x ) + δ ( s, x ) µ s + γ ( s, x ) σ s (cid:3) d s + (cid:90) t η ( s, x ) σ s d W θs , (3.11)where W θ a Brownian motion under P θ ∈ P Θ and θ = ( µ t , σ t ) t ≥ ∈ Γ Θ . Its smoothness plays a keyrole to construct the robust forward performance measures by specifying the structure of ( β, δ, γ, η ) . Lemma 3.1.
For P θ ∈ P Θ , let U be a L ( P θ ) -smooth stochastic flow as defined in (3.11). Let ussuppose that(i) the mapping x → U ( t, x ) is strictly concave and increasing;(ii) for an arbitrary π ∈ A ( x ) , there exists ( µ π , σ π ) ∈ Γ Θ , such that ess inf P (cid:48) ∈P ( t, P θ ) E P (cid:48) [ U ( s, X πs ) | F t ] = E P µπ,σπ [ U ( s, X πs ) | F t ] , t ≤ s, P θ - a.s. , and Y s = U ( s, X πs ) is a P µ π ,σ π -supermartingale; iii) there exists π ∗ ∈ A ( x ) such that Y ∗ s = U ( s, X π ∗ s ) is a P µ π ∗ ,σ π ∗ -martingale.Then π ∗ is the optimal investment strategy for Problem 2.1, associated with the worst-case scenario ( µ π ∗ , σ π ∗ ) of ( µ, σ ) .Proof. For each π ∈ A ( x ) , since Y s = U ( s, X πs ) is a P µ π ,σ π -supermartingale, ess inf P (cid:48) ∈P ( t, P θ ) E P (cid:48) [ U ( s, X πs ) | F t ] = E P µπ,σπ [ U ( s, X πs ) | F t ] ≤ U ( t, X πt ) , t ≤ s, P θ - a.s. Since there exists π ∗ ∈ A ( x ) such that Y ∗ s = U ( s, X π ∗ s ) is a P µ π ∗ ,σ π ∗ -martingale, we have ess inf P (cid:48) ∈P ( t, P θ ) E P (cid:48) [ U ( s, X πs ) | F t ] = E P µπ ∗ ,σπ ∗ [ U ( s, X π ∗ s ) | F t ] = U ( t, X π ∗ t ) , t ≤ s, P θ - a.s. Recalling the definition of robust forward performance (Definition 2.1), we can see that U is aforward performance, and the statement of this theorem is proved.Lemma 3.1 provides a method to find the worst-case scenario of the mean return and volatility ofthe risky asset, and the corresponding investment strategy, as stated in Theorem 3.2. Theorem 3.2.
Let U be a L ( P θ ) -smooth stochastic flow on (Ω , F , P θ ) with P θ ∈ P Θ and θ =( µ t , σ t ) t ≥ ∈ Γ Θ , and the mapping x → U ( t, x ) is strictly concave and increasing. We suppose thefollowing holds.(i) U satisfies the following equation sup π inf ( µ,σ ) ∈ Θ (cid:110) β ( t, x ) + δ ( t, x ) µ + γ ( t, x ) σ + U x ( t, x )( µ − r ) πx + η x ( t, x ) πxσ + 12 U xx ( t, x ) π σ x (cid:111) = 0 . (3.12) (ii) For any π ( t, x ) ∈ R , there exists (˜ µ t , ˜ σ t ) ∈ Θ such that inf ( µ,σ ) ∈ Θ (cid:110) δ ( t, x ) µ + γ ( t, x ) σ + U x ( t, x )( µ − r ) πx + η x ( t, x ) πxσ + 12 U xx ( t, x ) π σ x (cid:111) = δ ( t, x )˜ µ t + γ ( t, x )˜ σ t + U x ( t, x )(˜ µ t − r ) πx + η x ( t, x ) πx ˜ σ t + 12 U xx ( t, x ) π ˜ σ t x . et π ∗ ( t, x ) ∈ R satisfy π ∗ ( t, x ) = arg sup π inf ( µ,σ ) ∈ Θ (cid:110) δ ( t, x ) µ + γ ( t, x ) σ + U x ( t, x )( µ − r ) πx + η x ( t, x ) πxσ + 12 U xx ( t, x ) π σ x (cid:111) , and ( µ ∗ , σ ∗ ) satisfy sup π inf ( µ,σ ) ∈ Θ (cid:110) δ ( t, x ) µ + γ ( t, x ) σ + U x ( t, x )( µ − r ) πx + η x ( t, x ) πxσ + 12 U xx ( t, x ) π σ x (cid:111) = δ ( t, x ) µ ∗ t + γ ( t, x )( σ ∗ t ) + U x ( t, x )( µ ∗ t − r ) π ∗ x + η x ( t, x ) π ∗ x ( σ ∗ t ) + 12 U xx ( t, x )( π ∗ ) ( σ ∗ t ) x . (3.13) Let X ∗ be the unique solution of the stochastic differential equation d X ∗ t = ( µ ∗ t − r ) π ∗ t X ∗ t d t + π ∗ t X ∗ t σ ∗ t d W µ ∗ ,σ ∗ t , X ∗ = x. Then π ∗ ( t, X ∗ t ) solves the Problem 2.1.Proof. Under the regularity conditions on U , we apply the Itˆo-Ventzell formula to U ( t, X π ) for anyadmissible portfolio X π under each P θ ∈ P Θ d U ( t, X πt ) = (cid:8) β ( t, X πt ) + δ ( t, X πt ) µ t + γ ( t, X πt ) σ t (cid:9) d t + η ( t, X πt ) σ t d W θt + U x ( t, X πt )d X πt + 12 U xx ( t, X πt )d (cid:104) X π (cid:105) t + η x ( t, X πt ) σ t d (cid:104) X π , W θ (cid:105) t = (cid:110) β ( t, X πt ) + δ ( t, X πt ) µ t + γ ( t, X πt ) σ t + U x ( t, X πt )( µ t − r ) π t X πt + η x ( t, X πt ) π t X πt σ t + 12 U xx ( t, X πt ) π t σ t ( X πt ) (cid:111) d t + { η ( t, X πt ) σ t + U x ( t, X πt ) π t X πt σ t } d W θt . We denote by g ( t, µ t , σ t ) = β ( t, X πt ) + δ ( t, X πt ) µ t + γ ( t, X πt ) σ t + U x ( t, X πt )( µ t − r ) π t X π + η x ( t, X πt ) π t X πt σ t + U xx ( t, X πt ) π t σ t ( X πt ) .For t < s , ess inf P (cid:48) ∈P ( t, P θ ) E P (cid:48) [ U ( s, X πs ) | F t ] = ess inf P (cid:48) ∈P ( t, P θ ) E P (cid:48) [ U ( t, X πt ) + (cid:90) st g ( r, µ r , σ r )d r | F t ] ≥ ess inf P (cid:48) ∈P ( t, P θ ) E P (cid:48) [ U ( t, X πt ) + (cid:90) st inf µ,σ g ( r, µ, σ )d r | F t ] ess inf P (cid:48) ∈P ( t, P θ ) E P (cid:48) [ U ( t, X πt ) + (cid:90) st g ( r, ˜ µ, ˜ σ )d r | F t ]= E P µπ,σπ [ U ( t, X πt ) + (cid:90) st g ( r, ˜ µ, ˜ σ )d r | F t ]= E P µπ,σπ [ U ( s, X πs ) | F t ] . where ( µ π , σ π ) = ( µ, σ ) on [0 , t ] , and ( µ π , σ π ) = (˜ µ, ˜ σ ) on [ t, s ] .Therefore, ess inf P (cid:48) ∈P ( t,P θ ) E P (cid:48) [ U ( s, X πs ) | F t ] = E P µπ,σπ [ U ( s, X πs ) | F t ] . (3.14)It is obvious that U ( s, X πs ) is a P θ -supermartingale. From (3.13) it follows that U ( s, X π ∗ s ) is a P θ ∗ -martingale. Recalling Lemma 3.1 and (3.14), ( µ ∗ , σ ∗ ) represents the worst-case scenario of themean return and volatility of the risky asset, and π ∗ is the corresponding investment strategy.Theorem 3.2 provides a natural way to construct a robust forward performance measure, optimalinvestment strategy and the worst-case scenario of the mean return and volatility of risky assets. Wesummarize such results in Corollary 3.3. Corollary 3.3. (i) If the U is a robust forward performance measure and the worst-case ( µ ∗ , σ ∗ ) is selected, the optimal investment strategy is given in the feedback form ˜ π ( t, x ) = − η x ( t, x ) σ ∗ t + ( µ ∗ t − r ) U x ( t, x ) xσ ∗ t U xx ( t, x ) , (3.15) where the first and second term of the optimal strategy will be referred to as its non-myopic andmyopic part, respectively (Musiela & Zariphopoulou, 2010b).(ii) If the U is a robust forward performance measure, its characteristics ( β, δ, γ, η ) should satisfy. inf ( µ,σ ) ∈ Θ (cid:26) β + δµ + (cid:18) γ − η x U xx ( t, x ) (cid:19) σ − ( µ − r ) U x ( t, x )2 U xx ( t, x ) σ − ( µ − r ) U x η x U xx ( t, x ) (cid:27) = 0 , (3.16) for ( t, x ) ∈ [0 , ∞ ) × R + . The solution of condition (3.16) leads to the worst-case ( µ ∗ , σ ∗ ) . The constraint (3.16) on the local characteristics ( β, δ, γ, η ) implies that the forward performancemeasure is not unique for a given initial utility function. By specifying three of them, we can calculate14he fourth one. Hence, the investor in this framework has the freedom to specify her initial utility,as well as the additional characteristics of the utility field. However, in Merton’s framework, thedynamics and characteristics of the utility field are derived from the terminal utility function, whichis specified by the investor at the initial time. We note that the constraint (3.16) holds in the path-wisesense.The local characteristics ( β, δ, γ, η ) actually can be used to represent the investor’s attitude throughlocal risk tolerance τ U ( t, x ) = − U x ( t,x ) U xx ( t,x ) , utility risk premium (cid:37) U ( t, x, σ ) = η x ( t,x ) σ t U x ( t,x ) (El Karoui &Mrad, 2013), and market risk premium m ( µ, σ ) = µ − rσ . Actually, the optimal strategy ˜ π (3.15) canbe written as ˜ π ( t, x ) = µ ∗ − rxσ ∗ τ U − η x ( t, x ) xU xx ( t, x ) (3.17) = τ U σ ∗ x (cid:18) µ ∗ − rσ ∗ + η x ( t, x ) σ ∗ U x ( t, x ) (cid:19) = τ U σ ∗ x (cid:0) m ( µ ∗ , σ ∗ ) + (cid:37) U ( t, x, σ ∗ ) (cid:1) . (3.18)The first component of the investment strategy (3.17), known as myopic strategy, resembles the in-vestment policy followed by an investor in markets in which the investment opportunity set remainsconstant through time. The second one is called the excess hedging demand and represents the ad-ditional (positive or negative) investment generated by the volatility process ησ of the performanceprocess U (Musiela & Zariphopoulou, 2010b). Essentially, the investment strategy (3.18) reveals thatit is affected by the investor’s risk tolerance, market risk premium, and utility risk premium, as wellas the worst-case scenario of the mean return µ and the volatility σ of the risky asset. Obviously, itis the sum of utility risk premium and market risk premium that determines the trading direction ofan investor. Such statement holds regardless of the specification of the robust forward performancemeasure. Note that the worst-case scenario of ( µ, σ ) is characterized by (3.16). To analyze the impli-cation of ambiguity, we restrict ourself to the robust forward performance measure of special forms,and derive the analytical solution for (3.16). 15 Robust Forward Performance of the CRRA type
Utility function of the CRRA type is one of the commonly used utility function, which is a powerfunction of wealth. We assume an investor’s dynamic preference is characterized by utility functionof the CRRA type over the time t ∈ [0 , ∞ ) , with the initial utility function u ( x ) = x κ /κ , κ ∈ (0 , and time-varying coefficients. More specifically, we set such forward performance U of the followingform U ( t, x ) = exp( α ( t )) κ x κ , U (0 , x ) = x κ /κ , d α ( t ) = f ( t )d t + g ( t ) σ t d W θt , α (0) = 0 , (4.19)where κ ∈ (0 , and W θ = ( W θt ) t ≥ is a Brownian motion defined on a filtered probability space (Ω , F , P θ ) with P θ ∈ P Θ and θ = ( µ, σ ) ∈ Γ Θ . Without loss of generality. Its differential form is thengiven by d U ( t, x ) = U ( t, x ) (cid:18) f ( t )d t + 12 g ( t ) σ t d t + g ( t ) σ t d W θt (cid:19) , U (0 , x ) = x κ /κ , (4.20)and U x ( t, x ) = x κ − exp( α ( t )) , (4.21) U xx ( t, x ) = ( κ − x κ − exp( α ( t )) . (4.22)In this case, the utility risk premium (cid:37) U ( t, x, σ ) = (cid:37) U ( σ ) = g ( t ) σ .We can rewrite the forward performance measure (4.34) in the form of (3.11), where β = U ( t, x ) f ( t ) , γ = 12 U ( t, x ) g ( t ) ,δ = 0 , η = U ( t, x ) g ( t ) . (4.23)The characteristics ( β, δ, γ, η ) can be substituted into the constraints (3.16), to specify the structure ofthe forward performance (4.19).If there is no ambiguity on the mean return and volatility, the constraint (3.16) is reduced to f ( t ) = 12 g ( t ) σ t κ − κκ − (cid:40)
12 ( µ t − r ) σ t +( µ t − r ) g ( t ) (cid:41) , (4.24)16nd the corresponding investment strategy is given by π ∗ = g ( t ) σ t + ( µ t − r )(1 − κ ) σ t = 1(1 − κ ) σ t (cid:18) µ t − rσ t + g ( t ) σ t (cid:19) = 1(1 − κ ) σ t (cid:0) m ( µ t , σ t ) + (cid:37) U ( σ t ) (cid:1) . (4.25)The optimal investment strategy without ambiguity (4.25), as well as the optimal strategy with ambi-guity (3.18), implies that the market risk premium and utility risk premium play an important role inthe trading direction in both settings.In the following sub-sections, we will consider an investor’s conservative beliefs and the forwardperformance of the CRRA type in different settings: ambiguity on mean return µ , ambiguity on thevolatility σ , and ambiguity on both mean return and volatility. The structure of forward performancein these settings will involve optimizations with respect to µ and σ , as implied by the constraint (3.16). Ambiguity on the mean return is referred to as the case where the dynamics of mean return is am-biguous, with known dynamics of volatility. For the sake of simplicity, we assume σ t is known as aconstant σ . Proposition 4.1.
Assume an investor’s forward preference U is characterized by the initial utilityfunction u ( x ) = x κ /κ with κ ∈ (0 , , and propagates in the following form d U ( t, x ) = U ( t, x ) (cid:18) f ( t )d t + 12 g ( t ) σ d t + g ( t ) σ d W θt (cid:19) , U (0 , x ) = u ( x ) , (4.26) where f and g are deterministic functions of t , σ is the volatility of the risky asset, and W θ is aBrownian motion defined on a filtered probability space (Ω , F , P θ ) with P θ ∈ P Θ and θ = ( µ, σ ) ∈ Γ Θ .If the investor’s ambiguity is characterized by the lower bound µ and upper bound µ of µ , f should atisfy the following condition f ( t ) = 12 g ( t ) σ κ − κκ − (cid:40)
12 ( µ ∗ − r ) σ +( µ ∗ − r ) g ( t ) (cid:41) , (4.27) where µ ∗ = µ , if µ − rσ < − g ( t ) σ ,r − gσ , if µ − rσ ≤ − g ( t ) σ ≤ µ − rσ ,µ , if µ − rσ ≥ − g ( t ) σ . (4.28) Corresponding to the selection of worst-case mean return µ ∗ , the investment strategy π ∗ is given by π ∗ = g ( t ) σ + ( µ ∗ − r )(1 − κ ) σ . (4.29) Proof.
In this case, the constraint (3.16) is reduced to f ( t ) = 12 g ( t ) σ κ − κ sup µ ∈ [ µ,µ ] (cid:40)
12 ( µ − r ) ( κ − σ + ( µ − r ) g ( t ) κ − (cid:41) , (4.30)Assume the supermum is achieved at µ ∗ . Simple calculations lead to µ ∗ = µ , if µ − r < − g ( t ) σ ,r − gσ , if µ − r ≤ − g ( t ) σ ≤ µ − r,µ , if µ − r ≥ − g ( t ) σ . (4.31)Due to σ > , the belief on the worst-case return (4.31) is equivalent to that given by (4.28). Corre-spondingly, the optimal strategy (3.15) is reduced to π ∗ = g ( t ) σ + ( µ ∗ − r )(1 − κ ) σ . (4.32)We can interpret the selection rule (4.28) from the premium point of view. Recalling the definitionof the market risk premium m ( µ, σ ) and the utility risk premium (cid:37) U ( σ ) , i.e., m ( µ, σ ) = µ − rσ and (cid:37) U ( σ ) = g ( t ) σ ,
18e can rewrite (4.28) as µ ∗ = µ , if m ( µ, σ ) + (cid:37) U ( σ ) < ,r − gσ , if m ( µ, σ ) + (cid:37) U ( σ ) ≤ ≤ m ( µ, σ ) + (cid:37) U ( σ ) ,µ , if m ( µ, σ ) + (cid:37) U ( σ ) > . (4.33)It implies that the worst-case mean return and the trading direction depend on the total risk pre-mium that the investor can achieve in the setting of ambiguity on mean return, i.e., m ( µ, σ ) + (cid:37) U ( σ ) .When m ( µ, σ ) + (cid:37) U ( σ ) is positive, an investor will take µ as the worst-case mean return, and take along position ( π > . When m ( µ, σ ) + (cid:37) U ( σ ) is negative, an investor will take µ as the worst case,and take a short position ( π < . Otherwise, she will take r − gσ as the worst-case mean return,and do not invest on the risky asset ( π = 0) . From this point of view, it is the total risk premium thatcharacterizes the worst-case mean return and the investor’s trading direction.Such premium-based rule (4.28) on the conservative belief towards the mean return is consistentwith the rule proposed by Chong & Liang (2018) and Lin & Riedel (2014). Chong & Liang (2018)propose to select the worst-case scenario of the mean return in a feedback form associated with theposition on risky assets, i.e., the long and short positions correspond to µ and µ , respectively. In theclassical framework, the selection of worst-case mean return dependents on the investor’s position onthe risky asset, as argued by Lin & Riedel (2014) that nature decides for a low drift if an investortakes a long position, and for a high drift if an investor takes a long position. However, the rule(4.28) is not given in a feedback form associated with an investor’s position, but directly related tothe market situations and the investor’s utility risk premium. In this new framework, we highlight thecombination effect of the utility risk premium and the market risk premium on the worst-case meanreturn of the risky asset. We refer to volatility ambiguity as the case where the dynamics of volatility is unknown, but con-strained in the interval [ σ, σ ] with < σ ≤ σ . For the sake of simplicity, we suppose µ t to be a19onstant µ over the time. Proposition 4.2.
Assume an investor’s preference U is characterized by the initial utility function u ( x ) = x κ /κ with κ ∈ (0 , , and propagates in the following form d U ( t, x ) = U ( t, x ) (cid:18) f ( t )d t + 12 g ( t ) σ t d t + g ( t ) σ t d W θt (cid:19) , U (0 , x ) = u ( x ) , (4.34) where f and g are deterministic functions of t , σ is the volatility of the risky asset, and W θ is aBrownian motion defined on the filtered probability space (Ω , F , P θ ) with P θ ∈ P Θ and θ = ( µ, σ ) ∈ Γ Θ .If the investor ambiguity is characterized by the lower bound σ and upper bound σ of σ , f shouldsatisfy the following structure f ( t ) = κ ( µ − r ) g ( t ) κ − κ − (cid:32) g ( t ) σ ∗ + κ ( µ − r ) σ ∗ (cid:33) , (4.35) where σ ∗ = σ , if g ( t ) ≥ κ ( µ − r ) σ ,σ , if g ( t ) ≤ κ ( µ − r ) σ , | µ − r || g ( t ) | √ κ, if κ ( µ − r ) σ ≤ g ( t ) ≤ κ ( µ − r ) σ . (4.36) Correspondingly, the optimal investment strategy is π ∗ = g ( t ) σ ∗ + ( µ − r ) σ ∗ (1 − κ ) . Proof.
In this setting, the constraint (3.16) is reduced to f = κ ( µ − r ) g ( t ) κ − σ ∈ [ σ ,σ ] κ − (cid:32) g ( t ) σ + κ ( µ − r ) σ (cid:33) . (4.37)To solve the optimization problem, we denote θ := σ , and define a function h by h ( θ ) = − g ( t ) θ − κ ( µ − r ) θ , θ ∈ [ σ , σ ] . h reaches its maximum at θ ∗ , where θ ∗ = σ , if g ( t ) σ ≥ κ ( µ − r ) σ ,σ , if g ( t ) σ ≤ κ ( µ − r ) σ , | µ − r || g ( t ) | √ κ, if κ ( µ − r ) σ ≤ g ( t ) ≤ κ ( µ − r ) σ . (4.38)Due to θ = σ , we have the worst-case scenario σ ∗ of σ (4.38).The conservative belief on the volatility depends on the market risk premium and the utility riskpremium, as the case of the conservative belief on mean return (4.28) or (4.33). We will show thatit is the relative value of these two premiums that determines the conservative belief on volatility.Note that it is the sum of these two premium determines the conservative belief on mean return, asshown by (4.28) or (4.33). In our specific setting, ambiguity on mean return only affects the marketrisk premium, while ambiguity on volatility affects both the market risk premium and the utility riskpremium. It is then natural to consider the effects of their relative value.Define the relative value τ ( σ ) of the utility risk premium (cid:37) U ( σ ) with respect to the market priceof risk m ( µ, σ ) as τ ( σ ) = (cid:37) U ( σ ) m ( µ, σ ) := g ( t ) σ µ − rσ . Then, the worst-case volatility (4.38) can be rewritten as σ ∗ t = σ , if τ ( σ ) ≥ κ,σ , if τ ( σ ) ≤ κ, | µ − r || g ( t ) | √ κ, otherwise . (4.39)The rule (4.39) for worst-case volatility implies that if the relative value of the utility risk premiumover the market risk premium is large enough than the investor’s risk-averse attitude κ , the investorwill take σ as the worst-case volatility. Alternatively, if such relative value is smaller enough thanthe investor’s risk-averse attitude, the investor will take σ as the worst-case volatility. Otherwise, theworst-case volatility depends on her attitude toward risk and ambiguity about her future preferences.21verall, an ambiguity-averse investor will take her attitude toward risk and ambiguity into accountwhen ambiguous on the volatility of the driving force of market randomness. Empirical research shows that the mean return can be either positively or negatively related to thevolatility of risky assets (see e.g. Omori et al., 2007, Bandi & Ren`o, 2012, Yu, 2012). Without aconsensus of their relation, we employ a flexible model to capture the structured ambiguity on themean return and volatility of the driving force of market randomness (Epstein & Ji, 2013, 2014),
Θ = (cid:110) ( µ, σ ) | σ = σ + αz, µ − r = µ + z, z ∈ [ z , z ] (cid:111) , (4.40)where σ , µ > and α ∈ R such that σ > . α > implies that the return is positively related tothe volatility, and vice versa. The selection of worst-case value of mean return and volatility will bereduced to the selection of z ∗ ∈ [ z , z ] , where the spread z − z represents the size of an investor’sambiguity on the mean return and volatility.Recalling the constraints (3.16) and (4.23), we have f = sup µ,σ κ − (cid:26) κg ( t )( µ − r ) + 12 (cid:18) g ( t ) σ + κ ( µ − r ) σ (cid:19)(cid:27) = sup z κ − (cid:40) κ ( µ + z ) g ( t ) + 12 (cid:32) ( σ + αz ) g ( t ) + κ ( µ + z ) σ + αz (cid:33)(cid:41) = sup z ∈ [ z ,z ] κ − (cid:26) az + b σ + αz ) + c (cid:27) , (4.41)where a = κg ( t ) + 12 g ( t ) σ + κ α ,b = κ (cid:18) µ − σ α (cid:19) ,c = κµ g ( t ) + 12 σ g ( t ) + κσ α + κ ( αµ − σ ) α . (4.42)For any given set of the parameters ( σ , µ , κ, α, z , z , g ) , we can easily solve the problem (4.41)with respect to z ∈ [ z , z ] , and the optimal investment strategy is correspondingly given as the22xpression (3.15). The analytical expression for z ∗ is omitted here, since it is not very expressive, inthe sense that the solution for (4.41) does not provide a straightforward intuition for the determinantsof the conservative beliefs. Obviously, the value of z ∗ depends on the interval [ z , z ] and the shapeof (4.41). To derive more intuitional information on the conservative beliefs and its dependence on z ∗ , we define ˆ f ( z ) = 1 κ − (cid:26) az + b σ + αz ) + c (cid:27) , (4.43)where a, b, c are given in (4.42). The second order derivative ˆ f (cid:48)(cid:48) of ˆ f with respect to z is ˆ f (cid:48)(cid:48) ( z ) = α b ( κ − σ + αz ) . Since κ ∈ (0 , , σ + αz > , and b > , we have ˆ f (cid:48)(cid:48) ( z ) < for z ∈ [ z , z ] . That is, ˆ f is a concave function on [ z , z ] . Such property relates z ∗ to the model parameters and theconcavity of ˆ f , as shown in the Figure 1 with some toy examples.These toy examples show the concavity of ˆ f in the setting of α = 0 . and α = − . with thefollowing common parameters κ = 0 . , µ = 0 . , σ = 0 . , g ( t ) ≡ . . For each α , we denote by ˜ z the value of z ∈ [ − . , . at which ˆ f reaches its maximum. Then, wehave three cases of [ z , z ] ⊆ [ − . , . for each α , i.e. , z < ˜ z , z < ˜ z < z , and ˜ z < z . Takethe case of α = 0 . and z < ˜ z for example, ˆ f reaches its maximum at z ∗ = z if z ∈ [ z , z ] .Correspondingly, we have µ ∗ = µ and σ ∗ = σ . One can easily figure out z ∗ in the other cases fromFigure 1. We summarize these toy examples in Table 1. Generally speaking, z ∗ may take the upperor lower bound of the interval for z , or some value lying in the interval. When the mean return ispositively related to the volatility of the risky asset ( α > , the worst-case scenario of these twoparameters is ( µ, σ ) , ( µ, σ ) , or some intermediate value depending on some ˜ z ∈ [ z , z ] . Whenthey are negatively related, the conservative belief is ( µ, σ ) , ( µ, σ ) , or some intermediate valuedepending on some ˜ z ∈ [ z , z ] . 23able 1: Conservative belief on the mean return and the volatility α > α < z ∗ z ˜ z ∈ ( z , z ) z z ˜ z ∈ ( z , z ) z µ ∗ µ µ + r − ˜ z µ µ µ + r − ˜ z µσ ∗ σ σ + α ˜ z σ σ σ + α ˜ z σ By specifying the interval [ z , z ] , we can not only verify the conservative belief on ( µ, σ ) givenin Table 1 or Figure 1, but also the relation between trading direction and total risk premium. Somealternatives for [ z , z ] are given in Table 2. The worst-case scenario ( µ ∗ , σ ∗ ) is consistent with theimplications of Table 1 or Figure 1. The corresponding investment strategy and total risk premiumlisted in the last two columns show that the investor will take a long position on the risky assets if thetotal risk premium is position, and vice versa. This is consistent with our theoretical statements, asgiven by (3.18).Table 2: Utility parameters and the corresponding worst-case mean return and volatility z z α z ∗ u ∗ σ ∗ π ∗ Total Risk Premium-0.15 -0.08 0.5 z µ σ -0.0795 − -0.08 0.07 0.5 -0.0289 r − σ − − z µ σ + -0.15 -0.08 -0.5 z µ σ -0.5476 − -0.08 0.07 -0.5 -0.0311 r − σ + + z µ σ + z -0.35-0.3-0.25-0.2-0.15-0.1-0.050 ˆ f α =0.5 ˜ zz z z ∗ = z µ ∗ = µσ ∗ = σ (a) -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 z -0.4-0.35-0.3-0.25-0.2-0.15-0.1-0.050 ˆ f α =-0.5 ˜ zz z z ∗ = z µ ∗ = µσ ∗ = σ (b) -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 z -0.35-0.3-0.25-0.2-0.15-0.1-0.050 ˆ f α =0.5 ˜ zz z z ∗ = ˜ z µ ∗ = µ + r − ˜ z σ ∗ = σ + α ˜ z (c) -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 z -0.4-0.35-0.3-0.25-0.2-0.15-0.1-0.050 ˆ f α =-0.5 ˜ zz z z ∗ = ˜ z µ ∗ = µ + r − ˜ z σ ∗ = σ + α ˜ z (d) -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 z -0.35-0.3-0.25-0.2-0.15-0.1-0.050 ˆ f α =0.5 ˜ z z z z ∗ = z µ ∗ = µσ ∗ = σ (e) -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 z -0.4-0.35-0.3-0.25-0.2-0.15-0.1-0.050 ˆ f α =-0.5 ˜ z z z z ∗ = z µ ∗ = µσ ∗ = σ (f) Figure 1: optimal vale z ∗ in different settings.25 Conclusion
The complicated market confronts an investor to ambiguity on the driving force of market random-ness. Such ambiguity may take the form of ambiguity on the mean return rate and volatility of anrisky asset. It may also affect the investor’s preference when making investing decisions. That is,the investor may be ambiguous not only on the characteristics of risky assets but also on her futurepreference. We took these two types of ambiguity into account, and investigated the horizon-unbiasedinvestment problem.We proposed the robust forward performance measure by accounting for an investor’s ambiguityon the future preference, arising from the ambiguity on the driving force of market randomness.This robust forward performance measure is then applied to formulate the investment problem. Thesolution to such investment problem shows that the sum of the market risk premium and the utilityrisk premium jointly determines the optimal trading direction. If it is positive, the investor will takea long position on the risky asset. Otherwise, she will take a short position. This statement holdsregardless of the specific form of the forward performance measures.We then explored the worst-case scenarios of the mean return and volatility when the initial utilityis of the CRRA type. Specifically, we investigate the worst-case mean return and volatility in threesettings: ambiguity on mean return µ , ambiguity on the volatility σ , and ambiguity on both meanreturn and volatility. In the case of ambiguity on the mean return, it is the total value of the marketrisk premium and the utility risk premium that determines an investor’s conservative belief; In the caseof ambiguity on the volatility, it is the relative value of these two premiums that affects an investor’sconservative belief. In the case of ambiguity on both the mean return and volatility, the conservativebelief may not be directly associated with these two premiums. Note that, in all the three settings,the conservative beliefs may be the some intermediate values within their candidate value intervals,as well as boundaries.In conclusion, the results provide explanations on the mechanism of conservative belief selectionand robust portfolio choice when an investor propagates her preference in accordance with the marketevolution. 26 eferences Alghalith, M. (2012). Forward Dynamic Utility Functions: A New Model and New Results.
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