An expansion in the model space in the context of utility maximization
aa r X i v : . [ q -f i n . P M ] A ug AN EXPANSION IN THE MODEL SPACE IN THE CONTEXT OFUTILITY MAXIMIZATION
KASPER LARSEN, OLEKSII MOSTOVYI, AND GORDAN ŽITKOVI ´CA
BSTRACT . In the framework of an incomplete financial market where the stockprice dynamics are modeled by a continuous semimartingale (not necessarily Mar-kovian) an explicit second-order expansion formula for the power investor’s valuefunction - seen as a function of the underlying market price of risk process - isprovided. This allows us to provide first-order approximations of the optimal pri-mal and dual controls. Two specific calibrated numerical examples illustrating theaccuracy of the method are also given.
1. I
NTRODUCTION
In an incomplete financial setting with noise governed by a continuous martingaleand in which the investor’s preferences are modeled by a negative power utility func-tion, we provide a second-order Taylor expansion of the investor’s value functionwith respect to perturbations of the underlying market price of risk process. We showthat tractable models can be used to approximate highly intractable ones as long asthe latter can be interpreted as perturbations of the former. As a by-product of ouranalysis we explicitly construct first-order approximations of both the primal and thedual optimizers. Finally, we apply our approximation in two numerical examples.There are two different ways of looking at our contribution: as a tool to approximatethe value function and perform numerical computations, or as a stability result with
Mathematics Subject Classification.
Primary 91G10, 91G80; Secondary 60K35.
Journal of Economic Literature (JEL) Classification:
C61, G11.
Key words and phrases.
Continuous semimartingales, 2nd order expansion, incomplete markets,power utility, convex duality, optimal investment.The authors would like to thank Milica ˇCudina, Claus Munk, Mihai Sîrbu and Kim Weston fordiscussions. During the preparation of this work the first author has been supported by the National Sci-ence Foundation under Grant No. DMS-1411809 (2014 - 2017). The second author has been supportedby the National Science Foundation under grant No. DMS-1600307 (2015 - 2018). The third authorhas been supported by the NSF under Grants No. DMS-0706947 (2010 - 2015) and No. DMS-1107465(2012 - 2017). Any opinions, findings and conclusions or recommendations expressed in this materialare those of the author(s) and do not necessarily reflect the views of the National Science Foundation(NSF). applications to statistical estimation. Let us elaborate on these, and the related work,in order.
An approximation interpretation.
The conditions for existence and uniqueness of theinvestor’s utility optimizers are well-established (see [KLSX91] and [KS99]). How-ever, in general settings, the numerical computation of the investor’s value functionremains a challenging problem. Various existing approaches include:(1) In Markovian settings, the value function can typically be characterized by aHJB-equation. Its numerical implementation through a finite-grid approxima-tion is naturally subject to the curse of dimensionality. Many authors (see [KO96],[Wac02], [CV05], [Kra05], and [Liu07]) opt for affine and quadratic models forwhich closed-form solutions exist. Going beyond these specifications in high-dimensional settings by using PDE-techniques seems to be very hard computa-tionally.(2) In general (i.e., not necessarily Markovian) complete models, [CGZ03] and[DGR03] provide efficient Monte Carlo simulation techniques based on the mar-tingale method for complete markets developed in [CH89] and [KLS87].(3) Other approximation methods are based on various Taylor-type expansions. Theauthors of [Cam93] and [CV99] log-linearize the investor’s budget constraintas well as the investor’s first-order condition for optimality. [KU00] expand inthe investor’s risk-aversion coefficient around the log-investor (the myopic in-vestor’s problem is known to be tractable even in incomplete settings). Whensolving the HJB-equation numerically (using Longstaff-Schwartz type of tech-niques) [BGSCS05] expand the value function in the wealth variable to a forthdegree Taylor approximation.(4) Based on the duality results in [KLSX91], [HKW06] provide an upper boundon the error stemming from using sub-optimal strategies. [BKM13] proposea method based on minimizing over a subset of dual elements. This subset ischosen such that the corresponding dual utility can be computed explicitly andtransformed into a feasible primal strategy.(5) It is also important to mention the recent explosion in research in asymptoticmethods in a variety of different ares in mathematical finance (transaction costs,pricing, etc.). Since we focus on model expansion in utility maximization in thispaper, we simply point the reader to some of the most recent papers, namely[AMKS15], and [MKK15], and the references therein, for further information.In our work, no Markovian assumption is imposed and we deal with general, pos-sibly incomplete, markets with continuous price processes. We note that while ourresults apply only to p < , it is possible to extend them to p ∈ (0 , at the cost ofimposing additional integrability requirements. We do not pursue such an extension; N EXPANSION IN THE MODEL SPACE IN THE CONTEXT OF UTILITY MAXIMIZATION 3 the parameter range p ∈ (0 , which we leave out seems to lie outside the typicalrange of risk-aversion parameters observed in practice (see, e.g., [Szp86]). Moreover,we do not consider utility functions more general than the powers. While there areno significant additional mathematical difficulties in treating the general case underappropriate conditions on the relative risk-aversion coefficients, we do not believethat the added value justifies the corresponding notational and technical overhead.For example, all our results would become dependent on the agent’s initial wealth,and this dependence would permeate the entire analysis. A stability interpretation.
As we mentioned above, our contribution can also be seenas a stability result. It is well-known (see, e.g., [Rog01]) that even in Samuelson’smodel, estimating the drift is far more challenging than estimating the volatility.[LŽ07] identify the kinds of perturbations of the market price of risk process un-der which the value function behaves continuously. In the present paper we take thestability analysis one step further and provide a first-order Taylor expansion in aninfinite-dimensional space of the market price of risk processes. This way, we notonly identify the “continuous” directions, but also identify those features of the mar-ket price of risk process that affect the solution of the utility maximization problemthe most (at least locally). Any statistical procedure which is performed with utilitymaximization in mind should, therefore, focus on those, salient, features in order touse the scarce data most efficiently.Similar perturbations have been considered by [Mon13], but in a somewhat differentsetting. [Mon13] is based on Malliavin calculus and produces a first-order expansionfor the utility-indifference price of an exponential investor in an Itô-process drivenmarket; some of the ideas used can be traced to the related work [Dav06].
Mathematical challenges.
From a mathematical point of view, our approach is foundedon two ideas. One of them is to extend the techniques and results of [LŽ07]; indeed,the basic fact that the optimal dual minimizers converge when the market prices ofrisk process does is heavily exploited. It does not, however, suffice to get the full pic-ture. For that, one needs to work on the primal and the dual problems simultaneouslyand use a pair of bounds. The ideas used there are related to and can be interpretedas a nonlinear version of the primal-dual second-order error estimation techniquesfirst used in [Hen02] in the context of mathematical finance. The first-order expan-sion in the quantity of the unspanned contingent claim developed in [Hen02] wasgeneralized in [KS06b] (see also [KS06a]). The arguments in these papers rely onconvexity and concavity properties in the expansion parameter (wealth and numberof unspanned claims). This is not the case in the present paper; indeed, when seen asa function of the underlying market price of risk process, the investor’s value func-tion is neither convex nor concave and a more delicate, local, analysis needs to beperformed.
KASPER LARSEN, OLEKSII MOSTOVYI, AND GORDAN ŽITKOVI ´C
Numerical examples.
In Section 5 we use two examples to illustrate how our ap-proximation performs under realistic conditions. First, we consider the Kim-Ombergmodel (see [KO96]) which is widely used in the financial literature. Under a cal-ibrated set of parameters, we find that our approximation is indeed very accuratewhen compared to the exact values.Our second example belongs to a class of extended affine models introduced in[CFK07]. The authors show that this class of models has superior empirical prop-erties when compared to popular affine and quadratic specifications (such as thoseused, e.g., in [Liu07]). The resulting optimal investment problem for the extendedaffine models, unfortunately, does not seem to be explicitly solvable. Our approxima-tion technique turns out to be easily applicable and our error bounds are quite tightin the relevant parameter ranges.2. A
FAMILY OF UTILITY - MAXIMIZATION PROBLEMS
The setup.
We work on a filtered probability space (Ω , F , F = {F t } t ∈ [0 ,T ] , P ) ,with the finite time horizon T > . We assume that the filtration F is right-continuousand that the σ -algebra F consists of all P -trivial subsets of F .Let M be a continuous local martingale, and let R ( ε ) , ε ≥ be a family of continuous F -semimartingales given by R ( ε ) := M + Z · λ ( ε ) t d h M i t , on [0 , T ] , where λ ( ε ) := λ + ελ ′ , (2.1)for a pair λ, λ ′ ∈ P M , where P M denotes the collection of all progressively measur-able processes π with R T π t d h M i t < ∞ . As S ( ε ) := E ( R ( ε ) ) (where E denotes thestochastic exponential) will be interpreted as the price process of a financial asset, theassumption that λ ( ε ) ∈ P M can be taken as a minimal no-arbitrage-type condition.We remark right away that further integrability conditions on λ and λ ′ will need tobe imposed below for our main results to hold.2.2. The utility-maximization problem.
Given x > and ε ∈ [0 , ∞ ) , let X ( ε ) ( x ) denote the set of all nonnegative wealth processes starting from initial wealth x inthe financial market consisting of S ( ε ) := E ( R ( ε ) ) and a zero-interest bond, i.e., X ( ε ) ( x ) := n x E (cid:0) R T π t dR ( ε ) t (cid:1) : π ∈ P M o . Here, π is interpreted as the fraction of wealth invested in the risky asset S ( ε ) . Theinvestor’s preferences are modeled by a CRRA (power) utility function with the risk-aversion parameter p < : U ( x ) := x p p , x > . (2.2) N EXPANSION IN THE MODEL SPACE IN THE CONTEXT OF UTILITY MAXIMIZATION 5
The value function of the corresponding optimal-investment problem is defined by u ( ε ) ( x ) := sup X ∈X ( ε ) ( x ) E [ U ( X T )] , x > . (2.3)2.3. The dual utility-maximization problem.
As is usual in the utility-maximizati-on literature, a fuller picture is obtained if one also considers the appropriate versionof the optimization problem dual to (2.3). For that, we need to examine the no-arbitrage properties of the set of models introduced in Section 2.1 above.We observe, first, that the assumptions we placed on the market price of risk pro-cesses λ ( ε ) above are not sufficient to guarantee the existence of an equivalent martin-gale measure (NFLVR). They do preclude so-called “arbitrages of the first kind” andimply the related condition NUBPR. In particular, for all x, y > and ε ≥ thereexists a (strictly) positive càdlàg supermartingale Y with the property that Y = y and Y X is a supermartingale for each X ∈ X ( ε ) ( x ) ; we denote the set of all suchprocesses by Y ( ε ) ( y ) . While this is a consequence of the condition NUBPR in gen-eral, in this case an example of a process in Y ( ε ) ( y ) is given, explicitly, as yZ ( ε ) ,where Z ( ε ) is the minimal local martingale density: Z ( ε ) = E ( − Z · λ ( ε ) t dM t ) . (2.4)Having described the dual domain, we remind the reader that the conjugate utilityfunction V : (0 , ∞ ) → R is defined by V ( y ) := sup x> ( U ( x ) − xy ) = y − q q , where q := p − p ∈ ( − , . (2.5)We define the dual value function v ( ε ) : (0 , ∞ ) → R by v ( ε ) ( y ) := inf Y ∈Y ( ε ) ( y ) E [ V ( Y T )] , y > , ε ≥ . (2.6)Due to negativity (and, a fortiori, finiteness) of the primal value function u ( ε ) , the(abstract) Theorem 3.1 of [KS99] can now be applied (see also [Mos15]). Its mainassumption, namely the bipolar relationship between the primal and dual domains,holds due to the existence of the numéraire process, given explicitly by /Z ( ε ) (seeTheorem 4.12 in [KK07]). One can also use a simpler argument (see [Lar11]), whichapplies only to the case of a CRRA utility with p < , to obtain the following con-clusions for all ε ≥ :(1) both u ( ε ) and v ( ε ) are finite and the following conjugacy relationships hold v ( ε ) ( y ) = sup x> (cid:16) u ( ε ) ( x ) − xy (cid:17) , and u ( ε ) ( x ) = inf y> (cid:16) v ( ε ) ( y ) + xy (cid:17) . (2.7) KASPER LARSEN, OLEKSII MOSTOVYI, AND GORDAN ŽITKOVI ´C (2) For all x, y > there exist optimal solutions ˆ X ( ε ) ( x ) ∈ X ( ε ) ( x ) and ˆ Y ( ε ) ( y ) ∈Y ( ε ) ( y ) of (2.3) and (2.6), respectively, and are related by U ′ ( ˆ X ( ε ) T ( x )) = ˆ Y ( ε ) T ( y ( ε ) ( x )) where y ( ε ) ( x ) = ddx u ( ε ) ( x ) = px p − u ( ε ) (1) . (3) The product ˆ X ( ε ) ˆ Y ( ε ) is a uniformly-integrable martingale. In particular E [ ˆ X ( ε ) T ˆ Y ( ε ) T ] = xy. The homogeneity of the utility function U and its conjugate V transfers to the valuefunctions u ( ε ) and v ( ε ) and the optimal solutions ˆ X ( ε ) and ˆ Y ( ε ) : u ( ε ) ( x ) = x p u ( ε ) , v ( ε ) ( y ) = y − q v ( ε ) , ˆ X ( ε ) ( x ) = x ˆ X ( ε ) , ˆ Y ( ε ) ( y ) = y ˆ Y ( ε ) , (2.8)where, to simplify the notation, we write u ( ε ) , v ( ε ) , ˆ X ( ε ) and ˆ Y ( ε ) for u ( ε ) (1) , v ( ε ) (1) , ˆ X ( ε ) (1) and ˆ Y ( ε ) (1) , respectively.2.4. A change of measure.
For ε = 0 we denote by ˆ π (0) the primal optimizer, i.e.,the process in P M such that ˆ X (0) = E ( Z · ˆ π (0) u dR (0) u ) . We define the probability measure ˜ P (0) by d ˜ P (0) d P = ˆ X (0) T ˆ Y (0) T (cid:16) = v (0) V ( ˆ Y (0) T ) = u (0) U ( ˆ X (0) T ) (cid:17) , (2.9)where the last two equalities follow from the identities xU ′ ( x ) = pU ( x ) and yV ′ ( y ) = − qV ( y ) , and the relations between the value functions outlined above.The measure ˜ P (0) has been in the mathematical finance literature for a while (see,e.g., p. 911-2 in [KS99]). The explicit form of ˜ P (0) is not generally available, but,we note that, by Girsanov’s Theorem (see (4.1) and the discussion around it), theprocess ˜ M p := M + Z · (cid:16) λ t − ˆ π (0) t (cid:17) d h M i t (2.10)is a ˜ P (0) -local martingale; this fact will be used below in the proof of Proposition 4.3.3. T HE PROBLEM AND THE MAIN RESULTS
We first provide first-order expansions and error estimates of the primal and dualvalue functions. Secondly, we provide an expansion of the optimal controls in theBrownian setting.
N EXPANSION IN THE MODEL SPACE IN THE CONTEXT OF UTILITY MAXIMIZATION 7
Value functions.
At the basic level, we are interested in the first-order proper-ties of the convergence, as ε ց , of the value functions of the problems u ( ε ) and v ( ε ) to the value functions u (0) and v (0) of the “base” model (corresponding to ε = 0 ).To familiarize ourselves with the flavor of the results we can expect in the generalcase, we start by analyzing a similar problem for the logarithmic utility. It has theadvantage that it admits a simple explicit solution. Let u ( ε )log ( x ) and v ( ε )log ( y ) denote thevalue function of the utility maximization problem as in (2.3) and (2.6) above, butwith U ( x ) = log( x ) and V ( y ) = sup x ( U ( x ) − xy ) = − log( y ) − . It is a classicalresult that, as long as E [ R T ( λ t + ( λ ′ t ) ) d h M i t ] < ∞ , we have u ( ε )log ( x ) = log( x ) + E [ Z T ( λ ( ε ) t ) d h M i t ] and v ( ε )log = u ( ε )log − . The (exact) second-order expansion in ε of u ( ε )log ( x ) is thus given by u ( ε )log ( x ) = u (0)log ( x ) + ε E [ Z T λ t λ ′ t d h M i t ] + ε E [ Z T ( λ ′ t ) d h M i t ]= u (0)log ( x ) + ε E [ Z T λ ′ t dR (0) t ] + ε E [ Z T ( λ ′ t ) d h M i t ] , where R (0) is defined in (2.1). We cannot expect the value function to be a secondorder polynomial in ε in the case of a general power utility. We do obtain a formallysimilar first-order expansion in Theorem 3.1 below and an analogous error estimatein Theorem 3.2. Section 5 is devoted to their proofs. We remind the reader of thehomogeneity relationships in (2.8); they allow us to assume from now on that x = y = 1 . Theorem 3.1 (The Gâteaux derivative) . In the setting of Section 2, we assume that Z T ( λ ′ t ) d h M i t ∈ L − p ( P ) and Z T λ ′ t dR (0) t ∈ ∪ s> (1 − p ) L s ( P ) . (3.1) Then, with ∆ (0) := E ˜ P (0) [ R T λ ′ t dR (0) t ] , where ˜ P (0) is defined by (2.9) , we have ddε u ( ε ) (cid:12)(cid:12)(cid:12) ε =0+ := lim ε ց ε (cid:16) u ( ε ) − u (0) (cid:17) = pu (0) ∆ (0) , and (3.2) ddε v ( ε ) (cid:12)(cid:12)(cid:12) ε =0+ := lim ε ց ε (cid:16) v ( ε ) − v (0) (cid:17) = qv (0) ∆ (0) . (3.3) Theorem 3.2 (An error estimate) . In the setting of Section 2, we assume that Z T ( λ ′ t ) d h M i t , Z T λ ′ t dR (0) t ∈ L − p ) ( P ) and Φ e ε | p | Φ − ∈ L (˜ P (0) ) , (3.4) KASPER LARSEN, OLEKSII MOSTOVYI, AND GORDAN ŽITKOVI ´C for some ε > , where Φ := R T ˆ π (0) t λ ′ t d h M i t . Then there exist constants C > and ε ′ ∈ (0 , ε ] such that for all ε ∈ [0 , ε ′ ] we have (cid:12)(cid:12)(cid:12) u ( ε ) − u (0) − εpu (0) ∆ (0) (cid:12)(cid:12)(cid:12) ≤ Cε , and (3.5) (cid:12)(cid:12)(cid:12) v ( ε ) − v (0) − εqv (0) ∆ (0) (cid:12)(cid:12)(cid:12) ≤ Cε . (3.6) Remark . (1) It is perhaps more informative to think of the results in Theorems 3.1 and 3.2on the logarithmic scale. As is evident from (3.2) and (3.3), the functions u ( ε ) and v ( ε ) admit the right logarithmic derivative p ∆ (0) and q ∆ (0) , respectively, at ε = 0 . Moreover, we have the following small- ε asymptotics: u ( ε ) = u (0) e εp ∆ (0) + O ( ε ) and v ( ε ) = v (0) e εq ∆ (0) + O ( ε ) . If one takes one step further and uses the certainty equivalent CE ( ε ) , given by U (CE ( ε ) ) = u ( ε ) , we note that ∆ (0) is precisely the infinitesimal growth-rate of CE ( ε ) at ε = 0 -an ε -change of the market price of risk in the direction λ ′ yields to an e ε ∆ (0) -foldincrease in the certainty-equivalent of the initial wealth.(2) A careful analysis of the proof of Theorem 3.2 below reveals the following,additional, information:(a) The proof of Proposition 4.3 reveals that ∆ (0) = E ˜ P (0) [Φ] .(b) The condition involving Φ in (3.4) is needed only for the upper bound in(3.5) and the lower bound in (3.6). The other two bounds hold for all ε ≥ even if (3.4) holds with ε = 0 .(c) The constants C and ε ′ depend - in a simple way - on ε , p and the L − p ) (˜ P (0) ) -and L (˜ P (0) ) -bounds of the random variables in (3.4). For two one-sidedbounds, explicit formulas are given in Propositions 4.2 and 4.3. The othertwo bounds are somewhat less informative so we do not compute them ex-plicitly. The reader will find an example of how this can be done in a specificsetting in Subsection 5.3.(d) Even though we cannot claim that the functions u ( ε ) and v ( ε ) are convex orconcave, it is possible to show their local semiconcavity in ε (see [CS04]).This can be done via the techniques from the proof of Theorem 3.2.(3) The assumption of constant risk aversion (power utility) allows us to incorporatemany stochastic interest-rate models into our setting. Indeed, provided that c := E (cid:20) e p R T r t dt (cid:21) < ∞ , we can introduce the probability measure P r , defined by d P r d P := ce p R T r t dt , (3.7) N EXPANSION IN THE MODEL SPACE IN THE CONTEXT OF UTILITY MAXIMIZATION 9 on F T . For any admissible wealth process X we then have E [ U ( X T )] = c E P r (cid:20) U (cid:16) X T e − R T r u du (cid:17)(cid:21) . This way, the utility maximization under P r with a zero interest rate becomesequivalent to the utility maximization problem under P with the interest rate pro-cess { r t } t ∈ [0 ,T ] . [Žit05] and [Mos15] consider the setting of utility maximizationwith stochastic utility which embeds stochastic interest rates.Practical implementation of the above idea depends on how explicit one canbe about the Girsanov transformation associated with P r . It turns out, fortu-nately, that many of the widely-used interest-rate models, such as Vasiˇcek, CIR,or the quadratic normal models (see, e.g., [Mun13] for a textbook discussionof these models) allow for a fully explicit description (often due to their affinestructure). For example, in the Vasiˇcek model, the Girsanov drift under P r canbe computed quite explicitly, due to the underlying affine structure. Indeed, sup-pose that r has the Ornstein-Uhlenbeck dynamics of the form dr t := κ ( θ − r t ) dt + β dB t , r ∈ R , where B is a Brownian motion and κ > , θ, β ∈ R . Then the process B ( p ) := B − Z · b ( T − t ) dt, where b ( t ) = βpκ (1 − e − κt ) , is a P r -Brownian motion.3.2. Optimal controls.
The estimates (3.5) and (3.6) are of type O ( ε ) . A slightadjustment to the below proof of Proposition 4.3 shows that the wealth process ˜ X := E (cid:0) R ˆ π (0) dR ( ε ) ) satisfies (see 4.13) (cid:12)(cid:12) E [ U ( ˜ X T )] − u (0) (1 + εp ∆ (0) ) (cid:12)(cid:12) ≤ p ε | u (0) | E ˜ P (0) [Φ e ε | p | Φ − ] . Therefore, under the conditions of Theorem 3.2, ˆ π (0) is an O ( ε ) -optimal control forthe ε -model because the triangle inequality produces a constant C > such that (cid:12)(cid:12) E [ U ( ˜ X T )] − u ( ε ) (cid:12)(cid:12) ≤ Cε , for all ε > small enough. In this section we will provide a correction term to ˆ π (0) such that the resulting wealth process upgrades the convergence to o ( ε ) .For simplicity, we consider the (augmented) filtration generated by ( B, W ) where B ∈ R and W ∈ R d , d ∈ N , are two independent Brownian motions. In (2.1) wetake dM t := σ t dB t , M := 0 , (3.8) for a process σ ∈ P B with σ = 0 . We define ˜ P (0) by (2.9) and we denote by ( B ˜ P (0) , W ˜ P (0) ) the corresponding ˜ P (0) -Brownian motions. Provided that Φ := R T ˆ π (0) t λ ′ t σ t dt ∈ L (˜ P (0) ) , Φ has the unique martingale representation under ˜ P (0) Φ = E ˜ P (0) [Φ] + Z T γ Bt σ t dB ˜ P (0) t + Z T γ Wt dW ˜ P (0) t , (3.9)where we have used σ = 0 . Because Φ ∈ L (˜ P (0) ) the two processes γ W and γ B in(3.9) satisfy the integrability conditions E ˜ P (0) h Z T (cid:16) ( γ Bt σ t ) + ( γ Wt ) (cid:17) dt i < ∞ . These square integrability properties will be used in the proof of the next theorem.
Theorem 3.4 (2nd order expansion) . In the above Brownian setting, we assume Z T ( λ ′ t ) σ t dt ∈ L − p ( P ) ∩ L (˜ P (0) ) and Z T ˆ π (0) t λ ′ t σ t dt ∈ L (˜ P (0) ) , (3.10) as well as the existence of a constant ε > such that δ := λ ′ + pγ B − p satisfies e p R T (cid:0) ε ˆ π (0) λ ′ + ε ( δλ ′ − δ ) (cid:1) σ dt + pε R T δσdB ˜ P (0) t ∈ L (˜ P (0) ) , (3.11) for all ε ∈ (0 , ε ) . Then we have u ( ε ) − u (0) − εpu (0) ∆ (0) − ε pu (0) (cid:16) ∆ (00) + p (∆ (0) ) (cid:17) ∈ O ( ε ) , (3.12) v ( ε ) − v (0) − εqv (0) ∆ (0) − ε qv (0) (cid:16) ∆ (00) + q (∆ (0) ) (cid:17) ∈ O ( ε ) , (3.13) as ε ց . In (3.12) and (3.13) we have defined ∆ (00) := E ˜ P (0) "Z T p | γ Wt | + ( λ ′ t ) + pγ Bt ( γ Bt + 2 λ ′ t )1 − p σ t ! dt , (3.14) where the processes γ B and γ W are given by the martingale representation (3.9) .Remark . (1) The below proof of Theorem 3.4 shows that the process ˜ π := ˆ π (0) + ε λ ′ + pγ B − p , (3.15)is an O ( ε ) -optimal control for the ε -model in the sense that the wealth process ˜ X := E (cid:0) R ˜ πdR ( ε ) ) satisfies E [ U ( ˜ X T )] − u ( ε ) ∈ O ( ε ) as ε ց . (2) Because the filtration is generated by ( B, W ) , the optimizer ˆ H (0) for the dualproblem (2.6) can be written as ˆ H (0) = E ( − R ˆ ν (0) dW ) for a d -dimensionalprocess ˆ ν (0) in P W . The below proof of Theorem 3.4 also shows that the process ˜ ν := ˆ ν (0) − εpγ W , (3.16) N EXPANSION IN THE MODEL SPACE IN THE CONTEXT OF UTILITY MAXIMIZATION 11 is an O ( ε ) -optimal dual control in the ε -model.(3) Throughout the paper we have considered ε = 0 as the base model. Because wecan write λ + (¯ ε + ε ) λ ′ = λ + ¯ ελ ′ + ελ ′ , for any ¯ ε ∈ [ ε L , ε U ] with ε L < ε U , we can use Theorem 3.4 for the base model λ + ¯ ελ ′ to provide a 2nd order Taylor expansion around any point ¯ ε . Therefore,whenever ∆ (0) and ∆ (00) are bounded uniformly in ¯ ε ∈ [ ε L , ε U ] , Theorem 3 in[Oli54] ensures that u ( ε ) is twice differentiable in ε .(4) An easy way of eliminating the stochastic B ˜ P (0) -integral appearing in (3.11) isto use Hölder’s inequality with the exponents − /q and (1 − p ) ; see Section 5.3below for an example.4. P ROOFS OF THE MAIN THEOREMS
We start the proof with a short discussion of the special structure the dual domain Y ( ε ) has when the stock-price process S ( ε ) = E ( R ( ε ) ) is continuous. Indeed, it has beenshown in [LŽ07], Proposition 3.2, p. 1653, that in that case the maximal elements in Y ( ε ) (in the pointwise order) are precisely local martingales of the form Y = Z ( ε ) H, H ∈ H , where H denotes the set of all M -orthogonal positive local martingales H with H = 1 . We remark that even though the results in [LŽ07] were written underthe assumption NFLVR, a simple localization argument shows that they apply underthe present conditions, as well. Hence, we can write v ( ε ) = inf H ∈H E [ V ( Z ( ε ) T H T )] , and the minimizer ˆ Y ( ε ) always has the form ˆ Y ( ε ) = Z ( ε ) ˆ H ( ε ) , for some ˆ H ( ε ) ∈ H . (4.1)Finally, we introduce two shortcuts for expressions that appear frequently in theproof: η := Z T λ ′ t dR (0) t , Λ := Z T ( λ ′ t ) d h M i t , (4.2)and remind the reader that Φ := R T ˆ π (0) t λ ′ t d h M i t and ∆ (0) := E ˜ P (0) [ η ] . It will beuseful to keep in mind that (1 − p )(1 + q ) = 1 and that − /q and − p are conjugateexponents. A proof of Theorem 3.1.
The proof is based on the stability results of [LŽ07]and the following lemma:
Lemma 4.1.
Let { K ( ε ) } ε ≥ be a family of positive random variables such that(1) E [ Z ( δ ) T K ( ε ) ] ≤ for all ε, δ ≥ , and(2) K ( ε ) → K (0) in probability, as ε ց .Then, under the conditions of Theorem 3.1, we have lim ε ց ε E h V ( Z ( ε ) T K ( ε ) ) − V ( Z (0) T K ( ε ) ) i = q E h V ( Z (0) T K (0) ) η i . Proof.
The map ε Z ( ε ) T is almost surely continuously differentiable; indeed, wehave log( Z ( ε ) T ) = log( Z (0) T ) − ε Z T λ ′ t dR (0) t − ε Z T ( λ ′ t ) d h M i t , and, so, ddε Z ( ε ) T = − Z ( ε ) T (cid:16) η + ε Λ (cid:17) , a.s.Therefore, V ( Z ( ε ) T K ) − V ( Z (0) T K ) = Z ε qV ( Z ( δ ) T K )( η + δ Λ) dδ, (4.3)for each ε and each positive random variable K . Thus, V ( Z ( ε ) T K ( ε ) ) − V ( Z (0) T K ( ε ) ) − εqV ( Z (0) T K (0) ) η = A ε + B ε , (4.4)where A ε := Z ε q (cid:16) V ( Z ( δ ) T K ( ε ) ) − V ( Z (0) T K (0) ) (cid:17) η dδ, and B ε := Z ε qV ( Z ( δ ) T K ( ε ) )Λ δ dδ. (4.5)Hölder’s inequality implies that E [ B ε ] ≤ ε sup δ ∈ [0 ,ε ] (cid:16) E [ Z ( δ ) T K ( ε ) ] − q E [Λ − p ] q (cid:17) ≤ ε E [Λ − p ] q . (4.6)Thus, we have ε E [ B ε ] → , as ε ց . To show that ε E [ A ε ] → , we note that E [ A ε ] = R ε f ( ε, δ ) dδ , where the function f : [0 , ∞ ) → R is given by f ( ε, δ ) := q E h(cid:0) V ( Z ( δ ) T K ( ε ) ) − V ( Z (0) T K (0) ) (cid:1) η i . (4.7)Since f (0 ,
0) = 0 , it will be enough to show that f is continuous at (0 , . By theassumptions of the lemma and the definition of Z ( δ ) , we have V ( Z ( δ n ) T K ( ε n ) ) → V ( Z (0) T K (0) ) , in probability , for each sequence ( ε n , δ n ) ∈ [0 , ∞ ) such ( ε n , δ n ) → (0 , . Therefore, it sufficesto establish uniform integrability of the expression inside of the expectation in (4.7). N EXPANSION IN THE MODEL SPACE IN THE CONTEXT OF UTILITY MAXIMIZATION 13
For that we can use the theorem of de la Valleé-Poussin, whose conditions holdthanks to an application Hölder’s inequality as in (4.6) above, remembering that notonly η ∈ L − p , but also in L s , for some s > (1 − p ) . (cid:3) Proof of Theorem 3.1.
Thanks to the optimality of Z ( ε ) T ˆ H ( ε ) T , we have the upper esti-mate ε E h V ( Z ( ε ) T ˆ H ( ε ) T ) − V ( Z (0) T ˆ H (0) T ) i ≤ ε E h V ( Z ( ε ) T ˆ H (0) T ) − V ( Z (0) T ˆ H (0) T ) i (4.8)Similarly, we obtain the lower estimate ε E h V ( Z ( ε ) T ˆ H ( ε ) T ) − V ( Z (0) T ˆ H (0) T ) i ≥ ε E h V ( Z ( ε ) T ˆ H ( ε ) T ) − V ( Z (0) T ˆ H ( ε ) T ) i . (4.9)Our next task is to prove that the limits of the right-hand sides of (4.8) and (4.9)exist and both coincide with the right-hand side of (3.3). In each case, Lemma 4.1can be applied; in the first with K ( ε ) = ˆ H (0) T , and in the second with K ( ε ) = ˆ H ( ε ) T .In both cases the assumption (1) of Lemma 4.1 follows directly from that fact that Z ( ε ) T K ( ε ) ∈ Y ( ε ) . As for the assumption (2), it trivially holds in the first case. In thesecond case, we need to argue that ˆ H ( ε ) T → ˆ H (0) T in probability, as ε ց . That, inturn, follows easily from Lemma 3.10 in [LŽ07]; as mentioned above, the seeminglystronger assumption of NFLVR made in [LŽ07] is not necessary and its results holdunder the weaker condition NUBPR.Having proven (3.3), we turn to (3.2). Thanks to (2.8), the conjugacy relationship(2.7) takes the following, simple, form in our setting: pu ( ε ) = ( qv ( ε ) ) − p . (4.10)Therefore, u ( ε ) is right differentiable at ε = 0 , and we have p ddε u ( ε ) (cid:12)(cid:12)(cid:12) ε =0+ = (1 − p )( qv (0) ) − p q v (0) ∆ (0) = p u (0) ∆ (0) . (cid:3) Remaining proofs.Proposition 4.2.
Suppose that η ∈ L − p ) and Λ , Λ η ∈ L − p . Then for all ε ≥ we have v ( ε ) − v (0) − εqv (0) ∆ (0) ≤ C v ε + C ′ v ε , (4.11) where C v = | q |k η k / L − p ) + k Λ k L − p and C ′ v = | q |k η Λ k L − p .Proof. The upper estimate (4.8) and the representation (4.3) imply that E h V ( Z ( ε ) T ˆ H ( ε ) T ) − V ( Z (0) T ˆ H (0) T ) − εqV ( Z (0) T ˆ H (0) T ) η i ≤ E [ A ε ] + E [ B ε ] , where A ε and B ε are defined by (4.5), with K ( ε ) = K (0) = ˆ H (0) T . As in (4.6), wehave E [ B ε ] ≤ ε k Λ k L − p . To deal with A ε we note that its structure allows us to apply the representation from(4.3) once again to see q A ε = Z ε Z δ V ( Z ( β ) T ˆ H (0) T ) η ( η + β Λ) dβ dδ. This, in turn, can be estimated, via Hölder inequality, as in (4.6), as follows E [ A ε ] ≤ | q | ε sup β ∈ [0 ,ε ] E [( η ( η + β Λ)) − p ] q ≤ | q | ε (cid:16) k η k L − p + ε k η Λ k L − p (cid:17) , yielding the bound in (4.11). (cid:3) Unfortunately, the same idea cannot be applied to obtain a similar lower bound. In-stead, we turn to the primal problem and establish a lower bound for it.
Proposition 4.3.
Given ε > , assume that Λ ∈ L − p , and Φ e ε | p | Φ − ∈ L (˜ P (0) ) ,where ˜ P (0) is defined by (2.9) . Then, u ( ε ) − u (0) − εpu (0) ∆ (0) ≥ − C u ( ε ) ε for ε ∈ [0 , ε ] , where C u ( ε ) := p | u (0) | E ˜ P (0) [Φ e ε | p | Φ − ] .Proof. For ˜ X := E ( R · ˆ π (0) t dR ( ε ) t ) , we have ˜ X ∈ X ( ε ) so that, by optimality, u ( ε ) − u (0) − pε E [ U ( ˆ X (0) T )Φ] ≥ E [ U ( ˜ X T ) − U ( ˆ X (0) T ) − pεU ( ˆ X (0) T )Φ] . (4.12)Thanks to the form of ˜ X , the right-hand side of (4.12) above can be written as E [ U ( ˆ X (0) T ) D ε ] , where D ε = exp( pε Φ) − − pε Φ = R ε R δ p Φ e pβ Φ dβ dδ . Thus, E [ U ( ˆ X (0) T ) D ε ] = p Z ε Z δ E [ U ( ˆ X (0) T )Φ e pβ Φ ] dβ dδ ≥ p ε E [ U ( ˆ X (0) T )Φ e ε | p | Φ − ] . (4.13)Therefore, u ( ε ) − u (0) − εp E [ U ( ˆ X (0) T )Φ] ≥ − C u ( ε ) ε , for ε ∈ [0 , ε ] with C u as inthe statement.It remains to show that E [ U ( ˆ X (0) T )Φ] = E [ U ( ˆ X (0) T ) η ] which is equivalent to showing E ˜ P (0) [Φ] = E ˜ P (0) [ η ] by the definition of ˜ P (0) . We define the local ˜ P (0) -martingale ˜ M p by (2.10). Therefore, N = R · λ ′ t d ˜ M pt is also a local martingale. The desiredequality is therefore equivalent to the equality E ˜ P (0) [ N T ] = 0 by the definition of η and Φ . In turn, it is sufficient to show that N is an H -martingale under ˜ P (0) . Since h N i T = R T ( λ ′ t ) d h M i t = Λ , Hölder’s inequality implies that E ˜ P (0) [ h N i T ] = ( qv (0) ) − E [( ˆ Y (0) T ) − q Λ] ≤ ( qv (0) ) − E [Λ − p ] q < ∞ . (cid:3) N EXPANSION IN THE MODEL SPACE IN THE CONTEXT OF UTILITY MAXIMIZATION 15
Remark . If one is interested in an error estimate which does not feature theoptimal portfolio ˆ π (0) (through Φ ), one can adopt an alternative approach in theproof (and the statement) of Proposition 4.3. More specifically, by using ˜ X =ˆ X (0) E ( R · ελ ′ dR ( ε ) t ) as a test process (instead of E ( R · ˆ π (0) t dR ( ε ) t ) ), one obtains a con-stant C u ( ε ) which depends only on the primal and dual optimizers ˆ X (0) and ˆ Y (0) , inaddition to λ ′ , η and Λ . Proof of Theorem 3.2.
Two of the four inequalities in Theorem 3.2 have been estab-lished in Propositions 4.2 and 4.3. For the remaining two we use the special form(4.10) of the conjugacy relationship between u ( ε ) and v ( ε ) . Thanks to Proposition 4.3and the positivity of pu ( ε ) , qv ( ε ) and q , we have q (cid:16) v ( ε ) − v (0) − εqv (0) ∆ (0) (cid:17) = ( pu ( ε ) ) q − ( pu (0) ) q − εq ( pu (0) ) q ∆ (0) . The right-hand side above is further bounded from above, for ε in a (right) neighbor-hood of , by F ( ε ) := ( pu (0) + εpu (0) ∆ (0) − pCε ) q − ( pu (0) ) q − εq ( pu (0) ) q ∆ (0) , where C is the constant from Proposition 4.3. F is a C -function in some neighbor-hood of with F (0) = F ′ (0) = 0 ; hence, on each compact subset of that neighbor-hood it is bounded by a constant multiple of ε . In particular, we have v ( ε ) − v (0) − εqv (0) ∆ (0) ≥ − Cε , for some C > and ε in some (right) neighborhood of . A similar argument, butbased on Proposition 4.2, shows that (3.5) holds, as well. (cid:3) Proof of Theorem 3.4.
The first part of (3.10) means that Λ ∈ L − p ( P ) ; hence, thesecond half of the proof of Proposition 4.3 shows that E ˜ P (0) [Φ] = ∆ (0) . Therefore,the martingale representation (3.9) can be written as Φ = ∆ (0) + Z T γ Bt σ t dB ˜ P (0) t + Z T γ Wt dW ˜ P (0) t . (4.14)Because the filtration is generated by the Brownian motions ( B, W ) we can find ˆ ν (0) ∈ P W such that the dual optimizer ˆ H (0) can be represented as ˆ H (0) = E ( − Z ˆ ν (0) dW ) . Therefore, Girsanov’s Theorem ensures that under ˜ P (0) , the processes dB ˜ P (0) := dB + ( λ − ˆ π (0) ) σdt, and dW ˜ P (0) := dW + ˆ ν (0) dt, are independent Brownian motions. We start with the primal problem and define ˜ π := ˆ π (0) + εδ with δ := qγ B + λ ′ − p ∈ P B . Then we have ( ˜ X ) p := E (cid:0) Z ˜ πdR ( ε ) ) p = (cid:0) ˆ X (0) (cid:1) p e p R (cid:0) ε ˆ π (0) λ ′ + ε ( δλ ′ − δ ) (cid:1) σ dt + pε R δσdB ˜ P (0) . Consequently, by replacing e x with its Taylor expansion and using that the involved ˜ P (0) -expectation is finite (here we use the integrability requirement 3.11), we find afunction C u ( ε ) ∈ O ( ε ) such that E [ U ( ˜ X T )] = u (0) E ˜ P (0) (cid:20) e p R T (cid:0) ε ˆ π (0) λ ′ + ε ( δλ ′ − δ ) (cid:1) σ dt + pε R T δσdB ˜ P (0) (cid:21) = u (0) (cid:16) pε ∆ (0) + 12 pε n p (∆ (0) ) + ∆ (00) o(cid:17) + C u ( ε ) . (4.15)We then turn to the dual problem. For the perturbed dual control ˜ ν := ˆ ν (0) − εpγ W ∈P W we have (cid:16) Z ( ε ) E ( − Z ˜ νdW ) (cid:17) − q = e q R ( λ + ελ ′ ) σdB + q R (ˆ ν (0) − εpγ W ) dW + q R (cid:0) ( λ + ελ ′ ) σ + | ˆ ν (0) − εqγ W | (cid:1) dt = ( Z (0) ˆ H (0) ) − q e εq R λ ′ σdB ˜ P (0) − εqp R γ W dW ˜ P (0) + q R (cid:0) ε ( λ ′ ) σ + ε p | γ W | +2 ελ ′ π (0) σ (cid:1) dt . Since ˜ ν is admissible in the ε -problem we find v ( ε ) ≤ q E "(cid:16) Z ( ε ) T E ( − Z T ˜ νdW ) (cid:17) − q = v (0) E ˜ P (0) (cid:20) e εq R T λ ′ σdB ˜ P (0) − εqp R T γ W dW ˜ P (0) + q R T (cid:0) ε ( λ ′ ) σ + ε p | γ W | +2 ελ ′ π (0) σ (cid:1) dt (cid:21) . Finiteness of v ( ε ) ensures that the ˜ P (0) -expectation appearing on the last line is alsofinite (recall that q < ). As in the primal problem, this allows us to replace e x withits Taylor series and in turn implies that we can find a function C v ( ε ) ∈ O ( ε ) suchthat v (0) E ˜ P (0) (cid:20) e εq R T λ ′ σdB ˜ P (0) − εqp R T γ W dW ˜ P (0) + q R T (cid:0) ε ( λ ′ ) σ + ε p | γ W | +2 ελ ′ π (0) σ (cid:1) dt (cid:21) = v (0) (cid:16) qε ∆ (0) + 12 qε n q (∆ (0) ) + ∆ (00) o(cid:17) + C v ( ε ) . N EXPANSION IN THE MODEL SPACE IN THE CONTEXT OF UTILITY MAXIMIZATION 17
By combining this estimate and (4.15) with the primal-dual relation (4.10) we find u (0) (cid:16) pε ∆ (0) + 12 pε n p (∆ (0) ) + ∆ (00) o(cid:17) + C u ( ε ) ≤ u ( ǫ ) = 1 p ( qv ( ǫ ) ) − p ≤ p (cid:16) qv (0) h qǫ ∆ (0) + 12 qǫ n q (∆ (0) ) + ∆ (00) o + C v ( ǫ ) i(cid:17) − p . (4.16)The function x → x − p is real analytic on (0 , ∞ ) . Therefore, the fact that C v ∈ O ( ε ) ensures that the last line of (4.16) agrees with the first line of (4.16) up to O ( ε ) -terms. This establishes (3.12). A similar argument produces (3.13). (cid:3)
5. E
XAMPLES
First examples.
We start this section with a short list of trivial and extremecases. They are not here to illustrate the power of our main results, but simply tohelp the reader understand them better. They also tell a similar, qualitative, story:loosely speaking, the improvement in the utility (on the log scale) is proportionalboth to the base market price of risk process and to the size of the deviation. Locally,around λ , the value function of the utility maximization problem - parametrized bythe market price of risk process ˜ λ - is well approximated by an exponential functionof the form u (˜ λ ) ≈ u ( λ ) e h ˜ λ − λ, ˆ π (0) i , where h ρ, π i = E ˜ P (0) [ Z T ρ t π t dt ] , (5.1)where u (˜ λ ) and u ( λ ) denote the values of the utility-maximization problems withmarket price of risk processes ˜ λ and λ , respectively. Example 5.1 (Small market price of risk) . Suppose that λ ≡ so that we can think of S ( ε ) as the stock price in a market with a “small” market price of risk. Since Z (0) ≡ ,it is clearly the dual optimizer at ε = 0 and we have ˆ π (0) ≡ . Consequently, underthe assumptions of Theorem 3.2, we have ˜ P (0) = P and ∆ (0) = E ˜ P (0) [ Z T λ ′ t dM t ] = 0 . It follows that u ( ε ) = u (0) + O ( ε ) and v ( ε ) = v (0) + O ( ε ) , and the effects of ελ ′ are felt only in the second order, regardless of the risk-aversioncoefficient p < . Example 5.2 (Deviations from the Black-Scholes model) . Suppose that M = B isan F -Brownian motion and that λ = 0 is a constant process (we also use λ for thevalue of the constant). In that case, it is classical that the dual minimizer in the basemarket is Z (0) = E ( − λB ) and, consequently, that d ˜ P (0) d P = E ( qλB ) . It follows that ∆ (0) = λ − p E ˜ P (0) [ Z T λ ′ t dt ] . As we will see below, this form is especially convenient for computations.
Example 5.3 (Uniform deviations) . Another special case where it is particularly easyto compute the (logarithmic derivative) ∆ (0) is when the perturbation λ ′ is a constantprocess (whose value is also denoted by λ ′ ). Indeed, in that case ∆ (0) = λ ′ E ˜ P (0) [ Z T ˆ π (0) t dt ] . (5.2)It is especially instructive to consider the case where the base model is Black andScholes’ model since everything becomes explicit: the optimal portfolio is given bythe Merton proportion ˆ π (0) t = λ/ (1 − p ) , and the the values u ( ε ) and v ( ε ) are givenby pu (0) = exp( qλ T ) and qv (0) = exp( q − p λ T ) . Using (5.2) or by performing a straightforward direct computation, we easily get p ∆ (0) = qλ ′ λT, making the approximation in (5.1) exact.5.2. The Kim-Omberg model.
The Kim-Omberg model (see [KO96]) is one of themost widely used models for the market price of risk process. Because the Kim-Omberg model allows for explicit expressions for all quantities involved in CRRAutility maximization it serves as an excellent test case for the practical implementa-tion of our main results.We assume that F is the augmentation of the filtration generated by two indepen-dent one dimensional Brownian motions B and W and define λ KO be the Ornstein-Uhlenbeck process dλ KO t := κ ( θ − λ KO t ) dt + βdB t + γdW t , λ KO ∈ R , (5.3)where κ, θ, β and γ are constants. We define the volatility M t := B t in what follows.The following result summarizes the main properties in [KO96]: Theorem 5.4 (Kim and Omberg 1996) . Let the market price of risk process be de-fined by (5.3) , M := B , and let p < . Then there exist continuously differentiable N EXPANSION IN THE MODEL SPACE IN THE CONTEXT OF UTILITY MAXIMIZATION 19 functions a, b, c : [0 , ∞ ) → R such that for t ∈ [0 , T ) we have − a ′ ( t ) = α b ( t ) + α c ( t ) − α b ( t ) , a ( T ) = 0 , − b ′ ( t ) = α b ( t ) + α c ( t ) − α b ( t ) c ( t ) , b ( T ) = 0 , − c ′ ( t ) = − q + 2 α c ( t ) − α c ( t ) , c ( T ) = 0 , where α := θκ , α := (1 + q ) β + γ , α := β + γ and α := qβ − κ .Furthermore, the primal value function reads u KO ( x ) = x p p e − a (0) − b (0) λ KO − c ( T )( λ KO ) , x > , (5.4) and the corresponding primal optimizer is given by ˆ π KO t = b ( t ) β + (cid:0) c ( t ) β − (cid:1) λ KO t p − , t ∈ [0 , T ] . (5.5)For p < , the above Riccati equation describing c has the “normal non-explodingsolution” as defined in the appendix of [KO96]. Therefore, all three functions a, b ,and c are bounded on any finite time-interval [0 , T ] of (0 , ∞ ) .To illustrate our approximation we think of the Kim-Omberg model as a perturbationof a base model. As base model we will consider the following model with “totally-unhedgable-coefficients” (see Example 7.4, p. 305, in [KS98]): dλ t := κ ( θ − λ t ) dt + γ dW t , λ := λ KO . (5.6)This way, λ KO = λ + ελ ′ , where ε = β and dλ ′ t := − κλ ′ t dt + dB t , λ ′ := 0 . (5.7)The following result provides closed-form expressions for our correction terms: Lemma 5.5.
Let ( λ, λ ′ ) be defined by (5.6) - (5.7) and let p < . For the ε = 0 modelthe primal and dual optimizers are given by ˆ π (0) t = λ t − p , ˆ ν (0) t = γ (cid:16) b ( t ) + c ( t ) λ t (cid:17) , t ∈ [0 , T ] . (5.8) Furthermore, the processes ( γ B , γ W ) appearing in the martingale representation (3.9) of Φ are given by γ Bt = p − ( C ( t ) + C ( t ) λ t ) , (5.9) γ Wt = γp − ( C ( t ) + 2 C ( t ) λ t + C ( t ) λ ′ t ) , (5.10) where the functions C , C , C , C and C in (5.9) - (5.10) satisfy the ODEs − C ′ ( t ) = ˜ b ( t ) C ( t ) + γ C ( t ) , C ( T ) = 0 , − C ′ ( t ) = ˜ b ( t ) C ( t ) − κ C ( t ) , C ( T ) = 0 , − C ′ ( t ) = q C ( t ) − ˜ c ( t ) C ( t ) + 2˜ b ( t ) C ( t ) , C ( T ) = 0 , − C ′ ( t ) = q C ( t ) − c ( t ) C ( t ) , C ( T ) = 0 , − C ′ ( t ) = − ( κ + ˜ c ( t )) C ( t ) − , C ( T ) = 0 , on [0 , T ) , with ( a, b, c ) as in Theorem 5.4 (with β := 0 ), ˜ b ( t ) := κθ − γ b ( t ) and ˜ c ( t ) := κ + γ c ( t ) . Furthermore, for the measure ˜ P (0) defined by (2.9) and for all T > we have ∆ (0) := E ˜ P (0) "Z T λ ′ s ˆ π (0) s ds = − − p (cid:16) C ( T ) + C ( T ) λ + C ( T ) λ (cid:17) , (5.11) Proof.
The first part follows from Theorem 5.4 applied to the case β := 0 . To findthe martingale representation (3.9) we define the function f ( t, x, λ ) := x p p e − a ( t ) − b ( t ) λ − c ( t ) λ , t ∈ [0 , T ] , x > , λ ∈ R , where the functions ( a, b, c ) are as in Theorem 5.4. The martingale properties of f ( t, ˆ X (0) t , λ t ) and ˆ X (0) t ˆ Y (0) t as well as the proportionality property ( ˆ X (0) T ) p ∝ ˆ X (0) T ˆ Y (0) T produce pf ( t, ˆ X (0) t , λ t ) = p E [ f ( T, ˆ X (0) T , λ T ) |F t ] ∝ E [ ˆ Y (0) T ˆ X (0) T |F t ] = ˆ X (0) t ˆ Y (0) t . By computing the dynamics of the left-hand-side we see from Girsanov’s Theoremthat the two processes dB ˜ P (0) t := − qλ t dt + dB t ,dW ˜ P (0) t := (cid:16) b ( t ) + c ( t ) λ t (cid:17) γdt + dW t , are independent Brownian motions under ˜ P (0) . These dynamics and Itô’s Lemmaensure that N t := Z t λ ′ s λ s ds − C ( t ) − C ( t ) λ ′ t − C ( t ) λ t − C ( t ) λ t − C ( t ) λ t λ ′ t , is a ˜ P (0) -local martingale.Because the processes ( λ, λ ′ ) remain Ornstein-Uhlenbeck processes under ˜ P (0) andthe functions C - C are bounded, N is indeed a ˜ P (0) -martingale. Furthermore,thanks to the zero terminal conditions imposed on C - C , we see that Φ = − p Z T λ t λ ′ t dt = − p N T = − p N + Z T γ Bt dB ˜ P (0) t + Z T γ Wt dW ˜ P (0) t , N EXPANSION IN THE MODEL SPACE IN THE CONTEXT OF UTILITY MAXIMIZATION 21 for ( γ B , γ W ) defined by (5.9)-(5.10). (cid:3) Exact computations.
The proof of Lemma 5.5 shows that ∆ (0) := E ˜ P (0) "Z T λ ′ s ˆ π (0) s ds = 1 p − (cid:16) C (0) + C (0) λ + C (0) λ (cid:17) . This relation, a similar one (whose exact form and the derivation we omit) for thesecond-order term ∆ (00) of (3.14), and the availability of the exact expression (5.4)for the value function u KO allow for an efficient numerical computation of the zeroth-, first-, and second-order approximation, and their comparison with the exact values.The model parameters used in the below Table 1 are the calibrated model parametersfor the market portfolio reported in Section 4.2 in [LM12] (we ignore the constantinterest rate and constant volatility used in Section 4 in [LM12]). Moreover, we usenegative values of ε because the empirical covariation between excess return andthe stock’s return is typically negative (see, e.g., the discussion in Section 4.2 in[LM12]).Instead of hard-to-interpret expected utility values, we report their certainty equiv-alents (i.e., their compositions with the function CE := U − ; see Remark 3.3(1)).We set δ (0) := pu (0) ∆ (0) and δ (00) := pu (0) (cid:0) ∆ (00) + p (∆ (0) ) (cid:1) . ε λ CE ( u (0) ) CE ( u (0) + εδ (0) ) CE ( u (0) + εδ (0) + ε δ (00) ) CE ( u ( ε ) ) -0.01 0.1 1.046 1.047 1.048 1.048- 0.05 0.1 1.046 1.054 1.081 1.084- 0.10 0.1 1.046 1.063 1.181 1.206- 0.01 0.5 1.614 1.647 1.648 1.649- 0.05 0.5 1.614 1.794 1.850 1.846- 0.10 0.5 1.614 2.020 2.339 2.272 Table 1.
Certainty equivalents for the zeroth-, first-, and second-order approxi-mations and the exact values in the Kim-Omberg model with β := ε and unitinitial wealth. The model parameters used are γ := 0 . , κ := 0 . , θ :=0 . , p := − , and T := 10 . Monte-Carlo-based computations.
One of the advantages of our approach isthat it lends itself easily to computational methods based on Monte-Carlo (MC) sim-ulation. For the Kim-Omberg model we use the standard explicit Euler scheme fromMC simulation to compute the involved quantities of interest. In other words, wedo not rely on the availability of exact expressions for the value functions or thecorrection terms ∆ (0) and ∆ (00) . For a portfolio π and the model-perturbation parameter ε , the constant CE ( ε ) ( π ) ∈ (0 , ∞ ) is uniquely defined by U (cid:16) CE ( ε ) ( π ) (cid:17) = E " U E (cid:0) Z T π t dR ( ε ) t (cid:1)! . (5.12)In other words, CE ( ε ) ( π ) is the dollar amount whose utility value matches that ofthe expected utility an investor would obtain in the ε -model who uses the strategy π . We remind the reader that ˆ π (0) denotes the optimizer in the base ( ε = 0 ) model, ˜ π ( ε ) is the second-order improvement (as in 3.15 above) of ˆ π (0) , and ˆ π ( ε ) is the exactoptimizer in the ε -model. Both quantities CE ( ε ) (ˆ π (0) ) and CE ( ε ) (˜ π ( ε ) ) serve as lowerbounds for the exact value CE ( u ( ε ) ) . The second one, which we also denote byLB := CE ( ε ) (˜ π ( ε ) ) , (5.13)is second-order optimal and appears in our simulations. To obtain a correspondingupper bound, we simulate the dynamics of the dual process, based on (4.10) and thesecond-order optimal dual control ˜ ν defined by (3.16). We defineUB := U − p E "(cid:16) Z ( ε ) T E ( − Z T ˜ ν u dW u ) (cid:17) − q − p . (5.14)To quantify the simulation errors, we report the -confidence intervals based onMC simulated values of CE ( ε ) (ˆ π (0) ) , LB, and UB in the below Table 2. The valueCE ( u ( ε ) ) , computed without MC simulation and included for comparison only, isexact to decimal places. ε λ CE ( ε ) (ˆ π (0) ) LB UB CE ( u ( ε ) ) -0.01 0.10 [1 . , . [1.048, 1.049] [1.048, 1.049] 1.048-0.05 0.10 [1 . , . [1.083, 1.084] [1.083, 1.085] 1.084-0.10 0.10 [1 . , . [1.200, 1.201] [1.204, 1.208] 1.206-0.01 0.50 [1 . , . [1.647, 1.653] [1.646, 1.657] 1.649-0.05 0.50 [1 . , . [1.844, 1.850] [1.843, 1.857] 1.846-0.10 0.50 [1 . , . [2.248, 2.256] [2.266, 2.286] 2.272 Table 2. -confidence intervals for certainty equivalents for the upper and lowerbounds as well as the base model optimizer ˆ π (0) for the Kim-Omberg model. Thetrue exact values for the ε -model are included in the last column for comparison.Except for the last column, the numbers are based on MC simulation using Eu-ler’s scheme with one million paths each with time-step size . . The modelparameters are the same as in Table 1. In Table 2 we note the significant difference between the performance of the base-model optimizer ˆ π (0) and its second-order improvement ˜ π ( ε ) ; especially for largervalues of ε . Furthermore, the lower and upper bounds appear to be quite tight. N EXPANSION IN THE MODEL SPACE IN THE CONTEXT OF UTILITY MAXIMIZATION 23
Extended affine models.
We turn to a class of models for which no closed-form expressions for the value functions u and v seem to be available. It constitutesthe main example of the class of so-called extended-affine specifications of the mar-ket price of risk models introduced by [CFK07].As in the Kim-Omberg model above we let the augmented filtration be generatedby two independent Brownian motions B and W . The central role is played by thefollowing Feller process FdF t := κ ( θ − F t ) dt + p F t (cid:0) βdB t + γdW t (cid:1) , F > , (5.15)where κ, θ, β and γ are strictly positive constants such that the (strict) Feller condition κθ > β + γ holds. This ensures, in particular, that F is strictly positive on [0 , T ] , almost surely. Unlike in the Kim-Omberg model, the appropriate volatilitynormalization turns out to be √ F t ; that is, we define M := Z · p F t dB t . (5.16)A particular extended affine specification of the market price of risk process consid-ered in [CFK07] is given by λ CFK t := εF t + 1 , (5.17)where ε is a (positive or negative) constant. Unless ε = 0 , there is currently no knownclosed-form solution to the corresponding optimal investment problem (Theorem 4.5in [GR15] expresses the corresponding value function as an infinite sum of weightedgeneralized Laguerre polynomials). However, for ε = 0 , the resulting model iscovered by the analysis in [Kra05]. Therefore, we choose the constant market priceof risk process λ t := 1 for the base model whereas we define the perturbation process λ ′ by λ ′ t := 1 F t . (5.18) Theorem 5.6 (Kraft 2005) . For p < there exist continuously differentiable func-tions a, b : [0 , T ) → R such that − a ′ ( t ) = α b ( t ) , a ( T ) = 0 , − b ′ ( t ) = α b ( t ) − α b ( t ) − q, b ( T ) = 0 , where α := θκ , α := (1 + q ) β + γ , and α := qβ − κ . The value function ofthe utility-maximization problem with λ := 1 and M as in (5.16) is given by u (0) ( x ) = x p p e − a (0) − b (0) F , x > . The corresponding primal and dual optimizers are given by ˆ π (0) t = b ( t ) β − p − , ˆ ν (0) t = b ( t ) γ p F t , t ∈ [0 , T ] . (5.19)To check the conditions of our main theorems, we use the explicit expression in[HK08], Theorem 3.1, for the Laplace transform L ( a , a ) := E [exp( a Q + a Λ)] , Q := Z T F s ds, Λ := Z T F s ds. It is shown in [HK08] that L is finite in some neighborhood of under the strict Fellercondition κθ > β + γ . This implies that both Λ and Q have a finite exponentialmoment. In particular, Hölder’s inequality with exponents − /q and (1 − p ) impliesthat E ˜ P (0) [Λ] = 1 qv (0) E [( ˆ Y (0) T ) − q Λ] ≤ qv (0) E [Λ − p ] − p < ∞ . Thanks to the deterministic behavior of ˆ π (0) in (5.19), the martingale representation(3.9) of Φ holds with γ B = γ W = 0 . Consequently, we have Φ := Z T ˆ π (0) s ds = ∆ (0) , ∆ (00) := − p E ˜ P (0) [Λ] . To verify that (3.11) holds, we can use Hölder’s inequality (twice) with exponents − /q and (1 − p ) to see E ˜ P (0) " e − ε p (1 − p )2 Λ+ qε R T √ Ft dB ˜ P (0) t ≤ E ˜ P (0) h e − ε ( p + q )Λ i − p ≤ qv (0) E h e − ε (1 − p )( p + q )Λ i − p )2 , which is finite for ε > small enough. This allows Theorem 3.4 to be invoked for ε > small enough. The second-order optimal controls (˜ π, ˜ ν ) are then well definedby (3.15) and (3.16), and read ˜ π := ˆ π (0) + ε λ ′ − p , ˜ ν := ˆ ν (0) . (5.20)Table 3 is the analogue of Table 2 for the extended affine model with parameterstaken from Figure 4 in Section 3.3 in [LM12]. The methodology and the simulatedquantities are the same as for Table 2. N EXPANSION IN THE MODEL SPACE IN THE CONTEXT OF UTILITY MAXIMIZATION 25 ε F CE ( ε ) (ˆ π (0) ) LB UB .
10 0 .
01 [1 . , . . , . . , . .
05 0 .
01 [1 . , . . , . . , . .
01 0 .
01 [1 . , . . , . . , . .
10 0 .
05 [1 . , . . , . . , . .
05 0 .
05 [1 . , . . , . . , . .
01 0 .
05 [1 . , . . , . . , . Table 3. -confidence intervals for certainty equivalents for the upper and lowerbounds as well as the base model optimizer ˆ π (0) for the extended affine model. Theparameter values are κ := 5 , θ := 0 . , β := − . , γ := 0 . , p := − ,and T := 10 . The numbers are based on MC simulation using Euler’s scheme withone million paths each with time-step size . . The zeroth order approximation CE (0) (ˆ π (0) ) produces the certainty equivalent valuesCE (0) (ˆ π (0) ) = 1 .
043 ( F = 0 . , and CE (0) (ˆ π (0) ) = 1 .
045 ( F = 0 . . Perhaps even more than in the Kim-Omberg model, the numbers in Table 3 aboveillustrate the superiority of the second-order approximations (columns 4 and 5) overits first-order version (column 3) as well as the zeroth order values reported above.Again, the bounds in Table 3 appear quite tight when compared to the first-orderapproximations for moderate values of ε .R EFERENCES [AMKS15] Albert Altarovici, Johannes Muhle-Karbe, and H. Mete Soner,
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EPARTMENT OF M ATHEMATICAL S CIENCES , C
ARNEGIE M ELLON U NIVER - SITY
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EPARTMENT OF M ATHEMATICS , U
NIVERSITY OF C ONNECTICUT
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EPARTMENT OF M ATHEMATICS , U
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