A Primer on Portfolio Choice with Small Transaction Costs
AA Primer on Portfolio Choice with Small Transaction Costs ∗ Johannes Muhle-Karbe † Max Reppen ‡ H. Mete Soner § May 23, 2017
Abstract
This survey is an introduction to asymptotic methods for portfolio-choice problems with smalltransaction costs. We outline how to derive the corresponding dynamic programming equationsand simplify them in the small-cost limit. This allows to obtain explicit solutions in a widerange of settings, which we illustrate for a model with mean-reverting expected returns andproportional transaction costs. For even more complex models, we present a policy iterationscheme that allows to compute the solution numerically.
Starting with the work of Constantinides and Magill [64], Constantinides [23], Dumas and Luciano[34], Davis and Norman [29], and Shreve and Soner [81], models with transaction costs have beensubjected to intensive research. For example, much effort has been devoted to understandingliquidity premia in asset pricing [23, 53, 63, 25] or how transaction costs shape the trading volumein financial markets [80, 62, 42]. On a more practical level, transaction costs play a crucial role inthe design and implementation of trading strategies in the asset management industry, cf., e.g.,[45, 66, 30, 65].However, the quantitative analysis of models with trading costs is quite difficult. Unlike infrictionless models, the position in each asset becomes a state variable because it can no longer beadjusted immediately and for free. As a consequence, explicit solutions are no longer available evenin the simplest models with constant market and preference parameters and an infinite planninghorizon [34, 29, 83]. This is only compounded in more complex models with random and time-varyinginvestment opportunities. However, transaction costs become crucially important precisely in suchsettings where, for example, prices exhibit momentum or mean reversion [66, 65, 40, 41, 22, 30],switch between different regimes [53], or investors are exposed to idiosyncratic endowment shocks[62, 63]. Then, investors can no longer “accommodate large transaction costs by drastically reducingthe frequency and volume of trade” [23] as in simple models where portfolio rebalancing is the onlymotive to trade. Instead, striking the right balance between adjustments to optimize performanceand the induced implementation costs then becomes a central issue. ∗ The second and third authors were partly supported by the ETH Foundation, the Swiss Finance Institute, andSwiss National Foundation grant SNF 200021 153555. Parts of this paper were written while the first author wasvisiting the Forschungsinstitut Mathematik at ETH Z¨urich. † University of Michigan, Department of Mathematics, 530 Church Street, Ann Arbor, MI 48109, USA, email [email protected] . ‡ ETH Z¨urich, Department of Mathematics, R¨amistrasse 101, 8092 Z¨urich, Switzerland, email [email protected] . § ETH Z¨urich, Department of Mathematics, R¨amistrasse 101, 8092 Z¨urich, Switzerland, email [email protected] . a r X i v : . [ q -f i n . P M ] M a y o obtain tractable results in complex models with transaction costs, it is often very useful totake an asymptotic perspective and view them as small perturbations of a frictionless benchmarkmodel. The goal then is to obtain explicit asymptotic formulas for optimal trading policies and theassociated welfare effect of small transaction costs. Results of this kind were first obtained in simpleconcrete models that can be solved explicitly in the frictionless case [31, 81, 85, 52, 62, 12, 10, 42].In the last couple of years, there has been a lot of progress in extending these sensitivity analysesto much more general settings [11, 14, 82, 74, 2, 71, 65, 55, 54, 78, 17, 18, 1, 38]. These resultsare obtained using a range of different methods, ranging from analytic studies of the dynamicprogramming equation [82, 74, 2, 71, 65] to convex duality arguments [1], and weak-convergencetechniques [17, 18].In this survey, we review the homogenization approach put forward in [82] for models withproportional transaction costs. This approach based on partial differential equations is very flexibleand readily adapts to many variations of the model, e.g., different cost structures [2, 3, 71] orpreferences [14, 67]. Hence, similarly to the classical dynamic approach to frictionless controlproblems (see, e.g., [39] for an overview), this method has the potential to be a key tool for theanalysis of a wide range of complex models. The present review is written as a user’s guide forthe application of this method. We explain in detail both the basic underlying ideas and each stepof their application to a concrete problem. The goal is to provide a blueprint that will allow thereaders to apply the approach to a wide range of related problems.To illustrate the effects of transaction costs on active investment in a concrete example, we alsoprovide a detailed discussion of a model with mean-reverting returns. Starting from the frictionlesssolution of [57], we show how both the leading-order optimal trading policy and the correspondingperformance loss due to the trading friction can be computed explicitly. We show that – like formodels with constant investment opportunities [23] – this leads to wide no-trade regions and asevere reduction of trading volume. However, the corresponding welfare losses are greatly amplified– the opportunity cost of not being able to “time the market” is substantial. This underlines theimportance of correcting the performance of active investment strategies for trading costs (compare[6] and the references therein), for which the tractable asymptotic formulas reviewed here provide aconvenient analytical tool.This review is organized as follows. We first introduce our continuous-time model without andwith transaction costs in Section 2. Then, we derive the corresponding frictionless and frictionaldynamic programming equations. These partial differential equations for the value functions ofthe problems at hand are the starting point of the subsequent analysis. Afterwards, in Section 4,we outline the homogenization approach, and discuss in detail how to apply it to models withproportional transaction costs. This method allows to reduce the complexity of the problem athand by reducing the number of state variables, simplifying the underlying state dynamics, andpostponing finite time horizons to infinity. In Section 5, we explain how this allows to obtainexplicit solutions in the model of Kim and Omberg [57], where asset prices exhibit momentum.Subsequently, Section 6 provides references to several extensions of the homogenization results tomore general asset dynamics, preferences, and cost structures. Finally, in Section 7, we discuss anumerical scheme that allows computation of the solution to the simpler “homogenized” problemnumerically using a policy iteration algorithm. To focus on the main ideas and computational issues,mathematical formalism is treated liberally throughout this survey. Rigorous verification theoremsfor the results presented here can be found in [82, 78, 71, 2, 74, 1, 38, 17, 18, 68]. “Martingale methods” based on the duality theory introduced by [24] allow to go beyond the Markovian paradigm[55, 54, 1]. However, they are more difficult to adapt to new models and not applicable if the problem at hand is notconvex, e.g., due to the presence of fixed trading costs. otation We write D x ϕ (( t, x, y, f ) , D xf ϕ (( t, x, y, f ), etc. for the partial derivatives of a multi-variate function ϕ ( t, x, y, f ). When there is no confusion, we also use the more compact notation ϕ x , ϕ xf , etc. As is customary in asymptotic analysis, O ( δ ) denotes any function satisfying (cid:12)(cid:12) O ( δ ) (cid:12)(cid:12) ≤ Cδ for a constant C > δ ∈ [0 , ξ and a time point t ≥ E t [ ξ ] denotes the expectation of ξ conditional on the information up to time t . We consider a financial market with one safe and one risky asset with dynamics modulated by ageneral factor process. More precisely, the safe asset followsd B s B s = r ( F s )d s, and the risky dynamics are d S s S s = ( r ( F s ) + µ S ( F s ))d s + σ S ( F s ) d W Ss . (2.1)Here, ( W Ss ) s ∈ [0 ,T ] is a standard Brownian motion; the safe rate r ( f ), the expected excess return µ ( f ), and the volatility σ ( f ) are sufficiently smooth deterministic functions of the factor process( F s ) s ∈ [0 ,T ] . The latter follows an autonomous diffusion:d F s = µ F ( F s ) d s + σ F ( F s ) d W Fs . (2.2)Here, ( W Fs ) s ∈ [0 ,T ] is another standard Brownian motion that has constant correlation ρ ∈ [ − ,
1] withthe Brownian motion ( W Ss ) s ∈ [0 ,T ] driving the risky returns. Both µ F ( f ) and σ F ( f ) are sufficientlyregular deterministic functions. Examples.
All tractable models from the literature fit into this framework. Examples are:(i)
Black–Scholes Model : the standard example for the asset dynamics is the Black–Scholes model,where the safe rate, the expected risky return, and the volatility are all constants: r ( f ) ≡ r , µ S ( f ) ≡ µ S , and σ S ( f ) ≡ σ S .(ii) Kim–Omberg Model : to study the effects of transaction costs in a model where investmentopportunities vary randomly over time, we consider the model of Kim and Omberg [57]. Thismeans the safe rate and volatility remain constant ( r ( f ) ≡ r , σ S ( f ) ≡ σ S ), but the expectedexcess returns follow an Ornstein–Uhlenbeck process: µ S ( f ) = f andd F s = κ ( ¯ F − F s )d s + σ F d W Fs , (2.3)for constants κ , ¯ F and σ F describing the mean-reversion speed, the mean-reversion level, andthe volatility of the expected excess return. This is a standard model for the “predictability of asset returns”, which has been discussed extensively in theempirical literature [84, 21]. The importance of transaction costs in such environments that require to “time themarket” is evident and discussed in [30, 66, 65, 40, 71, 41, 22], for example. The framework in [57] also allow other choices of µ S and σ S for which the Sharpe ratio µ S /σ S remains anOrnstein–Uhlenbeck process, because all of these models span the same frictionless payoff spaces. As this invariancebreaks down with transaction costs, we focus on the present specification here. Heston–type models : in another widely used class of models, the volatility is assumed tobe a mean-reverting process. For example, Heston [49] proposes a constant interest rate r and excess return µ S , as well as a stochastic volatility σ S ( f ) = √ f where the factor F is asquare root process. Liu [61] instead sets µ S ( f ) = αf for some constant α , retaining the otherspecifications of Heston. Chacko and Viceira [19] keep Heston’s constant r and µ S , but theirvolatility is σ S ( f ) = (cid:112) /f . We now turn to trading and optimization in the above financial market. For concreteness, wefocus on a specific portfolio choice problem where consumption only takes place at the terminaltime. Extensions to more general settings do not pose any essential difficulties and are discussed inSection 6. We first briefly recall the frictionless case and then turn to models with transaction costs.
Frictionless Case
Starting with an initial endowment of x dollars in the safe account and arisky position worth y dollars, an agent can trade the safe and the risky asset continuously on[0 , T ]. Without trading costs, positions can be changed freely over time, so that the amount ofmoney Y t invested in the risky asset is therefore a suitable control variable. The wealth dynamicsgenerated by such a strategy is simply obtained by weighting the safe and risky returns accordingto the corresponding investments: d Z t,z,fs = Y s (cid:104) ( r ( F t,fs ) + µ S ( F t,fs ))d s + σ S ( F t,fs ) d W Ss (cid:105) + ( Z Y,x + y s − Y s ) r ( F t,fs )d s, s ≥ t,F t,ft = f, Z t,z,ft = z = x + y . (2.4)Without trading costs, the wealth z is the only state variable we need to keep track of, apartfrom to the factor f . In contrast, the decomposition of z into the risky position y and the safeposition x = z − y is irrelevant because it can be changed instantly and without cost, by updating thecontrol. If agents choose their trading strategies to maximize expected utility from terminal wealthat time T for some utility function U , we therefore expect the value function to be a deterministicfunction of the current time t , the current wealth z , and the current value f of the factor only: v ( t, z, f ) = sup ( Y s ) s ∈ [ t,T ] E t (cid:20) U (cid:16) Z t,z,fT (cid:17)(cid:21) . (2.5) Proportional Transaction Costs
Now, suppose that trades incur a cost λ proportional to thevalue traded. Then, the decomposition of the total wealth evidently matters, because this ratiocan no longer be adjusted for free. As a consequence, we need to keep track of the evolution ofthe safe and risky positions separately. These quantities now both become state variables thatcan only be adjusted gradually. To wit, agents now choose nondecreasing adapted process L and M that describe the cumulative transfers from the safe to the risky account and vice versa. Thecorresponding dynamics of the safe account are d X t,x,y,fs = r ( F t,fs ) X t,x,y,fs d s − (1 + λ ) d L s + (1 − λ ) d M s , s ≥ t,X t,x,y,ft = x. (2.6) The superscripts in our notation refer to the initial conditions Z t,z,ft = z and F t,ft = f . This means that transaction costs are always deducted from the safe account, both for purchases and sales of therisky asset. Y t,x,y,fs = Y t,x,y,fs (cid:104) ( r ( F t,fs ) + µ S ( F t,fs ))d s + σ S ( F t,fs ) d W Ss (cid:105) + d L s − d M s , s ≥ t,Y t,x,y,ft = y. (2.7)If agents maximize expected utility form terminal paper wealth, the corresponding frictional valuefunction will in turn depend on the current values of the safe and risky account in addition to timeand the current value of the factor process: v λ ( t, x, y, f ) = sup ( L s ,M s ) s ∈ [ t,T ] E t (cid:20) U (cid:16) X t,x,y,fT + Y t,x,y,fT (cid:17)(cid:21) . (2.8) Fixed and Proportional Transaction Costs
In this survey, we mostly focus on the abovemodel with proportional transaction costs in order to describe the main ideas of the homogenizationapproach for small transaction costs most clearly. However, the methods outlined here readilyadapt to more general settings [71, 2, 3]. For example, in a model with proportional costs λ p andadditional fixed costs λ f per trade, the cash dynamics (2.6) change to X t,x,y,fs = x + (cid:90) st r ( F t,fτ ) X t,x,y,fτ d τ − L s + M s − λ p ( L s + M s ) − λ f J s ( L, M ) , s ≥ t, where J s ( L, M ) is the total number of jumps of L and M up to time s . Since one pays a fixed,non-zero amount for each jump, J s ( L, M ) must be finite for any admissible strategy, unlike forproportional costs. Nevertheless, the value function depends on the same variables as in (2.8).
The key concept for studying the value functions (2.5), (2.8) and the corresponding optimal tradingstrategies is to describe them in terms of a partial differential equation derived from the dynamicprogramming principle of stochastic optimal control. Loosely speaking, the latter states that if wehave already determined the optimal policy on [ t + d t, T ], then fixing this policy and optimizingover the choice at time t leads to the same solution as optimizing over the entire interval [ t, T ]. Indiscrete time, this means that the optimal policy can be computed by backward induction; in thecontinuous-time limit, a partial differential equation is obtained. Let us illustrate this idea by first briefly recalling the frictionless case. Then, the dynamic program-ming principle suggests that v ( t, z, f ) = sup Y t E t (cid:20) v (cid:16) t + d t, z + d Z t,z,ft , f + d F t,ft (cid:17)(cid:21) . Now, apply Itˆo’s formula, insert the state dynamics (2.1–2.2), and cancel the stochastic integrals(because they are martingales and therefore have zero expectation if the integrands are sufficiently One could also consider expected utility from liquidation wealth X t,x,y,fT + (1 − λ ) Y t,x,y,fT , but this does not affectthe asymptotic results at the leading order. Here, we only provide a heuristic derivation of the dynamic programming equations. For a mathematically rigoroustreatment we refer the reader to the monograph [39]; more recent results can be found in [16, 15, 2, 3, 35]. t and send d t to zero. Dropping the arguments to ease notation,this leads to the following equation:0 = v t + µ F v f + 12 σ F v ff + sup Y t (cid:110) (cid:0) Y t ( r + µ S ) + ( z − Y t ) r (cid:1) v z + 12 Y t σ S v zz + Y t ρσ S σ F v zf (cid:111) . (3.1)The above equation incorporates the influence of the momentary choice for the evolution of thesystem into the value function. Hence, the problem at time t is reduced to a simple optimizationover instantaneous controls. In this particular case, we obtain a pointwise quadratic problem in Y t ,whose solution is given by Y t = θ ( t, z, f ) := − µ S ( f ) σ S ( f ) v z ( t, z, f ) v zz ( t, z, f ) − ρσ F ( f ) σ S ( f ) v zf ( t, z, f ) v zz ( t, z, f ) . (3.2)Plugging this expression back into the dynamic programming equation (3.1) yields A v := v t + µ F v f + 12 σ F v ff + rzv z + µ S θv z + 12 σ S θ v zz + θσ S σ F ρv zf = 0 . (3.3)This is the dynamic programming equation for the value function v . As the optimal portfolio (3.2)depends on v and its derivatives, this partial differential equation is fully nonlinear. Nevertheless,it can be solved in closed form for standard utility functions of power or exponential form and anumber of particular asset dynamics [70, 57, 19, 61]. This in turn leads to explicit expressions forthe optimal trading policy (3.2). This is illustrated in Section 5 for the model with mean-revertingreturns first studied by [57]; the closed-form solution for the corresponding dynamic programmingequation (3.3) is provided in Appendix A. Let us now pass to the value function (2.8) with proportional transaction costs. In this case, thevalue function depends not only on the total wealth z but rather on the full portfolio decomposition( x, y ), where x is the cash position and y is the dollar amount invested in the risky asset. Thedynamic programming principle then takes the following form: v λ ( t, x, y, f ) = sup d M t , d L t E t (cid:20) v λ (cid:16) t + d t, x + d X t,x,y,ft , y + d Y t,x,y,ft , f + d F t,ft (cid:17)(cid:21) . Like in the frictionless case, apply Itˆo’s formula, insert the state dynamics (2.2, 2.6–2.7), cancelthe stochastic integrals (because they are martingales and therefore have zero expectation if theintegrands are sufficiently integrable), divide by d t , and send d t to zero. This leads to0 = v λt + µ F v λf + y ( r + µ S ) v λy + xrv λx + σ F v λff + σ S y v λyy + ρσ S σ F yv λyf + sup d M t , d L t (cid:40) (cid:16) v λy − (1 + λ ) v λx (cid:17) d L t d t + (cid:16) (1 − λ ) v λx − v λy (cid:17) d M t d t (cid:41) . (3.4)Hence, if the marginal utility of increasing the risky position is neither too high nor too low at apoint ( t, x, y, f ), (1 + λ ) v λx ( t, x, y, f ) > v λy ( t, x, y, f ) > (1 − λ ) v λx ( t, x, y, f ) , M t = d L t = 0). We call the set of all such positions( t, x, y, f ) the no-trade region . In view of the dynamic programming equation (3.4), it follows thatthe following standard linear PDE is satisfied inside it:0 = v t + µ F v f + y ( r + µ S ) v y + xrv λx + σ F v ff + σ S y v yy + ρσ S σ F yv yf . (3.5)Outside the no-trade region, (3.4) shows that the right-hand side of this equation is less than orequal to zero, since one can always let the portfolio evolve uncontrolled. Moreover, if the marginalutility of increasing the risky position is high enough, v y ≥ (1 + λ ) v x , then the optimal control is to“buy risky shares at an infinite rate” (d L t / d t = ∞ ). Conversely, it is optimal to “sell risky shares atan infinite rate” (d M t / d t = ∞ ) whenever this marginal utility is low enough, v y ≤ (1 − λ ) v x . Thismeans that it is optimal to perform the minimal amount of trading that keeps the portfolio withinthe no-trade region, cf. the left panel of Figure 1 for an illustration. Moreover, it follows that inorder to satisfy the dynamic programming equation (3.4) with equality, we must have v y = (1 + λ ) v x in the “buying region” and v y = (1 − λ ) v x in the “selling region”. Together with (3.5), this leads tothe following “variational inequality” for the value function (2.8):0 = max (cid:110) v λt + µ F v λf + y ( r + µ S ) v λy + xrv λx + σ F v λff + σ S y v λyy + ρσ S σ F yv λyf ;(1 + λ ) v λx − v λy ; v λy − (1 − λ ) v λx (cid:111) . (3.6)Explicit solutions for this equation are not available even in the simplest models. The key difficultyis that the transaction costs increase the number of state variables by one and introduce a freeboundary—the no-trade region is unknown and needs to be determined as part of the solution. Inthe Black–Scholes model, this requires to solve for a time-dependent smooth curve [26]. In theKim–Omberg model, the tractability issue is further compounded, because the trading boundariesthen additionally depend on the mean-reverting expected return process. This lack of analyticaltractability can be overcome by passing to the small-cost limit, which we discuss in Section 4.
Before turning to the small-cost asymptotics, let us briefly sketch how to adapt the above derivationsfor models with additional fixed costs. Since only finitely many trades ∆ L t > M t > v λ ( t, x, y, f ) ≥ v λ ( t, x − ∆ L t + ∆ M t − λ p (∆ L t + ∆ M t ) − λ f , y + ∆ L t − ∆ M t , f ) . With H ( t, x, y, v λ ( t, · , · , f )) := sup (cid:96),m ≥ (cid:110) v λ (cid:0) t, x − (cid:96) + m − λ p ( (cid:96) + m ) − λ f , y + (cid:96) − m, f (cid:1)(cid:111) , it follows that the frictional value function v λ satisfies the following inequality: v λ ( t, x, y, f ) ≥ H ( t, x, y, v λ ( t, · , · , f )) . The states for which v λ > H again form a no-trade region, where the same argument as inSection 3.2 shows that the frictional value function solves the linear PDE (3.5). Combining this In the most tractable infinite horizon models, the trading boundaries are constant and can be characterized by ascalar nonlinear equation [34, 83, 44, 42]; see [46] for a survey of this literature. Figure 1: Simulation of the frictionless portfolio weight π t = Y t / ( X t + Y t ) and the boundaries ofthe corresponding (asymptotic) no-trade regions in the Kim–Omberg model, in solid lines. Thedashed lines depict the paths of optimal frictional portfolios for proportional transaction costs of1% (left panel) and for fixed costs of $1 for an investor with an initial wealth of $5000 (right panel).The parameters are γ = 3, r = 0 . σ S = 0 . κ = 0 . F = 0 .
041 and σ F = 0 . v λ = H which is binding outside the no-trade region, the followingvariational inequality for the frictional value function is obtained:0 = max (cid:110) v λt + µ F v λf + y ( r + µ S ) v λy + xrv λx + σ F v λff + σ S y v λyy + ρσ S σ F yv λyf ; H ( t, x, y, v λ ( t, · , · , f )) − v λ ( t, x, y, f ) (cid:111) . With only fixed costs ( λ p = 0), all trades are penalized equally, so that the corresponding optimalpolicy rebalances all the way back to the frictionless target, as shown in the right panel of Figure 1.Models with both fixed and proportional costs are intermediate between this regime and that ofproportional costs from Section 3.2 in the sense that the optimal policy is to trade to a pointin-between the boundary of the no-trade region and the frictionless target portfolio, compare [58, 3]. We now turn to the asymptotic analysis of models with small transaction costs. To ease theexposition, we focus a single risky asset traded with purely proportional costs. A multi-asset modelwith proportional costs is discussed in [74]; fixed and proportional costs are treated in [2, 3]; a studyof quadratic trading costs can be found in [71].As already pointed out above, explicit solutions for portfolio choice problems with transactioncosts are not available even in those settings that can be solved in closed form in the frictionless case.To overcome this lack of tractability, it is natural to study small transaction costs as a perturbationof the frictionless benchmark. The goal is to “reveal the salient features of the problem whileremaining a good approximation to the full but more complicated model” [85].The method developed in [82] that we present here has its roots in the homogenization literature[73, 59]. This class of problems contains an ergodic fast variable and the theory studies thelimit problem as this variable oscillates more and more quickly. This leads to a “homogenizedequation”. Interestingly, the latter is not simply the ergodic average of the original one. Instead, it8s obtained by a non-trivial coupling with a so-called corrector equation (sometimes also called the cell equation ). Models with small transaction costs are only loosely in analogy with these problemsas the dependence on the portfolio composition disappears in the limit and it does not immediatelyoffer a fast variable. However, [82] observed that—after a suitable rescaling—the deviation ofthe portfolio from the target position, ( ξ in (4.1) below) plays the same role as the fast variable.This observation allows to employ the similar formal calculations as in homogenization theory tocharacterize the asymptotic solution. Moreover, the powerful perturbed test function method ofEvans [37, 36] can be modified to obtain rigorous convergence results [82, 74, 71, 2, 3, 14]. The starting point for the asymptotic analysis is an appropriate ansatz for the value function v λ with small transaction costs λ . To this end, a key observation of [52, 77] is that two competingeffects needs to be balanced here. On the one hand, a narrower no-trade region leads to morefrequent trading, and whence also higher direct transaction costs. On the other hand, a widerno-trade region leads to larger oscillations around the frictionless target portfolio and thus higherindirect losses due to displacement from the optimal risk-return trade-off.With proportional transaction costs, the amount of trading required to remain inside a smallno-trade region with width ∆ scales with the inverse of ∆. Locally around the frictionless optimum,the first-order condition implies that value function is quadratic, so that the displacement lossshould scale with the squared width ∆ of the no-trade region. In summary, this suggests that ∆needs to minimize a total loss of the form C ∆ + C λ ∆ . As a consequence, the optimal no-trade region should be of order O ( λ / ) with a correspondingminimal utility loss of order O ( λ / ). As discussed in Section 3.2, the frictional value function v λ depends on time t , the current value f of the factor process, as well as the current safe and risky positions x and y . As the transaction cost λ tends to zero, both the risky and safe position converge to their frictionless counterparts. In orderto obtain nontrivial limiting quantities for the asymptotic analysis, we therefore re-parametrize themodel by switching from x and y to the frictionless state variable z = x + y and the normalized deviation ξ = y − θ ( t, z, f ) λ / (4.1)of the risky position from its frictionless target (3.2). In view of the discussion in Section 4.1,we then expect (4.1) to converge to a finite limit as λ →
0. To avoid fractional powers in the This is a property of reflected Brownian motion and the local time it accumulates at the boundaries. For fixed costs, the argument is similar: A trade is initiated whenever the portfolio reaches the boundary of theno-trade region. At such a point, the portfolio is rebalanced to the frictionless portfolio. The time it takes Brownianmotion to reach the boundary of the no-trade region is proportional to ∆ , so the number of trades per unit of time isproportional to 1 / ∆ . The corresponding utility loss should therefore be of the form C ∆ + C λ/ ∆ , which has aminimizer of order O ( λ / ) and a minimal value of order O ( λ / ); cf. [2] for more details. The definition of this fast variable depends on the scaling appropriate for the problem at hand. For example, forproblems with fixed rather than proportional costs, one needs to divide by λ / , cf. [2]. (cid:15) = λ / . With this notation and the above change of variables, the frictional value function can be written as v λ ( t, x, y, f ) =: v (cid:15) ( t, z, ξ, f ) . The considerations in Section 4.1 suggest that the leading-order term in the asymptotic expansionof the value function is of order O ( (cid:15) ). Since the impact of a single trade is of higher order O ( (cid:15) ),this term should not depend on the deviation (4.1) and should thus be a function (cid:15) u ( t, z, f ) ofthe frictionless state variables only. However, the deviation of the frictionless optimizer (i.e., theposition in the no-trade region) evidently plays a key role in determining the optimal trading policy(i.e., when to start trading). In order to take this into account and motivated by the homogenizationliterature, we introduce a second term (cid:15) w ( t, z, ξ, f ) in the asymptotic expansion. It is negligibleat the leading order in the value expansion, but via (4.1) its derivatives play a crucial role indetermining the optimal trading policy from the variational inequality (3.6).In summary, our ansatz for the asymptotic value function reads as follows: v (cid:15) ( t, z, ξ, f ) = v ( t, z, f ) − (cid:15) u ( t, z, f ) − (cid:15) w ( t, z, ξ, f ) + O ( (cid:15) ) . (4.2)The goal now is to determine u and w by plugging this expansion into the dynamic programmingequation (3.6) and matching terms sorted in powers of the asymptotic parameter λ = (cid:15) . In order to recast the variational inequality (3.6) in terms of the new variables z and ξ instead of x and y , we need to rewrite the corresponding differential operators. For an arbitrary function Ψ of( t, x, y, f ), or equivalently ( t, z, ξ, f ), we have D x Ψ( t, x, y, f ) = D z Ψ( t, z, ξ, f ) − θ z ( t, z, f ) (cid:15) D ξ Ψ( t, z, ξ, f ) ,D y Ψ( t, x, y, f ) = D z Ψ( t, z, ξ, f ) + 1 − θ z ( t, z, f ) (cid:15) D ξ Ψ( t, z, ξ, f ) , and in turn D yy Ψ( t, x, y, f ) = D z (cid:18) D z Ψ( t, z, ξ, f ) + 1 − θ z ( t, z, f ) (cid:15) D ξ Ψ( t, z, ξ, f ) (cid:19) + 1 − θ z ( t, z, f ) (cid:15) D ξ (cid:18) D z Ψ( t, z, ξ, f ) + 1 − θ z ( t, z, f ) (cid:15) D ξ Ψ( t, z, ξ, f ) (cid:19) . Likewise, D f Ψ( t, x, y, f ) = D f Ψ( t, z, ξ, f ) − θ f (cid:15) D ξ Ψ( t, z, ξ, f ) ,D ff Ψ( t, x, y, f ) = D f (cid:18) D f Ψ( t, z, ξ, f ) − θ f ( t, z, f ) (cid:15) D ξ Ψ( t, z, ξ, f ) (cid:19) − θ f ( t, z, f ) (cid:15) D ξ (cid:18) D f Ψ( t, z, ξ, f ) − θ f ( t, z, f ) (cid:15) D ξ Ψ( t, z, ξ, f ) (cid:19) ,D yf Ψ( t, x, y, f ) = D z (cid:18) D f Ψ( t, z, ξ, f ) − θ f ( t, z, f ) (cid:15) D ξ Ψ( t, z, ξ, f ) (cid:19) + 1 − θ z ( t, z, f ) (cid:15) D ξ (cid:18) D f Ψ( t, z, ξ, f ) − θ f ( t, z, f ) (cid:15) D ξ Ψ( t, z, ξ, f ) (cid:19) . O ( (cid:15) − ) arising in some of these expressions. These are the reason why (cid:15) w ( t, z, ξ, f ) cannot be absorbed in O ( (cid:15) ) in (4.2) but needs to be treated separately.With the above expressions, the ansatz (4.2) implies D x v (cid:15) = D z v − (cid:15) D z u + (cid:15) θ z w ξ + O ( (cid:15) ) = D z v − (cid:15) D z u + O ( (cid:15) ) ,D y v (cid:15) = D z v − (cid:15) D z u − (cid:15) (1 − θ z ) w ξ + O ( (cid:15) ) = D z v − (cid:15) D z u + O ( (cid:15) ) ,D yy v (cid:15) = D zz v − (cid:15) (cid:16) D z zu + (1 − θ z ) D ξξ w (cid:17) + O ( (cid:15) ) ,D f v (cid:15) = D f v − (cid:15) D f u + O ( (cid:15) ) ,D ff v (cid:15) = D ff v − (cid:15) (cid:16) D ff u + θ f D ξξ w (cid:17) + O ( (cid:15) ) ,D yf v (cid:15) = D zf v − (cid:15) (cid:0) u − (1 − θ z ) θ f D ξξ w (cid:1) + O ( (cid:15) ) . (4.3)We want to use these expressions to expand the frictional dynamic programming equation (3.6). Tothis end, recall its frictionless counterpart (3.3), A v := v t + µ F v f + 12 σ F v ff + rzv z + µ S θv z + 12 σ S θ v zz + θσ S σ F ρv zf = 0 , where the corresponding optimal risky position is θ = µ S v z + ρσ S σ F v fz − σ S v zz . (4.4)Together with (4.3), and using y = θ + (cid:15)ξ = θ + O ( (cid:15) ) to replace (cid:15) y with (cid:15) θ + O ( (cid:15) ), the PDE (3.5)in the no-trade region can now be expanded as follows: L v (cid:15) := v (cid:15)t + µ F v (cid:15)f + y ( v (cid:15)y ( µ S + r ) − v (cid:15)x r ) + rzv (cid:15)x + 12 σ S y v (cid:15)yy + σ S yσ F ρv (cid:15)fy + 12 σ F v (cid:15)ff = A v (cid:124)(cid:123)(cid:122)(cid:125) (3.3) = 0 + ( y − θ ) µ S v z + 12 σ S ( y − θ ) v zz + σ S ( y − θ ) σ F ρv zf (cid:124) (cid:123)(cid:122) (cid:125) I − (cid:15) (cid:18) u t + rzu z + µ S yu z + µ F u f + 12 σ F u ff + 12 σ S y u zz + σ S yσ F ρu zf (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) II − (cid:15) w ξξ (cid:16) σ S θ (1 − θ z ) − σ S σ F ρθ (1 − θ z ) θ f + σ F θ f (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) α + O ( (cid:15) ) (4.5)By (4.4), I = (cid:0) µ S v z + σ S σ F ρv fz (cid:1) ( y − θ ) + 12 σ S ( y − θ ) = − σ S v zz θ ( y − θ ) + 12 σ S ( y − θ )= 12 σ S v zz ( y − θ ) = 12 σ S v zz (cid:15) ξ . Next, note that y = θ + (cid:15)ξ = θ + O ( (cid:15) ) implies II = A u + O ( (cid:15) ) . Recall that A is a nonlinear operator in the frictionless dynamic programming equation (3.3), because thefrictionless control θ depends on the solution v of the equation. In contrast, θ is already determined in terms of v here, so that A acts as a linear operator on u .
11n summary, (4.5) simplifies to the following asymptotic expansion of the dynamic programmingequation in the no-trade region: L v (cid:15) = − (cid:15) (cid:18) − σ S ξ v zz + A u + 12 α w ξξ (cid:19) + O ( (cid:15) ) . It remains to derive expansions in the buy and sell regions. To this end, we rewrite the gradientconstraint from (3.6), v (cid:15)x − (1 − (cid:15) ) v (cid:15)y , using the expressions from (4.3), obtaining v (cid:15)x − (1 − (cid:15) ) v (cid:15)y = (cid:15) v y + ( v x − v y ) (cid:124) (cid:123)(cid:122) (cid:125) (cid:15) w ξ + O ( (cid:15) ) = (cid:15) ( v z + w ξ ) + O ( (cid:15) ) . Analogously, the second gradient constraint in (3.6) can be expanded as follows: v (cid:15)y − (1 − (cid:15) ) v (cid:15)x = (cid:15) ( v z − w ξ ) + O ( (cid:15) ) . Altogether, the asymptotic dynamic programming equation ismin (cid:40) (cid:15) (cid:18) − σ S ξ v zz + A u + 12 α w ξξ (cid:19) ; (cid:15) ( v z + w ξ ); (cid:15) ( v z − w ξ ) (cid:41) = 0 . Since factoring out (cid:15) and (cid:15) does not change this equation, it is equivalent tomin (cid:26) − σ S ξ v zz + A u + 12 α w ξξ ; v z + w ξ ; v z − w ξ (cid:27) = 0 . (4.6)The variational inequality (4.6) with two unknowns w and u turns out to effectively consist oftwo separate equations. To see why, write a ( t, z, f ) := A u ( t, z, f ) . Then (4.6) can be rewritten as an equation for w and a :min (cid:26) − σ S ξ v zz + a + 12 α w ξξ ; v z + w ξ ; v z − w ξ (cid:27) = 0 , (4.7)where α := σ S θ (1 − θ z ) − σ S σ F ρθ (1 − θ z ) θ f + σ F θ f , (4.8)is determined by the model parameters and the frictionless optimizer. For each value of t , z and f ,(4.7) has a solution ξ (cid:55)→ w ( t, z, ξ, f ) for precisely one value of a = a ( t, z, f ). Thus, by finding thesolution ( w, a ) to this equation, we have obtained a unique function a ( t, z, f ), from which we can inturn determine u as the solution of a linear PDE: A u = a. The key to this separation is the uniqueness of the solution ( w, a ) to (4.7). Any other choice offunction u (cid:48) would give another value a (cid:48) = A u (cid:48) , for which, by uniqueness of ( w, a ), there would notexist a solution w (cid:48) .In summary, the asymptotic expansion (4.2) of the frictional value function is determined bythe following equations: For any solution ( w, a ), ( w + C, a ) is also a solution for any constant C , so the uniqueness only concerns a . However,for a given choice of normalization, e.g., w ( z,
0) = 0, also w is uniquely determined. The choice of normalizationaffects neither the equation for u , nor the policy generated from w . A precise formulation of the uniqueness result ispresented in [74, Theorem 3.1]. Similar uniqueness results are proven in [51, 69]. efinition 4.1 (First corrector equation) . For any ( t, z, f ) , the first corrector equation for the pair ( w ( t, z, · , f ) , a ( t, z, f )) is min (cid:40) − σ S ξ v zz + a + 12 α w ξξ (cid:124) (cid:123)(cid:122) (cid:125) no trade region ; v z + w ξ ; v z − w ξ (cid:124) (cid:123)(cid:122) (cid:125) trade regions (cid:41) = 0 . (4.9) Since any constant can be added to a solution w to obtain another solution, we will impose thenormalization w ( t, z, , f ) = 0 which affects neither the value expansion nor the optimal policy atthe leading asymptotic order. Definition 4.2 (Second corrector equation) . Given a solution ( w, a ) of the first corrector equation,the second corrector equation for u ( · , · , · ) is A u ( t, z, f ) = a ( t, z, f ) , u ( T, z, f ) = 0 , where A is the generator of the frictionless optimal wealth process, appearing in (3.3) . The intuition for this separation into two equations is the following. In the first correctorequation, only the deviation ξ of the portfolio from its frictionless target is a variable. In contrast,the frictionless state variables ( t, z, f ) are treated as constants because they vary much more slowlythan ξ for small transaction costs. Conversely, the “fast variable” ξ is averaged out in the secondcorrector equation determining the leading order utility loss u , in that it does not enter directly butonly through the function a ( t, z, f ) computed from the first corrector equation. In the present setting, the first corrector equation can be solved explicitly. This allows us to under-stand the comparative statics of the asymptotically optimal no-trade region and the correspondingleading-order welfare effect of small transaction costs. As a byproduct, the calculations below alsoexplain why there is only a single value of a for which the first corrector equation has a solution.To find the solution, fix ( t, z, f ) and make the ansatz that (i) the no-trade region is a symmetricinterval ( − ∆ ξ, ∆ ξ ) around the frictionless optimizer ( ξ = 0), and (ii) the asymptotic value functionis of the following form: w ( ξ ) = w ( t, z, ξ, f ) = c ξ + c ξ if | ξ | ≤ ∆ ξ,w (∆ ξ ) + ( ξ − ∆ ξ ) if ξ ≥ ∆ ξ,w (∆ ξ ) − ( ξ − ∆ ξ ) if ξ ≤ ∆ ξ. Here, c , c are parameters to be determined along with a and ∆ ξ . In the no-trade region, pluggingthis ansatz into (4.9) leads to a = 12 σ S ξ v zz − α (6 c ξ + c ) = (cid:18) σ S v zz − α c (cid:19) − α c . Since this needs to hold for any value ξ ∈ ( − ∆ ξ, ∆ ξ ), comparison of the coefficients of ξ and 1yields c = σ S v zz α and c = − aα . (4.10) This is the lowest order symmetric polynomial in the deviation ξ with enough degrees of freedom to ensure valuematching and smooth pasting at the trading boundaries ± ∆ ξ .
13o pin down a and ∆ ξ , we impose that the value function is not only continuous but also twicecontinuously differentiable across the trading boundaries ± ∆ ξ . By symmetry, this leads to thefollowing two additional conditions: 0 = 12 c (∆ ξ ) + 2 c ,v z = 4 c (∆ ξ ) + 2 c ∆ ξ. These equations readily yield a = σ S v zz ξ , (4.11)with ∆ ξ = ∆ ξ ( t, z, f ) = (cid:32) − v z v zz α σ S (cid:33) . (4.12)Together with (4.10), this leads to a closed-form solution of the first corrector equation (4.9) interms of model parameters and inputs from the frictionless optimization problem. Recalling that λ = (cid:15) , we find that the asymptotically optimal no-trade region corresponding to theleading-order variational inequality (4.6) isNT λ ≈ { ( t, x, y, f ) : (cid:12)(cid:12) y − θ ( t, x + y, f ) (cid:12)(cid:12) ≤ λ / ∆ ξ ( t, z, f ) } . (4.13)In view of (4.12) and the representation (4.8) for α , this asymptotic no-trade region is determinedby (i) the diffusion coefficients σ S , σ F of the risky asset and the factor process, (ii) the frictionlessoptimal portfolio θ and its derivatives θ z , θ f , and (iii) the risk-tolerance − v z /v zz of the frictionlessvalue function. The comparative statics of this formula for general utilities are discussed in [54].Here, we focus on the case most relevant for applications: power utilities with constant relative riskaversion. Constant Relative Risk Aversion
As is well known (cf., e.g., [87]), power utilities U ( x ) = x − γ / (1 − γ ) imply that the optimal fraction of wealth invested in the risky asset, π ( t, f ) := θ ( t, z, f ) /z , is independent of the wealth level. Moreover, the value function inherits the homotheticity v ( t, z, f ) = z − γ v ( t, , f ), so − v zz v z ( t, z, f ) = γ/z . In view of (4.12) and (4.8), the asymptotic no-trade region can therefore be written in terms of risky weights (in contrast to monetary amounts)as NT λ ≈ (cid:26) ( t, x, y, f ) : (cid:12)(cid:12)(cid:12)(cid:12) yx + y − π ( t, f ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ λ / ∆ π ( t, f ) (cid:27) , (4.14)where ∆ π = γ (cid:32) π (1 − π ) − π (1 − π ) π f σ F σ S + π f σ F σ S (cid:33) / . (4.15)Hence, in this case, the halfwidth of the asymptotically optimal no-trade region is fully determinedby the volatilities σ S , σ F , the risk aversion γ , as well as the frictionless portfolio weight π andsensitivity π f with respect to the state variable f . These are the “smooth pasting conditions” of [9, 33]. .6 Welfare loss With the explicit solution of the first corrector equation from Section 4.4, we can also say moreabout the leading order term (cid:15) u ( t, z, f ) in the expansion (4.2) of the frictional value function. Tothis end, recall that A u = a , where a is given by (4.11) and the differential operator A is defined in(3.3). Now, note that A is the infinitesimal generator of the frictionless optimal wealth process Z t .Whence, the Feynman–Kac formula and (4.11) show that u ( t, z, f ) = E t (cid:34)(cid:90) Tt − a ( s, Z s , F s ) d s (cid:35) = v z ( t, z, f ) E Qt (cid:34)(cid:90) Tt − v zz ( s, Z s , F s ) v z ( s, Z s , F s ) σ S ( F s ) ξ ( s, Z s , F s ) d s (cid:35) , (4.16)where Q is the frictionless investor’s “marginal pricing measure”, whose density process is proportionalto the marginal indirect utility v z by the first-order condition of convex duality [28, 56]. With the representation (4.16) for u , Taylor’s theorem allows to rewrite the expansion (4.2) as v λ ( t, z, f ) = v (cid:16) t, z − CEL λ ( t, x, f ) , f (cid:17) + O ( λ ) . (4.17)Here, CEL λ ( t, z, f ) = λ / E Qt (cid:34)(cid:90) Tt − v zz ( s, Z s , F s ) v z ( s, Z s , F s ) σ S ( F s ) ξ ( s, Z s , F s ) d s (cid:35) (4.18)is the certainty equivalent loss due to small transaction costs—the amount of initial endowmentthe investor would forego in order to trade the risky asset without transaction costs. Like theasymptotic no-trade region, this measure for the welfare effect of the trading costs is determined by(i) the diffusion coefficients of the risky asset and the factor process, (ii) the frictionless optimalpolicy and its sensitivities, and (iii) the risk tolerance of the frictionless value function. As all ofthese quantities are generally random and time dependent, they are averaged both with respect totime and states. Constant Relative Risk Aversion
For power utilities with constant relative risk aversion γ > λ ( t, z, f ) = zλ / E ˜ Pt (cid:34)(cid:90) Tt γ σ S ( F s ) ∆ π ( s, F s ) d s (cid:35) , (4.19)where the expectation is computed under the measure ˜ P whose density process is proportionalto the value function v ( s, Z s , F s ) evaluated along the frictionless optimal wealth process. Byscaling out the current wealth z , this leads to the relative certainty equivalent loss —an appealingscale-invariant measure for the welfare effect of transaction costs, also used in the numerical workof [7], for example. Here, its small-cost approximation is obtained by averaging the frictionlessoptimizer, its sensitivities, and the volatilities of the risky asset and factor process in a suitable way. In complete markets, Q is simply the unique equivalent martingale measure. In any case, the correspondingdensity process is known explicitly if the value function of the problem at hand can be computed in closed form. This measure also plays an important role in the asymptotic analysis of small unhedgeable risks, compare [60]. F . To find its dynamics under the measure ˜ P , use Itˆo’sformula to compute the dynamics of the density process:d v ( s, Z s , F s ) v ( s, Z s , F s ) = A v ( s, Z s , F s ) v ( s, Z s , F s ) d s + v z ( s, Z s , F s ) v ( s, Z s , F s ) d Z s + v f ( s, Z s , F s ) v ( s, Z s , F s ) d F s =: d L t , (4.20)The d s -term vanishes by the frictionless dynamic programming equation (3.3). It follows that thedensity process of ˜ P is a stochastic exponential E ( L ) t = exp( L t − (cid:104) L (cid:105) t ), and the ˜ P dynamics of F can be readily computed using Girsanov’s theorem. If the frictionless value function is known inclosed form, the corresponding change of drift is once more fully explicit. Long Investment Horizons
The relative certainty equivalent loss (4.19) is a function of timeand the state variable only. If the planning horizon T is long, the time variable averages out andeven more tractable formulas obtain. To wit, for large T , the frictionless policy π ( t, f ) typicallyquickly converges to a steady-state value ¯ π ( f ) that only depends on the state variable (but not thecurrent time), and the frictionless value function approximately scales as follows [47]: v ( t, z, f ) ≈ z − γ − γ e (1 − γ )( T − t )ESR . (4.21)Here, the equivalent safe rate ESR is a fictitious interest rate which—in the long run—yields thesame growth rate of utility as trading in the original market. With small transaction costs, (4.17),(4.19), and (4.21) show that the corresponding leading-order expression is v λ ( t, x, y ) ≈ z − γ − γ e (1 − γ )( T − t ) ( ESR − ∆ESR λt ) . Here, the equivalent safe rate loss due to small transaction costs is∆ESR λt = λ / γ T − t ) E ˜ Pt (cid:34)(cid:90) Tt (∆ π ( s , F s ) σ S ( F s ) d s (cid:35) . On any finite time horizon T , this quantity is a function of time t and the value f of the statevariable. However, as the horizon grows, the ergodic theorem suggests that if the state variable F has a stationary distribution ν ˜ PF (d f ) under the measure ˜ P , then the equivalent safe rate lossconverges to a constant, like the frictionless equivalent safe rate: ∆ESR λ ≈ λ / γ (cid:90) ∞−∞ ∆¯ π ( f ) σ ( f ) ν ˜ PF (d f ) . (4.22)In summary, the welfare effect in infinite-horizon models with small transaction costs can becomputed by performing a simple numerical quadrature. We now illustrate the asymptotic results from Section 4 in two concrete examples. As a sanitycheck, we first consider the Black–Scholes model and verify that the formulas derived above indeedcoincide with the expressions directly obtained for this simple model (cf., e.g., [52, 10, 42, 82]).Afterwards, we turn to the model of Kim and Omberg, and discuss how the results change withstochastic investment opportunities. This requires that the effect of transaction costs is small even when compounded over a long horizon. To makethis argument precise, one can directly consider the infinite-horizon problem as in [42, 54, 68]. We denote by ∆¯ π ( f ) the halfwidth of the stationary no-trade region obtained from the long-run portfolio ¯ π ( f )via (4.15). .1 Black–Scholes Model For an investor with constant relative risk aversion γ >
0, the value function in the Black–Scholesmodel is v ( t, z ) = z − γ − γ exp (cid:16) (1 − γ )( r + µ γσ )( T − t ) (cid:17) and the corresponding risky weight is constant: π BS = µ/γσ . As a consequence, the halfwidth (4.15) of the asymptotic no-trade region simplifiesto ∆ π BS = (cid:18) γ π BS (1 − π BS ) (cid:19) / . The corresponding formula for the leading-order equivalent safe rate loss is∆ESR λ = γσ (cid:18) λ γ π BS (1 − π BS ) (cid:19) / . Both of these expressions vanish if zero or full investment is optimal in the frictionless model. Then,the respective frictionless optimal strategies never trade, and no transaction costs need to be paid.In contrast, transaction costs play a more important role if the frictionless target weight is close to1 /
2, but even then the quantitative effects are rather small [23]. If a leveraged portfolio is optimal( π BS > Let us now sketch how the above results change in the Kim–Omberg model, where the expectedexcess return follows an Ornstein–Uhlenbeck process with dynamics (2.3). For a power utilityfunction with constant relative risk aversion γ >
1, the frictionless value function v has the followingclosed-form expression [57]: v ( s, z, f ) = z − γ − γ exp (cid:18) A ( s ) + B ( s ) f + 12 C ( s ) f (cid:19) , (5.1)where the functions A , B , and C are the explicit solutions of some Riccati equations [57]. Thecorresponding optimal risky weight is linear in the state variable: π KO ( s, F s ) = F s γσ S + ρσ F γσ S ( B ( s ) + C ( s ) F s ) . In view of (4.15), this formula immediately yields a closed-form expression for the halfwidth ∆ π KO of the asymptotic no-trade region with transaction costs. Figure 1 shows a simulated sample pathof the frictionless portfolio π KO and the boundaries of the no-trade region π KO ± ∆ π KO .Since the frictionless risky weight is sensitive to changes in the state variable, the no-trade regionno longer vanishes if the frictionless risky weight is zero or one, but instead at two other levelsdetermined by the model parameters. For the parameters estimated from a long equity time seriesin [8], this is illustrated in Figure 2. There, the halfwidth of the no-trade region is plotted againstthe optimal frictionless risky weight for different values of the expected excess return.This is complemented by Figure 3, which shows how the optimal frictionless portfolio and thecorresponding no-trade region converge to their stationary long-run limits as the planning horizongrows. This stationary policy ¯ π KO corresponds to the stationary points ¯ B and ¯ C of the Riccatiequations for B ( s ) and C ( s ). The corresponding no-trade region is in turn derived from (4.15). For the convenience of the reader, we recall these formulas in Appendix A. - - - - Figure 2: Halfwidth of the no-trade region in the Kim–Omberg model plotted against the optimalfrictionless risky weight. (Yearly) Parameters are T = 40, γ = 3, and r = 0 . σ S = 0 . κ = 0 . F = 0 . σ F = 0 .
10 20 300.51.01.5
Figure 3: The frictionless position and no-trade region (solid) as a function of time (measured inyears), plotted alongside the infinite-horizon values (dashed). (Yearly) Parameters are γ = 3 and r = 0 . σ S = 0 . κ = 0 . F = 0 . σ F = 0 . .00 0.02 0.04 0.06 0.08 0.100102030405060 Figure 4: Relative loss in equivalent safe rate due to transaction costs for the Kim–Omberg model(solid) and a Black–Scholes model (dotted) with µ S ≡ ¯ F , plotted against the size of the cost. (Yearly)Parameters are γ = 3 and r = 0 . σ S = 0 . κ = 0 . F = 0 . σ F = 0 . B ( s ) and C ( s ) can also be used to simplify thecomputation of the leading-order relative certainty equivalent loss (4.22). In particular, Girsanov’stheorem and the long-run convergence show that the long term dynamics of the factor process F under the measure ˜ P with density process (4.20) ared F s = κ ( ¯ F − F s ) d s + σ F d W F = (cid:0) κ ( ¯ F − F s ) + σ F (1 − γ ) π KO ( s, F s ) σ S ρ + ( B ( s ) + C ( s ) F s ) σ F (cid:1) d s + σ F d ˜ W F ≈ (cid:0) κ ( ¯ F − F s ) + σ F (1 − γ )¯ π KO ( F s ) σ S ρ + ( ¯ B + ¯ CF s ) σ F (cid:1) d s + σ F d ˜ W F =: ˜ κ ( ˜ F − F s ) d s + σ F d ˜ W F , for a ˜ P -Brownian motion ˜ W F and suitably chosen constants ˜ κ and ˜ F . Hence, for a long planninghorizon, the factor process still has Ornstein–Uhlenbeck dynamics under the auxiliary measure ˜ P ,and its stationary law is ν ˜ PF ∼ N ( ˜ F , σ F / κ ). This allows computation of the relative certaintyequivalent loss (4.22) due to small transaction costs according to∆ESR λKO ≈ λ / γ (cid:113) σ F / ˜ κ ) π (cid:90) ∞−∞ ∆¯ π KO ( f ) σ ( f ) exp (cid:32) − ( f − ˜ F ) σ F / κ ) (cid:33) d f, (5.2)Since the integrand is known in closed form, (5.2) is easily evaluated by numerical quadrature. Thisis illustrated in Figure 4 which plots the relative certainty equivalence loss as a function of theproportional transaction cost. Compared to a Black–Scholes model with the same expected excessreturn, we observe that the welfare effect of transaction costs is indeed increased substantially byhaving to react to the time-varying investment opportunities.19 Extensions
So far, we have focused on the application of the homogenization approach to a portfolio choiceproblem with proportional transaction costs for purchases and sales of a single risky asset withMarkovian dynamics. Performance was measured in terms of expected utility from terminal wealthonly, i.e., intertemporal consumption was absent. This choice was made for concreteness and ease ofexposition. In this section, we survey results from the recent literature that show that the findingsoutlined in Section 4 remain true much more generally.
The results of the previous sections readily extend to models with intermediate consumption. Forexample, [82] is a study of an infinite-horizon model with preferences of the form E (cid:20)(cid:90) ∞ e − δt U ( c t ) d t (cid:21) → max! , where δ > U ( c t ) measures the utility from consumption (rate) c t at time t . The asymptotic no-trade region for this optimization criterion turns out to be of thesame form as in (4.13)—intermediate consumption is only reflected through the frictionless optimalpolicy. This remains true for more general “additive” preferences of the form E (cid:34)(cid:90) T U ( t, c t ) d t + U ( Z T ) (cid:35) → max! , (6.1)where U ( t, c t ) is the utility from intermediate consumption at time t ∈ [0 , T ] and U ( Z T ) is theutility from terminal wealth Z T at time T ; see [54, 1] for more details. Despite this robustness result,substantial intertemporal consumption can have a nonnegligible effect if the trading costs are notsmall enough. The intuition is that since costs are paid from the savings account, investors are willingto accept smaller risky positions before rebalancing [29]. In infinite-horizon Black–Scholes models,this manifests itself through a downward shift of the no-trade region at the second asymptotic order O ( λ / ), see [52, 62, 43]. For more general models, such results are not available.The effects of the transaction costs on the optimal consumption policy are of the simplestconceivable form: it is asymptotically optimal to simply adjust the frictionless rule for the (typicallylower) wealth with transaction costs [54]. For power utility, this implies that the consumption/wealthratio is unaffected by the trading costs [54, Section 4].Very recently, it has been shown [67] that even the additive structure in the preferences (6.1)is not crucial, in that the same leading-order results also remain true for recursive preferences asin [32] or models with habit formation such as [50]. At the leading asymptotic order, the optimaltrading strategy is again completely characterized by the local curvature of the agents’ preferences,measured by the risk-tolerance of their indirect utilities, and the consumption wealth ratio remainsunchanged. The fine structure of the preferences at hand only enters at the next-to-leading order. In previous sections, we have assumed that the joint dynamics (2.1–2.2) of the asset prices and afactor process are Markovian, i.e., all drift and diffusion coefficients are deterministic functions ofthe current state of the system. This assumption allows to apply PDE techniques, but is not crucialfor the validity of formulas (4.13) and (4.17). Indeed, these formulas are derived by “freezing” the20rictionless state variables for the analysis of the first corrector equation—a procedure that readilygeneralizes to general, not necessarily Markovian systems where drift and diffusion coefficientscan be arbitrary functionals of the information flow. Even in such more general models, formulas(4.13) for the asymptotically optimal no-trade region and (4.17) for the leading-order effect of smalltransaction costs remain valid, see [55, 54, 17, 18] for more details.Extending the approach presented here to models with jumps leads to the analysis of nonlocalintegro-differential equations. While this appears to be a daunting task, it is shown in [78], usingprobabilistic techniques, that asymptotic results similar to (4.13) and (4.17) can still be obtained.
Although we have focused on proportional transaction costs here, also this structure is not crucial.At least on a formal level, fixed costs [2], fixed and proportional costs [3], or quadratic costs [71]can be treated similarly.The fine structure of the optimal strategies crucially depends on the cost at hand. Withproportional costs, one performs the minimal amount of trading to remain in a no-trade intervalaround the frictionless target. With fixed costs, it is no longer possible to implement such a strategyinvolving infinitely many small trades; hence, one directly trades back to a target portfolio once theboundaries of the no-trade region is reached. Conversely, quadratic costs lead to smaller penaltiesfor very small trades but make large turnover rates prohibitively expensive. Thus, optimal strategiesalways trade towards the target at some finite, absolutely continuous rate.Despite these apparent differences, the “coarse” structure of all these models is neverthelessvery similar: In each case, the distance from the target is a trade-off against the specific tradingcost, balanced by an appropriate control. The corresponding expected displacement and averagetransaction costs display the same comparative statics in each case, up to a change of asymptoticconvergence rates and constants. In particular, the implications of transaction costs for welfare andaverage trading volume are very similar in each case. See [71] for more details.
The extensions sketched so far eventually lead to explicit asymptotic formulas of a similar complexityas for the benchmark model discussed in Section 4. In contrast, less is known about the case ofseveral risky assets. In this case, the homogenization approach still reduces the dimensionality ofthe problem [74] but the resulting corrector equations no longer admit an explicit solution. As aconsequence, numerical methods such as the policy iteration scheme in Section 7 are needed even forthe asymptotic analysis. Models with quadratic costs [40, 41, 48, 71] and fixed costs [4, 2] can stillbe solved in closed form in the multidimensional case, but the tractability issue is only exacerbatedfor general nonlinear costs.
The corrector equations obtained by passing to the small-cost limit are considerably simpler thanthe original dynamic programming equation. Even in situations with multiple risky assets [74] ornonlinear costs [3] where explicit solutions are not available, this simplifies the numerical analysisconsiderably. In this section, we present a variant of the classical “policy iteration algorithm” whichis tailored to the singular control problem at hand and works very well in practice. An attractive Alternatively methods, based on PDE techniques, can be found in [72, 27]. d -dimensional model with frictionless value function v and optimal risky positions θ .To simplify notation, all assets are traded with the same proportional transaction cost. Analogousarguments to those in Section 4 then show that the d -dimensional version of the first correctorequation (4.9) ismin i =1 ,...,d min (cid:40) − | σ (cid:62) S ξ | v zz + a + 12 Tr (cid:104) αα (cid:62) w ξξ (cid:105) ; v z + w ξ · e i ; v z − w ξ · e i (cid:41) = 0 , (7.1)where α is a model-dependent matrix analogous to (4.8), depending on θ and its derivatives.The corrector equation (7.1) can be interpreted as the dynamic programming equation of aninfinite-horizon control problem. To wit, let L i , M i , i = 1 , . . . , d be nondecreasing controls forΞ is = ξ i + d (cid:88) j =1 α i,j ( t, z, f ) W js + L is − M is . Then, (7.1) is the dynamic programming equation for the ergodic control problem a ( t, z, f ) := inf L,M J ( t, z, f ; L, M ) , (7.2)corresponding to the following infinite-horizon goal functional: J ( t, z, f ; L, M ) := lim sup s →∞ s E (cid:90) s − v zz ( t, z, f ) (cid:12)(cid:12)(cid:12) σ (cid:62) S Ξ τ (cid:12)(cid:12)(cid:12) d τ + v z ( t, z, f ) d (cid:88) i =1 ( L is + M is ) . This means that the controls are chosen so as to minimize the long-run average deviations ofthe controlled process Ξ from zero, subject to proportional adjustment costs. Note that in thisproblem, the state variables ( t, z, f ) of the original problem are frozen, so that the uncontrolled Ξ isa Brownian motion.To solve this problem numerically, we approximate the controls L , M by absolutely continuoustrading rates of the form L st = (cid:90) s (cid:96) i (Ξ τ ) d τ and M is = (cid:90) s m i (Ξ τ ) d τ, (cid:96) and m bounded by a finite constant K . With this restriction, the dynamics of thecontrolled process are dΞ is = ν i (Ξ; (cid:96), m ) d s + α d W s , where ν i ( ξ ; (cid:96), m ) = ( (cid:96) i − m i )( ξ ). The corresponding dynamic programming equation for the restrictedversion of (7.2) in turn is min i =1 ,...,d(cid:96) i ,m i ∈ [0 ,K ] (cid:16) L (cid:96),m w ( ξ ) + f ( (cid:96), m, ξ ) (cid:17) = − a, ∀ ξ ∈ R d , where L (cid:96),m w ( ξ ) = ν ( ξ ; (cid:96), m ) (cid:62) ∂ w∂ξ ( ξ ) + 12 Tr (cid:34) αα (cid:62) ∂ w∂ ( ξ i ) ∂ ( ξ j ) ( ξ ) (cid:35) (7.3)and f ( (cid:96), m, ξ ) = − (cid:12)(cid:12)(cid:12) σ (cid:62) S ξ (cid:12)(cid:12)(cid:12) v zz + v z d (cid:88) i =1 ( (cid:96) i + m i ) . Now, truncate the state space for the control problem to a large finite domain in R d , and consider adiscretization D ⊂ R d of this set. The approximation L (cid:96),m D : D (cid:55)→ R of the operator L (cid:96),m in (7.3)outlined in Appendix B can in turn be interpreted as the transition-rate matrix of a discrete controlproblem with dynamic programming equationmin i =1 ,...,d(cid:96) i ,m i ∈ [0 ,K ] (cid:88) ξ (cid:48) ∈D L (cid:96),m D ( ξ, ξ (cid:48) ) w ( ξ (cid:48) ) + f ( (cid:96), m, ξ ) = − a, ∀ ξ ∈ D . If the truncated domain is sufficiently large, the probability of Ξ reaching its boundary is small, sothat the corresponding boundary conditions can be chosen arbitrarily, as long as the discretizeddifferential operator can be interpreted as the transition-rate matrix of some discrete control problem.The key advantage of this scheme is that the bound K ensures that the transition probabilities arebounded away from 0, enabling us to represent the problem as a continuous time Markov decisionprocess for which standard policy iteration techniques apply. More specifically, this discrete problemcan be solved using the following policy iteration algorithm by choosing an initial policy ( (cid:96) , m ),e.g., (cid:96) , m ≡
0, and then iterating the following steps:(i) Compute ( w j , a j ) ∈ R |D| × R + as the solution of (cid:88) ξ (cid:48) ∈D L (cid:96) j ,m j D ( ξ, ξ (cid:48) ) w j ( ξ (cid:48) ) + f ( (cid:96) j , m j , ξ ) = − a ∀ ξ ∈ D . Note that these are |D| equations for |D| + 1 unknowns. The missing equation is obtained bynormalizing w as in Section 4.(ii) Find solutions (cid:96) j +1 and m j +1 to the |D| minimization problems (cid:96) j +1 ( ξ ) , m j +1 ( ξ ) ∈ arg min i =1 ,...,d(cid:96) i ,m i ∈ [0 ,K ] (cid:88) ξ (cid:48) ∈D L (cid:96),m D ( ξ, ξ (cid:48) ) w j ( ξ (cid:48) ) + f ( (cid:96), m, ξ ) , and return to the previous step. As the artificial constraint K tends to infinity, we then expect to approach the solution of the original problem. a j and a j − is small enough. It is knownthat this difference converges to 0 in finite time (cf., e.g., [75]). Although this bound is very largefor a general policy iteration scheme, it has been observed that policy iterations typically convergevery quickly—often in fewer than 20 iterations [79]. The fast convergence is attained thanks to thescheme’s close connections to Newton’s method. For more details on these connections, as well asthe convergence rate, compare [76, 79, 13].Solving the |D| optimization problems in the second step of each iteration may seem dauntingat first glance. However, when trading is only conducted through the safe account but not directlybetween risky assets, the solution of this problem is in fact explicit. Also in more general settings,the optimization problems are entirely independent of each other, so that their solution can be fullyparallelized.For simplicity, the chosen model is a Black–Scholes market consisting of two risky assets, wherean agent optimizes the power utility of consumption over an infinite horizon (impatience parameter δ ), like in Section 6.1 or [74]. Note that this choice only appears in the above problem through v zz , v z , and α , and in this case α = ( I d − θ (cid:62) z d ) diag[ θ ] σ S , where I d is the d -dimensional identity matrix, diag[ θ ] is the matrix with diagonal θ and otherelements zero, and 1 d = (1 , . . . , (cid:62) ∈ R d .Optimal strategies computed using this algorithm are depicted in Figure 5. These asymptoticno-trade regions should be interpreted as follows. Each axis represents the deviation of a riskyweight from its frictionless target. The white region indicates no trading, whereas the other regionsdescribe trading in the assets. Except in the corners, trading is only performed in one asset at atime, inducing vertical or horizontal movements of the portfolio position. For example, the top leftfigure has a white region of half-width 0.091, meaning that it is optimal to trade risky asset 2 (only)when its current weight from the frictionless optimizer by 9.1% of current wealth.The policy iteration scheme presented here can readily be generalized to more complex models.For example, details and justification for such generalizations can be found in [3] for a model withproportional and fixed costs like in Section 3.3. The output of the algorithm in this setting isillustrated in the bottom right panel in Figure 5. There, the solid line inside the no-trade regionis the rebalancing target, to which the portfolio is readjusted once the boundaries of the no-traderegion are breached. 24 . − .
05 0 0 .
05 0 . − . − . . . − . − .
05 0 0 .
05 0 . − . − . . . − . − . − .
05 0 0 .
05 0 . . − . − . − . . . . − . − . − . . . . − . − . − . . . . Figure 5: Asymptotic no-trade regions for different parameters ( µ i and σ i are the expected excessreturn and volatility of risky asset i = 1 , ρ is the correlation between their driving Brownianmotions). The axes and interpretation of the transaction cost is like in (4.14). rel. pos. δ γ r µ µ σ σ ρ λ f λ p z Top left 1 2 0.03 0.08 0.08 0.4 0.33 0.00 0 3% $50’000Top right 1 2 0.03 0.08 0.08 0.4 0.33 0.30 0 3% $50’000Bottom left 1 2 0.03 0.08 0.08 0.4 0.33 -0.30 0 3% $50’000Bottom right 1 3 0.03 0.08 0.08 0.4 0.4 0.30 $1 3% $50’000 Kim–Omberg value function
Consider a power utility function with constant relative risk aversion γ . As shown in [57], the valuefunction for the Kim–Omberg model (2.1, 2.3) then is of the following form: v ( u, z, f ) = z − γ − γ exp (cid:18) A ( u ) + B ( u ) f + 12 C ( u ) f (cid:19) . Define b = 2 (cid:18) − γγ σ F σ S ρ − κ (cid:19) , η = (cid:115) b − − γγ (cid:18) σ F σ S (cid:19) (cid:18) − γγ ρ (cid:19) . In the empirically relevant case when γ > ρ <
0, the discriminant η is positive, so that A , B ,and C can be identified as the “normal solution” of [57]: C ( u ) = 1 − γγ σ S − exp( − η ( T − u ))2 η − ( b + η ) (cid:18) − exp (cid:16) − η ( T − u ) (cid:17)(cid:19) ,B ( u ) = 4 1 − γγ κ ¯ Fσ S − exp( − η ( T − u ) / η − η ( b + η ) (cid:18) − exp (cid:16) − η ( T − u ) (cid:17)(cid:19) , and A ( u ) = 1 − γγ (cid:32) γr + 2 κ ¯ F σ S η + σ F σ S ( η − b ) (cid:33) ( T − u )+ 1 − γγ κ ¯ F σ S (2 b + η ) exp( − η ( T − u )) − b exp( − η ( T − u ) /
2) + 2 b − ηη (2 η − ( b + η )(1 − exp( − η ( T − u ))))+ 1 − γγ σ F σ S log (cid:12)(cid:12) η − ( b + η )(1 − exp( − η ( T − u ))) (cid:12)(cid:12) η ( η − b ) . Discretization scheme for policy iteration
For the numerical computations in Section 7, the differential operator of the ergodic control problemneeds to be discretized. To interpret the discretized operator as the transition rate matrix of some(continuous-time) Markov decision process, the following discretization scheme is used: ∂ w∂ ( ξ i ) ( ξ ) ≈ w ( ξ + e i h i ) − w ( ξ ) h i if ν i ( ξ ; (cid:96), m ) > ,w ( ξ ) − w ( ξ − e i h i ) h i if ν i ( ξ ; (cid:96) ) < ,∂ w∂ ( ξ i ) ( ξ ) ≈ w ( ξ + e i h i ) − w ( ξ ) + w ( ξ − e i h i ) h i ,∂ w∂ ( ξ i ) ∂ ( ξ i ) ( ξ ) ≈ w ( ξ ) + w ( ξ + e i h i + e j h j ) + w ( ξ − e i h i − e j h j )2 h i h j − w ( ξ + e i h i ) + w ( ξ − e i h i ) + w ( ξ + e j h j ) + w ( ξ − e j h j )2 h i h j if A i,j > , − w ( ξ ) + w ( ξ + e i h i − e j h j ) + w ( ξ − e i h i + e j h j )2 h i h j + w ( ξ + e i h i ) + w ( ξ − e i h i ) + w ( ξ + e j h j ) + w ( ξ − e j h j )2 h i h j if A i,j < , for i, j = 1 , . . . , d , i (cid:54) = j , and where h i is the grid size in the ξ i -direction. With A = αα (cid:62) , theapproximation L (cid:96) D : D (cid:55)→ R of the differential operator L (cid:96) is then given by L (cid:96) D ( ξ, ξ ) = − d (cid:88) i =1 (cid:32) A i,i h i − d (cid:88) j =1 ,j (cid:54) = i | A i,j | h i h j (cid:33) − d (cid:88) i =1 | ν i ( ξ ; (cid:96) ) | h i , L (cid:96) D ( ξ, ξ + e i h i ) = 12 (cid:32) A i,i h i − d (cid:88) j =1 ,j (cid:54) = i | A i,j | h i h j (cid:33) + d (cid:88) i =1 max { , ν i ( ξ ; (cid:96) ) } h i , L (cid:96) D ( ξ, ξ − e i h i ) = 12 (cid:32) A i,i h i − d (cid:88) j =1 ,j (cid:54) = i | A i,j | h i h j (cid:33) + d (cid:88) i =1 max { , − ν i ( ξ ; (cid:96) ) } h i , L (cid:96) D ( ξ, ξ ± e i h i ± e j h j ) = max { , A i,j } h i h j , L (cid:96) D ( ξ, ξ ± e i h i ∓ e j h j ) = max { , − A i,j } h i h j , for i, j = 1 , . . . , d and i (cid:54) = j . For a finite domain, we will also need the condition (cid:88) ξ (cid:48) ∈D L (cid:96) j D ( ξ, ξ (cid:48) ) = 0 . This ensures that the Markov decision process stays within the domain at all times. As long asthe domain is chosen large enough to contain the no-trade region this is not a constraint, since theoptimal strategy already ensures that the process does not exit the domain.27 eferences [1] L. Ahrens.
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