Transaction Costs, Trading Volume,
and the Liquidity Premium^{1}^{1}1For helpful comments, we thank Maxim Bichuch, George Constantinides, Aleš Černý, Mark Davis, Ioannis Karatzas, Ren Liu, Marcel Nutz, Scott Robertson, Johannes Ruf, Mihai Sirbu, Mete Soner, Gordan Zitković, and seminar participants at Ascona, MFO Oberwolfach, Columbia University, Princeton University, University of Oxford, CAU Kiel, London School of Economics, University of Michigan, TU Vienna, and the ICIAM meeting in Vancouver. We are also very grateful to two anonymous referees for numerous – and amazingly detailed – remarks and suggestions.
Abstract
In a market with one safe and one risky asset, an investor with a long horizon, constant investment opportunities, and constant relative risk aversion trades with small proportional transaction costs. We derive explicit formulas for the optimal investment policy, its implied welfare, liquidity premium, and trading volume. At the first order, the liquidity premium equals the spread, times share turnover, times a universal constant. Results are robust to consumption and finite horizons. We exploit the equivalence of the transaction cost market to another frictionless market, with a shadow risky asset, in which investment opportunities are stochastic. The shadow price is also found explicitly.
Mathematics Subject Classification: (2010) 91G10, 91G80.
JEL Classification: G11, G12.
Keywords: transaction costs, longrun, portfolio choice, liquidity premium, trading volume.
1 Introduction
If risk aversion and investment opportunities are constant — and frictions are absent — investors should hold a constant mix of safe and risky assets (Markowitz, 1952; Merton, 1969, 1971). Transaction costs substantially change this statement, casting some doubt on its farreaching implications.^{2}^{2}2Constantinides (1986) finds that “transaction costs have a firstorder effect on the assets’ demand.” Liu and Loewenstein (2002) note that “even small transaction costs lead to dramatic changes in the optimal behavior for an investor: from continuous trading to virtually buyandhold strategies.” Luttmer (1996) shows how small transaction costs help resolve asset pricing puzzles. Even the small spreads that are present in the most liquid markets entail wide oscillations in portfolio weights, which imply variable risk premia.
This paper studies a tractable benchmark of portfolio choice under transaction costs, with constant investment opportunities, summarized by a safe rate , and a risky asset with volatility and expected excess return , which trades at a bid (selling) price equal to a constant fraction of the ask (buying) price . Our analysis is based on the model of Dumas and Luciano (1991), which concentrates on longrun asymptotics to gain in tractability. In their framework, we find explicit solutions for the optimal policy, welfare, liquidity premium^{3}^{3}3 That is, the amount of excess return the investor is ready to forgo to trade the risky asset without transaction costs. and trading volume, in terms of model parameters, and of an additional quantity, the gap, identified as the solution to a scalar equation. For all these quantities, we derive closedform asymptotics, in terms of model parameters only, for small transaction costs.
We uncover novel relations among the liquidity premium, trading volume, and transaction costs. First, we show that share turnover (), the liquidity premium (), and the bidask spread satisfy the following asymptotic relation:
This relation is universal, as it involves neither market nor preference parameters. Also, because it links the liquidity premium, which is unobservable, with spreads and share turnover, which are observable, this relation can help estimate the liquidity premium using data on trading volume.
Second, we find that the liquidity premium behaves very differently in the presence of leverage. In the noleverage regime, the liquidity premium is an order of magnitude smaller than the spread (Constantinides, 1986), as unlevered investors respond to transaction costs by trading infrequently. With leverage, however, the liquidity premium increases quickly, because rebalancing a levered position entails high transaction costs, even under the optimal trading policy.
Third, we obtain the first continuoustime benchmark for trading volume, with explicit formulas for share and wealth turnover. Trading volume is an elusive quantity for frictionless models, in which turnover is typically infinite in any time interval.^{4}^{4}4The empirical literature has long been aware of this theoretical vacuum: Gallant, Rossi and Tauchen (1992) reckon that “The intrinsic difficulties of specifying plausible, rigorous, and implementable models of volume and prices are the reasons for the informal modeling approaches commonly used.” Lo and Wang (2000) note that “although most models of asset markets have focused on the behavior of returns […] their implications for trading volume have received far less attention.” In the absence of leverage, our results imply low trading volume compared to the levels observed in the market. Of course, our model can only explain trading generated by portfolio rebalancing, and not by other motives such as market timing, hedging, and lifecycle investing.
Moreover, welfare, the liquidity premium, and trading volume depend on the market parameters () only through the meanvariance ratio if measured in business time, that is, using a clock that ticks at the speed of the market’s variance . In usual calendar time, all these quantities are in turn multiplied by the variance .
Our main implication for portfolio choice is that a symmetric, stationary policy is optimal for a long horizon, and it is robust, at the first order, both to intermediate consumption, and to a finite horizon. Indeed, we show that the notrade region is perfectly symmetric with respect to the Merton proportion , if trading boundaries are expressed with trading prices, that is, if the buy boundary is computed from the ask price, and the sell boundary from the bid price.
Since in a frictionless market the optimal policy is independent both of intermediate consumption and of the horizon (Merton, 1971), our results entail that these two features are robust to small frictions. However plausible these conclusions may seem, the literature so far has offered diverse views on these issues (cf. Davis and Norman (1990); Dumas and Luciano (1991); Liu and Loewenstein (2002)). More importantly, robustness to the horizon implies that the longhorizon approximation, made for the sake of tractability, is reasonable and relevant. For typical parameter values, we see that our optimal strategy is nearly optimal already for horizons as short as two years.
A key idea for our results — and for their proof — is the equivalence between a market with transaction costs and constant investment opportunities, and another shadow market, without transaction costs, but with stochastic investment opportunities driven by a state variable. This state variable is the ratio between the investor’s risky and safe weights, which tracks the location of the portfolio within the trading boundaries, and affects both the volatility and the expected return of the shadow risky asset.
In this paper, using a shadow price has two related advantages over alternative methods: first, it allows us to tackle the issue of verification with duality methods developed for frictionless markets. These duality methods in turn yield the finitehorizon bounds in Theorem 3.1 below, which measure the performance of longrun policies over a given horizon – an issue that is especially important when an asymptotic objective funcion is used. The shadow price method was applied successfully by Kallsen and MuhleKarbe (2010); Gerhold, MuhleKarbe and Schachermayer (2012, 2011) for logarithmic utility, and this paper brings this approach to power utility, which allows to understand how optimal policies, welfare, liquidity premia and trading volume depend on risk aversion. The recent papers of Herczegh and Prokaj (2012); Choi, Sirbu and Žitković (2012) consider power utility from consumption in an infinite horizon.
The paper is organized as follows: Section 2 introduces the portfolio choice problem and states the main results. The model’s main implications are discussed in Section 3, and the main results are derived heuristically in Section 4. Section 5 concludes, and all proofs are in the appendix.
2 Model and Main Result
Consider a market with a safe asset earning an interest rate , i.e. , and a risky asset, trading at ask (buying) price following geometric Brownian motion,
Here, is a standard Brownian motion, is the expected excess return,^{5}^{5}5A negative excess return leads to a similar treatment, but entails buying as prices rise, rather than fall. For the sake of clarity, the rest of the paper concentrates on the more relevant case of a positive . and is the volatility. The corresponding bid (selling) price is , where represents the relative bidask spread.
A selffinancing trading strategy is a twodimensional, predictable process of finite variation, such that and represent the number of units in the safe and risky asset at time , and the initial number of units is . Writing as the difference between the cumulative number of shares bought () and sold () by time , the selffinancing condition relates the dynamics of and via
(2.1) 
As in Dumas and Luciano (1991), the investor maximizes the equivalent safe rate of power utility, an optimization objective that also proved useful with constraints on leverage (Grossman and Vila, 1992) and drawdowns (Grossman and Zhou, 1993).
Definition 2.1.
A trading strategy is admissible if its liquidation value is positive, in that:
An admissible strategy is longrun optimal if it maximizes the equivalent safe rate
(2.2) 
over all admissible strategies, where denotes the investor’s relative risk aversion.^{6}^{6}6The limiting case corresponds to logarithmic utility, studied by Taksar, Klass and Assaf (1988), Akian, Sulem and Taksar (2001), and Gerhold, MuhleKarbe and Schachermayer (2011). Theorem 2.2 remains valid for logarithmic utility setting .
Our main result is the following:
Theorem 2.2.
An investor with constant relative risk aversion trades to maximize (2.2). Then, for small transaction costs :

(Equivalent Safe Rate)
For the investor, trading the risky asset with transaction costs is equivalent to leaving all wealth in a hypothetical safe asset, which pays the higher equivalent safe rate:(2.3) where the gap is defined in below.

(Liquidity Premium)
Trading the risky asset with transaction costs is equivalent to trading a hypothetical asset, at no transaction costs, with the same volatility , but with lower expected excess return . Thus, the liquidity premium is(2.4) 
(Trading Policy)
It is optimal to keep the fraction of wealth held in the risky asset within the buy and sell boundaries(2.5) where the risky weights and are computed with ask and bid prices, respectively.^{7}^{7}7This optimal policy is not necessarily unique, in that its longrun performance is also attained by trading arbitrarily for a finite time, and then switching to the above policy. However, in related frictionless models, as the horizon increases, the optimal (finitehorizon) policy converges to a stationary policy, such as the one considered here (see, e.g., Dybvig, Rogers and Back (1999)). Dai and Yi (2009) obtain similar results in a model with proportional transaction costs, formally passing to a stationary version of their control problem PDE.

(Gap)
For , the constant is the unique value for which the solution of the initial value problemalso satisfies the terminal value condition:
In view of the explicit formula for in Lemma A.1 below, this is a scalar equation for . For , the gap vanishes.

(Trading Volume)
Let .^{8}^{8}8The corresponding formulas for are similar but simpler, compare Corollary C.3 and Lemma C.2. Then share turnover, defined as shares traded divided by shares held , has the longterm averageWealth turnover, defined as wealth traded divided by wealth held, has long termaverage^{9}^{9}9The number of shares is written as the difference of the cumulative shares bought (resp. sold), and wealth is evaluated at trading prices, i.e., at the bid price when selling, and at the ask price when buying.

(Asymptotics)
Setting , the following expansions in terms of the bidask spread hold:^{10}^{10}10Algorithmic calculations can deliver terms of arbitrarily high order.(2.6) (2.7) (2.8) (2.9) (2.10) (2.11)
In summary, our optimal trading policy, and its resulting welfare, liquidity premium, and trading volume are all simple functions of investment opportunities (, , ), preferences (), and the gap . The gap does not admit an explicit formula in terms of the transaction cost parameter , but is determined through the implicit relation in , and has the asymptotic expansion in , from which all other asymptotic expansions follow through the explicit formulas.
The frictionless markets with constant investment opportunities in items and of Theorem 2.2 are equivalent to the market with transaction costs in terms of equivalent safe rates. Nevertheless, the corresponding optimal policies are very different, requiring no or incessant rebalancing in the frictionless markets of and , respectively, whereas there is finite positive trading volume in the market with transaction costs.
By contrast, the shadow price, which is key in the derivation of our results, is a fictitious risky asset, with price evolving within the bidask spread, for which the corresponding frictionless market is equivalent to the transaction cost market in terms of both welfare and the optimal policy:
Theorem 2.3.
The policy in Theorem 2.2 and the equivalent safe rate in Theorem 2.2 are also optimal for a frictionless asset with shadow price , which always lies within the bidask spread, and coincides with the trading price at times of trading for the optimal policy. The shadow price satisfies
(2.12) 
for the deterministic functions and given explicitly in Lemma B.2. The state variable represents the logarithm of the ratio of risky and safe positions, which follows a Brownian motion with drift, reflected to remain in the interval , i.e.,
(2.13) 
Here, and are increasing processes, proportional to the cumulative purchases and sales, respectively (cf. (B.13) below). In the interior of the notrade region, that is, when lies in , the numbers of units of the safe and risky asset are constant, and the state variable follows Brownian motion with drift. As reaches the boundary of the notrade region, buying or selling takes place as to keep it within .
In view of Theorem 2.3, trading with constant investment opportunities and proportional transaction costs is equivalent to trading in a fictitious frictionless market with stochastic investment opportunities, which vary with the location of the investor’s portfolio in the notrade region.
3 Implications
3.1 Trading Strategies
Equation (2.5) implies that trading boundaries are symmetric around the frictionless Merton proportion . At first glance, this seems to contradict previous studies (e.g., Liu and Loewenstein (2002), Shreve and Soner (1994)), which emphasize how these boundaries are asymmetric, and may even fail to include the Merton proportion. These papers employ a common reference price (the average of the bid and ask prices) to evaluate both boundaries. By contrast, we express trading boundaries using trading prices (i.e., the ask price for the buy boundary, and the bid price for the sell boundary). This simple convention unveils the natural symmetry of the optimal policy, and explains asymmetries as figments of notation – even in their models. To see this, denote by and the buy and sell boundaries in terms of the ask price. These papers prove the bounds (Shreve and Soner (1994, equations (11.4) and (11.6)) in an infinitehorizon model with consumption and Liu and Loewenstein (2002, equations (22), (23)) in a finitehorizon model)
(3.1) 
With trading prices (i.e., substituting and ) these bounds become
(3.2) 
whence the Merton proportion always lies between and .
To understand the robustness of our optimal policy to intermediate consumption, we compare our trading boundaries with those obtained by Davis and Norman (1990) and Shreve and Soner (1994) in the consumption model of Magill and Constantinides (1976). The asymptotic expansions of Janeček and Shreve (2004) make this comparison straightforward.
With or without consumption, the trading boundaries coincide at the firstorder. This fact has a clear economic interpretation: the separation between consumption and investment, which holds in a frictionless model with constant investment opportunities, is a robust feature of frictionless models, because it still holds, at the first order, even with transaction costs. Put differently, if investment opportunities are constant, consumption has only a second order effect for investment decisions, in spite of the large notrade region implied by transaction costs. Figure 1 shows that our bounds are very close to those obtained in the model of Davis and Norman (1990) for bidask spreads below 1%, but start diverging for larger values.
3.2 Business time and MeanVariance Ratio
In a frictionless market, the equivalent safe rate and the optimal policy are:
This rate depends only on the safe rate and the Sharpe ratio . Investors are indifferent between two markets with identical safe rates and Sharpe ratios, because both markets lead to the same set of payoffs, even though a payoff is generated by different portfolios in the two markets. By contrast, the optimal portfolio depends only on the meanvariance ratio .
With transaction costs, Equation (2.6) shows that the asymptotic expansion of the gap per unit of variance only depends on the meanvariance ratio . Put differently, holding the meanvariance ratio constant, the expansion of is linear in . In fact, not only the expansion but also the exact quantity has this property, since in only depends on .
Consequently, the optimal policy in only depends on the meanvariance ratio , as in the frictionless case. The equivalent safe rate, however, no longer solely depends on the Sharpe ratio : investors are not indifferent between two markets with the same Sharpe ratio, because one market is more attractive than the other if it entails lower trading costs. As an extreme case, in one market it may be optimal lo leave all wealth in the risky asset, eliminating any need to trade. Instead, the formulas in , , and show that, like the gap per variance , the equivalent safe rate, the liquidity premium, and both share and wealth turnover only depend on , when measured per unit of variance. The interpretation is that these quantities are proportional to business time (Ané and Geman, 2000), and the factor of arises from measuring them in calendar time.
In the frictionless limit, the linearity in and the dependence on cancel, and the result depends on the Sharpe ratio alone. For example, the equivalent safe rate becomes^{11}^{11}11The other quantities are trivial: the gap and the liquidity premium become zero, while share and wealth turnover explode to infinity.
3.3 Liquidity Premium
The liquidity premium (Constantinides, 1986) is the amount of expected excess return the investor is ready to forgo to trade the risky asset without transaction costs, as to achieve the same equivalent safe rate. Figure 2 plots the liquidity premium against the spread (left panel) and risk aversion (right panel).
The liquidity premium is exactly zero when the Merton proportion is either zero or one. In these two limit cases, it is optimal not to trade at all, hence no compensation is required for the costs of trading. The liquidity premium is relatively low in the regime of no leverage (), corresponding to , confirming the results of Constantinides (1986), who reports liquidity premia one order of magnitude smaller than trading costs.
The leverage regime (), however, shows a very different picture. As risk aversion decreases below the fullinvestment level , the liquidity premium increases rapidly towards the expected excess return , as lower levels of risk aversion prescribe increasingly high leverage. The costs of rebalancing a levered position are high, and so are the corresponding liquidity premia.
The liquidity premium increases in spite of the increasing width of the notrade region for larger leverage ratios. In other words, even as a less risk averse investor tolerates wider oscillations in the risky weight, this increased flexibility is not enough to compensate for the higher costs required to rebalance a more volatile portfolio.
3.4 Trading Volume
In the empirical literature (cf. Lo and Wang (2000) and the references therein), the most common measure of trading volume is share turnover, defined as number of shares traded divided by shares held or, equivalently, as the value of shares traded divided by value of shares held. In our model, turnover is positive only at the trading boundaries, while it is null inside the notrade region. Since turnover, on average, grows linearly over time, we consider the longterm average of share turnover per unit of time, plotted in Figure 3 against risk aversion. Turnover is null at the fullinvestment level , as no trading takes place in this case. Lower levels of risk aversion generate leverage, and trading volume increases rapidly, like the liquidity premium.
Share turnover does not decrease to zero as the risky weight decreases to zero for increasing risk aversion . On the contrary, the first term in the asymptotic formula converges to a finite level. This phenomenon arises because more risk averse investors hold less risky assets (reducing volume), but also rebalance more frequently (increasing volume). As risk aversion increases, neither of these effects prevails, and turnover converges to a finite limit.
To better understand these properties, consider wealth turnover, defined as the value of shares traded, divided by total wealth (not by the value of shares held).^{12}^{12}12Technically, wealth is valued at the ask price at the buying boundary, and at the bid price at the selling boundary. Share and wealth turnover are qualitatively similar for low risk aversion, as the risky weight of wealth is larger, but they diverge as risk aversion increases and the risky weight declines to zero. Then, wealth turnover decreases to zero, whereas share turnover does not.
The levels of trading volume observed empirically imply very low values of risk aversion in our model. For example, Lo and Wang (2000) report in the NYSEAMEX an average weekly turnover of 0.78% between 19621996, which corresponds to an approximate annual turnover above 40%. As Figure 3 shows, such a high level of turnover requires a risk aversion below 2, even for a very small spread of . Such a value cannot be interpreted as risk aversion of a representative investor, because it would imply a leveraged position in the stock market, which is inconsistent with equilibrium. This phenomenon intensifies in the last two decades. As shown by Figure 4 turnover increases substantially from 1993 to 2010, with monthly averages of 20% typical from 2007 on, corresponding to an annual turnover of over 240%.
The overall implication is that portfolio rebalancing can generate substantial trading volume, but the model explains the trading volume observed empirically only with low risk aversion and high leverage. In a numerical study with risk aversion of six and spreads of 2%, Lynch and Tan (2011) also find that the resulting trading volume is too low, even allowing for labor income and predictable returns, and obtain a condition on the wealthincome ratio, under which the trading volume is the same order of magnitude reported by empirical studies. Our analytical results are consistent with their findings, but indicate that substantially higher volume can be explained with lower risk aversion, even in the absence of labor income.
3.5 Volume, Spreads and the Liquidity Premium
Liquidity  Share  Relative  

Period  Premium  Turnover  Spread 
19921995  0.066%  7%  1.20% 
19962000  0.083%  11%  0.97% 
20012005  0.038%  13%  0.37% 
20062010  0.022%  21%  0.12% 
The analogies between the comparative statics of the liquidity premium and trading volume suggest a close connection between these quantities. An inspection of the asymptotic formulas unveils the following relations:
(3.3) 
These two relations have the same meaning: the welfare effect of small transaction costs is proportional to trading volume times the spread. The constant of proportionality 3/4 is universal, that is, independent of both investment opportunities (, , ) and preferences ().
In the first formula, the welfare effect is measured by the liquidity premium, that is in terms of the risky asset. Likewise, trading volume is expressed as share turnover, which also focuses on the risky asset alone. By contrast, the second formula considers the decrease in the equivalent safe rate and wealth turnover, two quantities that treat both assets equally. In summary, if both welfare and volume are measured consistently with each other, the welfare effect approximately equals volume times the spread, up to the universal factor 3/4.
Figure 4 plots the spread, share turnover, and the liquidity premium implied by the first equation in (3.3). As in Lo and Wang (2000), the spread and share turnover are capitalizationweighted averages of all securities in the Center for Research on Security Prices (CRSP) monthly stocks database with share codes 10 and 11, and with nonzero bid, ask, volume and share outstanding. While turnover figures are available before 1992, separate bid and ask prices were not recorded until then, thereby preventing a reliable estimation of spreads for earlier periods.
Spreads steadily decline in the observation period, dropping by almost an order of magnitude after stock market decimalization of 2001. At the same time, trading volume substantially increases from a typical monthly turnover of 6% in the early 1990s to over 20% in the late 2000s. The implied liquidity premium also declines with spreads after decimalization, but less than the spread, in view of the increase in turnover. During the months of the financial crisis in late 2008, the implied liquidity premium rises sharply, not because of higher volumes, but because spreads widen substantially. Thus, although this implied liquidity premium is only a coarse estimate, it has advantages over other proxies, because it combines information on both prices and quantities, and is supported by a model.
3.6 Finite Horizons
The trading boundaries in this paper are optimal for a long investment horizon, but are also approximately optimal for finite horizons. The following theorem, which complements the main result, makes this point precise:
Theorem 3.1.
Fix a time horizon . Then the finitehorizon equivalent safe rate of any strategy satisfies the upper bound
(3.4)  
and the finitehorizon equivalent safe rate of our longrun optimal strategy satisfies the lower bound  
(3.5) 
In particular, for the same unlevered initial position (), the equivalent safe rates of and of the optimal policy for horizon differ by at most
(3.6) 
This result implies that the horizon, like consumption, only has a second order effect on portfolio choice with transaction costs, because the finitehorizon equivalent safe rate matches, at the order , the equivalent safe rate of the stationary longrun optimal policy. This result recovers, in particular, the firstorder asymptotics for the finitehorizon value function obtained by Bichuch (2011, Theorem 4.1). In addition, Theorem 3.1 provides explicit estimates for the correction terms of order arising from liquidation costs. Indeed, is the maximum rate achieved by trading optimally. The remaining terms arise due to the transient influence of the initial endowment, as well as the costs of the initial transaction, which takes place if the initial position lies outside the notrade region, and of the final portfolio liquidation. These costs are of order because they are incurred only once, and hence defrayed by a longer trading period. By contrast, portfolio rebalancing generates recurring costs, proportional to the horizon, and their impact on the equivalent safe rate does not decline as the horizon increases.
Even after accounting for all such costs in the worstcase scenario, the bound in (3.6) shows that their combined effect on the equivalent safe rate is lower than the spread , as soon as the horizon exceeds , that is four years in the absence of leverage. Yet, this bound holds only up to a term of order , so it is worth comparing it with the exact bounds in equations (B.20)(B.21), from which (3.4) and (3.5) are obtained.
The exact bounds in Figure 5 show that, for typical parameter values, the loss in equivalent safe rate of the longrun optimal strategy is lower than the spread even for horizons as short as 18 months, and quickly declines to become ten times smaller, for horizons close to ten years. In summary, the longrun approximation is a useful modeling device that makes the model tractable, and the resulting optimal policies are also nearly optimal even for horizons of a few years.
4 Heuristic Solution
This section contains an informal derivation of the main results. Here, formal arguments of stochastic control are used to obtain the optimal policy, its welfare, and their asymptotic expansions.
4.1 Transaction Costs Market
For a trading strategy , again write the number of risky shares as the difference of the cumulated units purchased and sold, and denote by
the values of the safe and risky positions in terms of the ask price . Then, the selffinancing condition (2.1), and the dynamics of and imply
Consider the maximization of expected power utility from terminal wealth at time ,^{13}^{13}13For a fixed horizon , one would need to specify whether terminal wealth is valued at bid, ask, or at liquidation prices, as in Definition 2.1. In fact, since these prices are within a constant positive multiple of each other, which price is used is inconsequential for a longrun objective. For the same reason, the terminal condition for the finite horizon value function does not have to be satisfied by the stationary value function, because its effect is negligible. and denote by its value function, which depends on time and the value of the safe and risky positions. Itô’s formula yields:
where the arguments of the functions are omitted for brevity. By the martingale optimality principle of stochastic control (cf. Fleming and Soner (2006)), the value function must be a supermartingale for any choice of the cumulative purchases and sales . Since these are increasing processes, it follows that and , that is
In the interior of this “notrade region”, where the number of risky shares remains constant, the drift of cannot be positive, and must become zero for the optimal policy:^{14}^{14}14Alternatively, this equation can be obtained from standard arguments of singular control, cf. Fleming and Soner (2006, Chapter VIII).
(4.1) 
To simplify further, note that the value function must be homogeneous with respect to wealth, and that — in the long run — it should grow exponentially with the horizon at a constant rate. These arguments lead to guess^{15}^{15}15This guess assumes that the cash position is strictly positive, , which excludes leverage. With leverage, factoring out leads to analogous calculations. In either case, under the optimal policy, the ratio always remains either strictly positive, or strictly negative, never to pass through zero. that for some to be found. Setting , the above equation reduces to
(4.2) 
Assuming that the notrade region coincides with some interval to be determined, and noting that at the left inequality in (4.2) holds as equality, while at the right inequality holds as equality, the following free boundary problem arises:
(4.3)  
(4.4)  
(4.5) 
These conditions are not enough to identify the solution, because they can be matched for any choice of the trading boundaries . The optimal boundaries are the ones that also satisfy the smoothpasting conditions (cf. Beneš, Shepp and Witsenhausen (1980); Dumas (1991)), formally obtained by differentiating (4.4) and (4.5) with respect to and , respectively:
(4.6)  
(4.7) 
In addition to the reduced value function , this system requires to solve for the excess equivalent safe rate and the trading boundaries and . Substituting (4.6) and (4.4) into (4.3) yields (cf. Dumas and Luciano (1991))
Setting , and factoring out , it follows that
Note that is the risky weight when it is time to buy, and hence the risky position is valued at the ask price. The same argument for shows that the other solution of the quadratic equation is , which is the risky weight when it is time to sell, and hence the risky position is valued at the bid price. Thus, the optimal policy is to buy when the “ask” fraction falls below , sell when the “bid” fraction rises above , and do nothing in between. Since and solve the same quadratic equation, they are related to via
It is convenient to set , because without transaction costs. We call the gap, since in a frictionless market, and, as increases, all variables diverge from their frictionless values. Put differently, to compensate for transaction costs, the investor would require another asset, with expected return and volatility , which trades without frictions and is uncorrelated with the risky asset.^{16}^{16}16Recall that in a frictionless market with two uncorrelated assets with returns and , both with volatility , the maximum Sharpe ratio is . That is, squared Sharpe ratios add across orthogonal shocks. With this notation, the buy and sell boundaries are just
In other words, the buy and sell boundaries are symmetric around the classical frictionless solution . Since are identified by in terms of , it now remains to find . After deriving and , the boundaries in the problem (4.3)(4.5) are no longer free, but fixed. With the substitution
the boundary problem (4.3)(4.5) reduces to a Riccati ODE
(4.8)  
(4.9)  
(4.10) 
where
(4.11) 
For each , the initial value problem (4.8)(4.9) has a solution , and the correct value of is identified by the second boundary condition (4.10).
4.2 Asymptotics
The equation (4.10) does not have an explicit solution, but it is possible to obtain an asymptotic expansion for small transaction costs () using the implicit function theorem. To this end, write the boundary condition (4.10) as , where:
Of course, corresponds to the frictionless case. The implicit function theorem then suggests that around zero follows the asymptotics , but the difficulty is that , because is not of order . Heuristic arguments (Shreve and Soner, 1994; Rogers, 2004) suggest that is of order ^{17}^{17}17Since is proportional to the width of the notrade region , the question is why the latter is of order . The intuition is that a notrade region of width around the frictionless optimum leads to transaction costs of order (because the time spent near the boundaries is approximately inversely proportional to the length of the interval), and to a welfare cost of the order (because the region is centered around the frictionless optimum, hence the linear welfare cost is zero). Hence, the total cost is of the order , and attains its minimum for . Thus, setting and , and computing the derivatives of the explicit formula for (cf. Lemma A.1) shows that:
As a result:
The asymptotic expansions of all other quantities then follow by Taylor expansion.
5 Conclusion
In a tractable model of transaction costs with one safe and one risky asset and constant investment opportunities, we have computed explicitly the optimal trading policy, its welfare, liquidity premium, and trading volume, for an investor with constant relative risk aversion and a long horizon.
The trading boundaries are symmetric around the Merton proportion, if each boundary is computed with the corresponding trading price. Both the liquidity premium and trading volume are small in the unlevered regime, but become substantial in the presence of leverage. For a small bidask spread, the liquidity premium is approximately equal to share turnover times the spread, times the universal constant 3/4.
Trading boundaries depend on investment opportunities only through the mean variance ratio. The equivalent safe rate, the liquidity premium, and trading volume also depend only on the mean variance ratio if measured in business time.
Appendix
Appendix A Explicit Formulas and their Properties
We now show that the candidate for the reduced value function and the quantity are indeed welldefined for sufficiently small spreads. The first step is to determine, for a given small , an explicit expression for the solution of the ODE (4.8), complemented by the initial condition (4.9).
Lemma A.1.
Let . Then for sufficiently small , the function
with
is a local solution of
(A.1) 
Moreover, is increasing (resp. decreasing) for (resp. ).
Proof.
The first part of the assertion is easily verified by taking derivatives, noticing that the case distinctions distinguish between the different signs of the discriminant
of the Riccati equation (A.1) for sufficiently small . Indeed, in the second case the discriminant is positive for sufficiently small . The first and third case correspond to a negative discriminant, as well as and , respectively, for sufficiently small , so that the function is welldefined in each case.
The second part of the assertion follows by inspection of the explicit formulas. ∎
Next, establish that the crucial constant , which determines both the notrade region and the equivalent safe rate, is welldefined.
Lemma A.2.
Let and be defined as in Lemma A.1, and set
Then, for sufficiently small , there exists a unique solution of
(A.2) 
As , it has the asymptotics