transaction costs, trading volume, and the liquidity premium · transaction costs, trading volume,...

Working Paper Series

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National Centre of Competence in Research Financial Valuation and Risk Management

Working Paper No. 727

Transaction Costs, Trading Volume, and the Liquidity

Premium

Stefan Gerhold Paolo Guasoni

Walter Schachermayer Johannes Muhle-Karbe

First version: August 2011 Current version: October 2011

This research has been carried out within the NCCR FINRISK project on

“Mathematical Methods in Financial Risk Management”

___________________________________________________________________________________________________________

Transaction Costs, Trading Volume,and the Liquidity Premium∗

Stefan Gerhold † Paolo Guasoni ‡ Johannes Muhle-Karbe §

Walter Schachermayer ¶

October 5, 2011

Abstract

In a market with one safe and one risky asset, an investor with a long horizon, constantinvestment opportunities, and constant relative risk aversion trades with small proportionaltransaction costs. We derive explicit formulas for the optimal investment policy, its impliedwelfare, liquidity premium, and trading volume. At the first order, the liquidity premiumequals the spread, times share turnover, times a universal constant. Results are robust toconsumption and finite horizons. We exploit the equivalence of the transaction cost market toanother frictionless market, with a shadow risky asset, in which investment opportunities arestochastic. The shadow price is also found explicitly.

Mathematics Subject Classification: (2010) 91G10, 91G80.JEL Classification: G11, G12.Keywords: transaction costs, long-run, portfolio choice, liquidity premium, trading volume.

∗For helpful comments, we thank Maxim Bichuch, George Constantinides, Aleš Černý, Mark Davis, IoannisKaratzas, Marcel Nutz, Scott Robertson, Johannes Ruf, Mihai Sirbu, Mete Soner, Gordan Zitković, and seminarparticipants at Ascona, MFO Oberwolfach, Columbia University, Princeton University, University of Oxford, CAUKiel, London School of Economics, University of Michigan, TU Vienna, and the ICIAM meeting in Vancouver.

†Technische Universität Wien, Institut für Wirtschaftsmathematik, Wiedner Hauptstrasse 8-10, A-1040 Wien,Austria, email [email protected]. Partially supported by the Austrian Federal Financing Agency and theChristian-Doppler-Gesellschaft (CDG).

‡Boston University, Department of Mathematics and Statistics, 111 Cummington Street, Boston, MA 02215, USA,and Dublin City University, School of Mathematical Sciences, Glasnevin, Dublin 9, Ireland, email [email protected] supported by NSF (DMS-0807994 and DMS-1109047), SFI (07/MI/008, 07/SK/M1189, 08/SRC/FMC1389),and the European Commission (RG-248896).

§ETH Zürich, Departement Mathematik, Rämistrasse 101, CH-8092, Zürich, Switzerland, [email protected]. Partially supported by the National Centre of Competence in Research “Fi-nancial Valuation and Risk Management” (NCCR FINRISK), Project D1 (Mathematical Methods in Financial RiskManagement), of the Swiss National Science Foundation (SNF).

¶Universität Wien, Fakultät für Mathematik, Nordbergstrasse 15, A-1090 Wien, Austria, [email protected]. Partially supported by the Austrian Science Fund (FWF) under grantP19456, the European Research Council (ERC) under grant FA506041, the Vienna Science and Technology Fund(WWTF) under grant MA09-003, and by the Christian-Doppler-Gesellschaft (CDG).

1

1 Introduction

If risk aversion and investment opportunities are constant — and frictions are absent — investorsshould hold a constant mix of safe and risky assets (Markowitz, 1952; Merton, 1969, 1971). Transac-tion costs substantially change this statement, casting some doubt on its far-reaching implications.1

Even the small spreads that are present in the most liquid markets entail wide oscillations in port-folio weights, which imply variable risk premia.

This paper studies a tractable benchmark of portfolio choice under transaction costs, withconstant investment opportunities, summarized by a safe rate r, and a risky asset with volatilityσ and expected excess return µ > 0, which trades at a bid (selling) price (1 − ε)St equal to aconstant fraction (1 − ε) of the ask (buying) price St. Our analysis is based on the model ofDumas and Luciano (1991), which concentrates on long-run asymptotics to gain in tractability. Intheir framework, we find explicit solutions for the optimal policy, welfare, liquidity premium,2 andtrading volume, in terms of model parameters, and of an additional quantity, the gap, identifiedas the solution to a scalar equation. For all these quantities, we derive closed-form asymptotics, interms 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 (ShTu), the liquidity premium (LiPr), and the bid-ask spread εsatisfy the following asymptotic relation:

LiPr ≈ 34ε ShTu . (1.1)

This relation is universal, as it involves neither market nor preference parameters. Also, becauseit links the liquidity premium, which is unobservable, with spreads and share turnover, which areobservable, 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 no-leverage 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 leveredpositions entails high transaction costs, even under the optimal trading policy.

Third, we obtain the first continuous-time benchmark for trading volume, with explicit formulasfor share and wealth turnover. Trading volume is an elusive quantity for frictionless models, inwhich turnover is typically infinite in any time interval.3 In the absence of leverage, our resultsimply low trading volume compared to the levels observed in the market. Of course, our model canonly explain trading generated by portfolio rebalancing, and not by other motives such as markettiming, hedging, and life-cycle investing.

Moreover, welfare, the liquidity premium, and trading volume depend on the market parameter(µ,σ) only through the mean-variance ratio µ/σ2 if measured in business time, that is, using a

1Constantinides (1986) finds that “transaction costs have a first-order effect on the assets’ demand.” Liu andLoewenstein (2002) note that “even small transaction costs lead to dramatic changes in the optimal behavior for aninvestor: from continuous trading to virtually buy-and-hold strategies.” Luttmer (1996) shows how small transactioncosts help resolve asset pricing puzzles.

2That is, the amount of excess return the investor is ready to forgo to trade the risky asset without transactioncosts.

3The empirical literature has long been aware of this theoretical vacuum: Gallant, Rossi and Tauchen (1992) reckonthat “The intrinsic difficulties of specifying plausible, rigorous, and implementable models of volume and prices arethe reasons for the informal modeling approaches commonly used.” Lo and Wang (2000) note that “although mostmodels of asset markets have focused on the behavior of returns [...] their implications for trading volume havereceived far less attention.”

2

clock that ticks at the speed of today’s market’s variance σ2. Thus, in usual calendar time, allthese quantities are functions of µ/σ2, multiplied by the variance σ2.

Our main implication for portfolio choice is that a symmetric, stationary policy is optimal fora long horizon, and it is robust, at the first order, both to intermediate consumption, and to afinite horizon. Indeed, we show that the no-trade region is perfectly symmetric with respect to theMerton proportion π∗ = µ/γσ2, if trading boundaries are expressed with trading prices, that is, ifthe buy boundary π− is computed from the ask price, and the sell boundary π+ from the bid price.

Since in a frictionless market the optimal policy is independent both of intermediate consump-tion and of the horizon (Merton, 1971), our results entail that these two features are robust tosmall frictions. However plausible these conclusions may seem, the literature so far has offereddiverse views on these issues (cf. Davis and Norman (1990); Dumas and Luciano (1991); Liu andLoewenstein (2002)). More importantly, robustness to the horizon implies that the long-horizonapproximation, made for the sake of tractability, is reasonable and relevant. For typical parametervalues, 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 withtransaction costs and constant investment opportunities, and another shadow market, withouttransaction costs, but with stochastic investment opportunities driven by a state variable. Thisstate variable is the ratio between the investor’s risky and safe weights, which tracks the location ofthe portfolio within the trading boundaries, and affects both the volatility and the expected returnof the shadow risky asset.

The paper is organized as follows: Section 2 introduces the portfolio choice problem and statesthe main results. The model’s main implications are discussed in Section 3, and the main resultsare 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 r, i.e. S0t = ert, and a risky asset,

trading at ask (buying) price St following geometric Brownian motion,

dSt/St = (µ+ r)dt+ σdWt.

Here, Wt is a standard Brownian motion, µ > 0 is the expected excess return,4 and σ > 0 is thevolatility. The corresponding bid (selling) price is (1− ε)St, where ε ∈ (0, 1) represents the relativebid-ask spread.

A self-financing trading strategy is two-dimensional, predictable process (ϕ0t ,ϕt) of finite vari-ation, such that ϕ0t and ϕt represent the number of units in the safe and risky asset at time t,and the initial number of units is (ϕ00− ,ϕ0−) ∈ R

2+\{0, 0}. Writing ϕt = ϕ

↑t − ϕ

↓t as the difference

between the cumulative number of shares bought (ϕ↑t ) and sold (ϕ↓t ) by time t, the self-financing

condition relates the dynamics of ϕ0 and ϕ via

dϕ0t = −

St

S0tdϕ

↑t + (1− ε)

St

S0tdϕ

↓t . (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).

4A negative excess return leads to a similar treatment, but entails buying as prices rise, rather than fall. For thesake of clarity, the rest of the paper concentrates on the more relevant case of a positive µ.

3

Definition 2.1. A trading strategy (ϕ0t ,ϕt) is admissible if its liquidation value is positive, in that:

Ξϕt = ϕ0tS

0t + (1− ε)Stϕ+t − ϕ−t St ≥ 0, a.s. for all t ≥ 0.

An admissible strategy (ϕ0t ,ϕt) is long-run optimal if it maximizes the equivalent safe rate

lim infT→∞

1

TlogE

�(Ξϕt )

1−γ� 11−γ (2.2)

over all admissible strategies, where 1 �= γ > 0 denotes the investor’s relative risk aversion.5

Our main result is the following:

Theorem 2.2. An investor with constant relative risk aversion γ > 0 trades to maximize (2.2).Then, for small transaction costs ε > 0:

i) (Equivalent Safe Rate)For the investor, trading the risky asset with transaction costs is equivalent to leaving all wealthin a hypothetical safe asset, which pays the higher equivalent safe rate:

ESR = r +µ2 − λ2

2γσ2, (2.3)

where the gap λ is defined in iv) below.

ii) (Liquidity Premium)Trading the risky asset with transaction costs is equivalent to trading a hypothetical asset, at notransaction costs, with the same volatility σ, but with lower expected excess return

�µ2 − λ2.

Thus, the liquidity premium isLiPr = µ−

�µ2 − λ2. (2.4)

iii) (Trading Policy)It is optimal to keep the fraction of wealth in the risky asset within the buy and sell boundaries

π− =µ− λγσ2

, π+ =µ+ λ

γσ2, (2.5)

where π− and π+ are computed with ask and bid prices, respectively.6

iv) (Gap)λ is the unique value for which the solution of the initial value problem

w�(x) + (1− γ)w(x)2 +

�2µ

σ2− 1

�w(x)− γ

�µ− λγσ2

��µ+ λ

γσ2

�= 0

w(0) =µ− λγσ2

,

5The limiting case γ → 1 corresponds to logarithmic utility, studied by Taksar, Klass and Assaf (1988) andGerhold, Muhle-Karbe and Schachermayer (2011b). Theorem 2.2 remains valid for logarithmic utility setting γ = 1.

6This optimal policy is not necessarily unique, in that its long-run performance is also attained by trading ar-bitrarily for a finite time, and then switching to the above policy. However, in related frictionless models, as thehorizon increases, the optimal (finite-horizon) 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 proportionaltransaction costs, formally passing to a stationary version of their control problem PDE.

4

also satisfies the terminal value condition:

w

�log

�u(λ)

l(λ)

��=

µ+ λ

γσ2, where

u(λ)

l(λ)=

1

(1− ε)(µ+ λ)(µ− λ− γσ2)(µ− λ)(µ+ λ− γσ2) .

In view of the explicit formula for w(x,λ) in Lemma B.1 below, this is a scalar equation for λ.

v) (Trading Volume)

Share turnover, defined as shares traded d||ϕ||t = dϕ↑t + dϕ↓t divided by shares held |ϕt|, has

the long-term average

ShTu = limT→∞

1

T

� T

0

d�ϕ�t|ϕt|

=σ2

2

�2µ

σ2− 1

��1− π−

(u(λ)/l(λ))2µσ2

−1 − 1− 1− π+

(u(λ)/l(λ))1−2µσ2 − 1

�.

Wealth turnover, defined as wealth traded divided by wealth held, has long term-average:7

WeTu = limT→∞

1

T

�� T

0

(1− ε)Stdϕ↓tϕ0tS

0t + ϕt(1− ε)St

+

� T

0

Stdϕ↑t

ϕ0tS0t + ϕtSt

�

=σ2

2

�2µ

σ2− 1

��π− (1− π−)

(u(λ)/l(λ))2µσ2

−1 − 1− π+ (1− π+)

(u(λ)/l(λ))1−2µσ2 − 1

�.

vi) (Asymptotics)Setting π∗ = µ/γσ2, the following expansions in terms of the bid-ask spread ε hold:8

λ = γσ2�

3

4γπ2∗ (1− π∗)

2�1/3

ε1/3 +O(ε). (2.6)

ESR = r +µ2

2γσ2− γσ

2

2

�3

4γπ2∗ (1− π∗)

2�2/3

ε2/3 +O(ε4/3). (2.7)

LiPr =µ

2π2∗

�3

4γπ2∗ (1− π∗)

2�2/3

ε2/3 +O(ε4/3). (2.8)

π± = π∗ ±�

3

4γπ2∗ (1− π∗)

2�1/3

ε1/3 +O(ε). (2.9)

ShTu =σ2

2(1− π∗)2π∗

�3

4γπ2∗ (1− π∗)

2�−1/3

ε−1/3 +O(ε1/3). (2.10)

WeTu =γσ2

3

�3

4γπ2∗(1− π∗)2

�2/3ε−1/3 +O(ε1/3). (2.11)

In summary, our optimal trading policy, and its resulting welfare, liquidity premium, and tradingvolume are all simple functions of investment opportunities (r, µ, σ), preferences (γ), and the gapλ. The gap does not admit an explicit formula in terms of the transaction cost parameter ε, butis determined through the implicit relation in iii), and has the asymptotic expansion in v), fromwhich all other asymptotic expansions follow through the explicit formulas.

7The number of shares is written as the difference ϕt = ϕ↑t −ϕ

↓t of the cumulative shares bought (resp. sold), and

wealth is evaluated at trading prices, i.e., at the bid price (1−ε)St when selling, and at the ask price St when buying.8Algorithmic calculations can deliver terms of arbitrarily high order.

5

The frictionless markets with constant investment opportunities in items i) and ii) of The-orem 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 incessant rebalancingin the frictionless markets but only finite trading volume with transaction costs.

By contrast, the shadow price, which is key in the derivation of our results, is a fictitious riskyasset, with price evolving within the bid-ask spread, that is equivalent to the transaction costmarket in terms of both welfare and the optimal policy:

Theorem 2.3. The policy in Theorem 2.2 iii) and the equivalent safe rate in Theorem 2.2 i)are also optimal for a frictionless asset with shadow price S̃, which always lies within the bid-askspread, and coincides with the trading price at times of trading for the optimal policy. The shadowprice satisfies

dS̃t/S̃t = (µ̃(Yt) + r)dt+ σ̃(Yt)dWt, (2.12)

for the deterministic functions µ̃(·) and σ̃(·) given explicitly in Lemma C.1. The state variableYt = log(ϕtSt/(lϕ0tS

0t )) represents the logarithm of the ratio of risky and safe positions, which

follows a Brownian motion with drift, reflected to remain in the interval [0, log(u(λ)/l(λ))], i.e.,

dYt = (µ− σ2/2)dt+ σdWt + dLt − dUt. (2.13)

Here, Lt and Ut are increasing processes, proportional to the cumulative purchases and sales,respectively (cf. (C.10) below). In the interior of the no-trade region, i.e., when Yt lies in(0, log(u(λ)/l(λ))), the numbers of units of the safe and risky asset are constant, and the statevariable Yt follows Brownian motion with drift. As Yt reaches the boundary of the no-trade region,buying or selling takes place as to keep it within [0, log(u(λ)/l(λ))].

In view of Theorem 2.3, trading with constant investment opportunities and proportional trans-action costs is equivalent to trading in a fictitious frictionless market with stochastic investmentopportunities, which vary with the location of the investor’s portfolio in the no-trade region.

3 Implications

3.1 Trading Strategies

Equation (2.5) implies that trading boundaries are symmetric around the frictionless Merton pro-portion π∗ = µ/γσ2. At first glance, this result seems to contradict previous studies (e.g., Liu andLoewenstein (2002)), which emphasize how these boundaries are asymmetric, and may even fail toinclude the Merton proportion. These papers employ a common reference price (the average of thebid and ask prices) to evaluate both boundaries. By contrast, we express trading boundaries usingtrading prices (i.e., the ask price for the buy boundary, and the bid price for the sell boundary). Thissimple convention unveils the natural symmetry of the optimal policy, and explains asymmetries aseither finite-horizon effects, or as figments of notation. Because our model excludes intermediateconsumption, 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). Theasymptotic expansions of Janeček and Shreve (2004) make this comparison straightforward.

With or without consumption, the trading boundaries coincide at the first-order. This fact hasa clear economic interpretation: the separation between consumption and investment, which holdsin a frictionless model with constant investment opportunities, is a robust feature of frictionlessmodels, because it still holds, at the first order, even with transaction costs. Put differently, ifinvestment opportunities are constant, consumption has only a second order effect for investment

6

0.00 0.02 0.04 0.06 0.08 0.100.50

0.55

0.60

0.65

0.70

0.75

0.0001 0.001 0.01 0.1

0.55

0.60

0.65

0.70

0.75

Figure 1: Buy (lower) and sell (upper) boundaries (vertical axis, as risky weights) as functionsof the spread ε, in linear scale (left panel) and cubic scale (right panel). The plot compares theapproximate weights from the first term of the expansion (dotted line), the exact optimal weights(solid line), and the boundaries found by Davis and Norman (1990) in the presence of consumption(dashed line). Parameters are µ = 8%,σ = 16%, γ = 5, and a zero discount rate for consumption(for the dashed line).

decisions, in spite of the large no-trade region implied by transaction costs. Figure 1 shows thatour bounds are very close to those obtained in the model of Davis and Norman (1990) for bid-askspreads below 1%, but start diverging for larger values.

3.2 Business time and mean-variance ratio

In a frictionless market, the equivalent safe rate and the optimal policy are:

ESR = r +1

2γ

�µ

σ

�2and π∗ =

µ

γσ2.

This rate depends only on the safe rate r and the Sharpe ratio µ/σ. Investors are indifferentbetween two markets with identical safe rates and Sharpe ratios, because both markets lead to thesame 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 mean-variance ratio µ/σ2.

With transaction costs, Equation (2.6) shows that the asymptotic expansion of the gap per unitof variance λ/σ2 only depends on the mean-variance ratio µ/σ2. Put differently, holding the mean-variance ratio µ/σ2 constant, the expansion of λ is linear in σ2. In fact, not only the expansionbut also the exact quantity has this property, since λ/σ2 in iv) only depends on µ/σ2.

Consequently, the optimal policy in iii) only depends on the mean-variance ratio µ/σ2, as inthe frictionless case. The equivalent safe rate, however, no longer solely depends on the Sharperatio µ/σ: investors are not indifferent between two markets with the same Sharpe ratio, becauseone market is more attractive than the other if it entails lower trading costs. As an extreme case, inone market it may be optimal lo leave all wealth in the risky asset, eliminating any need to trade.Instead, the formulas in i), ii), and v) show that, like the gap per variance λ/σ2, the equivalentsafe rate, the liquidity premium, and both share and wealth turnover only depend on µ/σ2, whenmeasured per unit of variance. The interpretation is that these quantities are proportional tobusiness time σ2t (Ané and Geman, 2000), and the factor of σ2 arises from measuring them incalendar time.

In the frictionless limit, the linearity in σ2 and the dependence on µ/σ2 confound each other,

7

0.00 0.01 0.02 0.03 0.04 0.050.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0 2 4 6 8 100.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

Figure 2: Left panel: liquidity premium (vertical axis) against the spread ε, for risk aversion γ equalto 5 (solid), 1 (long dashed), and 0.5 (short dashed). Right panel: liquidity premium (vertical axis)against risk aversion γ, for spread ε = 0.01% (solid), 0.1% (long dashed), 1% (short dashed), and10% (dotted). Parameters are µ = 8% and σ = 16%.

and the result depends on the Sharpe ratio alone. For example, the equivalent safe rate becomes9

σ2

2

�µ

σ2

�2=

1

2

�µ

σ

�2.

3.3 Liquidity Premium

The liquidity premium (Constantinides, 1986) is the amount of expected excess return the investoris ready to forgo to trade the risky asset without transaction costs, as to achieve the same equivalentsafe rate. Figure 2 plots the liquidity premium against the spread ε (left panel) and risk aversionγ (right panel).

The liquidity premium is exactly zero when either the risky or the safe weight become one,corresponding respectively to γ = µ/σ2 and γ = ∞. In these two limit cases, it is optimal not totrade at all, hence no compensation is required for the costs of trading. The liquidity premium isrelatively low in the regime of no leverage, (0 < π∗ < 1), corresponding to γ > µ/σ2, confirmingthe results of Constantinides (1986), who reports liquidity premia one order of magnitude smallerthan trading costs.

The leverage regime (γ < µ/σ2), however, shows a very different picture. As risk aversiondecreases below the full-investment level γ = µ/σ2, the liquidity premium increases rapidly toinfinity, as lower levels of risk aversion prescribe increasingly high leverage. The costs of rebalancinga levered position are high, and so are the corresponding liquidity premia.

The liquidity premium increases in spite of the increasing width of the no-trade region for largerleverage ratios. In other words, even as a less risk averse investor tolerates wider oscillations in therisky weight, this increased flexibility is not enough to compensate for the higher costs required torebalance a more volatile portfolio.

3.4 Trading Volume

In the empirical literature (cf. Lo and Wang (2000) and the references therein), the most commonmeasure of trading volume is share turnover, defined as number of shares traded divided by shares

9The other quantities are confounded to the point of being trivial: the gap and the liquidity premium becomezero, while share and wealth turnover explode to infinity.

8

0 2 4 6 8 100.0

0.1

0.2

0.3

0.4

0.5

0 2 4 6 8 100.0

0.1

0.2

0.3

0.4

0.5

Figure 3: Trading volume (vertical axis, annual fractions traded), as share turnover (left panel) andwealth turnover (right panel), against risk aversion (horizontal axis), for spread ε = 0.01% (solid),0.1% (long dashed), 1% (short dashed), and 10% (dotted). Parameters are µ = 8% and σ = 16%.

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 no-trade region. Sinceturnover, on average, grows linearly over time, we consider the long-term average of share turnoverper unit of time, plotted in Figure 3 against risk aversion. Turnover is null at the full-investmentlevel γ = µ/σ2, as no trading takes place in this case. Lower levels of risk aversion generate leverage,and trading volume increases rapidly, like the liquidity premium.

Unlike the liquidity premium, share turnover does not decrease to zero as the risky weightdecreases to zero, i.e., as risk aversion grows to infinity. On the contrary, the first term in theasymptotic formula converges to a finite level. This phenomenon arises because more risk averseinvestors hold less risky assets (reducing volume), but also rebalance more frequently (increasingvolume). As risk aversion increases, neither of these effects prevails, and turnover converges to afinite limit.

To better understand these properties, consider wealth turnover, defined as the value of sharestraded, divided by total wealth (not by the value of shares held).10 Share and wealth turnover arequalitatively similar for low risk aversion, as the risky weight of wealth is larger, but they divergeas risk aversion increases and the risky weight declines to zero. Then, wealth turnover decreases tozero, like the liquidity premium, whereas share turnover does not.

The levels of trading volume observed empirically imply very low values of risk aversion in ourmodel. For example, Lo and Wang (2000) report in the NYSE-AMEX an average weekly turnoverof 0.78% between 1962-1996, which corresponds to an approximate annual turnover above 40%. AsFigure 3 shows, such a high level of turnover requires a risk aversion below 2, even for a very smallspread of ε = 0.01%. This phenomenon intensifies in the last two decades. As shown by Figure4 turnover increases substantially from 1993 to 2010, with monthly averages of 20% typical from2007 on, corresponding to an annual turnover of over 240%.

The overall implication is that portfolio rebalancing can generate substantial trading volume,but not enough to explain all the trading volume observed empirically, except for implausibly lowrisk aversion and high leverage.

9

Liquidity Share RelativePeriod Premium Turnover Spread

1992-1995 0.066% 7% 1.20%1996-2000 0.083% 11% 0.97%2001-2005 0.038% 13% 0.37%2006-2010 0.022% 21% 0.12%

Figure 4: Left panel: share turnover (top), spread (center), and implied liquidity premium (bottom)in logarithmic scale, from 1992 to 2010. Right panel: monthly averages for share turnover, spread,and implied liquidity premium over subperiods. Spread and turnover are capitalization-weightedaverages across securities in the monthly CRSP database with share codes 10, 11 that have nonzerobid, ask, volume and shares outstanding.

3.5 Volume, Spreads and the Liquidity Premium

The analogies between the comparative statics of the liquidity premium and trading volume suggesta close connection between these quantities. An inspection of the asymptotic formulas unveils thefollowing relations:

LiPr =3

4εShTu +O(ε5/3) and

�r +

µ2

γσ2

�− ESR = 3

4εWeTu +O(ε5/3). (3.1)

These two relations have the same meaning: the welfare effect of small transaction costs is propor-tional to trading volume times the spread. The constant of proportionality 3/4 is universal, thatis, independent of both investment opportunities (r, µ, σ) and preferences (γ).

In the first formula, the welfare effect is measured by the liquidity premium, that is in terms ofthe risky asset. Likewise, trading volume is expressed as share turnover, which also focuses on therisky asset alone. By contrast, the second formula considers the decrease in the equivalent safe rateand wealth turnover, two quantities that treat both assets equally. In summary, if both welfare andvolume are measured consistently with each other, the welfare effect approximately equals volumetimes the spread, up to the universal factor 3/4.

Figure 4 plots the spread, share turnover, and the liquidity premium implied by the first equationin (3.1). As in Lo and Wang (2000), the spread and share turnover are capitalization-weightedaverages of all securities in the CRSP monthly stocks database with share codes 10 and 11, and withnonzero 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 estimationof spreads for earlier periods.

Spreads steadily decline in the observation period, dropping by almost an order of magnitudeafter stock market decimalization of 2001. At the same time, trading volume substantially increasesfrom a typical monthly turnover of 6% in the early 1990s to over 20% in the late 2000s. Theimplied liquidity premium also declines with spreads after decimalization, but less than the spread,

10Technically, wealth is valued at the ask price at the buying boundary, and at the bid price at the selling boundary.

10

0 2 4 6 8 10

10�4

0.001

0.01

0.1

1

Figure 5: Upper bound on the difference between the long-run and finite-horizon equivalent saferates (vertical axis), against the horizon (horizontal axis), for spread ε = 0.01% (solid), 0.1% (longdashed), 1% (short dashed), and 10% (dotted). Parameters are µ = 8%,σ = 16%, γ = 5.

in view of the increase in turnover. During the months of the financial crisis in late 2008, theimplied liquidity premium rises sharply, not because of higher volumes, but because spreads widensubstantially. Thus, although this implied liquidity premium is only a coarse estimate, it hasadvantages over other proxies, because it combines information on both prices and quantities, andis supported by a model.

3.6 Finite Horizons

The trading boundaries in this paper are optimal for a long investment horizon, but are alsoapproximately optimal for finite horizons. The following theorem, which complements the mainresult, makes this point precise:

Theorem 3.1. Fix a time horizon T > 0. Then the finite-horizon equivalent safe rate of anystrategy (φ0,φ) satisfies the upper bound

1

TlogE

�(ΞφT )

1−γ� 1

1−γ ≤ r + µ2 − λ2

2γσ2+

1

Tlog(φ00− + φ0−S0) + π∗

�

T+O(ε4/3), (3.2)

and the finite-horizon equivalent safe rate of our long-run optimal strategy (ϕ0,ϕ) satisfies the lowerbound

1

TlogE

�(ΞϕT )

1−γ� 11−γ ≥ r + µ2 − λ2

2γσ2+

1

Tlog(ϕ00− + ϕ0−S0)−

�2π∗ +

ϕ0−S0

ϕ00− + ϕ0−S0

�ε

T+O(ε4/3).

(3.3)

In particular, for the same unlevered initial position (φ0− = ϕ0− ≥ 0,φ00− = ϕ00− ≥ 0), the equivalent

safe rates of (ϕ0,ϕ) and of the optimal policy (φ0,φ) for horizon T differ by at most

1

T

�logE

�(ΞφT )

1−γ� 1

1−γ − logE�(ΞϕT )

1−γ� 11−γ�

≤ (3π∗ + 1)ε

T+O(ε4/3). (3.4)

This result implies that the horizon, like consumption, only has a second order effect on portfoliochoice with transaction costs, because the finite-horizon equivalent safe rate matches, at the order�2/3, the equivalent safe rate of the stationary long-run optimal policy. This result recovers, in

11

particular, the first-order asymptotics for the finite-horizon value function obtained by Bichuch(2011, Theorem 4.1). In addition, Theorem 3.1 provides explicit estimates for the correction terms

of order ε arising from liquidation costs. Indeed, r+ µ2−λ22γσ2 is the maximum rate achieved by trading

optimally. The remaining terms arise due to the transient influence of the initial endowment, as wellas the costs of the initial transaction, which takes place if the initial position lies outside the no-traderegion, and of the final portfolio liquidation. These costs are of order ε/T because they are incurredonly once, and hence defrayed by a longer trading period. By contrast, portfolio rebalancinggenerates recurring costs, proportional to the horizon, and their impact on the equivalent safe ratedoes not decline as the horizon increases.

Even after accounting for all such costs in the worst-case scenario, the bound in (3.4) showsthat their combined effect on the equivalent safe rate is lower than the spread ε, as soon as thehorizon exceeds 3π∗+1, that is four years in the absence of leverage. Yet, this bound holds only upa term of order ε4/3, so it is worth comparing it with the exact bounds in equations (C.17)-(C.18),from which (3.2) and (3.3) are obtained.

The exact bounds in Figure 5 show that, for typical parameter values, loss in equivalent safe rateof the long-run optimal strategy is lower than the spread ε 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 long-run approximation is a useful modeling device that makes the model tractable, and theresulting 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. First, formal arguments of stochas-tic control are used to obtain the optimal policy, its welfare, and their asymptotic expansions.

4.1 Transaction costs market

For a trading strategy (ϕ0t ,ϕt), again write the number of shares ϕt = ϕ↑t − ϕ

↓t as the difference of

the cumulated units purchased and sold, and denote by

X0t = ϕ

0tS

0t , Xt = ϕtSt,

the values of the safe and risky positions in terms of the ask price St. Then, the self-financingcondition (2.1), and the dynamics of S0t and St imply

dX0t =rX

0t dt− Stdϕ

↑t + (1− ε)Stdϕ

↓t ,

dXt =(µ+ r)Xtdt+ σXtdWt + Stdϕ↑t − Stdϕ↓.

Consider the maximization of expected power utility U(x) = x1−γ/(1 − γ) from terminal wealthat time T , and denote by V (t, x, y) its value function, which depends on time and the value of thesafe and risky positions. Itô’s formula yields:

dV (t,X0t , Xt) =Vtdt+ VxdX0t + VydXt +

1

2Vyyd�X,X�t

=

�Vt + rX

0t Vx + (µ+ r)XtVy +

σ2

2X

2t Vyy

�dt

+ St(Vy − Vx)dϕ↑t + St((1− ε)Vx − Vy)dϕ↓t + σXtVydWt,

12

where the arguments of the functions are omitted for brevity. Because V (t,X0t , Xt) must be a

supermartingale for any choice of the cumulative purchases and sales ϕ↑t ,ϕ↓t (which are increasing

processes), it follows that Vy − Vx ≤ 0 and (1− ε)Vx − Vy ≤ 0, that is

1 ≤ VxVy

≤ 11− ε .

In the interior of this region, the drift of V (t,X0t , Xt) cannot be positive, and must become zerofor the optimal policy,

Vt + rX0t Vx + (µ+ r)XtVy +

σ2

2X

2t Vyy = 0 if 1 <

Vx

Vy<

1

1− ε . (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 guess11 that V (t,X0t , Xt) = (X

0t )

1−γv(Xt/X0t )e−(1−γ)(r+β)t for some β to

be found. Setting z = y/x, the HJB equation reduces to

σ2

2z2v��(z) + µzv�(z)− (1− γ)βv(z) = 0 if 1 + z < (1− γ)v(z)

v�(z)<

1

1− ε + z. (4.2)

Assuming that the set {z : 1 + z ≤ (1−γ)v(z)v�(z) ≤1

1−ε + z} coincides with some interval l ≤ z ≤ uto be determined, and noting that at l the left inequality in (4.2) holds as equality, while at u theright inequality holds as equality, the following free boundary problem arises:

σ2

2z2v��(z) + µzv�(z)− (1− γ)βv(z) = 0 if l < z < u, (4.3)

(1 + l)v�(l)− (1− γ)v(l) = 0, (4.4)(1/(1− ε) + u)v�(u)− (1− γ)v(u) = 0. (4.5)

These conditions are not enough to identify the solution, because they can be matched for anychoice of the trading boundaries l, u. The optimal boundaries are the ones that also satisfy thesmooth-pasting conditions (cf. Dumas (1991)), formally obtained by differentiating (4.4) and (4.5)with respect to l and u, respectively:

(1 + l)v��(l) + γv�(l) = 0, (4.6)

(1/(1− ε) + u)v��(u) + γv�(u) = 0. (4.7)

In addition to the reduced value function v, this system requires to solve for the excess equivalentsafe rate β and the trading boundaries l and u. Substituting (4.6) and (4.4) into (4.3) yields (cf.Dumas and Luciano (1991))

−σ2

2(1− γ)γ l

2

(1 + l)2v + µ(1− γ) l

1 + lv − (1− γ)βv = 0.

Setting π− = l/(1 + l), and factoring out (1− γ)v, it follows that

−γσ2

2π2− + µπ− − β = 0.

11This guess assumes that the cash position is strictly positive, X0t > 0, which excludes leverage. With leverage,factoring out (−X0t )1−γ leads to analogous calculations.

13

Note that π− is the risky weight when it is time to buy, and hence the risky position is valued atthe ask price. The same argument for u shows that the other solution of the quadratic equation isπ+ = u(1− ε)/(1 + u(1− ε)), which is the risky weight when it is time to sell, and hence the riskyposition is valued at the bid price. Thus, the optimal policy is to buy when the “ask” fraction fallsbelow π−, sell when the “bid” fraction rises above π+, and do nothing in between. Since π− andπ+ solve the same quadratic equation, they are related to β via

π± =µ

γσ2±

�µ2 − 2βγσ2

γσ2.

It is convenient to set β = (µ2 − λ2)/2γσ2, because β = µ2/2γσ2 without transaction costs. Wecall λ the gap, since λ = 0 in a frictionless market, and, as λ increases, all variables diverge fromtheir frictionless values. Put differently, to compensate for transaction costs, the investor wouldrequire another asset, with expected return λ and volatility σ, which trades without frictions andis uncorrelated with the risky asset.12 With this notation, the buy and sell boundaries are just

π± =µ± λγσ2

.

In other words, the buy and sell boundaries are symmetric around the classical fricitionless solutionµ/γσ2. Since l(λ), u(λ) are identified by π± in terms of λ, it now remains to find λ. After derivingl(λ) and u(λ), the boundaries in the problem (4.3)-(4.5) are no longer free, but fixed. With thesubstitution

v(z) = e(1−γ)� log(z/l(λ))0 w(y)dy, i.e., w(y) =

l(λ)eyv�(l(λ)ey)

(1− γ)v(l(λ)ey) ,

the boundary problem (4.3)-(4.5) reduces to a Riccati ODE

w�(x) + (1− γ)w(x)2 +

�2µ

σ2− 1

�w(x)− γ

�µ− λγσ2

��µ+ λ

γσ2

�= 0, x ∈ [0, log u(λ)/l(λ)],

(4.8)

w(0) =µ− λγσ2

, (4.9)

w(log(u(λ)/l(λ))) =µ+ λ

γσ2, (4.10)

whereu(λ)

l(λ)=

1

(1− ε)π+(1− π−)π−(1− π+)

=1

(1− ε)(µ+ λ)(µ− λ− γσ2)(µ− λ)(µ+ λ− γσ2) . (4.11)

For each λ, the initial value problem (4.8)-(4.9) has a solution w(λ, ·), and the correct value of λ isidentified 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 asymptoticexpansion for small transaction costs (ε ∼ 0) using the implicit function theorem. To this end,write the boundary condition (4.10) as f(λ, ε) = 0, where:

f(λ, ε) = w(λ, log(u(λ)/l(λ)))− µ+ λγσ2

.

12Recall that in a market with two uncorrelated assets with returns µ1 and µ2, both with volatility σ, the maximumSharpe ratio is (µ21 + µ

22)/σ

2. That is, squared Sharpe ratios add across orthogonal shocks.

14

Of course, f(0, 0) = 0 corresponds to the frictionless case. The implicit function theorem thensuggests that around zero λ(ε) follows the asymptotics λ(ε) ∼ −εfε/fλ, but the difficulty is thatfλ = 0, because λ is not of order ε. Heuristic arguments (Shreve and Soner, 1994; Rogers, 2004)suggest that λ is of order ε1/3. Thus, setting λ = δ1/3 and f̂(δ, ε) = f(δ1/3, ε), and computing thederivatives of the explicit formula for w(λ, x) (cf. Lemma B.1) shows that:

f̂ε(0, 0) = −µ�µ− γσ2

�

γ2σ4, f̂δ(0, 0) =

4

3µ2σ2 − 3γµσ4 .

As a result:

δ(ε) ∼ −fεfδ

ε =3µ2

�µ− γσ2

�2

4γ2σ2ε whence λ(ε) ∼

�3µ2

�µ− γσ2

�2

4γ2σ2

�1/3ε1/3

.

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 investmentopportunities, we have computed explicitly the optimal trading policy, its welfare, liquidity pre-mium, 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 iscomputed with the corresponding trading price. Both the liquidity premium and trading volumeare small in the unlevered regime, but become substantial in the presence of leverage. For a smallbid-ask spread, the liquidity premium is approximately equal to share turnover times the spread,times the universal constant 3/4.

Trading boundaries depend on investement opportunities only through the mean variance ratio.The equivalent safe rate, the liquidity premium, and trading volume also depend only on the meanvariance ratio if measured in business time.

Appendix

A Derivation of the Shadow Market

The key to justify the above heuristic arguments is to reduce the portfolio choice problem withtransaction costs to another portfolio choice problem, without transaction costs, with the bid andask prices replaced by a single shadow price S̃t evolving within the bid-ask spread, which coincideswith either price at times of trading, and which yields the same optimal policy. In the case oflogarithmic utility, this approach was applied successfully by Kallsen and Muhle-Karbe (2010) andGerhold, Muhle-Karbe and Schachermayer (2011b,a).

Definition A.1. A shadow price is a frictionless price process S̃t, lying within the bid-ask spread((1− ε)St ≤ S̃t ≤ St a.s.), such that there is an optimal strategy for S̃t which is of finite variation,and entails buying only when the shadow price S̃t equals the ask price St, and selling only when S̃tequals the bid price (1− ε)St.

Once a candidate for such a shadow price is identified, long-run verification results for frictionlessmodels (cf. Guasoni and Robertson (2011)) deliver the optimality of the guessed policy. Further,

15

this method provides explicit upper and lower bounds on finite-horizon performance (cf. Lemma C.2below), thereby allowing to check whether the long-run optimal strategy is approximately optimalfor an horizon T . Put differently, it shows which horizons are long enough.

Finding a shadow price requires another heuristic argument. The idea is that S̃t/St, the ratiobetween the shadow and the ask price, should only depend on the ratio of risky and safe weights,i.e., on the same state variable as the reduced value function in the transaction cost problem.Consequently, we look for a shadow price of the form13

S̃t =St

eYtg(eYt),

where eYt = (Xt/X0t )/l is the ratio between the risky and safe positions at the ask price St, andcentered at the buying boundary

l =π−

1− π−=

µ− λγσ2 − (µ− λ) .

Inside the no-trade region, the numbers of units ϕ0t and ϕt remain constant so that Yt = log(ϕt/lϕ0t )+

log(St/S0t ) follows Brownian motion with drift. Since Yt must remain in [0, log(u/l)] by definition,it is reflected at the boundaries, that is,

dYt = (µ− σ2/2)dt+ σdWt + dLt − dUt,

for nondecreasing local time processes Lt, Ut that only increase on {Yt = 0} (resp. {Yt = log(u/l)}).The function g : [1, u/l] → [1, (1 − ε)u/l] is a C2-function satisfying the boundary conditions (cf.Gerhold, Muhle-Karbe and Schachermayer (2011b))

g(1) = 1, g(u/l) = (1− ε)u/l, g�(1) = 1, g�(u/l) = 1− ε. (A.1)

The first two conditions ensure that S̃t equals the ask price St (resp. the bid price (1− ε)St) whenYt sits at the buying boundary 0 (resp. at the selling boundary log(u/l)). The boundary conditionsfor g�, and Itô’s formula imply that S̃t is an Itô process with dynamics

dS̃t/S̃t = (µ̃(Yt) + r)dt+ σ̃(Yt)dWt,

where

µ̃(y) =µg�(ey)ey + σ

2

2 g��(ey)e2y

g(ey), and σ̃(y) =

σg�(ey)ey

g(ey).

To identify the function g, first derive the HJB equation for a generic g. Then, compare thisequation to the one obtained in the previous section for the market with transaction costs. Thefunction g is identified by the condition that the value function of the two problems must be thesame.

The wealth process corresponding to a policy π̃t in terms of the shadow price S̃t is

dX̃t = rX̃tdt+ π̃tµ̃(Yt)X̃tdt+ π̃tσ̃(Yt)X̃tdWt.

The usual ansatz for the value function Ṽ in frictionless markets driven by a state variable Yt, i.e.Ṽt = Ṽ (t, X̃t, Yt), and Itô’s formula yield

dṼ (t, X̃t, Yt) =

�Ṽt + rX̃tṼx + µ̃π̃tX̃tṼx +

σ̃2

2π̃2t X̃

2t Ṽxx +

�µ− σ

2

2

�Ṽy +

σ2

2Ṽyy + σσ̃π̃tX̃tṼxy

�dt

+ Ṽy(dLt − dUt) + (σ̃π̃tX̃tṼx + σṼy)dWt,13An equivalent guess is S̃t = Sth(Yt). With hindsight, the one in the text leads to simpler calculations, because

St is a multiple of eYt in the no-trade region.

16

where the arguments of the functions are omitted for brevity. Since Ṽ must be a supermartingalefor any strategy, and a martingale for the optimal strategy, the HJB equation reads as

supπ̃

�Ṽt + rxṼx + µ̃π̃xṼx +

σ̃2

2π̃2x2Ṽxx +

�µ− σ

2

2

�Ṽy +

σ2

2Ṽyy + σσ̃π̃xṼxy

�= 0,

with the Neumann boundary conditions

Ṽy(0) = Ṽy(log(u/l)) = 0.

The homogeneity of the value function (i.e., Ṽ (t, x, y) = x1−γ ṽ(t, y)), leads to the first-order con-dition:

π̃t =1

γ

�µ̃

σ̃2+

σ

σ̃

ṽy

ṽ

�.

Plugging this equality back into the HJB equation yields the nonlinear equation

ṽt + (1− γ)rṽ +�µ− σ

2

2

�ṽy +

σ2

2ṽyy +

1− γ2γ

�µ̃

σ̃+ σ

ṽy

ṽ

�2ṽ = 0.

Now, the equivalent safe rate β = (µ2 − λ2)/2γσ2 must be the same, both for the shadow marketand for the transaction cost market in the previous section. Thus, setting

ṽ(t, y) = e−(1−γ)(r+β)te(1−γ)� y0 w̃(z)dz,

which implies that ṽy/ṽ = (1− γ)w̃, the HJB equation reduces to the inhomogeneous Riccati ODE

w̃� + (1− γ)w̃2 +

�2µ

σ2− 1

�w̃ − 2β

σ2+

1

γσ2

�µ̃

σ̃+ σ(1− γ)w̃

�2= 0 (A.2)

with boundary conditionsw̃(0) = w̃(log(u/l)) = 0. (A.3)

For S̃t to be a shadow price, its value function

Ṽt = e−(1−γ)(r+β)t

X̃1−γt e

(1−γ)� y0 w̃(z)dz

must coincide with the value function

Vt = e−(1−γ)(r+β)t(X0t )

1−γe(1−γ)

� y0 w(z)dz

for the transaction cost problem derived above. By definition, the safe position X0t and the wealthX̃t in terms of S̃t = Ste−Ytg(eYt) are related via

X̃t

X0t=

ϕ0tS0t + ϕtS̃tϕ0tS

0t

= 1 + g(eYt)l.

Now, the condition Ṽ = V implies that 0 = log (1 + g(ey)l) +� y0 (w̃(z) − w(z))dz, which in turn

means that

w̃(y) = w(y)− g�(ey)eyl

1 + g(ey)l. (A.4)

Plugging this relation into the ODE (A.2) for w̃, using the ODE (4.8) for w, and simplifying leadsto �

(1− γ)w(y) + µ̃(y)σσ̃(y)

− g�(ey)eyl

1 + g(ey)l

�2= 0.

17

Inserting the definitions of µ̃(y) and σ̃(y), this relation is tantamount to the following ODE for g:14

g��(ey)ey

g�(ey)− 2g

�(ey)eyl

1 + g(ey)l+

2µ

σ2+ 2(1− γ)w(y) = 0. (A.5)

Next, the substitution

k(y) =1 + g(ey)l

g�(ey)eyl, i.e., g(ey) =

�1 +

1

l

�exp

�� y

0

1

k(z)dz

�− 1

l,

reduces this ODE to the inhomogeneous linear equation

k�(y) = k(y)

�2µ

σ2− 1 + 2(1− γ)w(y)

�− 1. (A.6)

Since 1/k(0) = w(0)− w̃(0) by (A.4), the boundary condition (A.3) for w̃ and its counterpart (4.9)for w imply that k(0) = γσ2/(µ − λ). The solution to (A.6) then follows from the variation ofconstants formula. In each of the three different cases of Lemma B.1, plugging in the respectiveexplicit formula for w (cf. Lemma B.1) and integrating leads to an explicit formula for k. In Case2, we have (with constants a and b as in Lemma B.1)

k(y) = cos2�tan−1

�b

a

�+ ay

��−1atan

�tan−1

�b

a

�+ ay

�+

b

a2+

(a2 + b2)γσ2

a2(µ− λ)

�,

The other two cases lead to analogous results (cf. Lemma B.4). Now the chain of substitutions isreversed starting from k, which is known explicitly up to the constant λ. First, set w̃(y) = w(y)−1/k(y); then w̃(0) = 0 by construction. To establish the other boundary condition w̃(log(u/l)) = 0,it suffices to check that 1/k(log(u/l)) = (µ+λ)/(γσ2). To this end, insert the boundary conditionsfor w,

µ+ λ

γσ2=w(log(u/λ)) =

a tan[tan−1( ba) + a log(ul )] + (

µσ2 −

12)

γ − 1 , (A.7)

µ+ λ

γσ2−�µ+ λ

γσ2

�2=w�(log(u/l)) =

a2

(γ − 1) cos2[tan−1( ba) + a log(ul )]

, (A.8)

into the explicit formula for 1/k(y).15 Now, observe that the function

g(ey) =

�1 +

1

l

�exp

�� y

0

1

k(z)dz

�− 1

l

= 1 +γσ2

µ− λ

1

1 + µ−λγσ2�

ba2+b2 −

aa2+b2 tan

�tan−1

�ba

�+ ay

�� − 1

satisfies g(1) = 1. Moreover, g(u/l) = (1 − ε)u/l, which again follows by inserting (A.8) intothe explicit expression for g. Finally, these boundary conditions for g and those for k implythat g�(1) = 1 and g�(u/l) = 1 − ε, i.e., g satisfies the smooth pasting conditions (A.1) and, byconstruction, also the ODE (A.5).

14For logarithmic utility (γ = 1) this ODE reduces to the one in Gerhold, Muhle-Karbe and Schachermayer (2011b,Equation (3.5)).

15The first equalities in (A.7) (resp. (A.8)) follow from Equations (4.6) and (4.7) respectively. The second equalitiesfollow from the explicit formula for w from Lemma B.1.

18

In summary, although the derivation and the formulas are more involved, the shadow price isdetermined as explicitly as in the case of logarithmic utility (Gerhold, Muhle-Karbe and Schacher-mayer, 2011b). That is, its dynamics are known explcitly in terms of λ, which is identified as thesolution of a scalar equation and can be developed into a fractional power series in terms of ε1/3.

With the heuristically derived candidate shadow price at hand, the proof of Theorem 2.2 isdivided into three parts. The first part establishes that the explicit formulas for the reduced valuefunction w, the auxiliary function k, and the function g parametrizing the shadow price are well-defined and indeed have the properties derived above. In the second part, these results are used toconstruct a shadow price, and to show that it satisfies the upper and lower finite-horizon bounds inLemma C.2, which in turn imply long-run optimality. Finally, the third part contains the explicitcalculation of the implied trading volume.

B Explicit formulas and their properties

The first step is to determine, for a given small λ > 0, an explicit expression for the solution w ofthe ODE (4.8), complemented by the initial condition (4.9).

Lemma B.1. Let 0 < µ/γσ2 �= 1. Then for sufficiently small λ > 0, the function

w(λ, x) =

a(λ) tanh[tanh−1(b(λ)/a(λ))−a(λ)x]+( µσ2

− 12 )γ−1 , if γ ∈ (0, 1) and

µγσ2 < 1 or γ > 1 and

µγσ2 > 1,

a(λ) tan[tan−1(b(λ)/a(λ))+a(λ)x]+( µσ2

− 12 )γ−1 , if γ > 1 and

µγσ2 ∈

�12 −

12

�1− 1γ ,

12 +

12

�1− 1γ

�,

a(λ) coth[coth−1(b(λ)/a(λ))−a(λ)x]+( µσ2

− 12 )γ−1 , otherwise,

with

a(λ) =

��(γ − 1)µ2 − λ2γσ4

−�12− µ

σ2

�2�� and b(λ) =1

2− µ

σ2+ (γ − 1)µ− λ

γσ2,

is a local solution of

w�(x) + (1− γ)w2(x) +

�2µ

σ2− 1

�w(x)− µ

2 − λ2

γσ4= 0, w(0) =

µ− λγσ2

. (B.1)

Moreover, x �→ w(λ, x) is increasing (resp. decreasing) for µ/γσ2 ∈ (0, 1) (resp. µ/γσ2 > 1).

Proof. The first part of the assertion is easily verified by taking derivatives. The second follows byinspection of the explicit formulas.

Next, establish that the crucial constant λ, which determines both the no-trade region and theequivalent safe rate, is well-defined.

Lemma B.2. Let 0 < µ/γσ2 �= 1 and w(λ, ·) be defined as in Lemma B.1, and set

l(λ) =µ− λ

γσ2 − (µ− λ) , u(λ) =1

(1− ε)µ+ λ

γσ2 − (µ+ λ) .

Then, for sufficiently small ε > 0, there exists a unique solution λ of

w

�λ, log

�u(λ)

l(λ)

��− µ+ λ

γσ2= 0. (B.2)

19

As ε ↓ 0, it has the asymptotics

λ = γσ2�

3

4γ

�µ

γσ2

�2�1− µ

γσ2

�2�ε1/3 + σ2

�(5− 2γ)

10

µ

γσ2

�1− µ

γσ2

�− 3

20

�ε+O(ε4/3).

Proof. The explicit expression for w in Lemma B.1 implies that w(λ, x) in Lemma B.1 is analyticin both variables at (0, 0). By the initial condition in (B.1), its power series has the form

w(λ, x) =µ− λγσ2

+∞�

i=1

∞�

j=0

Wijxiλj,

where expressions for the coefficients Wij are computed by by expanding the explicit expressionfor w. Hence, the left-hand side of the boundary condition (B.2) is an analytic function of ε and λ.Its power series expansion shows that the coefficients of ε0λj vanish for j = 0, 1, 2, so that thecondition (B.2) reduces to

λ3�

i≥0Aiλ

i = ε�

i,j≥0Bijε

iλj (B.3)

with (computable) coefficients Ai and Bij . This equation has to be solved for λ. Since

A0 =4

3µσ2(γσ2 − µ) and B00 =µ(γσ2 − µ)

γ2σ4

are non-zero, divide the equation (B.3) by�

i≥0Aiλi, and take the third root, obtaining that, for

some Cij ,

λ = ε1/3�

i,j≥0Cijε

jλj = ε1/3

�

i,j≥0Cij(ε

1/3)3jλj .

The right-hand side is an analytic function of λ and ε1/3, so that the implicit function theorem (Gun-ning and Rossi, 2009, Theorem I.B.4) yields a unique solution λ (for ε sufficiently small), which isan analytic function of ε1/3. Its power series coefficients can be computed at any order.

Henceforth, consider small transaction costs ε > 0, and let λ denote the constant in Lemma B.2.Moreover, set w(x) = w(λ, x), a = a(λ), b = b(λ), and u = u(λ), l = l(λ). By inspection, it followsthat the function w satisfies the following smooth pasting conditions:

Lemma B.3. Let 0 < µ/γσ2 �= 1. Then, in all three cases,

w�(0) =

µ− λγσ2

−�µ− λγσ2

�2, w

��log

�u

l

��=

µ+ λ

γσ2−�µ+ λ

γσ2

�2.

The next lemma states the properties of the function k.

Lemma B.4. Let 0 < µ/γσ2 �= 1 and define

k(y) =

cosh2�tanh−1

�ba

�− ay

� �1a tanh

�tanh−1

�ba

�− ay

�− ba2 +

(a2−b2)γσ2a2(µ−λ)

�,

cos2�tan−1

�ba

�+ ay

� �− 1a tan

�tan−1

�ba

�+ ay

�+ ba2 +

(a2+b2)γσ2

a2(µ−λ)

�,

sinh2�coth−1

�ba

�− ay

� �− 1a coth

�coth−1

�ba

�− ay

�+ ba2 +

(b2−a2)γσ2a2(µ−λ)

�,

20

with the same cases as in Lemma B.1. Then k satisfies the linear ODE

k�(y) = k(y)

�2µ

σ2− 1 + 2(1− γ)w(y)

�− 1, 0 ≤ y ≤ log

�u

l

�,

with boundary conditions

k(0) =γσ2

µ− λ , k�log

�u

l

��=

γσ2

µ+ λ.

Moreover, k is strictly decreasing (resp. increasing) and, in particular, strictly positive on [0, log(u/l)]for µ/γσ2 ∈ (0, 1) (resp. on [log(u/l), 0] for µ/γσ2 > 1).

Proof. That k satisfies the ODE follows by insertion. The identities cos2[tan−1(x)] = 1/(1 + x2)as well as cosh2[tanh−1(x)] = 1/(1 − x2) and sinh2[coth−1(x)] = 1/(x2 − 1) yield the boundarycondition at zero, whereas the boundary condition at log(u/l) follows by inserting

w

�log

�u

l

��=

µ+ λ

γσ2and w�

�log

�u

l

��=

µ+ λ

γσ2−�µ+ λ

γσ2

�2.

Finally, the ODE and a comparison argument yield that k is strictly decreasing (resp. increasingfor µ/γσ2 > 1).

Lemma B.5. Let 0 < µ/γσ2 �= 1 and define

g(ey) :=γσ2

µ− λ exp�� y

0

1

k(z)dz

�−�

γσ2

µ− λ − 1�,

for 0 ≤ y ≤ log(u/l) if µ/γσ2 ∈ (0, 1) resp. for log(u/l) ≤ y ≤ 0 if µ/γσ2 > 1. Then

g(ey) =

1 + γσ2

µ−λ

��1 + µ−λγσ2

�b

b2−a2 −a

b2−a2 tanh�tanh−1

�ba

�− ay

��−1− 1

�,

1 + γσ2

µ−λ

��1 + µ−λγσ2

�b

b2+a2 −a

a2+b2 tan�tan−1

�ba

�+ ay

��−1− 1

�,

1 + γσ2

µ−λ

��1 + µ−λγσ2

�b

b2−a2 −a

b2−a2 coth�coth−1

�ba

�− ay

��−1− 1

�,

with the same cases as in Lemma B.1, and g satisfies the boundary and smooth pasting conditions

g(1) = 1, g(u/l) = (1− ε)u/l, g�(1) = 1, g�(u/l) = 1− ε.

Moreover, g� > 0 such that g maps [1, u/l] (resp. [u/l, 1]) onto [1, (1− ε)u/l] (resp. [(1− ε)u/l, 1] ifµ/γσ2 > 1). Finally, g solves

g��(ey)ey

g�(ey)− 2g

�(ey)eyl

1 + g(ey)l+

2µ

σ2+ 2(1− γ)w(y) = 0. (B.4)

Proof. The explicit representation follows by elementary integration. Evidently, g(1) = 1. More-over, g(u/l) = (1−ε)u/l follows by inserting (µ+λ)/γσ2 = w(log(u/l)) once again. Next, g(1) = 1,g(u/l) = (1− ε)u/l, k(0) = γσ2/(µ− λ), and k(log(u/l)) = γσ2/(µ+ λ), and

g�(ey) =

e−y

k(y)

�g(ey) +

γσ2

µ− λ − 1�

(B.5)

imply g�(1) = 1 and g�(u/l) = 1 − ε. Hence g satisfies the smooth pasting conditions. Further-more, (B.5) and a comparison argument yield that g� > 0 because k > 0. Finally, computing thederivatives shows that g indeed satisfies the ODE (B.4).

21

C The shadow price and verification

To construct the shadow price as in Gerhold, Muhle-Karbe and Schachermayer (2011b,a), for y ∈[0, log(u/l)] (resp. [log(u/l), 0] if µ/γσ2 > 1), let Yt be Brownian motion with drift, reflected at 0 andlog(u/l), that is, the continuous, adapted process with values in [0, log(u/l)] (resp. in [log(u/l), 0]if µ/γσ2 > 1), such that

dYt = (µ− σ2/2)dt+ σdWt + dLt − dUt, Y0 = y, (C.1)

for nondecreasing (resp. nonincreasing if µ/γσ2 > 1) adapted processes Lt and Ut increasing (resp.decreasing if µ/γσ2 > 1) only on the sets {Yt = 0} and {Yt = log(u/l)}, respectively.

Lemma C.1. Define

y =

0, if lξ0S00 ≥ ξS0,log(u/l), if uξ0S00 ≤ ξS0,log[(ξS0/ξ0S00)/l], otherwise,

(C.2)

and let Yt be reflected Brownian motion with drift as in (C.1), started at Y0 = y. Then S̃t =Ste

−Ytg(eYt), with g as in Lemma B.5, is a positive Itô process with dynamics

dS̃t/S̃t = (µ̃(Yt) + r)dt+ σ̃(Yt)dWt, S̃0 = S0e−y

g(ey),

for

µ̃(y) =µg�(ey)ey + σ

2

2 g��(ey)e2y

g(ey), σ̃(y) =

σg�(ey)ey

g(ey),

and S̃t takes values in the bid-ask spread [(1− ε)St, St].

Note that the first (resp. second) case in (C.2) occurs if the initial ratio ξS0/ξ0S00 lies belowthe buying boundary l (resp. above the selling boundary l for µ/γσ2 > 1) or above the sellingboundary u (resp. below the buying boundary u for µ/γσ2 > 1). Then, there is a jump from theinitial position (ϕ00− ,ϕ0−) = (ξ

0, ξ), which moves the ratio to the nearest boundary of the interval[l, u] (cf. Lemma C.3 below). Since this initial trade involves the purchase (resp. sale) of shares,the initial value of S̃t is chosen to match the initial ask (resp. bid) price.

Proof of Lemma C.1. The first part of the assertion follows from the smooth pasting conditions forg and Itô’s formula. As for the second part, since g��(1) ≤ 0, a comparison argument yields that thederivative (g�(y)y − g(y))/y2 of g(y)/y is non-positive. Hence g(1)/1 = 1 and g(u/l)/(u/l) = 1− εyield that S̃t = Stg(eYt)e−Yt is indeed [(1− ε)St, St]-valued.

The long-run optimal portfolio in the frictionless “shadow market” with price process S̃t cannow be determined by adapting the argument in Guasoni and Robertson (2011).

Lemma C.2. The function w̃(y) = w(y) − g�(ey)eyl/(1 + g(ey)l) solves the system (A.2)-(A.3),that is:

σ2

2w̃

� + (1− γ)σ2

2w̃

2 +

�µ− σ

2

2

�w̃ +

1

2γ

�µ̃

σ̃+ σ(1− γ)w̃

�2= β, (C.3)

with boundary conditions w̃(0) = w̃(log(u/l)) = 0 and β = (µ2 − λ2)/2γσ2. Setting q̃(y) =� y0 w̃(z)dz, the shadow payoff X̃T corresponding to the policy π̃ =

1γ

�µ̃σ̃2 + (1− γ)

σσ̃ w̃

�(in terms of

22

S̃t) and the shadow discount factor M̃T = e−rTE(−� ·0

µ̃σ̃dWt)T satisfy the following bounds:

E

�X̃

1−γT

�=X̃1−γ0 e

(1−γ)(r+β)TÊ

�e(1−γ)(q̃(Y0)−q̃(YT ))

�, (C.4)

E

�M̃

1− 1γT

�γ=e(1−γ)(r+β)T Ê

�e( 1γ−1)(q̃(Y0)−q̃(YT ))

�γ. (C.5)

Here, Ê [·] denotes the expectation with respect to the myopic probability P̂ , defined by

dP̂

dP= exp

�� T

0

�− µ̃σ̃+ σ̃π̃

�dWt −

1

2

� T

0

�− µ̃σ̃+ σ̃π̃

�2dt

�.

Proof. First note that µ̃, σ̃, π̃, w̃ are functions of Yt, but the argument is omitted throughout toease notation. Next, it is readily verified by insertion that w̃ satisfies the boundary value problem.Now, to prove (C.4), notice that the shadow wealth process X̃t satisfies:

X̃1−γT = X̃

1−γ0 e

(1−γ)� T0 (r+µ̃π̃−

σ̃2

2 π̃2)dt+(1−γ)

� T0 σ̃π̃dWt .

Hence:

X̃1−γT =X̃

1−γ0

dP̂

dPe

� T0 ((1−γ)(r+µ̃π̃−

σ̃2

2 π̃2)+ 12 (−

µ̃σ̃+σ̃π̃)

2)dte

� T0 ((1−γ)σ̃π̃−(−

µ̃σ̃+σ̃π̃))dWt .

Substituting π̃ = 1γ (µ̃σ̃2 + (1 − γ)

σσ̃ w̃), the second integrand simplifies to −(1 − γ)σw̃. Similarly,

the first integrand first reduces to (1 − γ)r + 12µ̃2

σ̃2 + γσ̃2

2 π̃2 − γµ̃π̃, and then to the expression

(1− γ)r + (1− γ)2 σ22 w̃2 + 1−γ2γ (

µ̃σ̃ + σ(1− γ)w̃)

2. In summary:

X̃1−γT = X̃

1−γ0

dP̂

dPe(1−γ)

� T0 (r+(1−γ)

σ2

2 w̃2+ 12γ (

µ̃σ̃+σ(1−γ)w̃)

2)dt−(1−γ)

� T0 σw̃dWt . (C.6)

Now, Itô’s formula and the boundary conditions w̃(0) = w̃(log(u/l)) = 0 imply that:

q̃(YT )− q̃(Y0) =� T

0w̃(Yt)dYt +

1

2

� T

0w̃

�(Yt)d�Y, Y �t + w̃(0)LT − w̃(u/l)UT

=

� T

0

��µ− σ

2

2

�w̃ +

σ2

2w̃

��dt+

� T

0σw̃dWt.

Using this identity to replace the term� T0 σw̃dWt in (C.6) yields

X̃1−γT =X̃

1−γ0

dP̂

dPe(1−γ)

� T0 (r+

σ2

2 w̃�+(1−γ)σ

2

2 w̃2+(µ−σ

2

2 )w̃+12γ (

µ̃σ̃+σ(1−γ)w̃)

2)dt)e−(1−γ)(q̃(YT )−q̃(Y0)).

Since w̃ satisfies (C.3), the first bound follows:

E

�X̃

1−γT

�= X̃1−γ0 E

�dP̂

dPe(1−γ)(r+β)T−(1−γ)(q̃(YT )−q̃(Y0))

�= X̃1−γ0 e

(1−γ)(r+β)TÊ

�e(γ−1)(q̃(YT )−q̃(Y0))

�.

The argument for the second bound is similar. The shadow discount factor M̃T = e−rTE(−� ·0

µ̃σ̃dW )T

and the myopic probability P̂ satisfy

M̃1− 1γT = e

1−γγ

� T0

µ̃σ̃ dW+

1−γγ

� T0 (r+

µ̃2

2σ̃2)dt

,dP̂

dP= e

1−γγ

� T0 (

µ̃σ̃+σw̃)dWt−

(1−γ)2

2γ2

� T0 (

µ̃σ̃+σw̃)

2dt.

23

Hence:

M̃1− 1γT =

dP̂

dPe− 1−γγ

� T0 σw̃dWt+

1−γγ

� T0 r+

12 (

µ̃2

σ̃2+ 1−γγ (

µ̃σ̃+σw̃)

2)dt.

Since µ̃2

σ̃2 +1−γγ (

µ̃σ̃ + σw̃)

2 = (1− γ)σ2w̃2 + 1γ (µ̃σ̃ + σ(1− γ)w̃)

2, substituting again the equality

� T

0σw̃dWt = q̃(YT )− q̃(Y0)−

� T

0

��µ− σ

2

2

�w̃ +

σ2

2w̃

��dt

and the HJB equation (C.3) yields

M̃1− 1γT =

dP̂

dPe

1−γγ (r+β)T−

1−γγ (q̃(YT )−q̃(Y0)).

The second bound then follows by taking the expectation, and raising it to power of γ.

With the finite horizon bounds at hand, it is now straightforward to establish that the policyπ̃(Yt) is indeed long-run optimal in the frictionless market with price S̃t.

Lemma C.3. Let 0 < µ/γσ2 �= 1. Then, the policy

π̃(Yt) =1

γ

�µ̃(Yt)

σ̃2(Yt)+ (1− γ) σ

σ̃(Yt)w̃(Yt)

�=

g(eYt)l

1 + g(eYt)l(C.7)

is long-run optimal with equivalent safe rate r + β in the frictionless market with price process S̃t.The corresponding wealth process (in terms of S̃t), and the numbers of safe and risky units aregiven by

X̃t = (ξ0S00 + ξS̃0)E

�� ·

0(r + π̃(Yt)µ̃(Yt))dt+

� ·

0π̃(Yt)σ̃(Yt)dWt

�

t

,

ϕ0− = ξ, ϕt = π̃(Yt)X̃t/S̃t for t ≥ 0,ϕ00− = ξ

0, ϕ

0t = (1− π̃(Yt))X̃t/S0t for t ≥ 0.

Proof. The second representation for π̃(Yt) follows by inserting the definitions of µ̃(y), σ̃(y) fromLemma C.1, the ODE (B.4) for g(y), and the identity 1/k(y) = g�(ey)eyl/(1 + g(ey)l), which is adirect consequence of the definition of g. The formulas for the corresponding wealth process andthe numbers of safe and risky units follow from the standard frictionless definitions. Now let M̃t bethe shadow discount factor from Lemma C.2. Then, standard duality arguments for power utility(cf. Lemma 5 in Guasoni and Robertson (2011)) imply that the shadow payoff X̃φt correspondingto any admissible strategy φt satisfies the inequality

E

�(X̃φT )

1−γ� 1

1−γ ≤ E�M̃

γ−1γ

T

� γ1−γ

. (C.8)

This inequality in turn yields the following upper bound, valid for any admissible strategy φt inthe frictionless market with shadow price S̃t:

lim infT→∞

1

(1− γ)T logE�(X̃φT )

1−γ�≤ lim inf

T→∞

γ

(1− γ)T logE�M̃

γ−1γ

T

�. (C.9)

Since the function q̃ is bounded on the compact support of Yt, the second bound in Lemma C.2implies that the right-hand side equals r+ β. Likewise, the first bound in the same lemma impliesthat the shadow payoff X̃t (corresponding to the policy ϕt) attains this upper bound, concludingthe proof.

24

The next Lemma establishes that S̃t is a shadow price. Here the argument is similar to the onefor logarithmic utility (Gerhold, Muhle-Karbe and Schachermayer, 2011a).

Lemma C.4. Let 0 < µ/γσ2 �= 1. Then, the number of shares ϕt = π̃(Yt)X̃t/S̃t in the portfo-lio π̃(Yt) in Lemma C.3 has the dynamics

dϕt

ϕt=

�1− µ− λ

γσ2

�dLt −

�1− µ+ λ

γσ2

�dUt. (C.10)

Thus, for µ/γσ2 ∈ (0, 1) (resp. µ/γσ2 > 1), ϕt increases (resp. decreases) only when Yt = 0, thatis, when S̃t equals the ask (resp. bid) price, and decreases (resp. increases) only when Yt = log(u/l),that is, when S̃t equals the bid (resp. ask) price.

Proof. Itô’s formula applied to (C.7) yields

dπ̃t(Yt)

π̃(Yt)=µeYt(1 + g(eYt)l)g�(eYt)− σ2le2Ytg�(eYt)2 + 12σ

2e2Yt(1 + g(eYt)l)g��(eYt)

g(eYt)(1 + g(eYt)l)2dt

+σg�(eYt)eYt

g(eYt)(1 + g(eYt)l)dWt +

g�(eYt)eYt

g(eYt)(1 + g(eYt)l)(dLt − dUt).

Integrating ϕ = π̃(Yt)X̃t/S̃t by parts twice, using the dynamics of π̃(Yt), X̃t, and S̃t, and simplifying,it follows that

dϕt

ϕt=

�g�(eYt)eYt

g(eYt)(1 + g(eYt)l)

�d(Lt − Ut).

Since Lt and Ut only increase (resp. decrease when µ/γσ2 > 1) on {Yt = 0} and {Yt = log(u/l)},respectively, the assertion now follows from the boundary conditions for g and g�.

The optimal growth rate for any frictionless price within the bid-ask spread must be greater orequal than in the original market with bid-ask process ((1 − ε)St, St), because the investor tradesat more favorable prices. For a shadow price, there is an optimal strategy that only entails buying(resp. selling) stocks when S̃ coincides with the ask- resp. bid price. Hence, this strategy yieldsthe same payoff when executed at bid-ask prices, and thus is also optimal in the original modelwith transaction costs. The corresponding equivalent safe rate must also be the same, since thedifference due to the liquidation costs vanishes as the horizon grows in (2.2):

Proposition C.5. Suppose S̃t is a shadow price for a bid-ask process ((1− ε)St, St), with long-runoptimal portfolio (ϕ0t ,ϕt) as in Definition A.1, and the corresponding policy π̃t = ϕtS̃t/(ϕ

0tS

0t +ϕtS̃t)

is bounded by C(1− ε)/ε for some constant C ∈ (0, 1). Then, the portfolio (ϕ0t ,ϕt) is also long-runoptimal for ((1− ε)St, St), with the same equivalent safe rate as for the frictionless price S̃.

Proof. As ϕt only increases (resp. decreases) when S̃t = St (resp. S̃t = (1 − ε)St), (ϕ0t ,ϕt) is alsoself-financing for the bid-ask process ((1− ε)St, St). In addition, since St ≥ S̃t ≥ (1− ε)St,

ϕ0tS

0t + ϕtS̃t ≥ ϕ0tS0t + ϕ+t (1− ε)St − ϕ−t St ≥

�1− ε

1− ε |π̃t|�(ϕ0tS

0t + ϕtS̃t). (C.11)

Thus, if (ϕ0t ,ϕt) is admissible for S̃t, it is also admissible for ((1− ε)St, St), because |π̃t|ε/(1− ε) ∈(0, 1). Moreover, since |π̃t|ε/(1− ε) is even bounded away from 1 by assumption, (C.11) also yields

lim infT→∞

1

(1− γ)T logE�(ϕ0TS

0T + ϕ

+T (1− ε)ST − ϕ

−T ST )

1−γ�

= lim infT→∞

1

(1− γ)T logE�(ϕ0TS

0T + ϕT S̃T )

1−γ�,

(C.12)

25

that is, (ϕ0t ,ϕt) has the same growth rate, either with S̃t or with [(1− ε)St, St].For any admissible strategy (ψ0t ,ψt) for the bid-ask spread [(1 − ε)St, St], set ψ̃0t = ψ00− +� t

0 S̃s/S0sdψs. Then, (ψ̃

0t ,ψt) is a self-financing trading strategy for S̃t with ψ̃

0t ≥ ψ0t . Together with

S̃t ∈ [(1− ε)St, St], the long-run optimality of (ϕ0t ,ϕt) for S̃t, and (C.12), it follows that:

lim infT→∞

1

T

1

(1− γ) logE�(ψ0t S

0t + ψ

+t (1− ε)St − ψ−t St)1−γ

�

≤ lim infT→∞

1

T

1

(1− γ) logE�(ψ̃0t S

0t + ψtS̃t)

1−γ�

≤ lim infT→∞

1

T

1

(1− γ) logE�(ϕ0tS

0t + ϕtS̃t)

1−γ�

= lim infT→∞

1

T

1

(1− γ) logE�(ϕ0tS

0t + ϕ

+t (1− ε)St − ϕ−t St)1−γ

�.

Hence (ϕ0t ,ϕt) is also long-run optimal for ((1− ε)St, St).

Putting everything together, the main result now follows.

Theorem C.6. For ε > 0 small, and 0 < µ/γσ2 �= 1, the process S̃t in Lemma C.1 is a shadowprice. A long-run optimal policy — both for the frictionless market with price S̃t and in the marketwith bid-ask prices (1 − ε)St, St — is to keep the risky weight π̃t (in terms of S̃t) in the no-traderegion

[π−,π+] =

�g(1)l

1 + g(1)l,

g(u/l)l

1 + g(u/l)l

�=

�µ− λγσ2

,µ+ λ

γσ2

�.

As ε ↓ 0, its boundaries have the asymptotics

π± =µ

γσ2±

�3

4γ

�µ

γσ2

�2�1− µ

γσ2

�2�1/3ε1/3 ±

�(5− 2γ)10γ

µ

γσ2

�1− µ

γσ2

�2− 3

20γ

�ε+O(ε4/3).

The corresponding equivalent safe rate is:

r + β = r +µ2 − λ2

γσ2= r +

µ2

2γσ2− γσ

2

2

�3

4γ

�µ

γσ2

�2�1− µ

γσ2

�2�2/3ε2/3 +O(ε4/3).

If µ/γσ2 = 1, then S̃t = St is a shadow price, and it is optimal to invest all wealth in the riskyasset at time t = 0, never to trade afterwards. In this case, the equivalent safe rate is given by thefrictionless value r + β = r + µ2/2γσ2.

Proof. First let 0 < µ/γσ2 �= 1. Optimality of the strategy (ϕ0t ,ϕt) associated to π̃(Yt) for S̃t hasbeen shown in Lemma C.3. The second representation for π± follows from the boundary conditionsfor g and the definition of u in Lemma B.2, while the asymptotic expansions are an immediateconsequence of the fractional power series for λ (cf. Lemma B.2) and Taylor expansion.

Next, Lemma C.4 shows that S̃t is a shadow price process in the sense of Definition A.1. Inview of the asymptotic expansions for π±, Proposition C.5 shows that, for small transaction costsε, the same policy is also optimal, with the same equivalent safe rate, in the original market withbid-ask prices (1− ε)St, St.

Consider now the degenerate case µ/γσ2 = 1. Then the optimal strategy in the frictionlessmodel S̃t = St transfers all wealth to the risky asset at time t = 0, never to trade afterwards,

26

(ϕ0t = 0 and ϕt = ξ + ξ0S00/S0 for all t ≥ 0). Hence it is of finite variation and the number of

shares never decreases, and increases only at time t = 0, where the shadow price coincides with theask price. Thus, S̃t = St is a shadow price. For small ε, the remaining assertions then follow fromProposition C.5 as above.

Next, the proof of Theorem 3.1, which establishes asymptotic finite-horizons bounds. In fact,the proof yields exact bounds in terms of λ, from which the expansions in the theorem are obtained.

Proof of Theorem 3.1. Let (φ0,φ) be any admissible strategy. Then as in the proof of Proposi-

tion C.5, we have ΞφT ≤ X̃φT for the corresponding shadow payoff, that is, the terminal value of the

wealth process X̃φt = φ00 + φ0S̃0 +

� t0 φsdS̃s corresponding to trading φ in the frictionless market

with price process S̃t. Hence (Guasoni and Robertson, 2011, Lemma 5) and the second bound inLemma C.2 imply

1

(1− γ)T logE�(ΞφT )

1−γ�≤ r + β + 1

Tlog(ϕ00− + ϕ0−S0) +

γ

(1− γ)T log Ê�e( 1γ−1)(q̃(Y0)−q̃(YT ))

�.

(C.13)For the strategy (ϕ0,ϕ) from Lemma C.4, we have ΞϕT ≥ (1 −

ε1−ε

µ+λγσ2 )X̃

ϕT by the proof of Propo-

sition C.5. Hence the first bound in Lemma C.2 yields

1

(1− γ)T logE�(ΞϕT )

1−γ� ≥ r + β + 1Tlog(ϕ00− + ϕ0−S̃0) +

1

(1− γ)T log Ê�e(1−γ)(q̃(Y0)−q̃(YT ))

�

+1

(1− γ)T log�1− ε

1− εµ+ λ

γσ2

�.

(C.14)

To determine explicit estimates for these bounds, we first analyze the sign of w̃(y) and hence themonotonicity of q̃(y) =

� z0 w̃(z)dz. Whenever w̃(y) = 0, the ODE (C.3) yields

σ2γw̃

�(y) =µ2 − λ2

σ2− µ̃(y)

2

σ̃(y)2.

Plugging in the definitions of µ̃(y) and σ̃(y) (cf. Lemma C.1) and using the ODE (B.4) for g, thisequality reduces to

σ2γw̃

�(y) =µ2 − λ2

σ2−�σg�(ey)eyl

1 + g(ey)l

�2.

Now note that λ = O(ε1/3) and log(u/l) = O(ε1/3) for � ↓ 0. Hence a Taylor expansion and theboundary conditions for g show

σ2γw̃

�(y) =µ2

σ2

�1− 1

γ2

�+O(ε1/3).

Consequently — for a sufficiently small spread ε — the derivative of w̃ is always positive forw̃(y) = 0 if γ > 1, resp. negative when γ ∈ (0, 1). In view of w̃(0) = 0, a comparison argumentthen shows that w̃(y) ≥ 0 for γ > 1 resp. w̃ ≤ 0 for γ ∈ (0, 1). Hence the function q̃(y) is increasingin the first case and decreasing in the second.

Now, consider Case 2 of Lemma B.1; the calculations for the other two cases follow along thesame lines with minor modifications. Then γ > 1 such that q̃ is positive and increasing. Hence,

γ

(1− γ)T log Ê�e( 1γ−1)(q̃(Y0)−q̃(YT ))

�≤ 1

T

� log(u/l)

0w̃(y)dy (C.15)

27

and likewise1

(1− γ)T log Ê�e(1−γ)(q̃(Y0)−q̃(YT ))

�≥ − 1

T

� log(u/l)

0w̃(y)dy. (C.16)

Now, inserting w̃(y) = w(y)− g�(ey)eyl/(1 + g(ey)l) and integrating leads to� log(u/l)

0w̃(y)dy =

� log(u/l)

0w(y)dy − log

�µ− λ− γσ2

µ+ λ− γσ2

�, (C.17)

due to the boundary conditions for g and the definition of u, l. By elementary integration ofthe explicit formula in Lemma B.1 and using the boundary conditions from Lemma B.3 for theevaluation of the result at 0 resp. log(u/l), the integral of w can also be computed in closed form:� log(u/l)

0w(y)dy =

µσ2 −

12

γ − 1 log�

1

1− ε(µ+ λ)(µ− λ− γσ2)(µ− λ)(µ+ λ− γσ2)

�+

1

2(γ − 1) log�(µ+ λ)(µ+ λ− γσ2)(µ− λ)(µ− λ− γσ2)

�.

(C.18)

As � ↓ 0, a Taylor expansion and the power series for λ then yield� log(u/l)

0w̃(y)dy =

µ

γσ2ε+O(ε4/3).

Likewise,

log

�1− 1

1− εµ− λγσ2

�= − µ

γσ2ε+O(ε4/3),

and

log(ϕ00− + ϕ0−S̃0) ≥ log(ϕ00− + ϕ0−S0)−ϕ0−S0

ϕ00− + ϕ0−S0ε+O(ε2),

such that the claimed bounds follow from (C.13) and (C.15) resp. (C.14) and (C.16).

D Trading volume

As above, let ϕt = ϕ↑t − ϕ

↓t denote the number of risky units at time t, written as the difference of

the cumulated numbers of shares bought resp. sold until t. Relative share turnover, defined as themeasure d�ϕ�t/|ϕt| = dϕ↑t /|ϕt|+ dϕ

↓t /|ϕt|, is a scale-invariant indicator of trading volume (Lo and

Wang, 2000). The long-term average share turnover is defined as

limT→∞

1

T

� T

0

d�ϕ�t|ϕt|

.

Similarly, relative wealth turnover (1 − ε)Stdϕ↓t /(ϕ0tS0t + ϕt(1 − ε)St) + Stdϕ↑t /(ϕ

0tS

0t + ϕtSt) is

defined as the amount of wealth transacted divided by current wealth, where both quantities areevaluated in terms of the bid price (1− ε)St when selling shares resp. in terms of the ask price Stwhen purchasing them. As above, the long-term average wealth turnover is then defined as

limT→∞

1

T

�� T

0

(1− ε)Stdϕ↓tϕ0tS

0t + ϕt(1− ε)St

+

� T

0

Stdϕ↑t

ϕ0tS0t + ϕtSt

�.

Both of these limits admit explicit formulas in terms of the gap, which yield asymptotic ex-pansions for ε ↓ 0. The analysis starts with a preparatory result (cf. Janeček and Shreve (2004,Remark 4) for the case of driftless Brownian motion).

28

Lemma D.1. Let Yt be a diffusion on an interval [l, u], 0 < l < u, reflected at the boundaries, i.e.

dYt = b(Yt)dt+ a(Yt)1/2

dWt + dLt − dUt, (D.1)

where the mappings a(y) > 0 and b(y) are both continuous, and the continuous, nondecreasinglocal time processes Lt and Ut satisfy L0 = U0 = 0 and only increase on {Lt = l} and {Ut = u},respectively. Denoting by m(y) the invariant density of Yt, the following almost sure limits hold:

limT→∞

LT

T=

a(l)m(l)

2limT→∞

UT

T=

a(u)m(u)

2(D.2)

Proof. For f ∈ C2([l, u]), write Lf(y) := b(y)f �(y) + a(y)f ��(y)/2. Then, by Itô’s formula:

f(YT )− f(Y0)T

=1

T

� T

0Lf(Yt)dt+

1

T

� T

0f�(Yt)a(Yt)

1/2dWt + f

�(l)LT

T− f �(u)UT

T.

Now, take f such that f �(l) = 1 and f �(u) = 0, and pass to the limit T → ∞. The left-hand sidevanishes because f is bounded; the stochastic integral also vanishes by the Dambis-Dubins-Schwarztheorem, the law of the iterated logarithm, and the boundedness of f �. Thus, the ergodic theorem(Borodin and Salminen, 2002, II.36 and II.37) implies that

limT→∞

LT

T= −

� u

lLf(y)m(y)dy.

Now, the self-adjoint representation (Revuz and Yor, 1999, VII.3.12) Lf = (af �m)�/2m yields:

limT→∞

LT

T= −1

2

� u

l(af �m)�(y)dy =

a(l)m(l)f �(l)

2− a(u)m(u)f

�(u)

2=

a(l)m(l)

2.

The other limit follows from the same argument, using f such that f �(l) = 0 and f �(u) = 1.

Lemma D.2. Let 0 < µ/γσ2 �= 1 and, as in (C.1), let

Yt =

�µ− σ

2

2

�t+ σWt + Lt − Ut

be Brownian motion with drift, reflected at 0 and log(u/l). Then if µ �= σ2/2, the following almostsure limits hold:

limT→∞

LT

T=

σ2

2

�2µσ2 − 1

(u/l)2µσ2

−1 − 1

�and lim

T→∞

UT

T=

σ2

2

�1− 2µσ2

(u/l)1−2µσ2 − 1

�.

If µ = σ2/2, then limT→∞ LT /T = limT→∞ UT /T = σ2/(2 log(u/l)) a.s.

Proof of Lemma D.2. First let µ �= σ2/2. Moreover, suppose that µ/γσ2 ∈ (0, 1). Then the scalefunction and the speed measure of the diffusion Yt are

s(x) =

� x

0exp

�− 2

� ξ

0

µ− σ22σ2

dζ

�dξ =

1

1− 2µσ2e(1− 2µ

σ2)x,

m(dx) =1[0,log(u/l)](x)2dx

s�(x)σ2= 1[0,log(u/l)](x)

2

σ2e( 2µσ2

−1)xdx.

29

The invariant distribution of Yt is the normalized speed measure

ν(dx) =m(dx)

m([0, log(u/l)])= 1[0,log(u/l)](x)

2µσ2 − 1

(u/l)2µσ2

−1 − 1e( 2µσ2

−1)xdx.

For µ/γσ2 > 1, the endpoints 0 and log(u/l) exchange their roles, and the result is the same, up toreplacing [0, log(u/l)] with [log(u/l), 0] and multiplying the formula by �

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