Electronic copy available at: http://ssrn.com/abstract=2446817

The Limits of Leverage

⇤

Paolo Guasoni† Eberhard Mayerhofer‡

October 6, 2014

Abstract

When trading incurs proportional costs, leverage can scale an asset’s return only up to amultiple, which is sensitive to its volatility and liquidity. In a model with one safe and one riskyasset, with constant investment opportunities and proportional costs, we find strategies thatmaximize long term return given average volatility. As leverage increases, rising rebalancingcosts imply declining Sharpe ratios. Beyond a critical level, even returns decline. Holding theSharpe ratio constant, higher volatility leads to superior returns through lower costs. For fundsreplicating benchmark multiples, such as leveraged ETFs, our strategies optimally trade o↵alpha against tracking error.

JEL Classification: G11, G12.

Keywords: leverage, transaction costs, portfolio choice, performance evaluation, ETFs.

⇤For helpful comments, we thank Jaksa Cvitanic, Gur Huberman, Tim Leung, Johannes Muhle-Karbe, RonnieSircar, Peter Tankov, Walter Schachermayer, and seminar participants at ETH Zurich, Vienna Graduate School ofFinance, Quant Europe, Quant USA, and Ban↵ Workshop on Arbitrage and Portfolio Optimization.

†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 gua-soni@bu.edu. Partially supported by the ERC (278295), NSF (DMS-1412529), SFI (07/MI/008, 07/SK/M1189,08/SRC/FMC1389), and FP7 (RG-248896).

‡Dublin City University, School of Mathematical Sciences, Glasnevin, Dublin 9, Ireland, email eber-hard.mayerhofer@dcu.ie.

Electronic copy available at: http://ssrn.com/abstract=2446817

1 Introduction

If trading is costless, leverage can scale returns without limits. Using the words of Sharpe (2011),

“If an investor can borrow or lend as desired, any portfolio can be levered up ordown. A combination with a proportion k invested in a risky portfolio and 1� k in theriskless asset will have an expected excess return of k and a standard deviation equalto k times the standard deviation of the risky portfolio. Importantly, the Sharpe Ratioof the combination will be the same as that of the risky portfolio.”

In theory, this insight implies that the e�cient frontier is linear, that e�cient portfolios areidentified by their common maximum Sharpe ratio, and that any of them spans all the other ones.Also, if leverage can deliver any expected returns, then risk-neutral portfolio choice is meaningless,as it leads to infinite leverage.

In practice, hedge funds and high-frequency trading firms employ leverage to obtain high returnsfrom small relative mispricing of assets.1 Recent financial products such as leveraged mutual fundsand exchange traded funds (ETFs) closely follow the strategy described by Sharpe, rebalancingtheir exposure to an underlying asset, with the aim of replicating a multiple of its daily return.

This paper shows that trading costs undermine these classical properties of leverage and setsharp theoretical limits to its applications. We start by characterizing the set of portfolios thatmaximize long term expected returns for given average volatility, extending the familiar e�cientfrontier to a market with one safe and one risky asset, where both investment opportunities andrelative bid-ask spreads are constant. Figure 1 plots this frontier: expectedly, trading costs decreasereturns, with the exception of a full safe investment (the axes origin) or a full risky investment (theattachment point with unit coordinates), which lead to static portfolios without trading, and henceearn their frictionless return.2

But trading costs do not merely reduce expected returns below their frictionless benchmarks.Unexpectedly, in the leverage regime (the right of the full-investment point) rebalancing costs rise soquickly with volatility that returns cannot increase beyond a critical factor, the leverage multiplieror, briefly, the multiplier. The multiplier depends on the relative bid-ask spread ", on the expectedreturn µ and volatility �, and it is approximately equal to

0.3815⇣ µ

�2

⌘1/2"�1/2. (1.1)

Table 1 shows that even a modest bid-ask spread of 0.10% implies a multiplier of 23 for an assetwith 10% volatility and 5% expected return, while the multiplier declines to 10 for an asset withequal Sharpe ratio, but with a volatility of 50%. Leverage opportunities are much more limited forless liquid assets, with a spread of 1%, from less than 8 for 10% volatility to less than 4 for 50%volatility. Importantly, these limits on leverage hold even allowing for continuous trading, infinitemarket depth (any quantity trades at the bid or ask price), and zero capital requirements.

Our results have three broad implications. First, with a positive bid-ask spread even a risk-neutral investor who seeks to maximize expected long-run returns will take finite leverage, and infact a rather low leverage ratio in an illiquid market – risk-neutral portfolio choice is meaningful.

1A famous example is Long Term Capital Management, which used leverage of up to 30 to 40 times to increasereturns from convergence trades between on-the-run and o↵-the-run treasury bonds, see Edwards (1999).

2As we focus on long term investments, we neglect the one-o↵ costs of set up and liquidation, which are negligibleover a long holding period.

2

Electronic copy available at: http://ssrn.com/abstract=2446817

Bid-Ask Spread (")Volatility (�) 0.01% 0.10% 1.00%

10% 71.85 (71.22) 23.15 (22.58) 7.72 (7.12)20% 50.88 (50.36) 16.45 (15.92) 5.56 (5.04)50% 32.30 (31.85) 10.54 (10.07) 3.66 (3.18)

Table 1: Leverage multiplier (maximum factor by which a risky asset’s return can be scaled) fordi↵erent asset volatilities and bid-ask spreads, holding the Sharpe ratio at the constant level of 0.5.Multipliers are obtained from numerical solutions of (3.1), while their approximations from (1.1)are in brackets.

The resulting multiplier sets an endogenous level of risk that the investor chooses not to exceedregardless of risk aversion, simply to avoid reducing returns with trading costs. In this context,margin requirements based on volatility (such as value at risk and its variations) are binding onlywhen they reduce leverage below the multiplier, and are otherwise redundant. In addition, themultiplier shows that an exogenous increase in trading costs, such as a proportional Tobin tax onfinancial transactions, implicitly reduces the maximum leverage that any investor who seeks returnis willing to take, regardless of risk attitudes.

Second, two assets with the same Sharpe ratio do not generate the same e�cient frontierwith trading costs, and more volatility leads to a superior frontier. For example (Table 1) witha 1% spread the maximum leveraged return on an asset with 10% volatility and 5% return is7.72 ⇥ 5% ⇡ 39%. By contrast, an asset with 50% volatility and 25% return (equivalent to theprevious one from a classical viewpoint, since it has the same Sharpe ratio 0.5), leads to a maximumleveraged return of 3.66 ⇥ 25% ⇡ 92%. The reason is that a more volatile asset requires a lowerleverage ratio (hence lower rebalancing costs) to reach a certain return. Thus, an asset with highervolatility spans an e�cient frontier that achieves higher returns through lower costs.

Third, we obtain theoretical bounds on the potential returns of leveraged ETFs, and derive atestable restriction between the alpha and the tracking error of an optimally replicated ETF. In africtionless setting, an ETF can perfectly scale returns by any factor, without any tracking error:alpha is zero and the ETF return perfectly correlated with the benchmark’s. In reality, leveragedETFs have been introduced only since 2006, currently have leverage factors of up to three (minusthree for inverse funds), and funds on less liquid assets have significant tracking error.

Under optimal replication with trading costs, we show the following relation between the inter-cept ↵ and squared correlation R2 in the regression of the ETF return (net of management fees)on the benchmark’s return

↵ ⇡ �p3

12�2⇡⇤(1� ⇡⇤)

2 "p1�R2

, (1.2)

where ⇡⇤ is the desired scaling factor, � is the benchmark’s volatility, and " is its relative spread.The equation makes the optimal replication trade-o↵ clear: a higher R2 (lower tracking error) leadsto a more negative alpha through higher costs, and vice versa. More importantly, the equationo↵ers a testable relationship among observable quantities, without involving the expected returnµ, notoriously hard to estimate with precision.

This paper bears on the established literature on portfolio choice with frictions and on thenascent literature on leveraged ETFs. The e↵ect of transaction costs on portfolio choice is firststudied by Magill and Constantinides (1976), Constantinides (1986), and Davis and Norman (1990),

3

0

2

4

6

8

10

12

14

16

0 2 4 6 8 10 12 14 16

Figure 1: E�cient Frontier with trading costs, as expected return (vertical axis, in multiples of theasset’s return) against standard deviation (horizontal axis, in multiples of the asset’s volatility).The asset has expected return µ = 8%, volatility � = 16%, and bid-ask spread of 1%. The upperline denotes the classical e�cient frontier, with no transaction costs. The maximum height of thecurve corresponds to the leverage multiplier.

who identify a wide no-trade region, and derive the optimal trading boundaries through numericalprocedures. While these papers focus on the maximization of expected utility from intertemporalconsumption on an infinite horizon, Taksar, Klass and Assaf (1988), and Dumas and Luciano (1991)show that similar strategies are obtained in a model with terminal wealth and a long horizon –time preference has negligible e↵ects on trading policies. This paper adopts the same approach ofa long horizon, both for the sake of tractability, and because it focuses on the tradeo↵ betweenreturn, risk, and costs, rather than consumption.

Our asymptotic results for positive risk aversion are similar in spirit to the ones derived by Shreveand Soner (1994), Rogers (2004), Gerhold, Guasoni, Muhle-Karbe and Schachermayer (2014), andKallsen and Muhle-Karbe (2013), whereby transaction costs imply a no-trade region with widthof order O("1/3) and a welfare e↵ect of order O("2/3). We also find that the trading boundariesobtained from a local mean-variance criterion are equivalent at the first order to the ones obtainedfrom power utility. The risk-neutral expansions and the limits of leverage of order O("�1/2) are new,and are qualitatively di↵erent from the risk-averse case. These results are not regular perturbationsof a frictionless analogue, which is ill-posed. They are rather singular perturbations, which displaythe speed at which the frictionless problem becomes ill-posed as the crucial friction parametervanishes.

Our paper also contributes to the literature on leveraged ETFs. Tang and Xu (2013) observethat leveraged funds deviate significantly from their benchmarks even after management fees, andseparate tracking error into a compounding component, due to the convexity of levered returns anda rebalancing component, due to trading frictions (cf. Jarrow (2010); Lu et al. (2012); Avellaneda

4

and Zhang (2010); Cheng and Madhavan (2009)). Jiang and Yan (2012), Avellaneda and Dobi(2012), and Guo and Leung (2014) report that ETFs significantly underperform their benchmarkseven at daily frequencies, and Wagalath (2013) derives an asymptotic expression for the slippagethat results from rebalancing at fixed intervals. We incorporate trading costs explicitly in themodel, and derive optimal replication policies that trade o↵ alpha against tracking error.

Finally, this paper connects to the recent work of Frazzini and Pedersen (2012) on embeddedleverage. If di↵erent investors face di↵erent leverage constraints, they find that in equilibriumassets with higher factor exposures trade at a premium, thereby earning a lower return. Frazziniand Pedersen (2014) confirm this prediction across a range of markets and asset classes, and Asnesset al. (2012) use it to explain the performance risk-parity strategies. With exogenous asset prices,we find that assets with higher volatility generate a superior e�cient frontier by requiring lowerrebalancing costs for the same return. This observation suggests that the embedded leveragepremium may be induced by rebalancing costs in addition to leverage constraints, and should behigher for more illiquid assets.

The rest of the paper is organized as follows: section 2 introduces the model and the optimizationproblem. Section 3 contains the main results, which characterize the e�cient frontier in the risk-averse (Theorem 3.1) and risk-neutral (Theorem 3.2) cases. Section 4 discusses the implicationsof these results for the e�cient frontier, the trading boundaries of optimal policies, the embeddedleverage e↵ect, and performance evaluation, showing that the tradeo↵ between alpha and trackingerror is equivalent to the tradeo↵ between average return and volatility. The section concludes withtwo supporting results, which show that the risk-neutral solutions arise as limits of their risk-aversecounterparts for low risk-aversion, and that the risk-neutral solutions are not constrained by thesolvency condition. Section 5 o↵ers a derivation of the main free-boundary problems from heuristiccontrol arguments, and concluding remarks are in section 6. All proofs are in the appendix.

2 Model

The market includes one safe asset earning a constant interest rate of r � 0 and a risky asset withask (buying) price St given by

dSt

St= (µ+ r)dt+ �dBt, S0,�, µ > 0,

where B is a standard Brownian motion. The risky asset’s bid (selling) price is (1 � ")St, whichimplies a constant relative bid-ask spread of " > 0, or, equivalently, constant proportional transac-tion costs. A self-financed trading strategy is summarized by its initial capital x and the numberof shares 't of the risky asset held at time t. Denote by wt the fund’s wealth at time t, which isthe sum of the safe position x�

R t0 Ssd's � "

R t0 Ssd'

#s and the risky position St't evaluated at the

ask price3:

wt = x�Z t

0Ssd's � "

Z t

0Ssd'

#s + St't. (2.1)

We further require a strategy ' to by solvent, in that its corresponding wealth wt is strictly positiveat all times. (Admissible strategies are formally described in Definition A.1 below.)

3The convention of evaluating the risky position at the ask price is inconsequential. Using the bid price insteadwould lead to the same results.

5

We focus on objectives that trade o↵ a fund’s average return against its realized variance. Theportfolio return rt over the time-interval [t��t, t] is

rt =wt � wt��t

wt��t, (2.2)

while the annualized average return has the continuous-time approximation

rT =1

n�t

0tTX

t=k�

rt ⇡1

T

Z T

0

dwt

wt.

In the familiar setting of no trading costs, 1T

R T0

dwtwt

= r + 1T

R T0 µ⇡tdt +

1T

R T0 �⇡tdBt, where ⇡t =

'tSt/wt is the portfolio weight of the risky asset, hence the average return equals the average riskyexposure times its excess return, plus the safe rate.

Likewise, the average squared volatility on [0, T ] is obtained by the usual variance estimatorapplied to returns, and has the continuous-time approximation

1

n�t

0tTX

t=k�

(rt � rT )2 ⇡ 1

T

Z T

0

dhwitw2t

=�2

T

Z T

0⇡2t dt.

(The last equality holds because the trading cost term "R t0 Ssd'

#s in (2.1) is increasing and contin-

uous, hence its total sum of squares is negligible on a fine grid.) As a result, the usual risk-returntrade-o↵ is captured by maximizing

max'

1

TEZ T

0

dwt

wt� �

2

Z T

0

dhwitw2t

�, (2.3)

where the parameter � > 0 is interpreted as a proxy for risk-aversion. This mean-variance objectiveis relevant for two reasons: first, without trading costs and for a strategy with finite variance, itreduces to

max⇡

1

TEZ T

0

⇣µ⇡t �

�

2�2⇡2t

⌘dt

�(2.4)

which leads to the optimal constant-proportion portfolio ⇡ = µ��2 dating back to Markowitz and

Merton, and confirms that in a geometric Brownian motion market with costless trading, the objec-tive considered here is equivalent to the utility-maximization with constant relative risk aversion.Second, the objective (2.3) is equivalent to the maximization of alpha for a given tracking error,or, equivalently, to the minimization of tracking error for given alpha, which is relevant for fundsthat aim at replicating multiples of a benchmark’s return. This equivalence is shown in Section 4.4below.

To better understand the e↵ect of trading costs, note that the maximand in (2.3) reduces to amore concrete expression.

Lemma 2.1. For any T > 0 and admissible trading strategy ',

FT (') :=1

TEZ T

0

dwt

wt� �

2

Z T

0

dhwitw2t

�= r +

1

TE"Z T

0

✓µ⇡t �

��2

2⇡2t

◆dt� "

Z T

0⇡t

d'#t

't

#. (2.5)

6

The final extra term in (2.5) with trading costs hinders continuous portfolio rebalancing, makingconstant-proportion strategies unfeasible. The reason is that it is costly both to keep the exposureto the risky asset high enough to achieve the desired return, and to keep it low enough to limit thelevel of risk – trading costs reduce returns and increase risk.

To neglect the spurious, non-recurring e↵ects of portfolio set-up and liquidation, we focus onthe long run performance objective4

F1(') := lim supT!1

FT ('), (2.6)

which is akin to the one used by Dumas and Luciano (1991) in the context of utility maximization.As a consequence of the law of large numbers, the long run performance criterion entails thatex-ante and ex-post performances coincide, i.e.,

F1(') := lim supT!1

1

TEZ T

0

dwt

wt� �

2

Z T

0

dhwitw2t

�= lim sup

T!1

1

T

✓Z T

0

dwt

wt� �

2

Z T

0

dhwitw2t

◆. (2.7)

A few special cases are noteworthy. Without transaction costs, maximizing this objective isequivalent to maximizing power utility over any finite horizon T . With or without transactioncosts, in the risk-neutral case � = 0 the objective boils down to the average annualized return overa long horizon, while for � = 1 it coincides with logarithmic utility.

3 Main Results

The first result characterizes the optimal solution to the main objective in (2.6) in the usual case ofa positive aversion to risk (� > 0). In this setting, the next theorem shows that trading costs createa no-trade region around the frictionless portfolio ⇡⇤ = µ

��2 , and states the asymptotic expansions

of the resulting average return and standard deviation5, thereby extending the familiar e�cientfrontier to account for trading costs.

Theorem 3.1 (Risk Aversion and E�cient Frontier). Let � 6= µ/�2.

(i) For any � > 0 there exists "0 > 0 such that for all " < "0, the free boundary problem

1

2�2⇣2W 00(⇣) + (�2 + µ)⇣W 0(⇣) + µW (⇣)� 1

(1 + ⇣)2

✓µ� ��2

⇣

1 + ⇣

◆= 0, (3.1)

W (⇣�) = 0, (3.2)

W 0(⇣�) = 0, (3.3)

W (⇣+) ="

(1 + ⇣+)(1 + (1� ")⇣+), (3.4)

W 0(⇣+) ="((1� ")⇣2+ � 1)

(1 + ⇣+)2(1 + (1� ")⇣+)2(3.5)

has a unique solution (W, ⇣�, ⇣+) for which ⇣� < ⇣+.

4In this equation the lim sup is used merely to guarantee a good-definition a priori. A posteriori, we show thatoptimal strategies exist in which the lim sup is in fact a lim, hence the similar problem defined in terms of lim infleads to the same solution.

5The exact formulae for average return, standard deviation, and average trading costs are in Appendix C.

7

(ii) The trading strategy ', which buys at ⇡� := ⇣�/(1 + ⇣�) and sells at ⇡+ := ⇣+/(1 + ⇣+) aslittle as necessary to keep the risky weight ⇡t = ⇣t/(1 + ⇣t) within the interval [⇡�,⇡+], isoptimal.

(iii) The maximum performance is

max'2�

limT!1

1

T

Z T

0

dwt

wt� �

2

Z T

0

dhwitw2t

= r + µ⇡� � ��2

2⇡2�. (3.6)

where � is the set of admissible strategies in Definition A.1.

(iv) The trading boundaries ⇡� and ⇡+ have the asymptotic expansions

⇡± = ⇡⇤ ±✓

3

4�(⇡⇤)

2(1� ⇡⇤)2

◆1/3

"1/3 � (� � 1)(⇡⇤)2(1� ⇡⇤)

6

✓6

�⇡⇤(1� ⇡⇤)

◆2/3

"2/3 +O(").

(3.7)The long-run mean m, standard deviation s, average trading costs (ATC) and maximumperformance F (') have expansions (using the convention a1/n = sign(a) |a|1/n for any a 2 Rand odd integer n, and a2/n = (a2)1/n)

m := limT!1

1

T

Z T

0

dwt

wt= r +

µ2

��2+

µ(5⇡⇤ � 3)

�

✓�⇡⇤(1� ⇡⇤)

6

◆1/3

"2/3 +O("), (3.8)

s := limT!1

s1

T

⌧Z ·

0

dwt

wt

�

T

=µ

��+�(7⇡⇤ � 3)

4�

✓�⇡⇤(1� ⇡⇤)

6

◆1/3

"2/3 +O("), (3.9)

ATC := limT!1

1

T

Z T

0⇡t

d'#t

't=

3�2

�

✓�⇡⇤(1� ⇡⇤)

6

◆4/3

"2/3 +O("), (3.10)

F1(') = r +µ2

2��2� ��2

2

✓3

4�⇡2⇤(1� ⇡⇤)

2

◆2/3

"2/3 +O("). (3.11)

Proof. The proof is divided into Propositions B.1, B.4 and B.6 in Appendix B.

In contrast to the risk-averse setting, the risk-neutral objective leads to a novel solution, whichdoes not have a frictionless analogue: for small trading costs, both the optimal policy and itsperformance become unbounded as the optimal leverage increases arbitrarily. The next resultdescribes the solution to the risk-neutral problem, identifying the approximate dependence of theleverage multiplier and its performance on the asset’s risk, return, and liquidity.

Theorem 3.2 (Risk Neutrality and Limits of Leverage). Let � = 0.

(i) There exists "0 > 0 such that for all " < "0, the free boundary problem (3.1)–(3.5) has aunique solution (W, ⇣�, ⇣+) with ⇣� < ⇣+.

(ii) The trading strategy ' that buys at ⇡� := ⇣�/(1 + ⇣�) and sells at ⇡+ := ⇣+/(1 + ⇣+) to keepthe risky weight ⇡t = ⇣t/(1 + ⇣t) within the interval [⇡�,⇡+] is optimal.

(iii) The maximum expected return is

max'2�

limT!1

1

T

Z T

0

dwt

wt= r + µ⇡�. (3.12)

8

(iv) The trading boundaries have the series expansions

⇡� =(1� )1/2⇣ µ

�2

⌘1/2"�1/2 + 1 +O("1/2), (3.13)

⇡+ =1/2⇣ µ

�2

⌘1/2"�1/2 + 1 +O("1/2), (3.14)

where ⇡ 0.5828 is the unique solution of

3

2⇠ + log(1� ⇠) = 0, ⇠ 2 (0, 1).

The next section discusses how these results modify the familiar intuition about risk, return,and performance evaluation in the context of trading costs.

4 Implications

4.1 E�cient frontier

Theorem 3.1 extends the familiar e�cient frontier to account for trading costs. Compared to thelinear frictionless frontier, average returns decline because of rebalancing losses. Average volatilityincreases because more risk becomes necessary to obtain a given return net of trading costs.

To better understand the e↵ect of trading costs on return and volatility, consider the dynamicsof the portfolio weight in the absence of trading, which is

d⇡t = ⇡t(1� ⇡t)(µ� �2⇡t)dt+ �⇡t(1� ⇡t)dBt. (4.1)

The central quantity here is the portfolio weight volatility �⇡t(1�⇡t), which vanishes for the single-asset portfolios ⇡t = 0 or ⇡t = 1, remains bounded above by �/4 in the long-only case ⇡t 2 [0, 1],and rises quickly with leverage (⇡t > 1). This quantity is important because it measures the extentto which a portfolio, left to itself, strays from its initial composition in response to market shocksand, by reflection, the quantity of trading that is necessary to keep it within some region. Inthe long-only case, the portfolio weight volatility decreases as the no-trade region widens to span[0, 1], which means that a portfolio tends to spend more time near the boundaries. By contrast,with leverage portfolio weight volatility increases, which means that a wider boundary does notnecessarily mitigate trading costs.

Consistent with this intuition, equations (3.8), (3.9) show that the impact of trading costs issmaller on long-only portfolios, but rises quickly with leverage. Small trading costs reduce returnsand increase volatility at the order of "2/3 but, crucially, as leverage increases the error of thisapproximation also increases, and lower values of � make it precise for ever smaller values of ".

The performance (3.11) coincides at the first order with the equivalent safe rate from utilitymaximization with constant relative risk aversion � (Gerhold et al., 2014, Equation (2.4)), support-ing the interpretation of � as a risk-aversion parameter, and confirming that, for asymptoticallysmall costs, the e�cient frontier captures the risk-return trade-o↵ faced by a utility maximizer.

Figure 2 displays the e↵ect of trading costs on the e�cient frontier. As the bid-ask spreaddeclines, the frontier increases to the linear frictionless frontier, and the asymptotic results in thetheorem become more accurate. However, if the spread is held constant as leverage (hence volatility)increases, the asymptotic expansions become inaccurate, and in fact the e�cient frontier ceases toincrease at all after the leverage multiplier is reached.

9

0

2

4

6

8

10

12

14

16

18

20

0 5 10 15 20 25 30 35 40 45 50

Figure 2: E�cient Frontier with trading costs, as expected return (vertical axis, in multiples ofthe asset’s expected return) against standard deviation (horizontal axis, in multiples of the asset’svolatility). The asset has expected return µ = 8%, volatility � = 16%, and bid-ask spread of0.1%, 0.5%, 1%, 2%. The upper line is the frictionless e�cient frontier. The maximum of eachcurve is the leverage multiplier.

4.2 Trading Boundaries

Each point in the e�cient frontier corresponds to a rebalancing strategy that is optimal for somevalue of the risk-aversion parameter �. For small trading costs, equation (3.7) implies that thetrading boundaries corresponding to the e�cient frontier depart from the ones arising in utilitymaximization, which are (Gerhold et al., 2014)

⇡± = ⇡⇤ ±✓

3

4�(⇡⇤)

2(1� ⇡⇤)2

◆1/3

"1/3 +O("). (4.2)

The term of order "2/3 vanishes for � = 1 because this case coincides with the maximization oflogarithmic utility. For high levels of leverage (� < 1 and ⇡⇤ > 1), this term implies that thetrading boundaries that generate the e�cient frontier are higher than the trading boundaries thatmaximize utility. In Figure 3, � ! 1 corresponds to the safe portfolio in the origin (0,0), while� = µ/�2 to the risky investment (1,1), which has by definition the same volatility and returnas the risky asset. As � declines to zero, the trading boundaries converge to the right endpoints,which correspond to the strategy that maximizes average return with no regard for risk, therebyachieving the multiplier.

Observe that (Figure 3), as leverage increases, the sell boundary rises more quickly than thebuy boundary. For example, the risk-neutral portfolio tolerates leverage fluctuations from approxi-

10

0

2

4

6

8

10

12

14

0 1 2 3 4 5 6 7 8 9

Figure 3: Trading boundaries ⇡± (vertical axis, as multiples of wealth in risky assets) againstaverage portfolio volatility (horizontal axis, as multiples of �). Parameter values are µ = 8%,� = 16% and " = 1%. The middle curve depicts the implied Merton fraction, which equals the firstmedian.

mately 6, below which it will increase the risky position, up to approximately 14, above which it willreduce it. The locations of these boundaries trade o↵ the need to keep exposure to the risky assethigh to maximize return while also keeping rebalancing costs low. Risk aversion makes boundariescloser to each other by penalizing the high realized variance generated by the wide risk-neutralboundaries.

Importantly, these boundaries remain finite even as the frictionless Merton portfolio µ/(��2)diverges to infinity as � declines to zero. Thus the no-trade region is obviously not symmetricaround the frictionless portfolio, in contrast to the boundaries arising from utility maximization(Gerhold et al., 2014), which are always symmetric, and hence diverge when � is low. The di↵erenceis that here the risk-neutral objective is to maximize the expected return of the portfolio, while arisk-neutral utility maximizer focuses on expected wealth. In a frictionless setting this distinctionis irrelevant, and an investor can use a return-maximizing policy to maximize wealth instead.But trading costs drive a wedge between these two ostensibly equivalent risk-neutral criteria –maximizing expected return is not the same as maximizing expected wealth.

Theorem 3.2 (iii) describes in the risk-neutral case the optimal trading boundaries, which satisfy

11

the approximate relation⇡�⇡+

⇡ 0.4172 (4.3)

which is universal in that it holds for any asset, regardless of risk, return and liquidity. This relationmeans that an optimal risk-neutral rebalancing strategy should always tolerate wide variations inleverage over time, and that the maximum allowed leverage should be approximately 2.5 times theminimum. More frequent rebalancing cannot achieve the maximum return: it can be explainedeither by risk aversion or by elements that lie outside the model, such as jumps in asset prices.

The liquidation constraint (A.1) implies that

⇡t <1

"(4.4)

for every admissible trading strategy. Since ⇡t ⇡+ for the optimal trading strategy given byTheorem 3.1 and Theorem 3.2, the upper bound (4.4) is never binding for realistic bid-ask-spreads.

4.3 Embedded leverage

In frictionless markets, two perfectly correlated assets with equal Sharpe ratio generate the samee�cient frontier, and in fact the same payo↵ space. This equivalence fails in the presence of tradingcosts: the more volatile asset is superior, in that it generates an e�cient frontier that dominates theone generated by the less volatile asset. Figure 4 (top of the three curves) displays this phenomenon:for example, a portfolio with an average return of 50% net of trading costs is obtained from anasset with 25% return and 50% volatility at a small cost, as an average leverage factor of 2 entailsmoderate rebalancing.

Achieving the same 50% return from an asset with 20% volatility (and 10% return) is moreonerous: trading costs require leverage higher than 5, which in turn increases trading costs. Overall,the resulting portfolio needs about 120% rather than 100% volatility to achieve the desired 50%average return (middle curve in Figure 4).

From an asset with 10% volatility (and 5% return), obtaining a 50% return net of trading costsis impossible (bottom curve in Figure 4), because the leverage multiplier is less than 8 (Table 1, topright), and therefore the return can be scaled to less than 40%. The intuition is clear: increasingleverage also increases trading costs, calling in turn for more leverage to increase return, but alsofurther increasing costs. At some point, the marginal net return from more leverage becomes zero,and increasing it does more harm than good.

Because an asset with higher volatility is superior to another one, perfectly correlated andwith equal Sharpe ratio, but with lower volatility, the model suggests that in equilibrium theycannot coexist, and that the asset with lower volatility should o↵er a higher return to be held byinvestors. Indeed, Frazzini and Pedersen (2012, 2014) document significant negative excess returnsin assets with embedded leverage (higher volatility), and o↵er a theoretical explanation based onheterogeneous leverage constraints, which lead more constrained investors to bid up prices (andhence lower returns) of more volatile assets. This paper suggests that the same phenomenon mayarise even in the absence of constraints, as a result of rebalancing costs. In contrast to constraints-based explanations, our model suggests that the premium for embedded leverage should be higherfor more illiquid assets.

12

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

Figure 4: E�cient Frontier, as average expected return (vertical axis) against volatility (horizontalaxis), for an asset with Sharpe ratio µ/� = 0.5, for various levels of asset volatility, from 10%(bottom), 20%, to 50% (top), for a bid-ask spread " = 1%. The straight line is the frictionlessfrontier.

4.4 Performance Evaluation: Alpha vs. Tracking Error

Portfolio performance measures based on the regression of a fund’s return against its benchmark’sreturn are ubiquitous and, as a result, are closely monitored by managers who are evaluated withsuch performance measures.

A manager who seeks to replicate a multiple of benchmark’s return over each period faces thechoice between rebalancing often to match the target return closely, thereby achieving low trackingerror but high trading costs, or rebalancing infrequently to reduce costs, but at the expense ofsubstantial tracking error. The next argument shows that this trade-o↵ is equivalent to the one inTheorems 3.1 and 3.2, and explains the implications of these results for performance evaluation.

Denote by rFt and rBt the returns of a fund and its (unlevered) benchmark, respectively, overthe time step [t � �t, t]. If a fund aims at tracking a multiple ⇡⇤ of the benchmark’s return, itsaverage excess return ↵T per unit of time, also known as alpha, is

↵T =1

n�t

0tTX

t=k�t

(rFt � ⇡⇤rBt ) ⇡

1

T

Z T

0

✓dwF

t

wFt

� ⇡⇤dwB

t

wBt

◆. (4.5)

13

The corresponding realized tracking error, defined as realized error standard deviation per unit oftime and denoted by s, satisfies

s2 =1

n�t

0tTX

t=k�t

(rFt � ⇡⇤rBt � ↵T )

2 ⇡ 1

T

⌧Z ·

0

✓dwF

wF� ⇡⇤

dwB

wB

◆�

T

. (4.6)

Simplifying the continuous-time quantities in the right-hand sides, the objective of maximizingalpha with a penalty (or, equivalently, a constraint) for tracking error reduces to maximize

1

TE"Z T

0µ(⇡t � ⇡⇤)dt� "

Z T

0⇡t

d'#t

't� �

2

Z T

0�2(⇡t � ⇡⇤)

2dt

#. (4.7)

The first term in this expression represents the excess return that results from the di↵erence betweenthe fund’s average exposure and its target exposure. The second term accounts for the losses fromtrading costs and, added to the first one, yields the overall fund’s alpha. The last term is thepenalty for tracking error, which results from the cumulative squared di↵erence between the fundand the target exposure. With elementary manipulations, the previous expression simplifies to

1

TE"Z T

0

⇣(µ+ ��2⇡⇤)⇡t �

�

2�2⇡2t

⌘dt� "

Z T

0⇡t

d'#t

't

#� µ⇡⇤ �

�

2�2⇡2⇤ (4.8)

and therefore the trade-o↵ between alpha and tracking error is equivalent to the trade-o↵ betweenrisk and return, up to replacing µ with µ = µ+ ��2⇡⇤, where ⇡⇤ is the target leverage.

Such an equivalence is intuitive in hindsight, as in a market with excess return µ the optimalfrictionless portfolio is µ/(��2) = ⇡⇤ + µ/(��2). As a result, higher risk aversion � leads to aportfolio that is closer to the target ⇡⇤, thereby focusing on low tracking error while disregardingthe excess return. Vice versa, lower risk aversion implies a portfolio that overweighs the risky assetin order to generate alpha, while accepting higher tracking error. This intuition remains valid withtransaction costs, but the risk-return trade-o↵ is altered by rebalancing costs.

Figure 5 displays the optimal tracking error as a function of average leverage. Consistent withthe intuition underlying the previous results, higher average leverage entails wider boundaries,which result in higher tracking error. In particular, high tracking error is not necessarily evidenceof poor manager performance if the underlying asset is illiquid. On the contrary, a savvy manage-ment strategy must accept higher tracking error to achieve high returns, and low tracking error isconsistent only with moderate leverage levels.

A scale-free indicator of tracking error that is used in practice is the R2 of the time-seriesregression of the fund’s returns against the benchmark returns. First, note that the estimatedslope, or beta, of this regression has the continuous-time approximation

�T ⇡hR ·0

dwF

wF ,R ·0

dwB

wB iThR ·0

dwB

wB iT=

R T0 ⇡t�2dt

�2T=

1

T

Z T

0⇡tdt. (4.9)

As a result, the long horizon R2 of the regression, defined as the ratio between the variance of thepredicted return and the variance of the realized return, is

R2 = limT!1

⇣1T

R T0 ⇡tdt

⌘2

1T

R T0 ⇡2t dt

. (4.10)

14

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0 10 20 30 40 50 60

Figure 5: Tracking errorp1�R2 for di↵erent levels of leverage (horizontal axis), for spreads of

" = 0.01% (bottom), 0.1% (center), and 1% (top). The curves display values of � 2 [0,1].

The next result o↵ers an asymptotic approximation for this quantity, similar to the expansions inthe main results.

Proposition 4.1. The following expansions hold

↵ =�ATC = �3�2

�

✓�⇡⇤(1� ⇡⇤)

6

◆4/3

"2/3 +O("), (4.11)

p1�R2 =

p3

6

✓6(1� ⇡⇤)2

�⇡⇤

◆1/3

"1/3 +O("1), (4.12)

and therefore

↵ = �p3

12�2⇡⇤(1� ⇡⇤)

2 "p1�R2

+O("4/3). (4.13)

This result shows that the alpha of a levered portfolio, abstracting from management fees,equals minus the expected costs. The tracking error departs from zero as the spread " increases,and as the target leverage ⇡⇤ rises above the buy-and-hold level of one. Other things equal, a fundthat seeks to replicate a larger multiple ⇡⇤ of a benchmark’s return has a higher tracking error anda more negative alpha. Thus, it is misleading to compare two funds with di↵erent targets, and to

15

conclude that one is better managed than the other merely because its R2 is higher, or becauseits alpha is less negative. Even two funds with the same target may be optimally managed, andyet lead to di↵erent alpha and R2, as one may seek lower tracking error at the expense of morenegative alpha.

Equation (4.13) o↵ers an approximate relation in terms of observable quantities only, and can beused as a measure of replication performance that controls for the e↵ects of trading costs, volatility,target leverage, and tracking error. Thus, subtracting from the realized alpha the fund’s expenseratio and the right-hand side of (4.13) yields the amount of alpha that is unexplained by the model,and hence can be plausibly attributed to managerial skill – or lack thereof.

4.5 From risk aversion to risk neutrality

Theorems 3.1 and 3.2 are qualitatively di↵erent: while Theorem 3.1 with positive risk aversion leadsto a regular perturbation of the Markowitz-Merton solution, Theorem 3.2 with risk-neutrality leadsto a novel result with no meaningful analogue in the frictionless setting – a singular perturbation.Furthermore, a close reading of the statement of Theorem 3.1 shows that the existence of a solutionto the free-boundary problem, and the asymptotic expansions, hold for " less than some threshold"(�) that depends on the risk aversion �. In particular, if � approaches zero while " is held constant,Theorem 3.1 does not o↵er any conclusions on the convergence of the risk-averse to the risk-neutralsolution. Still, if the risk-neutral result it to be accepted as a genuine phenomenon rather than anartifact, it should be clarified whether the risk averse trading policy and its performance convergeto their risk neutral counterparts as risk aversion vanishes.

The next result resolves this point under some parametric restrictions.

Theorem 4.2. Let µ > �2, " > 0, and � > 0, and set

G(⇣) :="

(1 + ⇣)(1 + (1� ")⇣), h(⇣) = µ

⇣

1 + ⇣� ��2

2

✓⇣

1 + ⇣

◆2

.

Assume that, for any � 2 [0, �] the free boundary problem (3.1) has a unique solution (W (·; �), ⇣�(�), ⇣+(�))and that the function

W (⇣; �) :=

8><

>:

0, ⇣ < ⇣�(�)

W (⇣; �), ⇣ 2 [⇣�(�), ⇣+(�)]

G(⇣), ⇣ � ⇣+(�)

satisfies the equation

min

✓�2

2⇣2W 0(⇣; �) + µ⇣W (⇣; �)� h(⇣) + h(⇣�(�), G(⇣)� W (⇣; �), W (⇣; �)

◆= 0. (4.14)

Then, (4.14) is satisfied also for � = 0, and for each � 2 [0, �], the trading strategy that buys at

⇡�(�) =⇣�(�)

1+⇣�(�) and sells at ⇡+(�) =⇣+(�)

1+⇣+(�) to keep the risky weight ⇡t = ⇣t/(1 + ⇣t) within the

interval [⇡�(�),⇡+(�)] is optimal. Furthermore, ⇣±(�) ! ⇣±(0) and W (⇣; �) ! W (⇣; 0) as � # 0,each ⇣ 2 R.

In summary, this result confirms that, as the risk-aversion parameter � declines to zero, therisk-averse policy in Theorem 3.1 can only converge to the risk-neutral policy in Theorem 3.2, andthat the corresponding mean-variance objective in Theorem 3.1 converges to the average return inTheorem 3.2.

16

5 Heuristic Solution

This section o↵ers a heuristic derivation of the HJB equation. Consider the finite-horizon objective

max'2�

E"Z T

0

✓µ⇡t �

��2

2⇡2t

◆dt� "

Z T

0⇡t

d'#t

't

#(5.1)

From the outset, it is clear that this objective is scale-invariant, because doubling the initial numberof risky shares and safe units, and also doubling the number of shares 't held at time t has thee↵ect of keeping the objective functional constant. Thus, we conjecture that the residual valuefunction V depends on the calendar time t and on the variable ⇣t = ⇡t/(1 � ⇡t), which denotesthe number of shares held for each unit of the safe asset. In terms of this variable, the conditionalvalue of the above objective at time t becomes:

F'(t) =R t0

⇣µ ⇣s1+⇣s

� ��2

2⇣2s

(1+⇣s)2

⌘ds� "

R t0

⇣s1+⇣s

d'#s

's+ V (t, ⇣t). (5.2)

By Ito’s formula, the dynamics of F' is

dF'(t) =

✓µ

⇣t1 + ⇣t

� ��2

2

⇣2t(1 + ⇣t)2

◆dt� "

⇣t1 + ⇣t

d'#t

't+ Vt(t, ⇣t)dt+ V⇣(t, ⇣t)d⇣t +

1

2V⇣⇣(t, ⇣t)dh⇣it,

where subscripts of V denote respective partial derivatives. The self-financing condition (2.1)implies that

d⇣t⇣t

= µdt+ �dWt + (1 + ⇣t)d't

't+ "⇣t

d'#t

't, (5.3)

which in turn allows to simplify the dynamics of F' to (henceforth the arguments of V are omittedfor brevity)

dF'(t) =

✓µ

⇣t1 + ⇣t

� ��2

2

⇣2t(1 + ⇣t)2

+ Vt +�2

2⇣2t V⇣⇣ + µ⇣tV⇣

◆dt (5.4)

�⇣t✓V⇣(1 + (1� ")⇣t) +

"

1 + ⇣t

◆d'#

t

't+ ⇣t(1 + ⇣t)V⇣

d'"t

't+ �⇣tV⇣dWt. (5.5)

Now, by the martingale principle of optimal control (Davis and Varaiya, 1973) the process F'(t)above needs to be a supermartingale for any trading policy ', and a martingale for the opti-mal policy. Since '" and '# are increasing processes, the supermartingale condition implies theinequalities

� "

(1 + ⇣)(1 + (1� ")⇣) V⇣ 0, (5.6)

and the martingale condition prescribes that the left (respectively, right) inequality becomes anequality at the points of increase of '# (resp. '"). Likewise, it follows that

µ⇣

1 + ⇣� ��2

2

⇣2

(1 + ⇣)2+ Vt +

�2

2⇣2V⇣⇣ + µ⇣V⇣ 0 (5.7)

with the inequality holding as an equality whenever both inequalities in (5.6) are strict. To achievea stationary (that is, time-homogeneous) system, suppose that the residual value function is of the

17

form V (t, ⇣) = �(T � t) �R ⇣ W (z)dz for some � to be determined, which represents the average

optimal performance over a long period of time. Replacing this parametric form of the solution,the above inequalities become

0 W (⇣) "

(1 + ⇣)(1 + (1� ")⇣), (5.8)

µ⇣

1 + ⇣� ��2

2

⇣2

(1 + ⇣)2� �� �2

2⇣2W 0(⇣)� µ⇣W (⇣) 0, (5.9)

Assuming further that the the first inequality holds over some interval [⇣�, ⇣+], with each inequalityreducing to an equality at the respective endpoint, the optimality conditions become

�2

2⇣2W 0(⇣) + µ⇣W (⇣)� µ

⇣

1 + ⇣+��2

2

⇣2

(1 + ⇣)2+ � =0 for ⇣ 2 [⇣�, ⇣+], (5.10)

W (⇣�) =0, (5.11)

W (⇣+) ="

(⇣+ + 1)(1 + (1� ")⇣+), (5.12)

which lead to a family of candidate value functions, each of them corresponding to a pair orboundaries (⇣�, ⇣+). The optimal boundaries are identified by the smooth-pasting conditions,formally derived by di↵erentiating (5.11) and (5.12) with respect to their boundaries

W 0(⇣�) =0, (5.13)

W 0(⇣+) ="((1� ")⇣2+ � 1)

(1 + ⇣+)2(1 + (1� ")⇣+)2. (5.14)

These conditions allow to identify the value function. The four unknowns are the free parameter inthe general solution to the ordinary di↵erential equation (5.10), the free boundaries ⇣� and ⇣+, andthe optimal rate �. These quantities are identified by the boundary and smooth-pasting conditions(5.11), (5.12), (5.13), (5.14).

6 Conclusion

The costs of rebalancing a levered portfolio are substantial, and detract from its ostensible fric-tionless return. As leverage increases, such costs rise faster than the frictionless return, making itimpossible for an investor to lever an asset’s return beyond a certain multiple, net of trading costs.

In contrast to the frictionless theory, trading costs make the risk-return tradeo↵ nonlinear. Aninvestor who seeks high return prefers an asset with high volatility to another one with equal Sharperatio but lower volatility, because higher volatility makes leverage cheaper to realize. A risk-neutral,return-maximizing investor does not take infinite leverage, but rather keeps it within a band thatbalances high exposure with low rebalancing costs.

These findings have broad implications in portfolio choice, asset pricing, and financial interme-diation. For example, a bank that extends risky loans is akin to an investor trading in an illiquidrisky asset: in constrast to the frictionless common wisdom, our results imply that such a bank willnot increase its balance sheet without bounds, even if it is neutral to risk and regulatory capitalrequirements are absent. However, the endogenous finite leverage is sensitive to the volatility and

18

the liquidity of the loans, suggesting that attempts to encourage or discourage bank lending shouldaddress these factors.

A direct application of these results is in the performance evaluation of leveraged ETF. Optimalreplication strategies face a tradeo↵ between high correlation with the benchmark and lower alpha,and we derive a testable restriction that any optimal replication policy must satisfy.

A Admissible Strategies

A strategy is admissible if it is nonanticipative and solvent, up to a small increase in the spread:

Definition A.1. Let x > 0 (the initial capital) and let ('"t )t�0 and ('#

t )t�0 (the cumulative num-ber of shares bought and sold, respectively) be continuous, increasing processes, adapted to the

augmented natural filtration of B. (x, 't = '"t � '#

t ) is an admissible strategy if its liquidationvalue is strictly positive at all times: There exists "0 > " such that

x�Z t

0Ssd's + St't � "0

Z t

0Ssd'

#s � "0'+

t St > 0 a.s. for all t � 0. (A.1)

We denote the family of admissible trading strategies by �.

The following lemma shows that, without loss of generality, it is safe to exclude trading strategiesthat involve short selling or violate certain integrability conditions, because each such strategycannot be optimal.

Lemma A.2. Let ' 2 � be optimal for (2.6). Then:

(i) the strategy 't := 't1{'t�0} is also optimal; and

(ii) the following integrability conditions hold6

Z t

0⇡2udu < 1,

Z t

0⇡u

dk'uk'u

< 1 a.s. for all t � 0, (A.2)

where k'tk denotes the total variation of ' on [0, t].

Proof. Proof of (i): It is clear that 't is an admissible trading strategy if ' is. Furthermore ⇡t � ⇡tat all times t, and ⇡t = 0 whenever 't < 0, whence F (', T ) � F (', T ), each T > 0.

Proof of (ii): The first integrability condition in (A.2) is trivially satisfied, since it is a directconsequence of (A.1), which in turn implies ⇡t 1/", for all t, a.s.. To check the second one,suppose that some continuous, finite variation trading strategy ' satisfying ⇡t 1/" exists suchthat for some t > 0

Lt :=

Z t

0⇡u

d|'|u'u

= 1

with positive probability. Then on the same event one of the following integrals must be infinite,i.e.

L1t :=

Z t

0⇡u

d'#u

'u= 1, or L2

t :=

Z t

0⇡u

d'u

'u= 1.

6Note that ⇡t't

= Stwt

, therefore on the set {(!, t) : 't = 0} the quantity ⇡t't

is well-defined.

19

The former case leads to infinite average transaction costs (the third term times �1/" in (2.5)),hence the objective functional equals �1, which is outperformed by any buy-and-hold strategy.Likewise, if L2

t = 1, then

log('t)� log('0) =

Z t

0

d'u

'u= 1

which is impossible, since 't must be finite at all times due to proportional transaction costs.

The following lemma describes the dynamics of a the wealth process wt, the risky weight ⇡t,and the risky/safe ratio ⇣t.

Lemma A.3. For any admissible trading strategy ', we have7

d⇣t⇣t

= µdt+ �dBt + (1 + ⇣t)d'"

t

't� (1 + (1� ")⇣t)

d'#t

't, (A.3)

dwt

wt= rdt+ ⇡t(µdt+ �dBt � "

d'#t

't), (A.4)

d⇡t⇡t

= (1� ⇡t)(µdt+ �dBt)� ⇡t(1� ⇡t)�2dt+

d'"t

't� (1� "⇡t)

d'#t

't. (A.5)

For any such strategy, the functional

F (', T ) :=1

TEZ T

0

dwt

wt� �

2

Z T

0

dhwitw2t

�(A.6)

equals to

F (', T ) = r +1

TE"Z T

0

✓µ⇡t �

��2

2⇡2t

◆dt� "

Z T

0⇡t

d'#t

't

#. (A.7)

Proof. Denoting by Xt and Yt the wealth in the safe and risky positions respectively, the self-financing condition boils down to

dXt = rXtdt� Std'"t + (1� ")Std'

#t , (A.8)

dYt = Std'"t � Std'

#t + 'tdSt. (A.9)

and hence

dXt

Xt=rdt� ⇣t

d'"t

't+ (1� ")⇣t

d'#t

't, (A.10)

dYtYt

=d'"

t

't� d'#

t

't+

dSt

St, (A.11)

d(Yt/Xt)

Yt/Xt=dYtYt

� dXt

Xt+

dhXitX2

t

� dhX,Y itXtYt

=dYtYt

� dXt

Xt. (A.12)

Equation (A.3) follows from the last equation, and (A.4) holds in view of equation (A.10) and(A.11). For the derivation of equation (A.5), one uses the identity ⇡t = 1 � 1

1+⇣tand (A.3). The

expression in (A.7) for the objective functional follows from equation (A.4).

7The notation dXtXt

= dYt means Xt = X0 +R t

0XsdYs, hence the SDEs are well defined even for zero Xt.

20

Equation (A.7) implies an a priori upper bound on the objective function.

Corollary A.4. For any admissible strategy ' and any T > 0, the finite-horizon objective (A.6)satisfies

F (', T ) r +µ2

2��2, (A.13)

whence the long-run objective F (') r + µ2

2��2 .

Proof. Follows directly from (A.7) and the pointwise inequality µ⇡t � ��2

2 ⇡2t µ2

2��2 .

A.1 Proof of Lemma 2.1

Proof of Lemma 2.1. See Lemma A.3.

B Proof of Theorem 3.1

This section contains a series of propositions, which lead to the proof of Theorem 3.1 (i)–(iii). Part(iv) of the theorem is postponed to Appendix C. Set

G(⇣) :="

(1 + ⇣)(1 + (1� ")⇣)and h(⇣) := µ

✓⇣

1 + ⇣

◆� ��2

2

✓⇣

1 + ⇣

◆2

. (B.1)

Defining H := h0, the free boundary problem (3.1)–(3.5) reduces to

1

2�2⇣2W 00(⇣) + (�2 + µ)⇣W 0(⇣) + µW (⇣)�H(⇣) = 0, (B.2)

W (⇣�) = 0, (B.3)

W 0(⇣�) = 0, (B.4)

W (⇣+) = G(⇣+), (B.5)

W 0(⇣+) = G0(⇣+). (B.6)

Proposition B.1. Let � > 0. For su�ciently small ", the free boundary problem (B.2)–(B.6) hasa unique solution (W, ⇣�, ⇣+), with ⇣� < ⇣+. The free boundaries have the asymptotic expansion

⇣± =⇡⇤

1� ⇡⇤±✓

3

4�

◆1/3✓ ⇡⇤(1� ⇡⇤)2

◆2/3

"1/3 +(5� 2�)⇡⇤2�(1� ⇡⇤)2

✓�⇡⇤(1� ⇡⇤)

6

◆1/3

"2/3 +O("). (B.7)

Proof of Proposition B.1. Since ⇣� /2 {�1, 0}, any solution of the initial value problem (B.2)–(B.4)is of the form

W (⇣�, ⇣) =2

(�⇣)2

Z ⇣

⇣�

(h(y)� h(⇣�))

✓y

⇣

◆2�⇡⇤�2

dy. (B.8)

Suppose (W, ⇣�, ⇣+) is a solution of (B.2)–(B.6). Due to (B.8) W (·) ⌘ W (⇣�, ·). Let

J(⇣�, ⇣) :=�2⇣2�⇡⇤

2W (⇣�, ⇣). (B.9)

21

By the terminal conditions (B.5)–(B.6) at ⇣+, and setting � = "1/3, (⇣�, ⇣+) satisfy the followingsystem of non-linear equations,

1(⇣�, ⇣+) := W (⇣�, ⇣+)��3

(1 + ⇣+)(1 + (1� �3)⇣+)= 0, (B.10)

2(⇣�, ⇣+) :=2(h(⇣+)� h(⇣�))

�2⇣2+� 2�⇡⇤

⇣+W (⇣�, ⇣+)�

(1� �3)2

(1 + (1� �3)⇣+))2+

1

(1 + ⇣+)2= 0. (B.11)

Conversely, if (⇣�, ⇣+) solve (B.10)–(B.11), then the triplet (W (·; ⇣�), ⇣�, ⇣+) provides a solutionto the free boundary problem (B.2)–(B.6). Therefore, to provide a unique solution of the freeboundary problem, it su�ces to provide a unique solution of (B.10)–(B.11).

To obtain a guess for the asymptotic expansions of ⇣±, we develop 1,2 around

⇣⌥ = ⇣⇤ +B1,2� +O(�2), ⇣⇤ =⇡⇤

1� ⇡⇤,

which yields

1(⇣±(�)) = ��(1� ⇡⇤)6

3⇡2⇤

✓2B3

1 � 3B21B2 +B3

2 +3⇡2⇤

�(1� ⇡⇤)4

◆�3 +O(�4), (B.12)

2(⇣±(�)) =(B1 �B2)(B1 +B2)�(⇡⇤ � 1)6

⇡2⇤�2 +O(�3). (B.13)

By equating the coe�cients of the leading order terms to zero, we have the system

2B31 � 3B2

1B2 +B32 +

3⇡2⇤�(1� ⇡⇤)4

= 0, (B.14)

B1 +B2 = 0, (B.15)

which implies B1 = �B2 solves

B31 = � 3

4�

⇡2⇤(1� ⇡⇤)4

= 0,

an equation with a single, real-valued solution, namely

B1 = �✓

3

4�

◆1/3✓ ⇡⇤(1� ⇡⇤)2

◆2/3

. (B.16)

We claim that for su�ciently small � the system (B.10)–(B.11) has indeed a unique analytic solutionaround

⇣0,± :=⇡⇤

1� ⇡⇤±✓

3

4�

◆1/3✓ ⇡⇤(1� ⇡⇤)2

◆2/3

�.

This is equivalent to solving the corresponding system of equations � = (�1,�2) = 0 for (⌘�, ⌘+),around (B1, B2), where

⌘± :=⇣± � ⇡⇤

1�⇡⇤

�

and

�1 := 1(⇣�(⌘�), ⇣+(⌘+))

�3, �2 :=

2(⇣�(⌘�), ⇣+(⌘+))

�2. (B.17)

22

By Proposition B.2, there exists a unique solution for su�ciently small � > 0, which is analytic in�. Hence, also the original system (⇣�, ⇣+) = 0 has a unique solution (⇣�, ⇣+) around

⇡⇤1�⇡⇤

. As aconsequence, the free boundary problem (B.2)–(B.6) has a unique solution for su�ciently small ".

To derive the higher order terms of (B.7), it is useful to rewrite the integral (B.9) as

J(⇣�, ⇣+) =h(⇣�)(⇣

2�⇡⇤�1� � ⇣2�⇡⇤�1

+ )

2�⇡⇤ � 1| {z }=:I1

+

Z ⇣+

⇣�

h(y)y2�⇡⇤�2dy

| {z }=:I2

. (B.18)

The derivative of I2 with respect to � equals

dI2d�

= h(⇣+)⇣2�⇡⇤�2+

d⇣+d�

� h(⇣�)⇣2�⇡⇤�2�

d⇣�d�

, (B.19)

and shall be expanded as a power series in �. Integration with respect to � then yields an asymptoticexpansion of I2.

To obtain these expansions, we guess a solution of equations (B.10)–(B.11) of the form

⇣± =⇡⇤

1� ⇡⇤±✓

3

4�

◆1/3✓ ⇡⇤(1� ⇡⇤)2

◆2/3

� +A±�2 +O(�3),

for some unkowns A±, and substitute it into equations (B.10)–(B.11), using thereby (B.18) and(B.19). Comparing the coe�cients in the asymptotic expansion of the two equations reveals that

A� = A+ =(5� 2�)⇡⇤2�(1� ⇡⇤)2

✓�⇡⇤(1� ⇡⇤)

6

◆1/3

,

and thus we have derived (B.7).

Proposition B.2. For su�ciently small � > 0, the system �(⌘±(�), �) = 0 defined by (B.17), hasa unique solution (⌘�(�), ⌘+(�)) near (B1, B2), which is analytic in �.

Proof. Denote by D� the Frechet di↵erential of �. We claim that

det(D�)(⌘� = B1, ⌘+ = B2, � = 0) =6�(1� ⇡⇤)8(2�⇡⇤ � 1)

⇡2⇤6= 0, (B.20)

hence the implicit function theorem for analytic functions (Gunning and Rossi, 2009, TheoremI.B.4) ensures that for su�ciently small � there exists a unique solution (⌘�, ⌘+) of �(⌘�, ⌘+) = 0around (B1, B2) which is analytic in �.

So it remains to verify the determinant formula in (B.20). To this end, note that, by construc-tion,

2(⇣�, ⇣+) :=@ 1(⇣�, ⇣+)

@⇣+,

whence@�1(⌘�, ⌘+)

@⌘+|(B1,B2,0)= 0.

Thus

det(D�)(B1, B2, 0) =@�1(⌘�, ⌘+)

@⌘�|(B1,B2,0) ⇥

@�2(⌘�, ⌘+)

@⌘+|(B1,B2,0) .

23

We have@ 1

@⇣�= � 2h0(⇣�)

�2⇣2µ/�2

+

⇣2µ/�

2�2+

2µ/�2 � 1�

⇣2µ/�2�2

�2µ/�2 � 1

!

and since by the chain rule@�1(⌘�, ⌘+)

@⌘�=

1

�3

@ 1

@⇣�⇥ �

we obtain@�1(⌘�, ⌘+)

@⌘�|(B1,B2,0)=

62/3(1� ⇡⇤)3(�⇡⇤(1� ⇡⇤))1/3(1� 2�⇡⇤)

⇡⇤.

Similarly, we obtain

@�2(⌘�, ⌘+)

@⌘+|(B1,B2,0)= �61/3(1� ⇡⇤)4(�(1� ⇡⇤)⇡⇤)2/3

⇡2⇤,

from which (B.20), and hence the assertion, follows.

In the following, C2(A) denotes the space of twice continuously di↵erentiable functions on anopen set A ⇢ R.

Definition B.3. A solution of the HJB equation is a pair (V,�), where V 2 C2 and � 2 R), whichsatisfies

min(AV (x)� h(x) + �, G(x)� V 0(x), V 0(x)) = 0, x 2✓�1,� 1

1� "

◆[ (0,1), (B.21)

where A : C2(R) 7! C2(R) is the di↵erential operator

Af(x) :=�2

2x2f 00(x) + µxf 0(x).

Proposition B.4. Let (W, ⇣�, ⇣+) be the solution of the free boundary problem (B.5)–(B.6) (pro-vided by Proposition B.1) with asymptotic expansion (B.7). For su�ciently small ", the pair

V (·) :=Z ·

0W (⇣)d⇣, � := h(⇣�),

where

W (⇣) :=

8><

>:

0 for ⇣ < ⇣�,

W (⇣) for ⇣ 2 [⇣�, ⇣+],

G(⇣) for ⇣ � ⇣+,

(B.22)

is a solution of the HJB equation (B.21).

Proof of Proposition B.4. To check that (V,�) solves the HJB equation (B.21), consider seperatelythe domains [⇣�, ⇣+], ⇣ < ⇣� and ⇣ > ⇣+. In the following we use the decompositions

G(⇣) =1

1 + ⇣� 1� "

1 + (1� ")⇣and G0(⇣) =

✓1� "

1 + (1� ")⇣

◆2

� 1

(1 + ⇣)2.

24

First, note that on [⇣�, ⇣+], by construction it holds that

(AV (⇣)� h(⇣) + h(⇣�))0 =

1

2�2⇣2W 00(⇣) + (�2 + µ)⇣W 0(⇣) + µW (⇣)�H(⇣) = 0.

Furthermore, in view of the initial conditions (B.3)–(B.4),

(AV (⇣)� h(⇣) + h(⇣�)) |⇣=⇣�= AV (⇣) |⇣=⇣�= 0,

whenceAV (⇣)� h(⇣) + h(⇣�) ⌘ 0, ⇣ 2 [⇣�, ⇣+].

To see that 0 V 0 G on all of [⇣�, ⇣+], observe that

(h(⇣)� h(⇣�))0 = h0(⇣) = H(⇣) =

1

⇡⇤(1 + ⇣)2

✓⇡⇤ �

⇣

1 + ⇣

◆. (B.23)

Note that for ⇣� < ⇣ ⇣⇤, where ⇣⇤/(1 + ⇣⇤) = ⇡⇤, V 0(⇣) = W (⇣) > 0. It is shown that alsoW (·) � 0 on all of [⇣�, ⇣+]. This is equivalent to showing non-negativity of

w(⇣) := 2�2⇣2�⇡⇤W (⇣) =

Z ⇣

⇣�

(h(x)� h(⇣�))x2�⇡⇤�2dx. (B.24)

Now w0(⇣) = (h(⇣) � h(⇣�))⇣2�⇡⇤�2 = 0 if and only if h(⇣�) = h(⇣). Hence, either ⇣ = ⇣� or⇣ = ⇣ := 2⇡⇤ � ⇣�. By the first-order asymptotics of (B.7), we thus obtain ⇣ /2 [⇣�, ⇣+] forsu�ciently small ". Therefore w0 > 0 on (⇣�, ⇣+], and by (B.24) it follows that V 0 � 0 on all of[⇣�, ⇣+]. To conclude the validity of the HJB equation on [⇣�, ⇣+], it only remains to show theinequality V 0 G. To this end, notice that 1(⇣) = W (⇣)�G(⇣), (this is the function defined in(B.10), with fixed ⇣�) satisfies

1(⇣�) = �G(⇣�) = � "

(1 + ⇣�)(1 + (1� ")⇣�)= �(1� ⇡⇤)

2"+O("4/3),

hence for su�ciently small ", 1(⇣) < 0 on some interval [⇣�, ⇣), and 1(⇣) = 0. Therefore, ⇣ ⇣+.Since 1(⇣+) = 0 by construction, it su�ces to show that ⇣ = ⇣+ to prove non-negativity of V 0

on [⇣�, ⇣+]. Assume, for a contradiction, there exists a sequence �k # 0 such that for each k 2 N 1(⇣(�k)) = 0, and that ⇣�(�k) < ⇣(�k) < ⇣+(�k). We change to the variable u = ⇣�⇣⇤

� , and

introduce the notation u± = ⇣±�⇣⇤� , u = ⇣�⇣⇤

� . By selecting, if necessary, a subsequence, we mayassume without loss of generality that u(�k) converges, hence it must satisfy

limk!1

u(�k) =: B0 2 [B1, B2],

where B1 is defined in (B.16), and B2 = �B1. By the calculations leading to (B.16), we thereforeconclude that B0 must satisfy (B.14) in place of B2, i.e.

2B31 � 3B2

1B0 +B30 +

3⇡2⇤�(1� ⇡⇤)4

= 0. (B.25)

With B1 from (B.16) and after the change of variable ⇠ = �B0/B1, we obtain the equation

2� 3⇠ + ⇠3 = 0

25

which has the only solutions 1 and �2. Therefore, (B.25) has the only relevant solution

B0 = �B1 = B2.

By intertwining u+(�) and u(�k), we introduce

u⇤(�) =

(u(�k), k 2 Nu+(�), otherwise

.

Hence (u�(�), u⇤(�)) satisfies �(u�, u+) = 0 near (B1, B2), for su�ciently small �. By PropositionB.2, u⇤(�) = u+(�), which contradicts our assumption ⇣ 6= ⇣+.

Consider now ⇣ ⇣�. V solves the HJB equation, if

AV � h(⇣) + h(⇣�) = h(⇣�)� h(⇣) � 0, G(⇣) � 0.

Since h(⇣) � h(⇣�) = 0 for ⇣ = ⇣�, to obtain the first inequality it su�ces to show that (B.23) isnon-negative. Now for small " clearly ⇡� < ⇡⇤, hence for ⇣ = ⇣� (B.23) is indeed strictly positive.To settle the second inequality, recall that either ⇣ < �1/(1� ") or ⇣ > 0. On these domains, G isclearly a strictly positive function. Hence it is proved that V satisfies the HJB equation for ⇣ ⇣�.

Finally, consider ⇣ � ⇣+. Since G = W , it su�ces to show

L(⇣) := AV (⇣)� h(⇣) + h(⇣�) � 0, G(⇣) � 0. (B.26)

The second estimate is straightforward: Let ⇣+ > �1, then we have ⇣ > �1, and for su�cientlysmall ", (1 � ")⇣ > �1, hence G(⇣) > 0. The case ⇣+ < �1 can be dealt with similarly. For thefirst inequality in (B.26), note that

L(⇣) =�2⇣2

2G0(⇣) + µ⇣G(⇣)� h(⇣) + h(⇣�)

= h(⇣�)� h1((1� ")⇣) +(� � 1)�2

2

✓⇣

1 + ⇣

◆2

.

Therefore, it su�ces to show that

(⇣) := h(⇣�)� h1((1� ")⇣)� (1� �)�2

2

✓⇣

1 + ⇣

◆2

(B.27)

has no zeros on [⇣+,�1/(1� ")), besides ⇣+.The case � = 1 is simple as (⇣) = 0 can be reduced to solving a quadratic equation (see also

Taksar, Klass and Assaf (1988)). All other cases require investigating a fourth-order polynomial,as seen below. However, to demonstrate the strength and clarity of the asymptotic approach ofthis paper, we first discuss the case � = 1. The transformation z = ⇣

1+⇣ leads to

(1� ")⇣

1 + (1� ")⇣=

(1� ")z

1� "z

and thus one can rewrite (B.27) in terms of z, denoting it by

F (z, ") = µ⇡� � �2

2⇡2� � µ

✓(1� ")z

1� "z

◆+�2

2

✓(1� ")z

1� "z

◆2

.

26

It is proved next that F has no zeros on (⇡+, 1/"). Since F (⇡+) = 0, polynomial division by(z � ⇡+) yields

F (z, ") =(z � ⇡+)

(1� "z)2g(z), (B.28)

g(z) is a linear factor, and the following asymptotic expansions hold

g(⇡+) = �2✓

3

4�

⇣ µ

�2

⌘⇣1�

⇣ µ

�2

⌘⌘2◆1/3

"1/3 +O("2/3),

g(1/") =�2

2"+O(1).

It follows that g has no zeros on [⇡+, 1/"], for su�ciently small ". Hence F (z) > 0 for z 2 (⇡+, 1/").For the remainder of the proof, suppose � 6= 1. Using the transformation z = ⇣

1+⇣ one canrewrite, similarly as in the � = 1 case, function (B.27) in terms of z,

F (z, ") = µ⇡� � ��2

2⇡2� � µ

✓(1� ")z

1� "z

◆+�2

2

✓(1� ")z

1� "z

◆2

� (1� �)�2

2z2.

It is proved next that F has no zeros on (⇡+, 1/").Since F (⇡+) = 0, polynomial division by (z�⇡+) yields (B.28), where the third order polynomial

g has derivativeg0 = a0 + a1z + a2z

2,

with

a0 = ⇡�µ"2 � 1

2⇡2���

2"2 � 1

2⇡2+�

2(1� �)"2

+ ⇡+�2(1� �)"� µ"2 +

�2(� + "2)

2+ µ"� �2",

a1 = 2

✓�2(1� �)"� 1

2⇡+�

2(1� �)"2◆, (B.29)

a2 = �3

2�2(1� �)"2.

In view of (B.28), it is enough to show that g has no zeros on [⇡+, 1/"]. First, note the followingasymptotic expansions,

g(⇡+) = ��2✓

3

4�⇡2⇤(1� ⇡⇤)

2

◆1/3

"1/3 +O("2/3), (B.30)

g(1/") =�2

2"+O(1). (B.31)

Therefore, for su�ciently small ", g > 0 on both endpoints of [⇡+, 1/"]. It remains to show thatany local minimum of g in [⇡+, 1/"] is non-negative. In searching for local extrema, one obtainscomplex numbers z± where g0(z±) = 0. The asymptotic expansions of z± are given by

z± =2

3"± 1

3"

r� � 4

� � 1+O(1).

27

Obviously, there are no local extrema in [⇡+, 1/"] whenever � 2 [1, 4). Therefore g > 0 on all of[⇡+, 1/"], and thus F (z) � 0 on [⇡+, 1/"). The non-trivial case � /2 [1, 4) remains:

For 0 < � < 1 it holds that 4��1�� > 2, hence z± /2 [⇡+, 1/"]. It follows that g0 has no zeros in

this interval and thus g > 0 on [⇡+, 1/"].Next, consider � � 4: The local minimum z� of a third order polynomial with negative leading

coe�cient satisfies z� < z+ and g(z�) < g(z+). In view of (B.30) and (B.31), it remains to showg(z�) > 0. It holds that

g(z�) =

⇣�p�2 � 5� + 4 + 2� � 2

⌘⇣p�2 � 5� + 4 + � + 2

⌘�2

27(� � 1)"+O(1)

=3� + (� � 4)(2 + � +

p(� � 4)(� � 1))

27(� � 1)"+O(1),

whence g(z�) > 0 for su�ciently small ". We have shown that g > 0 on [⇡+, 1/"].Summarizing, (⇣) � 0 on ⇣ � ⇣+, which proves that the HJB equation (B.21) holds.

Lemma B.5. Let ⌘� < ⌘+ be such that either ⌘+ < �1/(1 � ") or ⌘� > 0. Then there existsan admissible trading strategy ' such that the risky/safe ratio ⌘t satisfies SDE (A.3). Moreover,

(⌘t, '"t , '

#t ) is a reflected di↵usion on the interval [⌘�, ⌘+]. In particular, ⌘t has stationary density

equals

⌫(⌘) :=2µ�2 � 1

⌘2µ�2�1

+ � ⌘2µ�2�1

�

⌘2µ�2�2, ⌘ 2 [⌘�, ⌘+], (B.32)

when ⌘� > 0, and otherwise equals

⌫(⌘) :=2µ�2 � 1

|⌘�|2µ�2�1 � |⌘+|

2µ�2�1

|⌘|2µ�2�2, ⌘ 2 [⌘�, ⌘+]. (B.33)

Proof. By the solution of the Skorohod problem for two reflecting boundaries Kruk et al. (2007),there exists a well-defined reflected di↵usion (⌘t, Lt, Ut) satisfying

d⌘t⌘t

= µdt+ �dBt + dLt � dUt,

where W is a standard Brownian motion, and L (resp. U) is a non-decreasing processes whichincreases only on the set {⌘ = ⌘�} (resp. {⌘ = ⌘+}). Also, ⌘� > 0 or ⌘+ < �1/(1� ") implies that⌘t > 0 or ⌘t < �1/(1� ") for all t, almost surely. Hence for each t > 0,

(1 + (1� ")⌘t), (1 + ⌘t)

are invertible, almost surely. Define the increasing processes ('", '#) by

d'"t

't= (1 + ⌘t)

�1dLt

andd'#

t

't= (1 + (1� ")⌘t)

�1dUt.

28

By construction, the associated measures d'", d'# are supported on ⌘t = ⌘� and ⌘t = ⌘+, respec-tively. Hence ' is a trading strategy, which by Lemma A.3 yields a risky/safe satisfying preciselythe stochastic di↵erential equation (A.3).

The admissibility of the trading strategy is clear, as ' is a continuous, finite variation tradingstrategy, and since it satisfies ⇡+ < 1/", which implies that there exists "0 > " such that ⇡t < 1/"0,for all t > 0, a.s.. Finally, the form of the stationary density ⌫(⌘), follows from the stationaryFocker-Planck equation: The infinitesimal generator of ⇣t is given by

Af(⇣) =�2

2⇣2f 00(⇣) + µ⇣f 0(⇣) =: a(⇣)/2'00(⇣) + b(⇣)'0(⇣).

The invariant density ⌫ solves the adjoint di↵erential equation

A⇤⌫(⌘) = (a(⌘)⌫(⌘))0 � 2b(⌘)⌫(⌘) = 0

and therefore equals

⌫(⌘) =c

a(⌘)exp

✓Z2b(⌘)

a(⌘)d⌘

◆, (B.34)

where the constant c > 0 depends on the boundaries ⇣�, ⇣+. By integration, and distinguishing thecases ⌘+ < 0 or ⌘� > 0, we obtain the explicity expressions (B.32) and (B.33).

The following constitutes the verification of optimality of the trading strategy of Lemma B.5with the trading boundaries given by Proposition B.1:

Proposition B.6. Let ⇣± be the free boundaries as derived in Proposition B.1, and denote by 'the trading strategy of Lemma B.5 associated with these free boundaries. Set

⇡± := ⇣±/(1 + ⇣±).

Then for all t > 0, the fraction of wealth ⇡t invested in the risky asset lies in the interval [⇡�,⇡+],almost surely, entails no trading whenever ⇡ 2 (⇡�,⇡+) (the no-trade region) and engages in tradingonly at the boundaries ⇡±. For su�ciently small ", ' is optimal, and the value function is given by

F1(') = max'2�

limT!1

1

TE[R'

T � �

2hR'iT ] = r + h(⇣�) = r + µ⇡� � ��2

2⇡2�. (B.35)

Proof of Proposition B.6. Recall from Proposition B.4 that � = h(⇣�) and (V,�), defined from theunique solution of the free boundary problem, is a solution of the HJB equation (B.21). For theverification, we use the proportion ⇡t of wealth in the risky asset instead of the risky/safe ratio ⇣t.The change of variables

⇣ = �1 +1

1� ⇡

amounts to a compactification of the real line, such that the two intervals [�1,�1/(1 � ")) and(0,1] are mapped onto the connected interval [0, 1/"). Denote by L the di↵erential operator

Lf(⇡) := �2

2f 00(⇡)⇡2(1� ⇡)2 + f 0(⇡)(µ� �2⇡)⇡(1� ⇡).

29

The function V (⇡) := V (⇣(⇡)) satisfies the HJB equation

min(LV (⇡)� h(⇡) + �, V 0(⇡), V 0(⇡)� "

1� "⇡) = 0, (B.36)

for 0 ⇡ < 1/", where h(⇡) = h(⇣(⇡)) = µ⇡ � ��2

2⇡2.

First, it is shown that F1(') � + r, for any admissible trading strategy '. By Lemma A.2we may without loss of generality assume ⇡t � 0, almost surely, for all t � 0. An application ofIto’s formula to the stochastic process V (⇡t), where V is the solution of the HJB equation (B.36),yields

V (⇡T )� V (⇡0) =

Z T

0V 0(⇡t)d⇡t +

1

2V 00(⇡t)dh⇡it (B.37)

=

Z T

0

⇣LV (⇡)� h(⇡t) + �

⌘dt+

Z T

0(h(⇡t)� �)dt (B.38)

+

Z T

0V 0(⇡t)⇡t(1� ⇡t)�dBt (B.39)

�Z T

0V 0(⇡t)(1� "⇡t)⇡t

d'#t

't(B.40)

+

Z T

0V 0(⇡t)⇡t

d'"t

't. (B.41)

The first term in line (B.38) is non-negative, due to (B.36). Furthermore, (A.1) implies theexistence of "0 > " such that ⇡t < 1/"0 < 1/", for all t, a.s., and using (B.36) we have

V 0(⇡t) ""0

"0 � ", a.s. for all t � 0. (B.42)

Hence (B.39) is a martingale with zero expectation. Again by (B.36) one has that

V 0(⇡t)⇡t(1� "⇡t) "⇡t,

which implies that for (B.40) we have

�Z T

0V 0(⇡t)(1� "⇡t)⇡t

d'#t

't� �"

Z T

0⇡t

d'#t

't.

Finally, (B.41) is non-negative, because V 0 � 0 due to (B.36).Thus, taking the expectation of (B.37) we obtain the estimate,

1

TE[V (⇡T )� V (⇡0)] � ��+

1

TE[Z T

0h(⇡t)dt]� "

1

T

Z T

0⇡t

d'#t

't. (B.43)

By (B.42)

|V (⇡t)� V (⇡0)| |⇡T � ⇡0| sup0<u1/"0

|V 0(u)| "

"0 � ",

30

therefore

limT!1

1

TE[V (⇡T )� V (⇡0)] = 0.

Hence letting T ! 1 in (B.43) reveals that for any admissible strategy

F1(') �+ r. (B.44)

Finally, it is shown that the bound � + r is attained by the admissible trading strategy ' definedby Lemma (B.5) in terms of the free boundaries (⇣�, ⇣+). Let ⇣t be the corresponding risky/saferatio. Using Ito’s formula, one has

dV (⇣t) = V 0(⇣t)⇣t�dBt + 0� "⇡td'#

t

't+ (h(⇣t)� �)dt.

Integration with respect to t and division by T yields, in view of (A.7),

1

TE[R'

T � �

2hR'iT ] = r + �+

1

TE[V (⇡T )� V (⇡0)].

Letting T ! 1 we have F1(') = �+ r. Due to (B.44), ' is an optimal trading strategy.

B.1 Proof of Theorem 3.1 (i)–(iii)

Theorem 3.1 (i) is proved in Proposition B.1, and Theorem 3.1 (ii) & (iii) are proved in PropositionB.6.

C Ergodic results

In this section, ergodic theorems are applied to derive closed-form expressions for average tradingcosts (ATC) and long-run mean and long-run variance of the optimal trading strategy. Theseformulas are then used to prove the asymptotic expansions of Theorem 3.1 (iv).

Let ⇣�, ⇣+ be the free boundaries obtained in Proposition B.1. Without loss of generality,assume that either ⇣� < ⇣+ < �1 (levered case) or ⇣� > ⇣+ > 0 throughout (non-levered case),and define the integral

I :=1

c

Z ⇣+

⇣�

h(⇣)|⇣|2�⇡⇤�2d⇣, (C.1)

where the normalizing constant is given by

c :=

Z ⇣+

⇣�

|⇣|2�⇡⇤�2d⇣ = sgn(⇣�)|⇣+|2�⇡⇤�1 � |⇣�|2�⇡⇤�1

2�⇡⇤ � 1.

The objective functional

Lemma C.1.

I = h(⇣�) +�2(2�⇡⇤ � 1)

2

0

B@G(⇣+)⇣+

1�⇣⇣�⇣+

⌘2�⇡⇤�1

1

CA . (C.2)

31

Proof. From equations (B.8) and (B.10) it follows thatZ ⇣+

⇣�

h(⇣)|⇣|2�⇡⇤�2d⇣ = h(⇣�) sgn(⇣�)|⇣+|2�⇡⇤�1 � |⇣�|2�⇡⇤�1

2�⇡⇤ � 1+�2⇣+

2�⇡⇤

2G(⇣+).

By normalizing, (C.2) follows.

Let now ⇣ be a geometric Brownian motion with parameters (µ,�), reflected at ⇣�, ⇣+ respec-tively, as in Lemma B.5. Recall the following ergodic result, of (Gerhold et al., 2014, LemmaC.1):

Lemma C.2. Let ⌘t be a di↵usion on an interval [l, u], 0 < l < u, reflected at the boundaries, i.e.

d⌘t = b(⌘t)dt+ a(⌘t)1/2dBt + dLt � dUt,

where the mappings a(⌘) > 0 and b(⌘) are both continuous, and the continuous, non-decreasingprocesses Lt and Ut satisfy L0 = U0 = 0 and increase only on {Lt = l} and {Ut = u}, respectively.Denoting by ⌫(⌘) the invariant density of ⌘t, the following almost sure limits hold:

limT!1

LT

T=

a(l)⌫(l)

2, lim

T!1

UT

T=

a(u)⌫(u)

2.

The next formula evaluates trading costs.

Lemma C.3. The average trading costs for the optimal trading policy are

ATC := " limT!1

1

T

Z T

0⇡t

d'#t

't=�2(2�⇡⇤ � 1)

2

0

B@G(⇣+)⇣+

1�⇣⇣�⇣+

⌘2�⇡⇤�1

1

CA . (C.3)

Proof. Apply Lemma C.2 to the stochastic process ⌘ := ⇣, u = ⇣+. Realizing that

"

Z T

0⇡t

d'#t

't= G(⇣+)

UT

T

and using the stationary density of ⇣t, Lemma B.5, which equals

⌫(⇣) := sgn(⇣�)2�⇡⇤ � 1

|⇣+|2�⇡⇤�1 � |⇣�|2�⇡⇤�1|⇣|2�⇡⇤�2, ⇣ 2 [⇣�, ⇣+],

and applying Lemma C.2, we obtain (C.3).

Remark C.4. An alternative proof provides a consistency check for the theory provided so far:By Lemma 2.1 one can rewrite the objective functional as

F (') = r + limT!1

1

T

Z T

0h(⇣t)dt�ATC.

Now by the ergodic theorem (Borodin and Salminen, 2002, II.35 and II.36),

limT!1

1

T

Z T

0h(⇣t)dt = I,

hence using Lemma C.1 it follows that

F (') = r + h(⇣�) +ATC �ATC = r + h(⇣�)

which is in agreement with the formula in Proposition B.6.

32

C.1 Long-run mean and variance

Define

Iµ :=

Z ⇣+

⇣�

✓⇣

1 + ⇣

◆|⇣|2�⇡⇤�2d⇣, Is2 :=

Z ⇣+

⇣�

✓⇣

1 + ⇣

◆2

|⇣|2�⇡⇤�2d⇣.

Since the long-run mean and long-run variance are given by

m := limT!1

1

TE[RT ] = r + µ lim

T!1

1

TE[Z T

0⇡tdt]�ATC

= r +µ

cIµ �ATC,

s2 := limT!1

1

TE[hRiT ] = �2 lim

T!1

1

TE[Z T

0⇡2t dt]

=�2

cIs2 ,

the following decomposition holds in view of the ergodic theorem (Borodin and Salminen, 2002,II.35 and II.36):

I =1

c

✓µIµ � ��2

2Is2

◆= (m� r +ATC)� �

2�2 (C.4)

= h(⇣�) +ATC.

Integration by parts yields

Iµ =

Z ⇣+

⇣�

⇣

1 + ⇣|⇣|2�⇡⇤�2d⇣ =

|⇣+|2�⇡⇤

2�⇡⇤(1 + ⇣+)� |⇣�|2�⇡⇤

2�⇡⇤(1 + ⇣�)+

Is2

2�⇡⇤. (C.5)

An application of this identity to (C.4) yields

I =�2

2c

✓|⇣+|2�⇡⇤

1 + ⇣+� |⇣�|2�⇡⇤

1 + ⇣�+ (1� �)Is2

◆.

Whenever � 6= 1, one may extract I�2 , and thus (C.5) and (C.3) yield a formula for �2. Therefore,the right side of equation (C.4) gives a formula for m in terms of s:

Lemma C.5. When � 6= 1, the following identities hold:

s2 =2

1� �

✓h(⇣�) +ATC � �2

2c

✓|⇣+|2�⇡⇤

1 + ⇣+� |⇣�|2�⇡⇤

1 + ⇣�

◆◆, (C.6)

m = r +�

2s2 + h(⇣�). (C.7)

C.2 Proof of Theorem 3.1 (iv)

Proof. The asymptotic expansion (3.7) for the trading boundaries ⇡± can be derived by developing⇣±

1+⇣±into a power series, thereby using the asymptotic expansions (B.7) of ⇣±.

Long-run mean m and long-run variance s2, as well as average trading costs ATC and the valuefunction � have closed form expressions in terms of the free boundaries ⇣�, ⇣+ (see equations (C.7),(C.6), and equations (C.3) and (B.35)). Using these formulas in combination with the asymptoticexpansions (B.7) of the free boundaries, the assertion follows.

33

D Proof of Theorem 3.2

In this section the free boundary problem (3.1)–(3.5) for � = 0 is solved for su�ciently small", it is shown that the solution (W, ⇣�, ⇣+) allows to construct a solution of the correspondingHJB equation, and similarly to the case � > 0, a verification argument reveals an optimal tradingstrategy.

Numerical experiments using � > 0 indicate that the trading boundaries ⇡± (hence the leveragemultiplier) satisfy

lim"#0

"1/2⇡± = 1/A±

for two constants A� > A+ > 0. Thus, we expect that the free boundaries in terms of the risky/saferatio obey for su�ciently small " the approximation

⇣± ⇡ �1�A±"1/2.

This insight lets us conjecture that ⇣± are analytic in � := "1/2.Using the free boundary problem, the system (B.10)–(B.11) is rewritten with � := "1/2 and the

second equation is multiplied by �.

W (⇣�, ⇣+)��2

(1 + ⇣+)(1 + (1� �2)⇣+)= 0, (D.1)

�

✓2(h(⇣+)� h(⇣�))

�2⇣2+� 2µ/�2

⇣+W (⇣�, ⇣+)�

(1� �2)2

(1 + (1� �2)⇣+))2+

1

(1 + ⇣+)2

◆= 0. (D.2)

Using the transformation u = �1�⇣� and noting that |⇣| = 1 + �u, one obtains

⌅(u�, u) := W (�1� u��,�1� u�) =2µ

�2(1 + u�)2

Z u

u�

✓1

u�� 1

⇠

◆✓1 + ⇠�

1 + u�

◆ 2µ�2�2

d⇠.

Accordingly, the system (D.1)–(D.2) transforms into

⌅(u�, u+)�1

u+((1� �2)u+ � �)= 0, (D.3)

2µ

�2

✓1

u+� 1

u�+

�

1 + u+�⌅(u�, u+)

◆�

2�1� �2

�u+ � �

u2+ (� + (�2 � 1)u+)2 = 0. (D.4)

Letting � ! 0 in (D.3)–(D.4), we get an equation for, say (A�, A+),

2µ

�2

✓log(A�/A+)�

A� �A+

A�

◆� 1

A2+

= 0, (D.5)

µ

�2

✓1

A+� 1

A�

◆� 1

A3+

= 0. (D.6)

Lemma D.1. There is a unique solution (A�, A+) of system (D.5)–(D.6), given by

A� =�1/2

1�

s�2

µ, A+ = �1/2

s�2

µ, (D.7)

34

where ⇡ 0.5828 is the unique solution of

f(⇠) :=3

2⇠ + log(1� ⇠) = 0, ⇠ 2 (0, 1). (D.8)

Proof. Equation (D.6) gives

A� =µA3

+

µA2+ � �2

. (D.9)

Hence substituting (D.9) into (D.5) gives the well-posed transcendental equation

� 3

A2++

2µ log⇣

µA2+

µA2+��2

⌘

�2= 0, A+ > 0. (D.10)

Therefore it is enough to establish that (D.10) has a unique solution given by the second formulain (D.7); the formula for A� then follows from (D.9). To this end, substitute

⇠ :=�2

µA2+

into (D.10) to obtain equation (D.8). Note that f(0) = 0, f 0 > 0 on (0, 1/3) and f 0 < 0 on (1/3, 1),while f(⇠) # �1 as ⇠ ! 1. This implies that f has a single zero on (1/3, 1).

Proposition D.2. For su�ciently small �, there exists a unique solution (u+, u�) of (D.3)–(D.4)near (A�, A+). This solution is analytic in � and satisfies the asymptotic expansion u± = A±+O(�),where A± are given by (D.7).

Proof. Denote the left sides of (D.3)–(D.4), by Fi((u�, u+), �), i = 1, 2 and F = (F1, F2). ByLemma D.1, F ((A�, A+), 0) = 0. Since

@⌅

@u�((A�, A+), 0) =

2µ

�2

✓A� �A+

A2�

◆,

@⌅

@u+((A�, A+), 0) =

2µ

�2

✓A+ �A�A�A+

◆,

one obtains

@F1

@u�((A�, A+), 0) =

2µ

�2

✓A� �A+

A2�

◆,

@F1

@u+((A�, A+), 0) =

2

A3++

2µ

�2

✓A+ �A�A�A+

◆= 0,

@F2

@u+((A�, A+), 0) =

6

A4+� 2µ

�2

✓1

A2+

◆,

where the second line vanishes due to (D.6), and therefore, the Jacobian DF of F satisfies

det(DF )((A�, A+), 0) =@F1

@u�((A�, A+), 0)⇥

@F2

@u+((A�, A+), 0)

= �4(µ/�2)7/2(� 1)5/2(3� 1) 6= 0,

because 2 (1/3, 1). Hence by the implicit function theorem for analytic functions (Gunning andRossi, 2009, Theorem I.B.4) the assertion follows.

35

Lemma D.3. Let be the solution of (D.8), and ✓ 2 [0, 1]. If

f(✓) = log(1� (1� ✓)) + (1� ✓)+1

2

(1� )2

(1� (1� ✓))2= 0 (D.11)

then ✓ = 0.

Proof. Clearly f(0) = 0 and also f(1) = 1/2(1� )2 > 0. There is a single local extremum of f ,in (0, 1), namely,

✓1 =0.5⇣3.2 +

p4.3 � 3.4 � 2.

⌘

2⇡ 0.7669,

but since f 0(0) = 0, and

f 00(0) =2��32 � 7+ 5

�� 1�

(1� )4> 0

✓1 must be the global maximum. Hence f > 0 on (0, 1].

Lemma D.4. Let A� be as in (D.7). The only solution of

2µ

�2

✓log(A�/⇠)�

A� � ⇠

A�

◆� 1

⇠2= 0 (D.12)

on [A+, A�] is ⇠ = A+.

Proof. Let ⇠ be a solution of (D.12). There exists ✓ 2 [0, 1] such that

⇠ = ✓A� + (1� ✓)A+ = A+

✓1 + (1� ✓)

1�

◆.

Hence A⇤+/A� = 1 + (✓ � 1), and therefore (D.12) can be rewritten as (D.11). An application of

Lemma D.3 yields ⇠ = A+.

Proof of Theorem 3.2

Proof. Arguing similarly as in the Proof of Proposition B.1 for the case � > 0, the solvability ofthe free boundary problem (3.1)–(3.5) for � = 0 is equivalent to solvability of the non-linear system(D.1)–(D.2). This, in turn, is equivalent to solving (D.3)–(D.4) for (u+(�), u�(�)). A uniquesolutions of the transformed system (D.3)–(D.4) near (A+, A�) is provided by Proposition D.2,and one has ⇣± = �1� u±�. In particular, one obtains

⇣± = �1�A±"1/2 +O(1). (D.13)

The solution of (3.1)–(3.5) is given by

W (⇣) :=2µ

�2|⇣|2µ�2

Z ⇣

⇣�

✓y

1 + y� ⇣�

1 + ⇣�

◆|y|2µ/�2�2dy. (D.14)

One defines exactly as in (B.22) a candidate (V,�) for the HJB equation (B.21). Next it is shownthat (V,�) solves the HJB equation (B.21) (for the intervals [⇣�, ⇣+], (�1, ⇣�] and finally for[⇣+,1)). In fact, the interval [�1/(1� "), 0) is excluded.

36

On [⇣�, ⇣+], we have

(AV (⇣)� h(⇣) + h(⇣�))0 =

1

2�2⇣2W 00(⇣) + (�2 + µ)⇣W 0(⇣) + µW (⇣)� µ

(1 + ⇣)2= 0

by construction. Furthermore, because of the initial conditions (3.2)–(3.3)

(AV (⇣)� h(⇣) + h(⇣�)) |⇣=⇣�= AV (⇣) |⇣=⇣�= 0

and thereforeAV (⇣)� h(⇣) + h(⇣�) ⌘ 0, ⇣ 2 [⇣�, ⇣+].

Next it is shown that 0 V 0 G on all of [⇣�, ⇣+]. Since

(h(⇣)� h(⇣�))0 = h0(⇣) =

µ

(1 + ⇣)2(D.15)

is strictly positive, h(⇣) � h(⇣�) > 0 for ⇣ 2 (⇣�, ⇣+]. From the explicit formula (D.14) onetherefore may conclude that V 0 = W � 0 for ⇣ 2 [⇣�, ⇣+]. It remains to show V 0 G. SinceV 0(⇣+)�G(⇣+) = 0, and since V 0(⇣�)�G(⇣�) = �G(⇣�) < 0, it su�ces to rule out any zero ⇣⇤+ ofV 0(⇣)�G(⇣) on (⇣�, ⇣+), for su�ciently small ". This is equivalent to ruling out any zeros of

(u, �) := V 0(⇣(u))�G(⇣(u)), u 2 (u+(�), u�(�)),

where ⇣(u) = �1 � u�, for su�ciently small �. Recall that u±(�) is implicitly defined by ⇣± =�1�u±(�)�, and we have lim�!0 u±(�) = A±. Assume, for a contradiction, there exists �k # 0 anda sequence u+(�k) satisfying u�(�k) < u⇤+(�k) < u+(�k) which is a solution of (u⇤+(�k), �k) = 0for each k 2 N. By taking a subsequence, if necessary, we may without loss of generality assumeu⇤+(�k) ! A⇤

+ 2 [A+, A�] as k ! 1. We distinguish two cases now.

• A⇤+ = A+: We define the map � 7! u⇤(�) by intertwining u+ and u⇤+ as follows:

u⇤+(�) =

(u⇤+(�k), k 2 Nu+(�), � 6= �k

.

Then for su�ciently small �, the pair (u�(�), u⇤+(�)) solves (D.3)–(D.4) near (A�, A+), henceby Proposition D.2, u⇤+ = u+, a contradiction to our previous assumption ⇣⇤+ 2 (⇣�, ⇣+).

• A⇤+ 2 (A+, A�]: By equation (D.3) we have

2µ

�2

✓log(A�/A

⇤+)�

A� �A⇤+

A�

◆� 1

(A⇤+)

2= 0.

Lemma D.4 states A⇤+ = A+, which is impossible.

Therefore V solves the HJB equation on [⇣�, ⇣+].Consider now ⇣ ⇣�. V solves the HJB equation, if

AV � h(⇣) + h(⇣�) = h(⇣�)� h(⇣) � 0, G(⇣) � 0.

The first inequality is clearly fulfilled. Also, since ⇣ < �1/(1� ") or ⇣ > 0, G is a strictly positivefunction on [�1, ⇣�], which finishes the proof for ⇣ ⇣�.

37

Finally, consider ⇣ � ⇣+. Since G = W , it su�ces to show that

L(⇣) := AV (⇣)� h(⇣) + h(⇣�) � 0, G(⇣) � 0. (D.16)

The second inequality has just been proved. So only the first inequality in (D.16) needs to beshown. Denoting h1(⇣) the function h for � = 1, one obtains

L(⇣) =�2⇣2

2G0(⇣) + µ⇣G(⇣)� h(⇣) + h(⇣�)

= h(⇣�)� h1((1� ")⇣) +(� � 1)�2

2

✓⇣

1 + ⇣

◆2

.

Therefore, by the boundary conditions at ⇣+,

L(⇣+) =�2⇣2

2W 0(⇣+) + µ⇣W (⇣+) + h(⇣�)� h(⇣+) = 0.

The last equality follows from our knowledge concerning the HJB equation on [⇣�, ⇣+].To show that L(⇣) � 0 for all ⇣, it su�ces to show that there are no solutions of the equation

(⇣) := h(⇣�)� h1((1� ")⇣)� �2

2

✓⇣

1 + ⇣

◆2

= 0 (D.17)

on ⇣ � ⇣+ except ⇣+. The transformation z = ⇣1+⇣ yields

(1� ")⇣

1 + (1� ")⇣=

(1� ")z

1� "z

and thus one can rewrite (D.17) in terms of z, denoting it by

F (z, ") = µ⇡� � µ

✓(1� ")z

1� "z

◆+�2

2

✓(1� ")z

1� "z

◆2

� �2

2z2.

It is proved next that F has no zeros on (⇡+, 1/"): Since F (⇡+) = 0, polynomial division by (z�⇡+)yields

F (z, ") =(z � ⇡+)

(1� "z)2g(z), (D.18)

where the third order polynomial g has derivative

g0 = a0 + a1z + a2z2,

with coe�cients given by (B.29) (setting � = 0). By the second formula of (D.7)

g(⇡+) = �µ+3�2

A2+

+O("1/2) (D.19)

is strictly positive for su�ciently small ", since > 1/3. The solutions z± of the equation

g0(z) = 0

38

are

z� = � 1

2A+"1/2+O(1)

which is negative for su�ciently small ", hence irrelevant, and

z+ =4

3"+O(1),

which is larger than 1/", hence also irrelevant. Since

g0(1/") = �2/2 +O("1/2)

it follows that g0(z) > 0 on all of [⇡+, 1/"]. Together with (D.19) it follows that g > 0 on [⇡+, 1/"].Hence F (z) > 0 for all z > ⇡+ which admit the conclusion that (V,�) solves the HJB equation(B.21).

Using the proof of Proposition B.6, one can obtain assertion (ii) and (iii). Finally, the expan-sions of the trading boundaries claimed in (iv) follow from the asymptotic expansions of the freeboundaries ⇣�, ⇣+ in (D.13).

D.1 Proof of Proposition 4.1

Proof of Proposition 4.1. While the first formula holds in view of (4.9), the second one is a resultof the asymptotic expansions provided by Theorem 3.1 (iv).

E Convergence

Lemma E.1. Let µ > �2. There exists �0 > 0 such that for all 0 � < �0 := µ�2 , � �0 the

objective functional for a trading strategy ' which only engages in buying at ⇡� = 1+ � and sellingat ⇡� < ⇡+ = (1� �)/" outperforms a buy and hold strategy. More precisely, for all � < �0, � �0

F1(') � r + µ� ��2

2+

✓µ� ��2

2

◆� > r + µ� ��2

2.

Proof. Using the stationary density ⇢(d⇡) of ⇡t on [⇡�,⇡+] (which can be derived from LemmaB.5), one obtains

F (') = r +

Z ⇡+

⇡�

✓µ⇡ � ��2

2⇡2◆⇢(d⇡)�ATC

� r + µ(1 + �)� ��2

2(1 + �)2 �

(� + 1)(2✏� 1)3�2µ� �2

�

4✏

✓�⇣�2(�+1)✏+�+1

�

⌘ 2µ�2

+ (� + 1)(2✏� 1)

◆

� r + µ� ��2

2+ (µ� ��2)� �O(�min(2, 2µ

�2�1)). (E.1)

where Lemma C.3 has been invoked to calculate and estimate the average trading costs ATC. Theasymptotic expansion holds for su�ciently small �. Since µ > ��2, the claim follows.

39

E.1 Proof of Theorem 4.2

Proof. By equation (4.4), the curves (0, �] ! R : � 7! ⇡±(�) range in a relatively compact set,namely [1, 1" ). Consider therefore a sequence �k, k = 1, 2, . . . which satisfies

1 ⇡0� := limi!1

⇡�(�k) limi!1

⇡+(�k) =: ⇡0+ = 1/".

Define accordingly ⇣0±. Then �1 ⇣0� ⇣0+ � 11�" . First, three elementary facts are proved:

(i) ⇡0� > 1, which is equivalent to ⇣0� > �1. Assume, for a contradiction, ⇡0� = 1. Then⇡�(�k) ! 1 and thus �(�k) ! r + µ, as k ! 1. Hence, the objective functional eventuallyminorizes the uniform bound provided by Lemma E.1, a mere impossibility. We concludethat ⇡0� > 1.

(ii) ⇡0� < ⇡0+: This holds due to the fact that, by observing limits for the initial and terminalconditions of zero order in (3.1),

W (⇣0�) = 0 < G(⇣0�).

(iii) Also, ⇡0+ < 1" . Assume, for a contradiction, that ⇡0+ = 1

" . Then G(⇣+(�k)) ! 1, as k ! 1,and, since ⇣0� < ⇣0+, the average trading costs corresponding to �k satisfy (by Lemma C.3)

ATC(k) :=�2⇣2µ�2 � 1

⌘

2

G(⇣+(�k))⇣+(�k)

1�⇣⇣�(�k)⇣+(�k)

⌘2µ/�2�1! 1,

as k ! 1. Denote by 'k the trading strategy which only buys (resp. sells) ath ⇡�(�k) (resp.⇡+(�k)). By the results of Appendix C the value function satisfies for each k

F1('k) = r +

Z ⇡+(�k)

⇡�(�k)(µ⇡ � ��2

2⇡2)⇢(d⇡)�ATC(k) r +

µ

"�ATC(k) ! �1

as k ! 1. In particular, for su�ciently large k � k0, a buy-and-hold strategy ' satisfies

F1(') = r + µ� �k�2

2> F1('k),

which contradicts the assumption concerning optimality of the trading strategy [⇡�(�k),⇡+(�k)].Hence ⇡0+ < 1/".

Since the sequence ⇣�(�k) converges, by (Keller-Ressel et al., 2010, Lemma 9) the solutionsof the initial value problem associated with (3.1) and �k, namely W (⇣; ⇣�(�k)), converges to thesolution of the initial value problem (3.1) (where � = 0),

W 0(⇣) = � 2

�2⇣2

Z ⇣

⇣0�

(µ⇣

1 + ⇣� µ

⇣0�1 + ⇣0�

)(⇣/⇣0�)2µ/�2�2d⇣.

The terminal conditions are met by W 0, because G is continuous on (�1,� 11�"). Also, for each

k, k = 1, 2, . . . , by assumption the HJB equation (B.21) is satisfied. Non-negativity is preserved

40

by taking limits, hence, (W (⇣; 0),�(0)) satisfies the HJB equation as well. Using the verificationarguments of the proof of Proposition B.6 it follows that the trading strategies associated with theintervals [⇡�(�),⇡+(�)] are optimal for risk-aversion levels � 2 [0, �], but also [⇡0�,⇡

0+] is optimal

for a risk-neutral investor.⇣�(�) can have only one accumulation point for � # 0, because �0 is the value function. Unique-

ness of ⇣0� is therefore clear and it follows that ⇣0� = ⇣�(0). By assumption, the free bound-ary problem has a unique solution, hence it follows that ⇡+(0) = ⇡0+. In particular, the curves(0, �] ! R : � 7! ⇡±(�) each have a unique limit ⇡0± as � # 0, which equals ⇡±(0), the solution ofthe free boundary problem.

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