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Finance and Stochastics
Published in: Finance and Stochastics 1/2014

01-01-2014

Transaction costs, trading volume, and the liquidity premium

Authors: Stefan Gerhold, Paolo Guasoni, Johannes Muhle-Karbe, Walter Schachermayer

Published in: Finance and Stochastics | Issue 1/2014

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Abstract

In a market with one safe and one risky asset, an investor with a long horizon, constant investment opportunities and constant relative risk aversion trades with small proportional transaction costs. We derive explicit formulas for the optimal investment policy, its implied welfare, liquidity premium, and trading volume. At the first order, the liquidity premium equals the spread, times share turnover, times a universal constant. The results are robust to consumption and finite horizons. We exploit the equivalence of the transaction cost market to another frictionless market, with a shadow risky asset, in which investment opportunities are stochastic. The shadow price is also found explicitly.
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Appendix
Available only for authorised users
Footnotes
1
Constantinides [7] finds that “transaction costs have a first-order effect on the assets’ demand”. Liu and Loewenstein [25] note that “even small transaction costs lead to dramatic changes in the optimal behavior for an investor: from continuous trading to virtually buy-and-hold strategies”. Luttmer [27] shows how small transaction costs help resolve asset pricing puzzles.
 
2
That is, the amount of excess return the investor is ready to forgo to trade the risky asset without transaction costs.
 
3
The empirical literature has long been aware of this theoretical vacuum: Gallant et al. [15] reckon that “The intrinsic difficulties of specifying plausible, rigorous, and implementable models of volume and prices are the reasons for the informal modeling approaches commonly used”. Lo and Wang [26] note that “although most models of asset markets have focused on the behavior of returns […] their implications for trading volume have received far less attention”.
 
4
A negative excess return leads to a similar treatment, but entails buying as prices rise, rather than fall. For the sake of clarity, the rest of the paper concentrates on the more relevant case of a positive μ.
 
5
The limiting case γ→1 corresponds to logarithmic utility, studied by Taksar et al. [37], Akian et al. [1], as well as Gerhold et al. [17]. Theorem 2.2 remains valid for logarithmic utility by setting γ=1.
 
6
This optimal policy is not necessarily unique, in that its long-run performance is also attained by trading arbitrarily for a finite time, and then switching to the above policy. However, in related frictionless models, as the horizon increases, the optimal (finite-horizon) policy converges to a stationary policy, such as the one considered here (see e.g. Dybvig et al. [13]). Dai and Yi [8] obtain similar results in a model with proportional transaction costs, formally passing to a stationary version of their control problem PDE.
 
7
The corresponding formulas for μ=σ 2/2 are similar but simpler; cf. Corollary C.3 and Lemma C.2.
 
8
The number of shares is written as the difference \(\varphi_{t}=\varphi^{\uparrow}_{t}-\varphi ^{\downarrow}_{t}\) of the cumulative shares bought (resp. sold), and wealth is evaluated at trading prices, i.e., at the bid price (1−ε)S t when selling, and at the ask price S t when buying.
 
9
Algorithmic calculations can deliver terms of arbitrarily high order.
 
10
The other quantities are trivial: the gap and the liquidity premium become zero, while share and wealth turnover explode to infinity.
 
11
Technically, wealth is valued at the ask price at the buying boundary, and at the bid price at the selling boundary.
 
12
For a fixed horizon T, one would need to specify whether terminal wealth is valued at bid, ask, or at liquidation prices, as in Definition 2.1. In fact, since these prices are within a constant positive multiple of each other, which price is used is inconsequential for a long-run objective. For the same reason, the terminal condition for the finite-horizon value function does not have to be satisfied by the stationary value function, because its effect is negligible.
 
13
Alternatively, this equation can be obtained from standard arguments of singular control; cf. Fleming and Soner [14, Chap. VIII].
 
14
This guess assumes that the cash position is strictly positive, X t >0, which excludes leverage. With leverage, factoring out (−X t )1−γ leads to analogous calculations. In either case, under the optimal policy, the ratio Y t /X t always remains either strictly positive, or strictly negative, never to pass through zero.
 
15
Recall that in a frictionless market with two uncorrelated assets with returns μ 1 and μ 2, both with volatility σ, the maximum Sharpe ratio is \((\mu_{1}^{2}+\mu _{2}^{2})/\sigma^{2}\). That is, squared Sharpe ratios add across orthogonal shocks.
 
16
Since λ is proportional to the width δ of the no-trade region, the question is why the latter is of order ε 1/3. The intuition is that a no-trade region of width δ around the frictionless optimum leads to transaction costs of order ε/δ (because the time spent near the boundaries is approximately inversely proportional to the length of the interval), and to a welfare cost of the order δ 2 (because the region is centered around the frictionless optimum, hence the linear welfare cost is zero). Hence, the total cost is of the order ε/δ+δ 2, and attains its minimum for δ=O(ε 1/3).
 
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Metadata
Title
Transaction costs, trading volume, and the liquidity premium
Authors
Stefan Gerhold
Paolo Guasoni
Johannes Muhle-Karbe
Walter Schachermayer
Publication date
01-01-2014
Publisher
Springer Berlin Heidelberg
Published in
Finance and Stochastics / Issue 1/2014
Print ISSN: 0949-2984
Electronic ISSN: 1432-1122
DOI
https://doi.org/10.1007/s00780-013-0210-y

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