- Email: [email protected]
Market making with convex quotes
Market Making with Convex Quotes
Journal Pre-proof
Market Making with Convex Quotes Hae-shin Hwang, Paan Jindapon PII: DOI: Reference:
S1544-6123(18)30632-9 https://doi.org/10.1016/j.frl.2019.101361 FRL 101361
To appear in:
Finance Research Letters
Received date: Revised date: Accepted date:
10 September 2018 6 November 2019 10 November 2019
Please cite this article as: Hae-shin Hwang, Paan Jindapon, Market Making with Convex Quotes, Finance Research Letters (2019), doi: https://doi.org/10.1016/j.frl.2019.101361
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Inc.
Highlights • We investigate equilibrium properties of a quote-driven market where price is convex in order size. • As the degree of convexity increases, the distribution of incoming order contracts and becomes bimodal. • All of these equilibria tolerate the same level of information asymmetry between traders and market makers. • All of these equilibria induce the same distribution of equilibrium price.
1
Market Making with Convex Quotes Hae-shin Hwang∗
Paan Jindapon†
Texas A&M University
University of Alabama
November 14, 2019
Abstract We investigate equilibrium properties of a quote-driven market where price is convex in order size. As the degree of convexity increases, the distribution of incoming order contracts and becomes bimodal. Nonetheless, all of these equilibria tolerate the same level of information asymmetry between traders and market makers, and also induce the same distribution of equilibrium price.
Keywords: Market microstructure; Market making; Adverse selection; Risk aversion JEL Codes: D8; G1
∗
Email: [email protected] Corresponding author, Department of Economics, Finance and Legal Studies, University of Alabama, Tuscaloosa, AL 35487-0224, USA, Phone: +1 (205) 348-7841, Email: [email protected] †
2
1
Introduction
Trading mechanisms in security markets have different institutional attributes. In the literature, trading mechanisms are often classified by the availability of price information at the time of order submission. In a quote-driven mechanism (QDM hereafter), traders obtain price quotations from market makers (or dealers) before submitting orders and transaction prices are as quoted. In an order-driven mechanism (ODM hereafter), traders submit orders for execution through an auction process, also known as limit order book, and transaction prices are often unknown to the traders at the time of order submission. While most stock exchanges in the world are operated with limit order books, the QDM has been adopted in NASDAQ, London SEAQ, and many foreign exchange markets. Both QDM and ODM have been studied extensively.1 Using Glosten’s (1989) and Kyle’s (1989) rational expectations models, Madhavan (1992) theoretically shows that, when transactions are executed continuously, the the two mechanisms are equivalent under perfect competition. In contrast, Xing and Xue (2017) propose a unified framework and find that the QDM performs better when the order size is small while the ODM performs better when the order size is large. A key assumption is these comparisons is the linearity of equilibrium price function in the QDM. In this paper, we adopt Glosten’s (1989) model of the QDM and show that, in fact, there are infinitely many nonlinear equilibria where price is convex in order size, i.e., convex for selling quotes (positive orders) and concave for buying quotes (negative orders). Due to a larger price impact in these nonlinear equilibria, traders find the QDM less attractive. We also analyze market tolerance for information asymmetry in the QDM. Like the aforementioned papers, we assume that each trader is endowed with a risky asset and a private noisy signal about its return. Since all of our market makers neither receive information about the return nor observe each trader’s endowment of the risky asset, they cannot distinguish between the liquidity-motivated and the information-motivated components of an incoming order. If the information asymmetry between the traders and the market makers is so severe that the market makers are unable to protect themselves 1 For example, see Madhavan (1992), Biais, Foucault, and Salani´e (1998), Viswanathan and Wang (2002), and Malinova and Park (2013), Xing and Xue (2017). For survey of the literature, see Madhavan (2000), Biais, Glosten, and Spatt (2005) and Foucault, Pagano, and R¨ oell (2013).
3
from the informed traders, the market will shut down. We find that all of these equilibria, linear and nonlinear, can tolerate the same level of information asymmetry. In a model of one informed trader versus multiple uninformed traders, Bhattacharya and Spiegel (1991) also identify both linear and nonlinear equilibrium price schedules, and show that the maximum degree of information asymmetry the uninformed traders can tolerate is the same in linear and nonlinear equilibria. Existence of nonlinear equilibria is also discussed in Biais, Martimort, and Rochet (2000).2 While Biais, Martimort, and Rochet complement Glosten’s (1994) equilibrium in a limit order market by deriving an equilibrium under imperfect competition with convex marginal selling quotes, we generalize Glosten (1989) by allowing for convexity in the market makers’ average selling quotes. In fact, our convex average selling quotes implies convex marginal selling quotes as well.3 Convex price functions are documented in various empirical research. For example, see Niemeyer and Sand˚ as (1994), Maslov and Mills (2001), Weber and Rosenow (2005), Visaltanachoti, Charoenwong, and Ding (2008), and Malo and Pennanen (2012). While recent empirical research on the QDM seems to be overshadowed by abundant evidence of the ODM adopted by most stock exchanges, liquidity and stability of the QDM have been examined in the betting industry—see Flepp, N¨ uesch, and Franck (2017). In addition to the convex price-order relationship in dealers’ quotes, we also suggest other empirical predictions. First, as the degree of convexity increases, the corresponding distribution of incoming order contracts while preserving the mean as in mean-preserving contraction defined by Rothschild and Stiglitz (1970). However, buy and sell orders can be clustered away from zero and generate two peaks, one at a buying order and one at a selling order. This bimodal distribution is also predicted by Ro¸su’s (2009) theoretical model of the limit order book. Finally, our theory predicts that the distribution of equilibrium price is the same for all equilibria regardless of the degree of convexity. 2
See additional analysis in Biais, Martimort, and Rochet (2013) and also a generalization in Back and Baruch (2013). 3 Given average price equation P = m + bQ + cQγ in Section 3, the corresponding marginal price is P = m + 2bQ + c(γ + 1)Qγ . Since γ > 1, both price equations are convex.
4
2
The Model
There are two assets: a risk-free asset with zero return and a risky asset with a random return per unit X. Two types of agents are present in the market: informed traders and uninformed market makers who provide liquidity by trading with the traders. Trader i is endowed with Ai units of the risk-free asset and Wi units of the risky asset, and given a private signal, Si = X + εi , which is personalized noisy information about the risky asset before arriving at the market. Given P (Q), unit price of the risky asset dependent on incoming order, trader i determines his demand for the risky asset, denoted by Qi , to maximize his expected utility derived from his final wealth, Yi = (Wi + Qi )X + Ai − P (Qi )Qi ,
(1)
conditional on information set Φi which consists of realized values of Ai , Wi , and Si . Note that a positive (negative) Qi means that trader i’s is buying (selling) the risky asset. We assume that each trader has a CARA utility function Ui (Yi ) = −e(−ρi Yi ) , where
ρi > 0 is the absolute risk-aversion parameter.4 Without loss of generality, we set each individual’s cash endowment Ai to zero. Moreover, all agents have common prior believes about the return of the risky asset, the initial endowment of the risky asset, and the noise in the private information. Following Glosten (1989), we assume that X is distributed as N (m, 1/πx ), Wi is distributed as N (0, 1/πw ), and εi is distributed as N (0, 1/πε ) and that these three random variables are mutually independent. The values of πx , πw , and πε can be interpreted as precision parameters of X, Wi , and εi respectively. We analyze a quote-driven market in which risk-neutral market makers provide bid-ask quotations to a trader as a set of price-quantity combinations before receiving an order. The market makers can revise their quotes only after a transaction is complete. If traders arrive at the market sequentially and the market makers execute one order at a time, then we can limit our framework to a single period model with one trader so the subscript i can 4 This assumption is standard in the literature due to its tractability. While experimental findings on appropriate functional forms for financial decisions are mixed—Levy (1994) supports CARA and DARA and rejects IRRA while Brocas, Carrillo, Giga, and Zapatero (2019) strongly support DARA and IRRA and do not reject CARA—we use this CARA-Normal framework so the readers can compare our results to the linear equilibrium previously derived by Glosten (1989) and Madhavan (1992). Recent theoretical papers on insider trading also adopt this framework—see, for example, Efstathios (2016) and Lenkey (2019).
5
be dropped. There is no inventory cost for the market makers. They do not have access to private information about the risky asset, nor do they observe each trader’s endowments or type. Therefore, each market maker’s informational disadvantage arises from two sources: unknown endowment of the risky asset and unobserved private signal. Given a CARA utility function with parameter ρ, a normally distributed final wealth Y , and an information set Φ, an expected utility maximizing trader will maximize his certainty equivalent which can be written as ρ L = E[Y |Φ] − V ar[Y |Φ] 2
ρ = (W + Q)E[X|Φ] − P (Q)Q − (W + Q)2 V ar[X|Φ], 2
where E[X|Φ] = E[X|S] = m +
πε (S πx +πε
− m) and V ar[X|Φ] = V ar[X|S] =
(2) 1 . πx +πε
We
generalize Glosten’s linearity by assuming that equilibrium price function has the form P = m + λ(Q)Q, where λ(Q) is twice differentiable. The trader’s first-order condition for expected-utility maximization can be written as dL πε (S − m) − ρW ρ = − 2λ(Q) + Q − λ0 (Q)Q2 = 0. dQ πx + πε πx + πε
(3)
For convenience, we define a variable Z to summarize the trader’s private information, i.e., S and W , as Z :=
πε (S − m) − ρW πx + πε
(4)
so the first-order condition in (3) implies that the trader chooses Q such that Z=
ρ λ (Q)Q + 2λ(Q) + πx + π ε 0
Q.
(5)
Upon receiving a market order from the trader, each market maker forms E(X|Q), a conditional expected value of the risky asset given the trader’s order. How does each market maker rationally derive E(X|Q)?
While S and W are unknown to all mar-
ket makers, they can derive Z from the trader’s order Q since (5) implies that Z is a strictly increasing function of Q when the trader’s second-order condition is satisfied (i.e., d2 L/dQ2 = −dZ/dQ < 0). Thus E[X|Q] = E[X|Z]. Moreover, the market makers know 6
from (4) that Z is normally distributed with E(Z) = 0 and V ar[Z] =
πε ρ2 (γ + 1)πε + = 2 πx (πx + πε ) πw (πx + πε ) πx (πx + πε )
(6)
where γ= Since Cov[X, Z] =
πε , πx (πx +πε )
ρ 2 πx . πw πε (πx + πε )
(7)
Z we find that E[X|Z] = m + γ+1 . Using (5), we can write each
market maker’s conditional expected value of X given an order Q as E[X|Q] = m +
3
ρ λ0 (Q)Q + 2λ(Q) + ( πx +π ) ε
γ+1
Q.
(8)
Equilibrium
The following conditions are required in equilibrium: Condition 1. Expected utility maximization: the trader chooses to trade Q units of the risky asset to maximize his expected utility given his private signal and endowment of the risky asset. Condition 2. No arbitrage opportunities: the trader cannot make a risk-free profit from arbitraging. Condition 3. Zero expected profit: market makers compete under perfect competition so that P (Q) = E[X|Q]. We identify each equilibrium by the equilibrium price function P (Q) = m + λ(Q)Q. Given the price equation and the conditional expectation E[X|Q] in (8), we find from setting P (Q) = E[X|Q] that λ(Q) =
ρ λ0 (Q)Q + 2λ(Q) + ( πx +π ) ε
γ+1
.
(9)
We solve the above differential equation and state a condition for existence of equilibrium in the following proposition. Proposition 1. If γ > 1, there exists a symmetric equilibrium in which equilibrium price
7
quote is given by P =
m + bQ + cQγ
m + bQ − c(−Q)γ
if Q ≥ 0
(10)
if Q < 0
and the trader’s order Q given S and W satisfies (b + c|Q|(γ−1) )Q =
πε (S − m) − ρW , (γ + 1)(πx + πε )
(11)
where b=
ρ , (γ − 1)(πx + πε )
(12)
for any c ≥ 0. For example, we assume ρ = 2, πx = 3, and πw = πε = 1. Given these parameters, we find that γ = 3, b = 0.25, and θ = 0.02 and we can plot equilibrium price quotes given c = 0, 0.2, and 0.4 in Figure 1. Note that the linear equilibrium derived in Glosten (1989) is our special case when c = 0. If c > 0, the price equation is strictly convex in order size |Q|. Glosten (1989) also uses γ as a measure of information symmetry, and for an equilibrium to exist γ must be greater than 1. Note that γ increases with the proportion of the variation in the private signal that is due to its noise because the ratio V ar(ε) : V ar(S) is equal to
πx (πx +πε )
on the right-hand side of (7). Moreover, it can be shown that
∂γ ∂πε
< 0,
so γ increases as the private signal becomes noisier. If γ is lower than 1, no market makers are willing to trade because of such a great informational disadvantage and the market will shut down. Next we derive the distributions of Q and P in equilibrium. Using the definition of Z in (4), we see that the right-hand side of (11) is
Z . (γ+1)
Since the distribution of Z is known,
we can derive the distribution of Q and its properties as follows. Proposition 2. The distribution of Q in equilibrium has the following properties: (i) If c = 0, Q is distributed as N (0, bθ2 ) where θ=
πε . πx (γ + 1)(πx + πε )
(ii) An increase in c causes a mean-preserving contraction in the distribution of Q. 8
(13)
1.0 0.8
2.0
Density
0.6
1.5
0.0
0.0
0.2
0.5
0.4
1.0
Price
-1.0
-0.5
0.0
0.5
1.0
-1.0
-0.5
Order
0.0
0.5
1.0
Order
Figure 1: Left: Equilibrium price quotes given c = 0 (solid curve), c = 0.2 (dashed curve), and c = 0.4 (dotted curve). Right: Probability density functions of Q in equilibrium given c = 0 (solid curve), c = 0.2 (dashed curve), and c = 0.4 (dotted curve) (iii) When c is large enough, the density of Q becomes bimodal. We can plot the density of Q given c = 0, 0.2, and 0.4 in Figure 1. Note that as c increases, the density shifts towards the mean and becomes bimodal. However, as we show in next result, the distribution of P does not depend on c. Proposition 3. In equilibrium, P is distributed as N (m, θ) which is independent of c.
4
Conclusion
In this paper we theoretically investigate equilibrium properties of an asset market given perfect competition between market makers and nonlinear price quotes. We propose a class of convex equilibria that includes Glosten’s (1989) linear equilibrium as a special case. All equilibria can tolerate the same degree of information asymmetry, that is, if the adverse selection problem is so severe that a linear equilibrium fails to exist, no other nonlinear equilibria will exist given the same set of parameters. All equilibria also induce the same distribution of equilibrium price. However, as equilibrium becomes “more convex,” the distribution of equilibrium quantity has a smaller variance and may become bimodal.
9
Appendix Proof of Proposition 1 We rewrite the zero-profit condition in (9) as λ0 (Q)Q +b (γ − 1)
λ(Q) =
(14)
where b is given by (12). We can solve the above differential equation as λ(Q) = b + c|Q|γ−1
(15)
for any constant c. By substituting λ0 (Q)Q = c(γ − 1)|Q|(γ−1) and λ00 (Q)Q2 = c(γ − 1)(γ − 2)|Q|(γ−1) in (5), we find that the first-order condition is equivalent to Z = (γ + 1)λ(Q)Q
(16)
and thus (11) is obtained. We find that d2 L dZ =− = −(γ + 1) dQ2 dQ
ρ (γ−1) + cγ|Q| . (γ − 1)(πx + πε )
(17)
Therefore, the second-order condition, d2 L/dQ2 < 0, holds for all Q if c ≥ 0 and γ > 1.
Proof of Proposition 2 (i) Given the variance of Z in (6), we standardize Z by letting
V := p
Z V ar[Z]
=Z
s
πx (πx + πε ) . (γ + 1)πε
(18)
Substituting Z from (16) in (18) yields V =
λ(Q)Q √ θ
10
(19)
where θ is given by (13). Using the solution of λ(Q) in (15), we can write realization of V as v=
√ (bq + cq γ )/ θ
√ (bq − c(−q)γ )/ θ
if q ≥ 0
(20)
if q < 0
where b is given by (12). Since V is a standard normal random variable, it follows that the pdf of Q can be written as
2
v 0 e(−v /2) fQ (q) = √ , 2π √ where v 0 = dv/dq. If c = 0, then v = bq/ θ and it follows that
(21)
2
be(−(bq) /(2θ)) √ . fQ (q) = 2πθ
(22)
(ii) We can estimate the effect of a change in c on the density function as ∂fQ (q) |q|(γ−1) e(−v √ = (γ − v 0 vq) ∂c 2πθ
2 /2)
(23)
where v0 =
dv b + cγ|q|(γ−1) √ = . dq θ
(24)
Notice that the sign of ∂fQ (q)/∂c is the same as the sign of γ − v 0 vq. Given v and v 0 as in (20) and (24) respectively, we find that v 0 vq = a1 q 2 + a2 |q|(γ+1) + a3 |q|2γ
(25)
where a1 , a2 , a3 > 0 if γ > 1 and c > 0. Thus v 0 vq is an increasing function of |q| and takes the value of zero only when |q| is zero. If we let q¯ > 0 be the value of |q| such that v 0 vq = γ, then we find that
> 0 if q ∈ (−¯ q , 0) ∪ (0, q¯) ∂fQ (q) = 0 if q ∈ {−¯ q , 0, q¯} ∂c < 0 if q ∈ (−∞, −¯ q ) ∪ (¯ q , ∞). 11
(26)
Since fQ (q) is symmetric and the mean of Q is zero for all c, the change in the density function due to an increase in c is a mean-preserving contraction.5 (iii) We find that ∂fQ (q) = ∂q
cγ(γ − 1) v 0 v 0 vq √ − γ−1 |q| θ
|q|(γ−2) e(−v √ 2π
2 /2)
(27)
where v 0 v 0 vq = a4 |q|(3−γ) + a5 q 2 + a6 |q|(γ+1) + a7 |q|2γ . |q|γ−1
(28)
Note that a4 , ..., a7 > 0 if γ > 1 and c > 0. If 1 < γ < 3, v 0 v 0 vq/|q|γ−1 is an increasing function of |q| and takes the value of zero only when |q| is zero. Thus there are two values of q > 0 such that ∂fQ (q)/∂q = 0. If γ > 3, there will be two values of q > 0 such that ∂fQ (q)/∂q = 0 when c is high enough.
Proof of Proposition 3 We define p = m + λ(q)q =: φ(q). Given equilibrium price function in (10) and distribution of Q in equilibrium in (21), we can derive the distribution of P as 0 v fP (p) = p0
2
e(−v /2) √ 2π
!
,
(29)
where v is defined in (20), v 0 = dv/dq, p0 = dp/dq, and q = φ−1 (p). Instead of solving for φ−1 (p) directly, we write
p−m=
and find that (20) and (30) imply
bq + cq γ
bq − c(−q)γ
if q ≥ 0
(30)
if q < 0
p−m √ θ
(31)
p0 √ v = . θ
(32)
v≡ and, hence, 0
5
See Rothschild and Stiglitz (1970).
12
By substituting (31) and (32) into (29), we can write fP (p) as 2
e(−(p−m) /(2θ)) √ . fP (p) = 2πθ
(33)
References [1] Back, Kerry, and Shmuel Baruch, 2013, Strategic liquidity provision in limit order markets, Econometrica 81, 363-392. [2] Bhattacharya, Utpal, and Matthew Spiegel, 1991, Insiders, outsiders, and market breakdowns, Review of Financial Studies 4, 255-282. [3] Biais, Bruno, Thierry Foucault, and Fran¸cois Salani´e, 1998, Floors, dealer markets and limit order markets, Journal of Financial Markets 1, 253-284. [4] Biais, Bruno, Larry, Glosten, and Chester Spatt, 2005, Market microstructure: A survey of microfoundations, Journal of Financial Markets 8, 217-264. [5] Biais, Bruno, David Martimort, and Jean-Charles Rochet, 2000, Competing mechanisms in a common value environment, Econometrica 68, 799-837. [6] Biais, Bruno, David Martimort, and Jean-Charles Rochet, 2013, Corrigendum to “Competing mechanisms in a common value environment,” Econometrica 81, 393406. [7] Brocas, Isabelle, Juan D. Carrillo, Aleksandar Giga and Fernando Zapatero, 2019, Risk aversion in a dynamic asset allocation experiment, Journal of Financial and Quantitative Analysis 54, 2209-2232. [8] Efstathios, Avdis, 2016, Information tradeoffs in dynamic financial markets, Journal of Financial Economics 122, 568-584. [9] Flepp Raphael, Stephan N¨ uesch, and Egon Franck, 2017, The liquidity advantage of the quote-driven market: Evidence from the betting industry, Quarterly Review of Economics and Finance 64, 3 06-317. [10] Foucault, Thierry, Marco Pagano, and Ailsa R¨oell, 2013, Market Liquidity: Theory, Evidence, and Policy, Oxford: Oxford University Press. [11] Glosten, Lawrence R., 1989, Insider trading, liquidity, and the role of the monopolist specialist, Journal of Business 62, 211-235. [12] Glosten, Lawrence R., 1994, Is the electronic open limit order book inevitable?, Journal of Finance 49, 1127-1161. [13] Kyle, Albert S., 1989, Informed speculation with imperfect competition, Review of Economic Studies 56, 317-355. 13
[14] Lenkey, Stephen L., 2019, Cancellable insider trading plans: An analysis of SEC Rule 10b5-1, Review of Financial Studies, forthcoming. [15] Levy, Haim, 1994, Absolute and relative risk aversion: An experimental study, Journal of Risk and Uncertainty 8, 289-307. [16] Madhavan, Ananth, 1992, Trading mechanisms in securities markets, Journal of Finance 47, 607-641. [17] Madhavan, Ananth, 2000, Market microstructure: A survey, Journal of Financial Markets 3, 205-258. [18] Malinova, Katya, and Andreas Park, 2013, Liquidity, volume and price efficiency: The impact of order vs. quote driven trading, Journal of Financial Markets 16, 104–126. [19] Malo, Pekka, and Teemu Pennanen, 2012, Reduced form modeling of limit oder markets, Quantitative Finance 12, 1025-1036. [20] Maslov, Sergei, and Mark Mills, 2001, Price fluctuations from the order book perspective—empirical facts and a simple model, Physica A 299, 234-246. [21] Niemeyer, Jonas, and Patrik Sand˚ as, 1994, An empirical analysis of the trading structure at the Stockholm Stock Exchange, Journal of Multinational Finance Management 3, 63-101. [22] Ro¸su, Ioanid, 2009, A dynamic model of limit order book, Review of Financial Studies 22, 4601-4641. [23] Rothschild, Michael, and Joseph E. Stiglitz, 1970, Increasing risk: I. A definition, Journal of Economic Theory 2, 225-243. [24] Visaltanachoti, Nuttawat, Charlie Charoenwong, and David K. Ding, 2008, Liquidity distribution in the limit order book on the stock exchange of Thailand, International Review of Financial Analysis 17, 291-311. [25] Viswanathan, S., and James J.D. Wang, 2002, Market architecture: limit-order books versus dealership markets, Journal of Financial Markets 5, 127–167. [26] Weber, Philipp, and Bernd Rosenow, 2005, Order book approach to price impact, Quantitative Finance 5, 357-364. [27] Xing, Xiaochuan, and Yi Xue, 2017, Trading mechanisms and market quality: limitorder books versus dealership markets, Economics Letters 154, 35-44.
14
Journal Pre-proof
Market Making with Convex Quotes Hae-shin Hwang, Paan Jindapon PII: DOI: Reference:
S1544-6123(18)30632-9 https://doi.org/10.1016/j.frl.2019.101361 FRL 101361
To appear in:
Finance Research Letters
Received date: Revised date: Accepted date:
10 September 2018 6 November 2019 10 November 2019
Please cite this article as: Hae-shin Hwang, Paan Jindapon, Market Making with Convex Quotes, Finance Research Letters (2019), doi: https://doi.org/10.1016/j.frl.2019.101361
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Inc.
Highlights • We investigate equilibrium properties of a quote-driven market where price is convex in order size. • As the degree of convexity increases, the distribution of incoming order contracts and becomes bimodal. • All of these equilibria tolerate the same level of information asymmetry between traders and market makers. • All of these equilibria induce the same distribution of equilibrium price.
1
Market Making with Convex Quotes Hae-shin Hwang∗
Paan Jindapon†
Texas A&M University
University of Alabama
November 14, 2019
Abstract We investigate equilibrium properties of a quote-driven market where price is convex in order size. As the degree of convexity increases, the distribution of incoming order contracts and becomes bimodal. Nonetheless, all of these equilibria tolerate the same level of information asymmetry between traders and market makers, and also induce the same distribution of equilibrium price.
Keywords: Market microstructure; Market making; Adverse selection; Risk aversion JEL Codes: D8; G1
∗
Email: [email protected] Corresponding author, Department of Economics, Finance and Legal Studies, University of Alabama, Tuscaloosa, AL 35487-0224, USA, Phone: +1 (205) 348-7841, Email: [email protected] †
2
1
Introduction
Trading mechanisms in security markets have different institutional attributes. In the literature, trading mechanisms are often classified by the availability of price information at the time of order submission. In a quote-driven mechanism (QDM hereafter), traders obtain price quotations from market makers (or dealers) before submitting orders and transaction prices are as quoted. In an order-driven mechanism (ODM hereafter), traders submit orders for execution through an auction process, also known as limit order book, and transaction prices are often unknown to the traders at the time of order submission. While most stock exchanges in the world are operated with limit order books, the QDM has been adopted in NASDAQ, London SEAQ, and many foreign exchange markets. Both QDM and ODM have been studied extensively.1 Using Glosten’s (1989) and Kyle’s (1989) rational expectations models, Madhavan (1992) theoretically shows that, when transactions are executed continuously, the the two mechanisms are equivalent under perfect competition. In contrast, Xing and Xue (2017) propose a unified framework and find that the QDM performs better when the order size is small while the ODM performs better when the order size is large. A key assumption is these comparisons is the linearity of equilibrium price function in the QDM. In this paper, we adopt Glosten’s (1989) model of the QDM and show that, in fact, there are infinitely many nonlinear equilibria where price is convex in order size, i.e., convex for selling quotes (positive orders) and concave for buying quotes (negative orders). Due to a larger price impact in these nonlinear equilibria, traders find the QDM less attractive. We also analyze market tolerance for information asymmetry in the QDM. Like the aforementioned papers, we assume that each trader is endowed with a risky asset and a private noisy signal about its return. Since all of our market makers neither receive information about the return nor observe each trader’s endowment of the risky asset, they cannot distinguish between the liquidity-motivated and the information-motivated components of an incoming order. If the information asymmetry between the traders and the market makers is so severe that the market makers are unable to protect themselves 1 For example, see Madhavan (1992), Biais, Foucault, and Salani´e (1998), Viswanathan and Wang (2002), and Malinova and Park (2013), Xing and Xue (2017). For survey of the literature, see Madhavan (2000), Biais, Glosten, and Spatt (2005) and Foucault, Pagano, and R¨ oell (2013).
3
from the informed traders, the market will shut down. We find that all of these equilibria, linear and nonlinear, can tolerate the same level of information asymmetry. In a model of one informed trader versus multiple uninformed traders, Bhattacharya and Spiegel (1991) also identify both linear and nonlinear equilibrium price schedules, and show that the maximum degree of information asymmetry the uninformed traders can tolerate is the same in linear and nonlinear equilibria. Existence of nonlinear equilibria is also discussed in Biais, Martimort, and Rochet (2000).2 While Biais, Martimort, and Rochet complement Glosten’s (1994) equilibrium in a limit order market by deriving an equilibrium under imperfect competition with convex marginal selling quotes, we generalize Glosten (1989) by allowing for convexity in the market makers’ average selling quotes. In fact, our convex average selling quotes implies convex marginal selling quotes as well.3 Convex price functions are documented in various empirical research. For example, see Niemeyer and Sand˚ as (1994), Maslov and Mills (2001), Weber and Rosenow (2005), Visaltanachoti, Charoenwong, and Ding (2008), and Malo and Pennanen (2012). While recent empirical research on the QDM seems to be overshadowed by abundant evidence of the ODM adopted by most stock exchanges, liquidity and stability of the QDM have been examined in the betting industry—see Flepp, N¨ uesch, and Franck (2017). In addition to the convex price-order relationship in dealers’ quotes, we also suggest other empirical predictions. First, as the degree of convexity increases, the corresponding distribution of incoming order contracts while preserving the mean as in mean-preserving contraction defined by Rothschild and Stiglitz (1970). However, buy and sell orders can be clustered away from zero and generate two peaks, one at a buying order and one at a selling order. This bimodal distribution is also predicted by Ro¸su’s (2009) theoretical model of the limit order book. Finally, our theory predicts that the distribution of equilibrium price is the same for all equilibria regardless of the degree of convexity. 2
See additional analysis in Biais, Martimort, and Rochet (2013) and also a generalization in Back and Baruch (2013). 3 Given average price equation P = m + bQ + cQγ in Section 3, the corresponding marginal price is P = m + 2bQ + c(γ + 1)Qγ . Since γ > 1, both price equations are convex.
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2
The Model
There are two assets: a risk-free asset with zero return and a risky asset with a random return per unit X. Two types of agents are present in the market: informed traders and uninformed market makers who provide liquidity by trading with the traders. Trader i is endowed with Ai units of the risk-free asset and Wi units of the risky asset, and given a private signal, Si = X + εi , which is personalized noisy information about the risky asset before arriving at the market. Given P (Q), unit price of the risky asset dependent on incoming order, trader i determines his demand for the risky asset, denoted by Qi , to maximize his expected utility derived from his final wealth, Yi = (Wi + Qi )X + Ai − P (Qi )Qi ,
(1)
conditional on information set Φi which consists of realized values of Ai , Wi , and Si . Note that a positive (negative) Qi means that trader i’s is buying (selling) the risky asset. We assume that each trader has a CARA utility function Ui (Yi ) = −e(−ρi Yi ) , where
ρi > 0 is the absolute risk-aversion parameter.4 Without loss of generality, we set each individual’s cash endowment Ai to zero. Moreover, all agents have common prior believes about the return of the risky asset, the initial endowment of the risky asset, and the noise in the private information. Following Glosten (1989), we assume that X is distributed as N (m, 1/πx ), Wi is distributed as N (0, 1/πw ), and εi is distributed as N (0, 1/πε ) and that these three random variables are mutually independent. The values of πx , πw , and πε can be interpreted as precision parameters of X, Wi , and εi respectively. We analyze a quote-driven market in which risk-neutral market makers provide bid-ask quotations to a trader as a set of price-quantity combinations before receiving an order. The market makers can revise their quotes only after a transaction is complete. If traders arrive at the market sequentially and the market makers execute one order at a time, then we can limit our framework to a single period model with one trader so the subscript i can 4 This assumption is standard in the literature due to its tractability. While experimental findings on appropriate functional forms for financial decisions are mixed—Levy (1994) supports CARA and DARA and rejects IRRA while Brocas, Carrillo, Giga, and Zapatero (2019) strongly support DARA and IRRA and do not reject CARA—we use this CARA-Normal framework so the readers can compare our results to the linear equilibrium previously derived by Glosten (1989) and Madhavan (1992). Recent theoretical papers on insider trading also adopt this framework—see, for example, Efstathios (2016) and Lenkey (2019).
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be dropped. There is no inventory cost for the market makers. They do not have access to private information about the risky asset, nor do they observe each trader’s endowments or type. Therefore, each market maker’s informational disadvantage arises from two sources: unknown endowment of the risky asset and unobserved private signal. Given a CARA utility function with parameter ρ, a normally distributed final wealth Y , and an information set Φ, an expected utility maximizing trader will maximize his certainty equivalent which can be written as ρ L = E[Y |Φ] − V ar[Y |Φ] 2
ρ = (W + Q)E[X|Φ] − P (Q)Q − (W + Q)2 V ar[X|Φ], 2
where E[X|Φ] = E[X|S] = m +
πε (S πx +πε
− m) and V ar[X|Φ] = V ar[X|S] =
(2) 1 . πx +πε
We
generalize Glosten’s linearity by assuming that equilibrium price function has the form P = m + λ(Q)Q, where λ(Q) is twice differentiable. The trader’s first-order condition for expected-utility maximization can be written as dL πε (S − m) − ρW ρ = − 2λ(Q) + Q − λ0 (Q)Q2 = 0. dQ πx + πε πx + πε
(3)
For convenience, we define a variable Z to summarize the trader’s private information, i.e., S and W , as Z :=
πε (S − m) − ρW πx + πε
(4)
so the first-order condition in (3) implies that the trader chooses Q such that Z=
ρ λ (Q)Q + 2λ(Q) + πx + π ε 0
Q.
(5)
Upon receiving a market order from the trader, each market maker forms E(X|Q), a conditional expected value of the risky asset given the trader’s order. How does each market maker rationally derive E(X|Q)?
While S and W are unknown to all mar-
ket makers, they can derive Z from the trader’s order Q since (5) implies that Z is a strictly increasing function of Q when the trader’s second-order condition is satisfied (i.e., d2 L/dQ2 = −dZ/dQ < 0). Thus E[X|Q] = E[X|Z]. Moreover, the market makers know 6
from (4) that Z is normally distributed with E(Z) = 0 and V ar[Z] =
πε ρ2 (γ + 1)πε + = 2 πx (πx + πε ) πw (πx + πε ) πx (πx + πε )
(6)
where γ= Since Cov[X, Z] =
πε , πx (πx +πε )
ρ 2 πx . πw πε (πx + πε )
(7)
Z we find that E[X|Z] = m + γ+1 . Using (5), we can write each
market maker’s conditional expected value of X given an order Q as E[X|Q] = m +
3
ρ λ0 (Q)Q + 2λ(Q) + ( πx +π ) ε
γ+1
Q.
(8)
Equilibrium
The following conditions are required in equilibrium: Condition 1. Expected utility maximization: the trader chooses to trade Q units of the risky asset to maximize his expected utility given his private signal and endowment of the risky asset. Condition 2. No arbitrage opportunities: the trader cannot make a risk-free profit from arbitraging. Condition 3. Zero expected profit: market makers compete under perfect competition so that P (Q) = E[X|Q]. We identify each equilibrium by the equilibrium price function P (Q) = m + λ(Q)Q. Given the price equation and the conditional expectation E[X|Q] in (8), we find from setting P (Q) = E[X|Q] that λ(Q) =
ρ λ0 (Q)Q + 2λ(Q) + ( πx +π ) ε
γ+1
.
(9)
We solve the above differential equation and state a condition for existence of equilibrium in the following proposition. Proposition 1. If γ > 1, there exists a symmetric equilibrium in which equilibrium price
7
quote is given by P =
m + bQ + cQγ
m + bQ − c(−Q)γ
if Q ≥ 0
(10)
if Q < 0
and the trader’s order Q given S and W satisfies (b + c|Q|(γ−1) )Q =
πε (S − m) − ρW , (γ + 1)(πx + πε )
(11)
where b=
ρ , (γ − 1)(πx + πε )
(12)
for any c ≥ 0. For example, we assume ρ = 2, πx = 3, and πw = πε = 1. Given these parameters, we find that γ = 3, b = 0.25, and θ = 0.02 and we can plot equilibrium price quotes given c = 0, 0.2, and 0.4 in Figure 1. Note that the linear equilibrium derived in Glosten (1989) is our special case when c = 0. If c > 0, the price equation is strictly convex in order size |Q|. Glosten (1989) also uses γ as a measure of information symmetry, and for an equilibrium to exist γ must be greater than 1. Note that γ increases with the proportion of the variation in the private signal that is due to its noise because the ratio V ar(ε) : V ar(S) is equal to
πx (πx +πε )
on the right-hand side of (7). Moreover, it can be shown that
∂γ ∂πε
< 0,
so γ increases as the private signal becomes noisier. If γ is lower than 1, no market makers are willing to trade because of such a great informational disadvantage and the market will shut down. Next we derive the distributions of Q and P in equilibrium. Using the definition of Z in (4), we see that the right-hand side of (11) is
Z . (γ+1)
Since the distribution of Z is known,
we can derive the distribution of Q and its properties as follows. Proposition 2. The distribution of Q in equilibrium has the following properties: (i) If c = 0, Q is distributed as N (0, bθ2 ) where θ=
πε . πx (γ + 1)(πx + πε )
(ii) An increase in c causes a mean-preserving contraction in the distribution of Q. 8
(13)
1.0 0.8
2.0
Density
0.6
1.5
0.0
0.0
0.2
0.5
0.4
1.0
Price
-1.0
-0.5
0.0
0.5
1.0
-1.0
-0.5
Order
0.0
0.5
1.0
Order
Figure 1: Left: Equilibrium price quotes given c = 0 (solid curve), c = 0.2 (dashed curve), and c = 0.4 (dotted curve). Right: Probability density functions of Q in equilibrium given c = 0 (solid curve), c = 0.2 (dashed curve), and c = 0.4 (dotted curve) (iii) When c is large enough, the density of Q becomes bimodal. We can plot the density of Q given c = 0, 0.2, and 0.4 in Figure 1. Note that as c increases, the density shifts towards the mean and becomes bimodal. However, as we show in next result, the distribution of P does not depend on c. Proposition 3. In equilibrium, P is distributed as N (m, θ) which is independent of c.
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Conclusion
In this paper we theoretically investigate equilibrium properties of an asset market given perfect competition between market makers and nonlinear price quotes. We propose a class of convex equilibria that includes Glosten’s (1989) linear equilibrium as a special case. All equilibria can tolerate the same degree of information asymmetry, that is, if the adverse selection problem is so severe that a linear equilibrium fails to exist, no other nonlinear equilibria will exist given the same set of parameters. All equilibria also induce the same distribution of equilibrium price. However, as equilibrium becomes “more convex,” the distribution of equilibrium quantity has a smaller variance and may become bimodal.
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Appendix Proof of Proposition 1 We rewrite the zero-profit condition in (9) as λ0 (Q)Q +b (γ − 1)
λ(Q) =
(14)
where b is given by (12). We can solve the above differential equation as λ(Q) = b + c|Q|γ−1
(15)
for any constant c. By substituting λ0 (Q)Q = c(γ − 1)|Q|(γ−1) and λ00 (Q)Q2 = c(γ − 1)(γ − 2)|Q|(γ−1) in (5), we find that the first-order condition is equivalent to Z = (γ + 1)λ(Q)Q
(16)
and thus (11) is obtained. We find that d2 L dZ =− = −(γ + 1) dQ2 dQ
ρ (γ−1) + cγ|Q| . (γ − 1)(πx + πε )
(17)
Therefore, the second-order condition, d2 L/dQ2 < 0, holds for all Q if c ≥ 0 and γ > 1.
Proof of Proposition 2 (i) Given the variance of Z in (6), we standardize Z by letting
V := p
Z V ar[Z]
=Z
s
πx (πx + πε ) . (γ + 1)πε
(18)
Substituting Z from (16) in (18) yields V =
λ(Q)Q √ θ
10
(19)
where θ is given by (13). Using the solution of λ(Q) in (15), we can write realization of V as v=
√ (bq + cq γ )/ θ
√ (bq − c(−q)γ )/ θ
if q ≥ 0
(20)
if q < 0
where b is given by (12). Since V is a standard normal random variable, it follows that the pdf of Q can be written as
2
v 0 e(−v /2) fQ (q) = √ , 2π √ where v 0 = dv/dq. If c = 0, then v = bq/ θ and it follows that
(21)
2
be(−(bq) /(2θ)) √ . fQ (q) = 2πθ
(22)
(ii) We can estimate the effect of a change in c on the density function as ∂fQ (q) |q|(γ−1) e(−v √ = (γ − v 0 vq) ∂c 2πθ
2 /2)
(23)
where v0 =
dv b + cγ|q|(γ−1) √ = . dq θ
(24)
Notice that the sign of ∂fQ (q)/∂c is the same as the sign of γ − v 0 vq. Given v and v 0 as in (20) and (24) respectively, we find that v 0 vq = a1 q 2 + a2 |q|(γ+1) + a3 |q|2γ
(25)
where a1 , a2 , a3 > 0 if γ > 1 and c > 0. Thus v 0 vq is an increasing function of |q| and takes the value of zero only when |q| is zero. If we let q¯ > 0 be the value of |q| such that v 0 vq = γ, then we find that
> 0 if q ∈ (−¯ q , 0) ∪ (0, q¯) ∂fQ (q) = 0 if q ∈ {−¯ q , 0, q¯} ∂c < 0 if q ∈ (−∞, −¯ q ) ∪ (¯ q , ∞). 11
(26)
Since fQ (q) is symmetric and the mean of Q is zero for all c, the change in the density function due to an increase in c is a mean-preserving contraction.5 (iii) We find that ∂fQ (q) = ∂q
cγ(γ − 1) v 0 v 0 vq √ − γ−1 |q| θ
|q|(γ−2) e(−v √ 2π
2 /2)
(27)
where v 0 v 0 vq = a4 |q|(3−γ) + a5 q 2 + a6 |q|(γ+1) + a7 |q|2γ . |q|γ−1
(28)
Note that a4 , ..., a7 > 0 if γ > 1 and c > 0. If 1 < γ < 3, v 0 v 0 vq/|q|γ−1 is an increasing function of |q| and takes the value of zero only when |q| is zero. Thus there are two values of q > 0 such that ∂fQ (q)/∂q = 0. If γ > 3, there will be two values of q > 0 such that ∂fQ (q)/∂q = 0 when c is high enough.
Proof of Proposition 3 We define p = m + λ(q)q =: φ(q). Given equilibrium price function in (10) and distribution of Q in equilibrium in (21), we can derive the distribution of P as 0 v fP (p) = p0
2
e(−v /2) √ 2π
!
,
(29)
where v is defined in (20), v 0 = dv/dq, p0 = dp/dq, and q = φ−1 (p). Instead of solving for φ−1 (p) directly, we write
p−m=
and find that (20) and (30) imply
bq + cq γ
bq − c(−q)γ
if q ≥ 0
(30)
if q < 0
p−m √ θ
(31)
p0 √ v = . θ
(32)
v≡ and, hence, 0
5
See Rothschild and Stiglitz (1970).
12
By substituting (31) and (32) into (29), we can write fP (p) as 2
e(−(p−m) /(2θ)) √ . fP (p) = 2πθ
(33)
References [1] Back, Kerry, and Shmuel Baruch, 2013, Strategic liquidity provision in limit order markets, Econometrica 81, 363-392. [2] Bhattacharya, Utpal, and Matthew Spiegel, 1991, Insiders, outsiders, and market breakdowns, Review of Financial Studies 4, 255-282. [3] Biais, Bruno, Thierry Foucault, and Fran¸cois Salani´e, 1998, Floors, dealer markets and limit order markets, Journal of Financial Markets 1, 253-284. [4] Biais, Bruno, Larry, Glosten, and Chester Spatt, 2005, Market microstructure: A survey of microfoundations, Journal of Financial Markets 8, 217-264. [5] Biais, Bruno, David Martimort, and Jean-Charles Rochet, 2000, Competing mechanisms in a common value environment, Econometrica 68, 799-837. [6] Biais, Bruno, David Martimort, and Jean-Charles Rochet, 2013, Corrigendum to “Competing mechanisms in a common value environment,” Econometrica 81, 393406. [7] Brocas, Isabelle, Juan D. Carrillo, Aleksandar Giga and Fernando Zapatero, 2019, Risk aversion in a dynamic asset allocation experiment, Journal of Financial and Quantitative Analysis 54, 2209-2232. [8] Efstathios, Avdis, 2016, Information tradeoffs in dynamic financial markets, Journal of Financial Economics 122, 568-584. [9] Flepp Raphael, Stephan N¨ uesch, and Egon Franck, 2017, The liquidity advantage of the quote-driven market: Evidence from the betting industry, Quarterly Review of Economics and Finance 64, 3 06-317. [10] Foucault, Thierry, Marco Pagano, and Ailsa R¨oell, 2013, Market Liquidity: Theory, Evidence, and Policy, Oxford: Oxford University Press. [11] Glosten, Lawrence R., 1989, Insider trading, liquidity, and the role of the monopolist specialist, Journal of Business 62, 211-235. [12] Glosten, Lawrence R., 1994, Is the electronic open limit order book inevitable?, Journal of Finance 49, 1127-1161. [13] Kyle, Albert S., 1989, Informed speculation with imperfect competition, Review of Economic Studies 56, 317-355. 13
[14] Lenkey, Stephen L., 2019, Cancellable insider trading plans: An analysis of SEC Rule 10b5-1, Review of Financial Studies, forthcoming. [15] Levy, Haim, 1994, Absolute and relative risk aversion: An experimental study, Journal of Risk and Uncertainty 8, 289-307. [16] Madhavan, Ananth, 1992, Trading mechanisms in securities markets, Journal of Finance 47, 607-641. [17] Madhavan, Ananth, 2000, Market microstructure: A survey, Journal of Financial Markets 3, 205-258. [18] Malinova, Katya, and Andreas Park, 2013, Liquidity, volume and price efficiency: The impact of order vs. quote driven trading, Journal of Financial Markets 16, 104–126. [19] Malo, Pekka, and Teemu Pennanen, 2012, Reduced form modeling of limit oder markets, Quantitative Finance 12, 1025-1036. [20] Maslov, Sergei, and Mark Mills, 2001, Price fluctuations from the order book perspective—empirical facts and a simple model, Physica A 299, 234-246. [21] Niemeyer, Jonas, and Patrik Sand˚ as, 1994, An empirical analysis of the trading structure at the Stockholm Stock Exchange, Journal of Multinational Finance Management 3, 63-101. [22] Ro¸su, Ioanid, 2009, A dynamic model of limit order book, Review of Financial Studies 22, 4601-4641. [23] Rothschild, Michael, and Joseph E. Stiglitz, 1970, Increasing risk: I. A definition, Journal of Economic Theory 2, 225-243. [24] Visaltanachoti, Nuttawat, Charlie Charoenwong, and David K. Ding, 2008, Liquidity distribution in the limit order book on the stock exchange of Thailand, International Review of Financial Analysis 17, 291-311. [25] Viswanathan, S., and James J.D. Wang, 2002, Market architecture: limit-order books versus dealership markets, Journal of Financial Markets 5, 127–167. [26] Weber, Philipp, and Bernd Rosenow, 2005, Order book approach to price impact, Quantitative Finance 5, 357-364. [27] Xing, Xiaochuan, and Yi Xue, 2017, Trading mechanisms and market quality: limitorder books versus dealership markets, Economics Letters 154, 35-44.
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