Dynamic Market Making and Asset Pricing

Dynamic Market Making and Asset Pricing Wen Chen 1 Yajun Wang 2 1 The Chinese University of Hong Kong, Shenzhen 2 Baruch College Institute of Financial Studies Southwestern University of Finance and Economics July 19, 2018 Chen and Wang Dynamic Market Making and Asset Pricing 1 / 37

Trading Games We think of trading a stock as playing a trading game by two groups of traders (Jack Treynor, 1971): Long-term traders: care about fundamental, heterogeneous in information and hedging motives, e.g., institutional traders Intermediaries with short-term strategies: limited risk-bearing and inventory-carrying capacities, heterogeneous in trading speed and networks, e.g., designated market makers, high frequency traders Chen and Wang Dynamic Market Making and Asset Pricing 2 / 37

Key Features Our model captures key features in many financial markets: Imperfect competition among market makers Dealer markets Exchanges: Designated market makers Market makers have limited risk-bearing capacity (risk averse) and inventory-carrying capacity (solvency constraint) Market makers optimally make offsetting trades in bid and ask markets Market makers tend to hold close-to-zero inventories at the end of the day Long-term traders have dual trading motives: information and hedging Chen and Wang Dynamic Market Making and Asset Pricing 3 / 37

Key Features Graphical Illustration Sell shares to market makers at the bid Sellers Market Makers Simulataneously choose optimal number of shares to buy at the bid and sell at the ask Cournot competition for the orders Ask buy shares from market makers at the ask Buyers Bid Market clearing at bid and ask. The posted bid and ask prices are the prices to achieve market makers optimal amount to trade. Chen and Wang Dynamic Market Making and Asset Pricing 4 / 37

Key Differences from Existing Models Multi-period dynamic model: Liu and Wang (2016) Endogenize both bid-ask spread and trading volume Glosten and Milgrom (1980), Easley and O Hara (1987) Wang (1993, 1994), He and Wang (1995) Market makers have market power, and they care about the round-trip profit Competition is imperfect among long-term traders: Kyle (1985,1989), Foster and Viswanathan (1996), Kyle, Obizhaeva and Wang (2017) Chen and Wang Dynamic Market Making and Asset Pricing 5 / 37

Overview of Main Results Information and hedging trading motives have different impacts on the dynamics of bid-ask spread and trading volume When there is pure information after market opening Trading volume spikes on information arrival Spread decreases on information arrival When there is a pure liquidity shock after market opening Trading volume increases before and on a liquidity shock Spread increases on a liquidity shock In the presence of both information and liquidity shocks, both trading volume and bid-ask spread exhibit U-shaped patterns against the trading duration (Harris (1986), Mclnish and Wood (1992), Foster and Viswanathan (1993)) Chen and Wang Dynamic Market Making and Asset Pricing 6 / 37

Overview of Main Results (Cont.) When market makers have significant market power, other traders optimally smooth out their trading even though they are not strategic Market makers market power dampens and spreads out the trading spikes due to information arrivals or liquidity shocks Information ripple effect on trading volume: trading persists after the information is incorporated into price (Morse (1980), Campbell, Grossman, and Wang (1993), Fleming et. al. (1999)) Market makers market power makes other traders hedge earlier, since it is more costly for long-term traders to hedge near the market closure Chen and Wang Dynamic Market Making and Asset Pricing 7 / 37

Overview of Main Results (Cont.) Impacts of trading frequency on traders welfare and profit Adding trading rounds can undermine market makers market power by decreasing bid-ask spread, thus always benefits long-term traders Adding trading rounds has two opposite effects on market makers profit Spread effect: bid-ask spread decreases (dominate when trading is driven by information) Volume effect: total trading volume increases (can dominate when trading is driven by liquidity shocks) Chen and Wang Dynamic Market Making and Asset Pricing 8 / 37

The Model! " Informed Traders! % Market Makers & ) + & ' & * + + & + + ( ) ( ' ( * ( +! # Uninformed Traders t=0 1 2 T T+1 A multi-period model of trading with finite time horizon and T + 1 rounds of trading with trading periods t = 0, 1,..., T There are N I of identical informed investors (I), N U of identical uninformed investors (U), and N M market makers (M) Informed and uninformed traders are perfectly competitive and market makers are imperfectly competitive Chen and Wang Dynamic Market Making and Asset Pricing 9 / 37

The Model (Cont.)! " Informed Traders & ' & ) & + &, V! % Market Makers ( ' + ( ) ( + + + (, + * ' * ) * + *,! # Uninformed Traders t=0 1 2 T T+1 One risk-free asset with zero net supply and one risky security with a supply of θ shares Risk-free rate is normalized to 0. The stock pays a liquidation value of V N ( V, τv 1 ) at the final date T + 1 Every informed trader has a private information v t = V + ε t at time t, where ε t N (0, τ 1 ε,t ) Chen and Wang Dynamic Market Making and Asset Pricing 10 / 37

The Model (Cont.). ' =& '. * =. ' + & *., =. * + &, & ' & * &, -. - = 01' & 0 & - N! " Informed Traders ( ' ( * (, ( - V! % Market Makers ) ' + ) * ), + + ) - + + ' + * +, + -! # Uninformed Traders t=0 1 2 T T+1 At each trading period t, every informed trader also receives η t shares of an illiquidity asset with final payoff N, where η t N (0, τ 1 η,t ) N = V V, X t = X t 1 + η t, Chen and Wang Dynamic Market Making and Asset Pricing 11 / 37

Long-term Traders Problem Given A t, B t, for i {I, U}, trader i s problem is to solve max E[ e λi WT i +1 F i t ], θt i where λ i is the risk aversion coefficient of type i traders and W I T +1 = W I 1 + W U T +1 = W U 1 + T [(θt I θt 1) I B t (θt I θt 1) I + A t ] + θt I V + X T N, t=0 T [(θt U θt 1) U B t (θt U θt 1) U + A t ] + θt U V, t=0 where x + = x if x > 0 otherwise 0, and similarly x = x if x < 0 otherwise 0. Chen and Wang Dynamic Market Making and Asset Pricing 12 / 37

Market Makers Problem Let α j t and β j t be the number of shares that market maker j {1, 2,..., N M } sells at ask (i.e., ask depth) and buys at bid (i.e., bid depth) respectively. For j = 1, 2,..., N M, the designated market maker j s problem is ] lim max E [ e λm W M,j T +1 F M t, s.t. Wt M,j W (1) where λ M α j t,βj t W M,j T +1 = W M,j 1 T + [αta j t (αt, j βt) j βtb j t (αt, j βt)] j + θ M,j T V. t=0 The assumptions that market makers are extremely risk averse and capital constrained correspond to the fact that intermediaries have limited risk-bearing and inventory-carrying capacities Chen and Wang Dynamic Market Making and Asset Pricing 13 / 37

Market Makers Problem (Cont.) Under these assumptions, market maker j s optimization problem is equivalent to [ max α j t β j t =αj t θm,j t 1 E αta j t βtb j t Ft M ]. This implies that market makers who trade in both the ask market and the bid market optimally make offsetting trades with other traders to maximize the profit from each round of trading. Chen and Wang Dynamic Market Making and Asset Pricing 14 / 37

Market Clearing Conditions Given the demand schedules of the informed and the uninformed (θ I t(a t, B t ) and θ U t (A t, B t )), the bid price B t (α j t, β j t) and the ask price A t (α j t, β j t) can be determined by N M j=1 N M j=1 α j t = i=i, U β j t = i=i, U N i (θ i t(a t, B t ) θ i t 1(A t 1, B t 1 )) +, N i (θ i t(a t, B t ) θ i t 1(A t 1, B t 1 )). Chen and Wang Dynamic Market Making and Asset Pricing 15 / 37

Definition of Equilibrium A subgame perfect Nash equilibrium (θ I t(a t, B t ), θ U t (A t, B t ), A t, B t, α j t, β j t) is such that given any A t and B t, θ i t(a t, B t ) solves a type i investor s problem under the information set F i t for i {I, U}; given θ I t(a t, B t ) and θ U t (A t, B t ), α j t and β j t solve the market maker s problem, under the information set of the market maker F M t ; A t := A t (α j t, β j t) and B t := B t (α j t, β j t) clear both the risky security and the risk-free asset markets. Chen and Wang Dynamic Market Making and Asset Pricing 16 / 37

Information Sets Informed traders: F I t = {η 0, η 1,..., η t, v 0, v 1,..., v t, A 0, A 1,..., A t, B 0, B 1,..., B t } Uninformed traders and market makers: F U t = F M t = {A 0, A 1,..., A t, B 0, B 1,..., B t } In a linear equilibrium, the ask and bid prices are informationally equivalent to composite signals {S 0, S 1,..., S t }, where S t = E[V Ft I ] }{{} µ t X }{{} t expected payoff hedging demand, µ t = λ I Var[V F I t ] λ I τ V + t s=0 τ. ε,s The information revealing is unaffected by imperfect competition among market makers, since informed traders have the same information Chen and Wang Dynamic Market Making and Asset Pricing 17 / 37

Investors Reservation Prices The reservation price Pt IR for informed traders (Pt UR for uninformed traders) is the critical price such that an investor I (U) buys the stock if and only if the ask is lower than this critical price. Informed traders reservation price is: P IR t = δ I t ( E[V F I t ] µ t X t ) + (1 δ I t ) V + g I t E[X F U t ] + f I t θ γi t N I Uninformed traders reservation price is: P UR t = δt U E[V Ft U ] + (1 δt U ) V + gt U E[X t Ft U ] + ft U θ γu t N U N U n=1 N I n=1 θ I,n t 1 θ U,n t 1 All the coefficients can be computed recursively, and depend on NM and T In the presence of liquidity shocks: δt I > 1, δt U > 1 Chen and Wang Dynamic Market Making and Asset Pricing 18 / 37

The Equilibrium The difference in the reservation prices is t := P IR t The equilibrium ask and bid prices for t are A t = Pt UR + N M N M + 1 B t = Pt UR + N M N M + 1 The bid-ask spread is The equilibrium trading volume is P UR t γt U /N U γt/n I I + γt U t + /N U γt U /N U γt/n I I + γt U t /N U A t B t = t N M + 1 Vol t = N M t N M + 1 γt/n I I + γt U. /N U + t N M + 1 t N M + 1 Chen and Wang Dynamic Market Making and Asset Pricing 19 / 37

The Equilibrium Graphical Illustration., /0 7 8 +, =. /0, 1 / 6, /3 / 4 5, 69:! " #$! " &$., <0! " #$! " &$ ' ( + * 7 8 ;, =. <0, + 1 < 6, /3 < 4 =, 69: 5, 6 = =, 6 5, 6, =, 6 The higher the bid price B t, the more a market maker can buy from other investors, and the lower the ask price A t, the more a market maker can sell to other investors. Facing the demand and supply functions of other investors, an oligopolistic market maker optimally trades off the prices and quantities. Chen and Wang Dynamic Market Making and Asset Pricing 20 / 37

Impacts of Information on Volume and Spread Market opens with information and a liquidity shock, after that only information arrives at the middle of trading duration 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 2.2 2 1.8 1.6 1.4 1.2 1 0.8 10-3 0 0 0.2 0.4 0.6 0.8 1 0.6 0 0.2 0.4 0.6 0.8 1 Parameters are N M = 1, N I = 10, N U = 100, T = 10, λ I = λ U = 1, θ = 1.1, V = 1, τ V = 0.9, τ η,t = 10 6,and τ ε,t = 10 6 except for τ η,0 = 3, τ ε,0 = τ ε,t /2 = 1. Chen and Wang Dynamic Market Making and Asset Pricing 21 / 37

Impacts of Information on Volume and Spread Market opens with information and a liquidity shock, after that only information arrives at a constant rate 0.8 0.7 0.6 0.5 0.025 0.02 0.015 0.4 0.3 0.01 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 0.005 0 0 0.2 0.4 0.6 0.8 1 Parameters are N M = 1, N I = 10, N U = 100, T = 20, λ I = λ U = 1, θ = 1.1, V = 1, τ V = 0.9, τ η,t = 10 6 except for τ η,0 = 10, and τ ε,t = 1. Chen and Wang Dynamic Market Making and Asset Pricing 22 / 37

Impacts of Liquidity shocks on Volume and Spread Market opens with information and a liquidity shock, after that only a liquidity shock arrives at the middle of trading duration 7 6 5 4 3 2 1 0 0 0.2 0.4 0.6 0.8 1 0.018 0.016 0.014 0.012 0.01 0.008 0.006 0.004 0.002 0 0 0.2 0.4 0.6 0.8 1 Parameters are N M = 1, N I = 10, N U = 100, T = 10, λ I = λ U = 1, θ = 1.1, V = 1, τ V = 0.9, τ η,t = 10 6,and τ ε,t = 10 6 except for τ η,0 = τ η,t /2 = 3, τ ε,0 = 1. Chen and Wang Dynamic Market Making and Asset Pricing 23 / 37

Impacts of Liquidity shocks on Volume and Spread Market opens with information and a liquidity shock, after that only liquidity shocks arrive at a constant rate 60 50 40 30 20 10 0 0 0.2 0.4 0.6 0.8 1 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 0 0.2 0.4 0.6 0.8 1 Parameters are N M = 1, N I = 10, N U = 100, T = 20, λ I = λ U = 1, θ = 1.1, V = 1, τ V = 0.9, τ ε,t = 10 6 except for τ ε,0 = 1, and τ η,t = 10. Other traders tend to hedge ealier in order to avoid higher transaction cost due to market makers market power Chen and Wang Dynamic Market Making and Asset Pricing 24 / 37

Information and Liquidity shocks Both trading volume and bid-ask spread exhibit U-shaped patterns in the presence of both information and liquidity shocks (Harris (1986), Mclnish and Wood (1992), Foster and Viswanathan (1993)) 6 5 4 3 2 1 0 0 0.2 0.4 0.6 0.8 1 0.11 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 0.2 0.4 0.6 0.8 1 Parameters are N M = 1, N I = 10, N U = 100, T = 20, λ I = λ U = 1, θ = 1.1, V = 1, τ V = 0.9, τ ε,t = 1, and τ η,t = 10. Other traders tend to delay their hedging demands in the presence of information Chen and Wang Dynamic Market Making and Asset Pricing 25 / 37

Imperfect Competition among Market Makers Market makers market power dampens and spreads out the trading spikes due to information (Information ripple effect on trading volume: Morse (1980), Fleming et. al. (1999)) 4 3.5 3 2.5 2 1.5 1 0.5 0 0 0.2 0.4 0.6 0.8 1 Parameters are N I = 10, N U = 100, T = 20, λ I = λ U = 1, θ = 1.1, V = 1, τ V = 0.9, τ η,t = 10 6 except for τ η,0 = 3, and τ ε,t = 10 6, except for τ ε,0 = τ ε,t /4 = τ ε,t /2 = τ ε,3t /4 = 1. The variation of trading volume tends to be smaller for stocks with less competitive market makers during periods of news events Chen and Wang Dynamic Market Making and Asset Pricing 26 / 37

Imperfect Competition among Market Makers Market makers market power induces traders to hedge more in advance of the liquidity shocks 120 100 80 60 40 20 0 0 0.2 0.4 0.6 0.8 1 Parameters are T = 20, N I = 10, N U = 100, λ I = 0.8, λ U = 1, θ = 1.1, V = 1, τ V = 0.9, τ X = 1.1. τ ε,t = 10 6 except for τ ε,1 = τ ε,8 = τ ε,15 = 1, and τ η,t = 10 6 except for τ η,1 = 10. The U-shaped pattern of the trading volume tends to be more pronounced for stocks with less amount of private information, more hedgers, or more market makers Chen and Wang Dynamic Market Making and Asset Pricing 27 / 37

What affects Spread The bid-ask spread at time t is proportional to the absolute value of reservation price difference at t, A t B t = t N M + 1. Given the structure of information flow and liquidity shocks Bid-ask spread decreases with the number of market makers N M. Bid-ask spread decrease with the number of trading rounds T. Chen and Wang Dynamic Market Making and Asset Pricing 28 / 37

Impacts of Trading Frequency on Spread Adding trading rounds can undermine the market power of maker makers by reducing the bid-ask spread 0.02 0.015 0.01 0.005 0 0 0.2 0.4 0.6 0.8 1 Parameters are N I = 10, N U = 100, λ I = 0.8, λ U = 1, θ = 1.1, V = 1, τ V = 0.9. τ ε,t = 3/T and τ η,t = 10 6 except for τ η,1 = 10. The U-shaped pattern of the bid-ask spread tends to be more pronounced for stocks with fewer market makers Chen and Wang Dynamic Market Making and Asset Pricing 29 / 37

The Impact of Trading Frequency on Traders Welfare Market opens with information and a liquidity shock, after that only information arrives at certain times -1.578-1.58-1.582-1.584-1.586-1.588-1.59-1.592 4 6 8 10 12 14-0.989-0.9891-0.9892-0.9893-0.9894-0.9895-0.9896-0.9897-0.9898-0.9899-0.99 4 6 8 10 12 14 Parameter values are N M = 1, N I = 10, N U = 100, λ I = λ U = 1, θ = 1.1, τ V = 0.9, τ ε,t = 10 6 expect for τ ε,0 = τ ε,t /2 = 1, and τ η,t = 10 6 except for τ η,0 = 3. Chen and Wang Dynamic Market Making and Asset Pricing 30 / 37

The Impact of Trading Frequency on Traders Welfare Market opens with information and a liquidity shock, after that only liquidity shocks arrive at certain times -1.8-0.987-2 -2.2-0.9875-2.4-2.6-0.988-2.8-3 -0.9885-3.2-3.4-0.989-3.6-3.8 0 5 10 15 20 25-0.9895 0 5 10 15 20 25 Parameter values are N M = 1, N I = 10, N U = 100, λ I = λ U = 1, θ = 1.1, τ V = 0.9, τ ε,t = 10 6 expect for τ ε,1 = 1, and τ η,t = 10 6 except for τ η,1 = τ η,t /2 = 3. Adding trading rounds benefits long-term traders, no matter the trading is driven by information or liquidiy shocks Chen and Wang Dynamic Market Making and Asset Pricing 31 / 37

Impacts of Trading Frequency on Market Maker s Profit Adding trading rounds has two opposite effects on market makers profit Spread effect: adding trading rounds decreases bid-ask spread Volume effect: adding trading rounds increases the total trading volume Chen and Wang Dynamic Market Making and Asset Pricing 32 / 37

Impacts of Trading Frequency on Market Maker s Profit Market opens with information and a liquidity shock, after that only information arrives at certain times: spread effect dominates 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 4 6 8 10 12 14 Parameter values are N M = 1, N I = 10, N U = 100, λ I = λ U = 1, θ = 1.1, τ V = 0.9, τ ε,t = 10 6 expect for τ ε,0 = τ ε,t /2 = 1, and τ η,t = 10 6 except for τ η,0 = 3. Chen and Wang Dynamic Market Making and Asset Pricing 33 / 37

Impacts of Trading Frequency on Market Maker s Profit Market opens with information and a liquidity shock, after that only liquidity shocks arrive at certain times: volume effect dominates when T is small, while spread effect dominates when T is large 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0 5 10 15 20 25 Parameter values are N M = 1, N I = 10, N U = 100, λ I = λ U = 1, θ = 1.1, τ V = 0.9, τ ε,t = 10 6 expect for τ ε,1 = 1, and τ η,t = 10 6 except for τ η,1 = τ η,t /2 = 3. Chen and Wang Dynamic Market Making and Asset Pricing 34 / 37

Impacts of Trading Frequency on Market Maker s Profit Both information and liquidity shocks arrive at certain times 0.7 1.1 0.6 1 0.5 0.9 0.4 0.8 0.3 0.7 0.2 0.6 0.1 0.5 0 4 6 8 10 12 14 (a) mainly information driven 0.4 4 6 8 10 12 14 (b) mainly liquidity driven Parameter values are N M = 1, N I = 10, N U = 100, λ I = λ U = 1, θ = 1.1, τ V = 0.9, τ ε,t = 10 6 expect for τ ε,1 = τ ε,t /2 = 1, and τ η,t = 10 6 except for τ η,1 = τ η,t /2 = 3. Chen and Wang Dynamic Market Making and Asset Pricing 35 / 37

Practical Implications The magnitude of the U-shaped patterns across different stocks The U-shaped pattern of the bid-ask spread tends to be more pronounced for stocks with fewer market makers The U-shaped pattern of the trading volume tends to be more pronounced for stocks with less amount of private information, more hedgers, or more market makers The variation of trading volume tends to be smaller for stocks with less competitive market makers during periods of news events Trading platforms regulation on traders welfare Supports continuous-time trading if the tick size can be arbitrarily small Chen and Wang Dynamic Market Making and Asset Pricing 36 / 37

Concluding Remarks Information and liquidity shocks together give rise to the U-shaped patterns of trading volume and bid-ask spread against trading duration Due to market makers market power, other traders tend to smooth out their trading even though they are not strategic Adding trading rounds leads to smaller bid-ask spreads, thus undermines market makers market power and always benefits long-term traders has two opposite effects on market maker s profit Chen and Wang Dynamic Market Making and Asset Pricing 37 / 37