# Goal Market Maker Pricing and Information about Prospective Order Flow

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2 The Model: Fundamentals Simple Lucas tree. At time t, the fundamental value v t is the discounted NPV of expected future dividends {d t+s : s=,..., }, discounted at rate r. v t is common knowledge at time of trade t for all market participants. At time t all future dividends have an expected value of d and a variance of Σ w.l.o.g. dividends are assumed to be paid out each period before trading This means that: v t = d/r v for all t. Public Order Flow Two components in baseline model order flow noise that is i.i.d. normally distributed ε t N(0,Ψ), ε t and ε s independent for all t s, Responsiveness to market maker inventory so that total demand for the stock is: u t = γz t + ε t where γ (0,]. z t is market makers inventory Ideally γ should be endogenous, so response is proportional to the distance of the current market price from fundamental value e.g. γ = kδ for some exogenous k Think of public as risk neutral long term investors overall, but only a small fraction visits the market in any period 7 8 Market Makers Market makers are myopic in the sense that they trade to maximise mean variance utility, with collective CARA coefficient φ, defined over their wealth at the end of the upcoming period But rational and fully aware of how the market works Competitive, that is, price takers. End of period wealth computed by marking inventory to market at the end of period equilibrium price. Solve for equilibrium Conjecture that the equilibrium market price can be written as: p t = v δz t for some parameter value δ>0 to be determined endogenously Look for stationary solution Derive associated distribution of p t+ Supply = demand at time t Equate coefficients 9 0 p t = v δz t Solution: Base Case where Focusing on smallest root (most liquid market solution) Public order flow: alternative models. This talk (2,2) u t = γ z t + ε t where γ (0,] and z t is market makers inventory 2. Better model harder to analyse (3,5) Can talk about potential u t = γ(v t p t ) + ε t sunshine trading advantages where γ>0 of anticipatory trading 3. OK model intermediate (2,4) u t = γ(v t p t ) + ε t Last two models simplify drastically in limit towards continuous me (trading interval 0) but then two period order placement strategy becomes meaningless 2 2

3 Multiperiod Order Placement Now we will split up the noise trading demand into two parts, one of which (ε t ) is one shot, the other (η t ) is carried out over multiple periods: order flow shock πη t at time t followed by long run order placement strategy: π ( π) t+s η t in period t+s, s=0,,. That is, each period the order flow is: u t = γz t + ε t + π sϵ(0, ) ( π ) t s η t s with ε t N(0,Ψ) and η t N(0,Ω). Market makers know that π( π)η t is still to come the next period, π( π) 2 η t the period after that, etc. Simpler Alternative Assumption: Two Period Order Flow Order flow shock η t is carried out over two consecutive periods t and t+ in proportions π : π for some π [0,]. That is, each period the order flow is: u t = γz t + ε t + πη t + ( π)η t with ε t N(0,Ψ) and η t N(0,Ω). The next period, market makers know that ( π)η t is still to come. 3 4 Possible Regimes I. Market makers can distinguish one shot and two period order flow shocks II. Market makers cannot distinguish the current shocks but learn about the past Past is perfectly revealed or not? III. Mixed: some market makers ( HFT ) can, and others cannot I. Omniscient Market Makers.. At time t they know: ε t, η t and η t Price is of the form: p t = v δz t + βη t Where 5 6 Time Line t pay out dividend d t+ t+ discover η t+ and ε t+ II. Market makers cannot distinguish different types of order flow.. At time t they cannot distinguish ε t from η t (but they do know η t for simplicity) They observe s t = ε t + πη t And we conjecture p t = v δz t + βs t Inventory impact δ solves: which is higher than in previous regime. And 7 8 3

4 Comparison of Regimes I and II Inventory has stronger impact on price in regime II One shot traders are better off if market makers are informed Two shot traders worse off in equilibrium? Questions to be addressed Nail down precise impact of anticipatory trading on liquidity, volatility, welfare of participants Model with both informed and uninformed market makers (regime III) More price volatility informed market makers (HFT) have shorter holding periods Examine optimal order placement Continuous time model more tractable Requires long run order placement model rather than twoperiod version 9 20 Conclusion Explicit model of multi period order flow placement Permits analysis of anticipatory trading by market makers pros and cons of slow order placement strategies Market makers information about future order flow affects price formation Many issues still to be examined 2 4

5 Abstract Market maker pricing and information about prospective order flow The paper considers the impact of order placement strategies on market pricing. In the model, some orders are submitted for immediate execution in their entirety, while others are part of an ongoing strategy to trickle in orders slowly so as to reduce price pressure by taking advantage of the resiliency of the market. If market makers can perfectly distinguish the two types of order flow, their prices are different than if they can only guess at the type of orders they face. A common complaint is that exchanges sell such detailed order flow information to high frequency traders that these are able to identify orders that are the tip of an iceberg, generating adverse price pressure. Gainers and losers from exchange policies to sell such information are considered, in a model of risk-averse market making in which order flow exerts price pressure despite the absence of asymmetries of information about fundamental risks. Introduction: Main Features of Model: market making with inventory risk averse market makers temporary deviations of price from fundamental value due to limited market participation distinguish fundamental risk and order flow risk no asymmetries of information about fundamentals Simplifying features not a GE asset pricing model: fundamental value is exogenous (wlog, constant), no " price discovery" market makers: myopic preferences competitive (price takers) exogenous behavior of general universe of investors CARA-normal Use model to investigate determinants of volatility, liquidity and serial correlation of returns speed of inventory adjustments

6 front running or anticipatory trading impact of potential asymmetries of information about future order flow shocks (high frequency traders) optimal order placement strategies 2 Baseline Model Repeated OLG or "relay race" model (no analysis of intertemporal consumption choices) Fundamentals (simple Lucas tree). At time, the fundamental value is the discounted NPV of expected future dividends { + : = }, discounted at rate = +. " # X = E + = there is no asymmetric information about fundamental valuation, and so is common knowledge at time of trade for all market participants. At time t all future dividends have an expected value of and a variance of Σ; for simplicity dividends are assumed to be paid out each period before trading, though as periods are short (this is a microstructure model!) the underlying idea is that is actually a series of time- before-interest earnings which, if retained, earn interest each period until they are paid out. This means that: = for all Order flow from the general public. There is order flow noise that is i.i.d. normally distributed (0 Ψ), and independent for all 6=, as well as a public order flow response proportional to the distance of the current market price fundamental value, so that total demand for the stock is: = + where (0 ] The underlying idea is that if all the world were present on the stock market, then and the stock price would equal fundamental value at all times. But only a very limited set of risk averse members of the general public are present at any time ; and they trade in proportion to the amount by which the price deviates from fundamental value and then leave the market, to be replaced by a fresh set of public traders in the next period. Market makers. Market makers are rational and fully aware of how the market works, but myopic in the sense that they trade to maximise meanvariance utility, with collective CARA coefficient, defined over their wealth The analysis would be identical except for notation, as one would have to distinguish cum-dividend and ex-dividendend valuations and prices. 2

7 at the end of the upcoming period. They are assumed to be competitive, that is, price takers. To compute their end-of-period wealth, they mark their stock to market using the end-of-period equilibrium price. That is, a market maker contemplating supplying an amount to the market at time who starts out with cash and stock inventory computes his end-of-period wealth as: e + = e + + e + where e + = (+)( + )+ e + = i.e. e + = (+)( + )+ ³ e+ + e + ( ) where quantities that are uncertain at time of trading at time are decorated with a superscript. Thus the market makers expected utility is E + =(+) + + E [e + ] ( + ) ( ) 2 ( ) 2 Σ + 2 () where two conjuctures, to be verified in equilibrium, have been made regarding the uncertainty about the end-of-period price: (i) it is independent of the earnings surprise and (ii) it is stationary, that is, the per-period price volatility 2 is constant over time. This price volatility is endogenous, an unknown to be determined by solving the model. To solve this model we first write down the solution to the market makers supply choice problem, then impose equilibrium (demand equal to supply) and then look for a stationary solution. The market makers FOC, choosing supply to maximize their expected utility (expression ), is: + E [e + ] ( + ) = ( ) Σ + 2 = + ( + ) + E [e + ] Σ + 2 The market clears when: =,thatis + ( + ) + E [e + ] = Σ + 2 so that the market clearing price satisfies: 3

8 + ( Σ + 2 ) = (E Σ + 2 [e + ] ) ( ) (2) = Σ (E [e + ] ) (3) We will conjecture that the equilibrium market price can be written as: = (4) for some parameter value 0 to be determined endogenously. Then where e + = e + e + = e + = + e + e + = + ( e + ) so that e + = + ( e + ) Thus expectations about the next market price satisfy, using (4) to substitute out for : = + E [e + ] = + (5) µ 2 2 = Ψ (6) + Inserting equations (5) and (6) into the expession for the market clearing price (3), = Σ (E [e + ] ) = Σ + 2 Ψ + ( + )(+) Σ + 2 Ψ + (7) + The two undetermined endogenous parameters, and 2,cannowbederived. Equating the coefficient in equation (7) with that of the conjectured price equation (4), and substituting out for 2 using equation(6), 4

9 " Ã = µ # 2 Σ + Ψ! (8) which yields: 0 = Ψ 2 ( + )( + + ) +(+) 2 Σ q = ( + + ) ( + + ) ΨΣ (+) 2Ψ 2 = Note: = µ 2 Ψ = ( + + )2 2 2 ΨΣ ( + + ) Ψ ( + + )2 2 2 Ψ s Σ p + + ( + + ) Ψ 2 2 2Σ check this! Ψ q ( + + ) ΨΣ There are two solutions. Argue that the lower solution for makes sense, as limit of finite-horizon problem For a solution to exist, it must be that ΨΣ if not, there is no steady state. As the time interval between trading periods shrinks, if we take Σ Ψ proportional to interval size then in the limit q + ( + ) ΨΣ 2Ψ 5

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