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Relevant Questions: Regulators How to ensure robust / fair markets: Are markets more or less efficient than in the past? Why do we need to trade on a millisecond timescale? Should we have such a fragmented market? Are market structures like dark pools beneficial? How can we ensure market stability? Circuit breakers? Minimum order life span? Transaction taxes? Periodic auctions? 10
Outline There are many interesting and important open questions — there is room for innovation in modeling / problem formulation as well as methodology. 11
Outline There are many interesting and important open questions — there is room for innovation in modeling / problem formulation as well as methodology. We will consider in detail a handful of specific problems: The Cost of Latency Order Routing and Fragmented Markets Dark Pools 11
The Cost of Latency Joint work with Mehmet Sa˘ glam. 12
Does Speed Pay? July 23, 2009 “Stock Traders Find Speed Pays, in Milliseconds” Powerful computers, some housed right next to the machines that drive marketplaces like the New York Stock Exchange, enable high-frequency traders to transmit millions of orders at lightning speed. … High-frequency traders often confound other investors by issuing and then canceling orders almost simultaneously … And their computers can essentially bully slower investors into giving up profits. … “It’s become a technological arms race, and what separates winners and losers is how fast they can move,” said Joseph M. Mecane of NYSE Euronext, which operates the New York Stock Exchange. 13
Latency and Its Significance Introduction to latency: Delay between a trading decision and its implementation 14
Latency and Its Significance Introduction to latency: Delay between a trading decision and its implementation 2 minutes (NYSE, pre-1980) ⇒ 20 seconds (NYSE, 1980) ⇒ 100 s of milliseconds (NYSE, 2007) ⇒ 1 millisecond (NYSE Arca, 2009) “low latency” ⇒ 10 – 100 microseconds (current state-of-the-art) “ultra-low latency” 14
Latency and Its Significance Introduction to latency: Delay between a trading decision and its implementation 2 minutes (NYSE, pre-1980) ⇒ 20 seconds (NYSE, 1980) ⇒ 100 s of milliseconds (NYSE, 2007) ⇒ 1 millisecond (NYSE Arca, 2009) “low latency” ⇒ 10 – 100 microseconds (current state-of-the-art) “ultra-low latency” Driven by technological innovation and competition between exchanges 14
Latency and Its Significance Introduction to latency: Delay between a trading decision and its implementation 2 minutes (NYSE, pre-1980) ⇒ 20 seconds (NYSE, 1980) ⇒ 100 s of milliseconds (NYSE, 2007) ⇒ 1 millisecond (NYSE Arca, 2009) “low latency” ⇒ 10 – 100 microseconds (current state-of-the-art) “ultra-low latency” Driven by technological innovation and competition between exchanges Why is low-latency trading important for market participants? Contemporaneous decision making Competitive advantage/disadvantage Time priority rules in microstructure 14
Latency and Its Significance Introduction to latency: Delay between a trading decision and its implementation 2 minutes (NYSE, pre-1980) ⇒ 20 seconds (NYSE, 1980) ⇒ 100 s of milliseconds (NYSE, 2007) ⇒ 1 millisecond (NYSE Arca, 2009) “low latency” ⇒ 10 – 100 microseconds (current state-of-the-art) “ultra-low latency” Driven by technological innovation and competition between exchanges Why is low-latency trading important for market participants? Contemporaneous decision making Competitive advantage/disadvantage Time priority rules in microstructure 14
Our Contributions Quantify the cost of latency in a benchmark stylized execution problem Provide a closed-form expression in well-known market parameters Latency is important to all investors with low cost structures 15
Literature Review Empirical work on the effects of improvements in trading technology [Easley, Hendershott, Ramadorai, 2008] [Hendershott, Jones, Menkveld, 2008] [Hasbrouck & Saar, 2008, 2010; Stoikov, 2009; Kearns, et al., 2010; others] Latency of the price ticker [Cespa & Foucault, 2007] Optionality embedded in limit orders [Angel, 1994; Harris, 1998; others] [Copeland & Galai, 1983; Chacko, Jurek, Stafford, 2008; others] Discrete-time hedging of contingent claims [Boyle & Emanuel, 1980; Bertsimas, Kogan, Lo, 2000; others] 16
Benchmark Model: No Latency An investor must sell 1 share of stock over the time horizon [0 , T ] 17
Benchmark Model: No Latency An investor must sell 1 share of stock over the time horizon [0 , T ] Option A: place a market order to sell S t = bid price at time t , S t ∼ S 0 + σ B t 17
Benchmark Model: No Latency An investor must sell 1 share of stock over the time horizon [0 , T ] Option A: place a market order to sell S t = bid price at time t , S t ∼ S 0 + σ B t Option B: place a limit order to sell L t = limit order price at time t (decision variable) 17
Benchmark Model: No Latency L t S t t 0 T 18
Benchmark Model: No Latency A limit order executes if either: Market buy order arrives and L t ≤ S t + δ . L t S t t 0 T 18
Benchmark Model: No Latency A limit order executes if either: Market buy order arrives and L t ≤ S t + δ . S τ 2 + δ limit order executes L t S τ 1 + δ S t t τ 1 τ 2 0 T impatient buyers arrive, Poisson ( µ ) 18
Benchmark Model: No Latency A limit order executes if either: Market buy order arrives and S t ≥ L t L t ≤ S t + δ . S τ 2 + δ limit order executes L t S τ 1 + δ S t t τ 1 τ 2 0 T impatient buyers arrive, Poisson ( µ ) 18
Benchmark Model: No Latency A limit order executes if either: Market buy order arrives and S t ≥ L t L t ≤ S t + δ . S τ 2 + δ limit order executes L t S τ 1 + δ S t t τ 1 τ 2 τ 3 0 T impatient buyers arrive, Poisson ( µ ) 18
Benchmark Model: No Latency Objective: maximize E [ sale price ] − S 0 19
Benchmark Model: No Latency Objective: maximize E [ sale price ] − S 0 Optimal strategy: ‘Pegging’ Use limit orders, L t = S t + δ If not executed by time T , sell with a market order at S T τ = arrival time of next L t impatient buyer L t = S t + δ � � Value = E S τ ∧ T + δ I { τ ≤ T } − S 0 � 1 − e − µ T � S t = δ t 0 T 19
Latency Model ∆ t = Latency ℓ i +1 ℓ i +2 ℓ 0 ℓ i · · · · · · T 0 = 0 T i = i ∆ t T i +1 T i +2 T = n ∆ t 20
Latency Model ∆ t = Latency ℓ i − 1 ℓ i ℓ i +1 ℓ i +1 ℓ i +2 ℓ 0 ℓ i · · · · · · T 0 = 0 T i = i ∆ t T i +1 T i +2 T = n ∆ t 20
Latency Model ∆ t = Latency ℓ i − 1 ℓ i ℓ i +1 ℓ i +1 ℓ i +2 ℓ 0 ℓ i · · · · · · T 0 = 0 T i = i ∆ t T i +1 T i +2 T = n ∆ t sup E [ sale price ] − S 0 (when latency is ∆ t ) h 0 (∆ t ) � ℓ 0 ,ℓ 1 ,... 20
Latency Model ∆ t = Latency ℓ i − 1 ℓ i ℓ i +1 ℓ i +1 ℓ i +2 ℓ 0 ℓ i · · · · · · T 0 = 0 T i = i ∆ t T i +1 T i +2 T = n ∆ t sup E [ sale price ] − S 0 (when latency is ∆ t ) h 0 (∆ t ) � ℓ 0 ,ℓ 1 ,... � � h i (∆ t ) � sup E sale price | F T i , no trade in [0 , T i +1 ) − S T i ℓ i ,ℓ i +1 ,... 20
Main Result Lemma. ( Consistency ) � 1 − e − µ T � lim ∆ t → 0 h 0 (∆ t ) = δ � �� � ¯ h 0 21
Main Result Lemma. ( Consistency ) � 1 − e − µ T � lim ∆ t → 0 h 0 (∆ t ) = δ � �� � ¯ h 0 Theorem. ( Asymptotic Value ) As ∆ t → 0 , � � √ δ 2 1 − σ � h 0 (∆ t ) = ¯ ∆ t log + o h 0 · ∆ t δ 2 πσ 2 ∆ t 21
Latency Cost Definition. ¯ h 0 − h 0 (∆ t ) Latency Cost � ¯ h 0 Interpretation: The cost of latency as a percentage of ‘cost of immediacy’ in the absence of any latency 22
Latency Cost Definition. ¯ h 0 − h 0 (∆ t ) Latency Cost � ¯ h 0 Interpretation: The cost of latency as a percentage of ‘cost of immediacy’ in the absence of any latency Corollary. As ∆ t → 0 , √ � � √ δ 2 Latency Cost = σ ∆ t � log 2 πσ 2 ∆ t + o ∆ t δ 22
Latency Cost Definition. ¯ h 0 − h 0 (∆ t ) Latency Cost � ¯ h 0 Interpretation: The cost of latency as a percentage of ‘cost of immediacy’ in the absence of any latency Corollary. As ∆ t → 0 , √ � � √ δ 2 Latency Cost = σ ∆ t � log 2 πσ 2 ∆ t + o ∆ t δ Does not depend on µ or T √ Increasing function of σ ∆ t /δ Increasing marginal benefits to reductions in latency 22
Empirical Application: GS Latency Cost Parameters estimated for Goldman Sachs Group, Inc., January 2010 30 % exact 25 % approximation 20 % Latency cost 15 % 10 % 5 % 0 % 0 0 50 100 150 200 250 300 350 400 450 500 Latency ∆ t (ms) 23
Interpretation Normalize cost of immediacy to $0 . 01 per share Value of decreasing latency from 500 ms to 1 ms: ≈ $0 . 0020 24
Interpretation Normalize cost of immediacy to $0 . 01 per share Value of decreasing latency from 500 ms to 1 ms: ≈ $0 . 0020 Typical high frequency trading profits: $0 . 0010 – $0 . 0020 sources: Tradeworx, Inc.; Knight Trading 2009 10K filing; AQR Capital; Traders Magazine 11/2005, Q&A With Dave Cummings ; etc. 24
Interpretation Normalize cost of immediacy to $0 . 01 per share Value of decreasing latency from 500 ms to 1 ms: ≈ $0 . 0020 Typical high frequency trading profits: $0 . 0010 – $0 . 0020 sources: Tradeworx, Inc.; Knight Trading 2009 10K filing; AQR Capital; Traders Magazine 11/2005, Q&A With Dave Cummings ; etc. Average algorithmic trading execution fee: $0 . 0033 (“non-idea driven”) source: TABB Group, 2010 24
Interpretation Normalize cost of immediacy to $0 . 01 per share Value of decreasing latency from 500 ms to 1 ms: ≈ $0 . 0020 Typical high frequency trading profits: $0 . 0010 – $0 . 0020 sources: Tradeworx, Inc.; Knight Trading 2009 10K filing; AQR Capital; Traders Magazine 11/2005, Q&A With Dave Cummings ; etc. Average algorithmic trading execution fee: $0 . 0033 (“non-idea driven”) source: TABB Group, 2010 Other trading costs for a large investor: $0 . 0005 – $0 . 0015 (e.g., brokerage & SEC fees, etc.) 24
Conclusion Contributions: Quantify the cost of latency in a benchmark stylized execution problem Provide a closed-form expression in well-known market parameters Implications: Latency can be important to any investor using limit orders How important depends on the rest of the investor’s trading costs (commissions, etc.) Extensions: Empirical analysis of the historical evolution of latency cost and implied latency for U.S. equities More complex price dynamics (jump diffusions) and fill models 25
Order Routing and Fragmented Markets Joint work with Costis Maglaras and Hua Zheng. 26
Motivation / Contributions How to analyze fragmented LOBs? each a multi-class queueing system with complex dynamics in addition, agents exert control via “smart order routing” 27
Motivation / Contributions How to analyze fragmented LOBs? each a multi-class queueing system with complex dynamics in addition, agents exert control via “smart order routing” We propose a tractable model to analyze decentralized LOBs, incorporates: limit order routing capability market order routing capability time-money tradeoff heterogeneity 27
Motivation / Contributions How to analyze fragmented LOBs? each a multi-class queueing system with complex dynamics in addition, agents exert control via “smart order routing” We propose a tractable model to analyze decentralized LOBs, incorporates: limit order routing capability market order routing capability time-money tradeoff heterogeneity In this framework, we: characterize the equilibrium through a fluid model 27
Motivation / Contributions How to analyze fragmented LOBs? each a multi-class queueing system with complex dynamics in addition, agents exert control via “smart order routing” We propose a tractable model to analyze decentralized LOBs, incorporates: limit order routing capability market order routing capability time-money tradeoff heterogeneity In this framework, we: characterize the equilibrium through a fluid model establish that routing decisions simplify dynamics: state space collapse 27
Motivation / Contributions How to analyze fragmented LOBs? each a multi-class queueing system with complex dynamics in addition, agents exert control via “smart order routing” We propose a tractable model to analyze decentralized LOBs, incorporates: limit order routing capability market order routing capability time-money tradeoff heterogeneity In this framework, we: characterize the equilibrium through a fluid model establish that routing decisions simplify dynamics: state space collapse provide empirical findings supporting state space collapse 27
Related Literature Market microstructure: Kyle; Glosten-Milgrom; Glosten; … Empirical analysis of limit order books: Bouchaud et al.; Hollifield et al.; Smith et al.; … Dynamic models and optimal execution in LOB: Obizhaeva & Wang; Cont, Stoikov, Talreja; Rosu; Alfonsi et al.; Foucault et al.; Parlour; Stoikov, Avellaneda, Reed; Cont & de Larrard; Predoiu et al.; Guo & de Larrard … Transaction cost modeling, adverse selection, …: Madhavan; Dufour & Engle; Holthausen et al.; Huberman & Stanzl; Almgren et al.; Gatheral; Sofianos; … Make/take fees, liquidity cycles, etc.: Foucault et al.; Malinova & Park Stochastic models of multi-class queueing networks 28
The Limit Order Book (LOB) ASK price BID 29
The Limit Order Book (LOB) ASK buy limit order arrivals cancellations market sell orders price market buy orders cancellations sell limit order arrivals BID 29
Multiple Limit Order Books exchange 1 exchange 2 . . . exchange N national best bid/ask (NBBO) 30
Multiple Limit Order Books exchange 1 exchange 2 We consider the evolution of: one side of the market . . . the ‘top-of-the-book’, i.e., national best bid queues across all exchanges exchange N national best bid/ask (NBBO) 30
Time Scales Three relevant time scales: Events: order / trade / cancellation interarrival times ( ∼ ms – sec) Delays: waiting times at different exchanges ( ∼ sec – min) Rates: time-of-day variation of flow characteristics ( ∼ min – hrs) 31
Time Scales Three relevant time scales: Events: order / trade / cancellation interarrival times ( ∼ ms – sec) Delays: waiting times at different exchanges ( ∼ sec – min) Rates: time-of-day variation of flow characteristics ( ∼ min – hrs) Order placement decisions depend on queueing delays in LOBs (our focus) assume constant arrival rates of limit orders and trades order sizes are small relative to overall flow over relevant time scale overall limit order and trade volumes are high 31
One-sided Multi LOB Fluid Model Fluid model: Continuous & deterministic arrivals of infinitesimal traders 32
One-sided Multi LOB Fluid Model Fluid model: Continuous & deterministic arrivals of infinitesimal traders exchange 1 exchange 2 exchange N 32
One-sided Multi LOB Fluid Model Fluid model: Continuous & deterministic arrivals of infinitesimal traders exchange 1 market order exchange 2 ? Λ exchange N optimized limit order flow 32
One-sided Multi LOB Fluid Model Fluid model: Continuous & deterministic arrivals of infinitesimal traders exchange 1 market order λ 1 exchange 2 λ 2 ? Λ exchange N λ n dedicated optimized limit order flow limit order flow 32
One-sided Multi LOB Fluid Model Fluid model: Continuous & deterministic arrivals of infinitesimal traders exchange 1 market order µ 1 λ 1 exchange 2 µ 2 λ 2 ? Λ exchange N µ N λ n dedicated optimized market order flow limit order flow limit order flow (attraction model) 32
The Limit Order Placement Decision Factors affecting limit order placement: 33
The Limit Order Placement Decision Factors affecting limit order placement: Expected delay ( ≈ 1 to 1000 seconds) 33
The Limit Order Placement Decision Factors affecting limit order placement: Expected delay ( ≈ 1 to 1000 seconds) Rebates ( ≈ − $0 . 0002 to $0 . 0030 per share) 33
The Limit Order Placement Decision Factors affecting limit order placement: Expected delay ( ≈ 1 to 1000 seconds) Rebates ( ≈ − $0 . 0002 to $0 . 0030 per share) r i = rebate ED i = expected delay 33
The Limit Order Placement Decision Factors affecting limit order placement: Expected delay ( ≈ 1 to 1000 seconds) Rebates ( ≈ − $0 . 0002 to $0 . 0030 per share) r i = rebate ED i = expected delay Traders choose to route their order to exchange i given by argmax γ r i − ED i i γ ∼ F i.i.d. across traders, captures delay tolerance / rebate tradeoff ⇒ γ ∼ 10 1 to 10 4 seconds per $0 . 01 allows choice amongst Pareto efficient ( r i , ED i ) pairs Implicit option for a market order: r 0 ≪ 0 , ED 0 = 0 33
The Market Order Routing Decision Market orders execute immediately, no queueing Market orders incur fees ( ≈ r i ) Natural criterion is to route an infinitesimal order according to argmin { r i : Q i > 0 , i = 1 , . . . , N } i 34
The Market Order Routing Decision Market orders execute immediately, no queueing Market orders incur fees ( ≈ r i ) Natural criterion is to route an infinitesimal order according to argmin { r i : Q i > 0 , i = 1 , . . . , N } i Routing decision differs from “fee minimization” due to Order sizes are not infinitesimal; may have to be split across exchanges Latency to exchange introduces notion of P ( fill ) when Q i are small Not all flow is “optimized”, or has other economic considerations Traders avoid “clearing” queues to avoid increased price slippage 34
The Market Order Routing Decision Attraction Model: Bounded rationality and model intricacies motivate fitting a probabilistic model of the form f i ( Q i ) µ i ( Q ) � µ � f j ( Q j ) j f i ( · ) captures “attraction" of exchange i : ↑ in Q i and ↓ in r i Remainder of this talk uses: f i ( Q i ) � β i Q i (we imagine β i ∼ 1 / r i ) 35
Transient Dynamics & Flow Equilibrium Dynamics: Coupled ODEs describe ˙ Q ( t ) dynamics, depend on ( λ, Λ , µ ) 36
Transient Dynamics & Flow Equilibrium Dynamics: Coupled ODEs describe ˙ Q ( t ) dynamics, depend on ( λ, Λ , µ ) Equilibrium: Fixed point Q ∗ 36
Transient Dynamics & Flow Equilibrium Dynamics: Coupled ODEs describe ˙ Q ( t ) dynamics, depend on ( λ, Λ , µ ) Equilibrium: Fixed point Q ∗ Downstream analysis: (not in talk) ( λ, Λ , µ ) vary stochastically at slower time scale ( ∼ min–hrs) than queue delays ( ∼ sec–min) Q evolves stochastically over such “steady state configurations” Point-wise stochastic fluid model 36
Transient Dynamics & Flow Equilibrium Dynamics: Coupled ODEs describe ˙ Q ( t ) dynamics, depend on ( λ, Λ , µ ) Equilibrium: Fixed point Q ∗ Downstream analysis: (not in talk) ( λ, Λ , µ ) vary stochastically at slower time scale ( ∼ min–hrs) than queue delays ( ∼ sec–min) Q evolves stochastically over such “steady state configurations” Point-wise stochastic fluid model We will analyze the structure of equilibria Q ∗ as a function of ( λ, Λ , µ ) . 36
Fluid Model Equilibrium π i ( γ ) = fraction of type γ investors who send orders to exchange i Definition. An equilibrium ( π ∗ , Q ∗ ) must satisfy 37
Fluid Model Equilibrium π i ( γ ) = fraction of type γ investors who send orders to exchange i Definition. An equilibrium ( π ∗ , Q ∗ ) must satisfy (i) Individual rationality: for all γ , π ∗ ( γ ) optimizes N Q ∗ � � � maximize i π 0 ( γ ) γ r 0 + π i ( γ ) γ r i − µ i ( Q ∗ ) π ( γ ) i =1 N ED i � subject to π ( γ ) ≥ 0 , π i ( γ ) = 1 . i =0 37
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