Diffusion scaling of a limit order book model Steven E. Shreve - PowerPoint PPT Presentation

Diffusion scaling of a limit order book model Steven E. Shreve Department of Mathematical Sciences Carnegie Mellon University shreve@andrew.cmu.edu Ongoing work with Christopher Almost John Lehoczky

Outline 1. What is a limit-order book? 2. “Zero-intelligence” Poisson models of the limit order book. 3. Partial history. 4. Our model. 5. Diffusion scaling of our model. ◮ Split Brownian motion ◮ Snapped Brownian motion 6. The diffusion limit of our model 1 / 30

Limit-order book: Order types Orders to buy and sell an asset arrive at an exchange. 1. Market buy/sell order — specifies number of shares to be bought/sold at the best available price, right away. 2. Limit buy/sell order — specifies a price and a number of shares to be bought/sold at that price, when available. 3. Order cancellation — agents who have submitted a limit order may cancel the order before it is executed. ◮ Market orders are executed immediately. ◮ Limit orders are queued for later execution. ◮ The Limit Order Book is the collection of queued limit orders awaiting execution or cancellation. 2 / 30

Limit-order book: Bid and ask prices Limit buy orders bid ask Limit sell orders ◮ The (best) bid price is the highest limit buy order price in the book. It is the best available price for a market sell. ◮ The (best) ask price is the lowest limit sell order price in the book. It is the best available price for a market buy. 3 / 30

Limit-order book: MMM on June 30, 2010, at 0.1 second intervals Thanks to: Mark Schervish, Head Department of Statistics Carnegie Mellon University 4 / 30

Zero-intelligence model: Build a “zero-intelligence” Poisson model of the limit-order book and determine its diffusion limit. ◮ “Zero-intelligence” — No strategic play by the agents submitting orders. ◮ Poisson — Arrivals of buy and sell limit and market orders are Poisson processes. Cancellations are also governed by Poisson processes. ◮ Maybe a little intelligence — Arrival and cancellation rates depend on the state of the limit-order book. ◮ Diffusion limit — Accelerate time by a factor of n , divide volume by √ n , and pass to the limit as n → ∞ . 5 / 30

Partial history: ◮ Garman, M. (1976) “Market microstructure,” J. Financial Economics 3 , 257–275 Poisson arrivals of buy and sell orders, rather a model based on utility maximization. Arrival rates depend on price, which is set by a market maker to maximize his profit. ◮ Smith, E., Farmer, J.D., Gillemot, L. & Krishnamurthy, S. (2003) “Statistical theory of the continuous double auction,” Quant. Finance 3 , 481–514. Poisson arrivals of buy and sell orders, with arrival rates independent on the state of the order book. Simulation studies. 6 / 30

Partial history: ◮ Bayraktar, E., Horst, U. & Sircar, R. (2008) “Queueing theoretic approaches to financial price fluctuations,” in Handbooks in OR & MS , Vol. 15, J. R. Birge and V. Linetsky, eds., pp. 637–677. A survey of “zero-intelligence modeling,” including work by the same authors in which long-range dependence of prices is obtained as a diffusion-scaled limit. ◮ Cont, R., Stoikov, S. & Talreja, R. (2010) “A stochastic model for order book dynamics,” Operations Research 58 , 549–563. Poisson arrivals of buy and sell orders keyed off the opposite best price. Compute statistics of the order-book behavior by Laplace transforms analysis. 7 / 30

Our model: Arrivals and cancellations of buy orders 1 1 λ c c ◮ All arriving and departing orders are of size 1. ◮ Poisson arrivals of market buys at rate λ > 1. These execute at the (best) ask price. ◮ Poisson arrivals of limit buys at one and two ticks below the (best) ask price, both at rate 1. ◮ Cancellations of limit buys two ticks below the (best) bid price, at rate θ/ √ n per order. 8 / 30

Our model: Arrivals and cancellations of sell orders 1 1 λ c c λ 1 1 ◮ All orders are of size 1. ◮ Poisson arrivals of market sells at rate λ > 1. These execute at the (best) bid price. ◮ Poisson arrivals of limit sells at one and two ticks above the (best) bid price, both at rate 1. ◮ Cancellations of limit sells two ticks above the (best) ask price, at rate θ/ √ n per order. 9 / 30

Our model: Poisson transitions at rates indicated c c 1 1 λ 1 1 λ c c λ 1 1 λ 1 1 c 1 1 λ 1 1 1 1 λ λ c λ 1 1 λ 1 1 λ 1 1 c c c 1 1 λ 1 1 λ 1 1 λ c c c λ 1 1 λ 1 1 λ 1 1 10 / 30

Our model: Simulation √ Parameters: n = 10 6 , θ = 12, λ = (1 + 5) / 2. Thanks to Christopher Almost Ph.D. student Department of Mathematical Sciences Carnegie Mellon University 11 / 30

Our model: Limit-order book arrivals and departures c c 1 1 λ 1 1 λ U V W X Y Z c c λ 1 1 λ 1 1 c 1 1 λ 1 1 1 1 λ λ c λ 1 1 λ 1 1 λ 1 1 c c c 1 1 λ 1 1 λ 1 1 λ c c c λ 1 1 λ 1 1 λ 1 1 12 / 30

Split Brownian motion: Transitions of ( W , X ) X 1 1 1 1 λ λ 1 1 1 1 1 1 1 1 1 λ λ W λ λ 1 1 λ 1 1 1 λ 1 1 13 / 30

Split Brownian motion: Theorem The diffusion scaling of a generic process Q is defined to be 1 � Q n ( t ) := √ nQ ( nt ) . Theorem √ Assume λ = (1 + 5) / 2 . Conditional on V and Y remaining nonzero, ( � W n , � X n ) converges in distribution to the split Brownian motion √ (d) ( W ∗ , X ∗ ) = 2 λ (max { B ∗ , 0 } , min { B ∗ , 0 } ) where B ∗ is a standard one-dimensional Brownian motion. 14 / 30

Split Brownian motion: Transitions of ( W , X ) G = X X 1 1 ( W , X ) is “globally balanced” √ ⇔ λ = 1+ 5 1 1 λ λ 2 1 1 1 1 1 1 1 1 1 λ λ W G = X λ λ 1 1 λ 1 1 1 λ 1 1 G = − W G = − W 15 / 30

Split Brownian motion: Analysis of G λ λ 1 1 − 2 − 1 0 1 2 1 1 λ λ Essentially the dynamics of G ◮ Show � G n ⇒ 0. Similar to analysis of queueing system with traffic intensity < 1. ◮ ( W ∗ , X ∗ ) is confined to green path (state-space collapse). ◮ (For later) proportion of time G spends: 1 zero: λ − 1 1 positive: negative: λ + 1 λ + 1 λ + 1 16 / 30

Split Brownian motion: Transitions of ( W , X ) G = X X H = W + λ X 1 1 ( W , X ) is “globally balanced” √ ⇔ λ = 1+ 5 1 1 λ λ 2 1 1 1 1 1 1 1 1 1 λ λ W G = X H = W + X λ λ 1 1 λ 1 1 1 λ 1 1 G = − W G = − W H = λ W + X H = W + X 17 / 30

Split Brownian motion: Analysis of H 1 Lemma H is a martingale. λ 1 Proof (in the first quadrant): H = W + λ X . Drift in H is (0 − λ ) · λ + (0 + λ ) · 1 + (1 + 0) · 1 = − λ 2 + λ + 1 , √ which is zero when λ = 1+ 5 . � 2 Proof is similar in other quadrants. 18 / 30

Split Brownian motion: Rate of growth of � H , H � X 1 1 √ λ = 1+ 5 ⇔ 2 λ 2 − λ − 1 = 0 G = X > 0 1 1 λ λ H = W + λ X d � H , H � t = (3 λ + 3) dt 1 1 G = 0 , H = W d � H , H � t = (2 λ + 3) dt 1 1 1 1 1 1 1 λ λ W G = X < 0 H = W + X λ λ d � H , H � t = (2 λ + 2) dt 1 1 λ 1 1 1 λ 1 1 19 / 30

Split Brownian motion: Diffusion limit of H Using proportions of time G spends positive, zero, and negative, we average to get λ + 1 id +(2 λ + 3) λ − 1 1 1 � � H n , � H n � ⇒ (3 λ + 3) λ + 1 id +(2 λ + 2) λ + 1 id = 4 λ id . Martingale central limit theorem (see Ethier & Kurtz ) implies √ � H n ⇒ 2 λ B ∗ , where B ∗ is a standard Brownian motion. 20 / 30

Split Brownian motion: Diffusion limit of ( W , X ) G n = � � X X n ⇒ 0 H n = � � W n + λ � X n √ 1 1 ⇒ 2 λ B ∗ 1 1 λ λ 1 1 1 1 1 1 1 1 1 λ λ W G n = � � X n ⇒ 0 H n = � � W n + � λ λ X n √ ⇒ 2 λ B ∗ 1 1 λ 1 1 1 λ 1 1 G n = − � � G n = − � � W n ⇒ 0 W n ⇒ 0 H n = λ � � W n + � H n = � � W n + � X n X n √ √ ⇒ 2 λ B ∗ ⇒ 2 λ B ∗ 21 / 30

Limit order book arrivals c c 1 1 λ 1 1 λ U V W X Y Z c c λ 1 1 λ 1 1 c 1 1 λ 1 1 1 1 λ λ c λ 1 1 λ 1 1 λ 1 1 c c c 1 1 λ 1 1 λ 1 1 λ c c c λ 1 1 λ 1 1 λ 1 1 22 / 30

Snapped Brownian motion: Conjecture Conjecture √ 5) / 2 . Conditional on Y remaining nonzero, � Assume λ = (1 + V n converges to a snapped Brownian motion, controlled by the split Brownian motion ( W ∗ , X ∗ ) . Informally, the snapped Brownian motion V ∗ , the limit of � V n , behaves as follows. ◮ V ∗ is a Brownian motion when X ∗ is negative (and W ∗ = 0). ◮ V ∗ is equal to the constant 1 /θ when W ∗ is positive (and X ∗ = 0). ◮ V ∗ is “snapped” to the value 1 /θ when the split Brownian motion switches from negative to positive. 23 / 30

Snapped Brownian motion: Transitions of V X C C 1 1 What is happening to V ? 1 1 λ λ 1 1 1 − λ 0 1 1 1 1 1 1 1 λ λ W λ λ 1 1 λ 1 1 1 λ 1 1 1 − λ 1 − λ 1 24 / 30

Snapped Brownian motion: Transitions of V X C C 1 1 What is happening to V ? 1 1 λ λ 1 1 1 − λ 0 1 1 1 1 1 1 1 λ λ W λ λ 1 1 1 λ λ 1 1 1 1 1 λ λ 1 1 1 1 − λ 1 − λ 1 25 / 30