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

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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.

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Limit-order book: Bid and ask prices

Limit sell orders Limit buy orders ask bid

◮ 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.

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Limit-order book:

MMM on June 30, 2010, at 0.1 second intervals Thanks to: Mark Schervish, Head Department of Statistics Carnegie Mellon University

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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 → ∞.

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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.

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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.

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Our model: Arrivals and cancellations of buy orders

c c 1 1 λ

◮ 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.

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Our model: Arrivals and cancellations of sell orders

1 1 λ c c 1 1 λ

◮ 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.

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Our model: Poisson transitions at rates indicated

1 λ 1 c 1 1 λ c c 1 1 λ λ 1 1 1 λ 1 λ 1 1 c c λ 1 1 1 1 λ c 1 λ 1 1 λ 1 c λ 1 1 c c λ 1 1 1 1 λ λ 1 1 c 1 1 λ 1 c c 1 λ

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Our model: Simulation

Parameters: n = 106, θ = 12, λ = (1 + √ 5)/2. Thanks to Christopher Almost Ph.D. student Department of Mathematical Sciences Carnegie Mellon University

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Our model: Limit-order book arrivals and departures

1 λ 1 c 1 1 λ c c 1 1 λ λ 1 1 1 λ 1 λ 1 1 c c λ 1 1 1 1 λ c 1 λ 1 1 λ 1 c λ 1 1 c c λ 1 1 1 1 λ λ 1 1 c 1 1 λ 1 c c 1 λ U V W X Y Z

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Split Brownian motion: Transitions of (W , X)

X W λ 1 1 λ 1 1 1 1 1 λ 1 1 λ λ 1 1 λ 1 1 λ 1 1 λ 1 1 1 1 1

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Split Brownian motion: Theorem

The diffusion scaling of a generic process Q is defined to be

• Qn(t) :=

1 √nQ(nt).

Theorem

Assume λ = (1 + √ 5)/2. Conditional on V and Y remaining nonzero, ( Wn, Xn) converges in distribution to the split Brownian motion (W ∗, X ∗)

(d)

= 2 √ λ(max{B∗, 0}, min{B∗, 0}) where B∗ is a standard one-dimensional Brownian motion.

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Split Brownian motion: Transitions of (W , X)

X W λ 1 1 λ 1 1 1 1 1 λ 1 1 λ λ 1 1 λ 1 1 λ 1 1 λ 1 1 1 1 1 (W , X) is “globally balanced” ⇔ λ = 1+

√ 5 2

G = X G = X G = −W G = −W

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Split Brownian motion: Analysis of G

−1 −2 1 2 λ λ 1 1 1 1 λ λ Essentially the dynamics of G

◮ Show

Gn ⇒ 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:

positive: 1 λ + 1 zero: λ − 1 λ + 1 negative: 1 λ + 1

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Split Brownian motion: Transitions of (W , X)

X W λ 1 1 λ 1 1 1 1 1 λ 1 1 λ λ 1 1 λ 1 1 λ 1 1 λ 1 1 1 1 1 (W , X) is “globally balanced” ⇔ λ = 1+

√ 5 2

G = X G = X G = −W G = −W H = W + λX H = W + X H = W + X H = λW + X

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Split Brownian motion: Analysis of H

Lemma

H is a martingale. Proof (in the first quadrant): H = W + λX.

1 λ 1

Drift in H is (0 − λ) · λ + (0 + λ) · 1 + (1 + 0) · 1 = −λ2 + λ + 1, which is zero when λ = 1+

√ 5 2

.

• Proof is similar in other quadrants.

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Split Brownian motion: Rate of growth of H, H

X W λ 1 1 λ 1 1 1 1 1 λ 1 1 λ λ 1 1 λ 1 1 λ 1 1 λ 1 1 1 1 1 λ = 1+

√ 5 2

⇔ λ2 − λ − 1 = 0 G = X > 0 H = W + λX dH, Ht = (3λ + 3)dt G = 0, H = W dH, Ht = (2λ + 3)dt G = X < 0 H = W + X dH, Ht = (2λ + 2)dt

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Split Brownian motion: Diffusion limit of H

Using proportions of time G spends positive, zero, and negative, we average to get

• Hn,

Hn ⇒ (3λ + 3) 1 λ + 1 id +(2λ + 3)λ − 1 λ + 1 id +(2λ + 2) 1 λ + 1 id = 4λ id . Martingale central limit theorem (see Ethier & Kurtz) implies

• Hn ⇒ 2

√ λB∗, where B∗ is a standard Brownian motion.

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Split Brownian motion: Diffusion limit of (W , X)

X W λ 1 1 λ 1 1 1 1 1 λ 1 1 λ λ 1 1 λ 1 1 λ 1 1 λ 1 1 1 1 1

• Gn =

Xn ⇒ 0

• Hn =

Wn + λ Xn ⇒ 2 √ λB∗

• Gn =

Xn ⇒ 0

• Hn =

Wn + Xn ⇒ 2 √ λB∗

• Gn = −

Wn ⇒ 0

• Hn =

Wn + Xn ⇒ 2 √ λB∗

• Gn = −

Wn ⇒ 0

• Hn = λ

Wn + Xn ⇒ 2 √ λB∗

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Limit order book arrivals

1 λ 1 c 1 1 λ c c 1 1 λ λ 1 1 1 λ 1 λ 1 1 c c λ 1 1 1 1 λ c 1 λ 1 1 λ 1 c λ 1 1 c c λ 1 1 1 1 λ λ 1 1 c 1 1 λ 1 c c 1 λ U V W X Y Z

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Snapped Brownian motion: Conjecture

Conjecture

Assume λ = (1 + √ 5)/2. Conditional on Y remaining nonzero, Vn converges to a snapped Brownian motion, controlled by the split Brownian motion (W ∗, X ∗). Informally, the snapped Brownian motion V ∗, the limit of Vn, 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.

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Snapped Brownian motion: Transitions of V

X W λ 1 1 λ 1 1 1 1 1 λ 1 1 λ λ 1 1 λ 1 1 λ 1 1 λ 1 1 1 1 1 C C 1 1 − λ 1 − λ 1 − λ What is happening to V ?

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Snapped Brownian motion: Transitions of V

X W λ 1 1 λ 1 1 1 1 1 λ 1 1 λ λ 1 1 λ 1 1 λ 1 1 λ 1 1 1 1 1 1 λ 1 1 λ 1 C C 1 1 − λ 1 − λ 1 − λ What is happening to V ?

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Snapped Brownian motion: Dynamics of V ∗

◮ Recall dynamics of G:

−1 −2 1 2 λ λ 1 1 1 1 λ λ

◮ Proportion of time spent:

positive: 1 λ + 1 zero: λ − 1 λ + 1 negative: 1 λ + 1

◮ When split BM is negative, net drift of V ∗ is

(1 − λ) 1 λ + 1 + (1 − λ)λ − 1 λ + 1 + (1) 1 λ + 1 = 1 λ + 1(1 + λ − λ2) = 0 Hence V ∗ is diffusing.

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Snapped Brownian motion: Transitions of V

X W λ 1 1 λ 1 1 1 1 1 λ 1 1 λ λ 1 1 λ 1 1 λ 1 1 λ 1 1 1 1 1 λ 1 1 1 1 λ C C 1 1 − λ 1 − λ 1 − λ What is happening to V ?

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Snapped Brownian motion: Dynamics of V ∗ (continued)

◮ When split BM is positive, Vn is cancelling at rate θ/√n or

growing at rate 1.

◮ Let ¯

vn be unscaled average height of Vn when split BM is

• positive. Cancellation is stabilizing, so net drift is zero.

0 set =

• −¯

vn θ √n

• 1

λ + 1 + (0)λ − 1 λ + 1 + (1) 1 λ + 1 ¯ vn = √n θ Hence Vn hangs around the level 1/θ.

◮ In fact, V ∗ is snapped to the level 1/θ. ◮ V ∗ is a process that is diffusing or snapped according to

whether another diffusion (split BM) is negative or positive.

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Diffusion limit of limit-order book

V ∗ W ∗ X ∗ Y ∗ Y ∗ W ∗ X ∗ V ∗ V ∗ W ∗ X ∗ Y ∗

◮ Let B∗ be a standard Brownian motion ◮ W ∗ = 2

√ λ max{B∗, 0}

◮ X ∗ = 2

√ λ min{B∗, 0} This is correct only as long as the “bracketing processes” V ∗ and Y ∗ do not vanish.

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Diffusion limit of limit-order book

U∗ V ∗ W ∗ X ∗ Y ∗ Consider the case W ∗ = 0 and X ∗ < 0:

◮ V ∗ evolves as 2

√ λ times a Brownian motion;

◮ X ∗ evolves as 2

√ λ times an independent Brownian motion;

◮ Y ∗ is pinned at −1/θ; ◮ U∗ is pinned at 1/θ; ◮ If X ∗ reaches zero before V ∗ reaches zero, then V ∗ and Y ∗

remain the ”bracketing processes.”

◮ If V ∗ reaches zero before X ∗ reaches zero, then U∗ and X ∗

become the “bracketing processes”. The proofs of the claims on this page are still under construction.

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