Diffusion scaling of a limit-order book - qfl-berlin.de Shreve.pdf · Di usion scaling of a...

Diffusion scaling of a limit-order book

Steven E. ShreveDepartment of Mathematical Sciences

Carnegie Mellon Universityshreve@andrew.cmu.edu

Joint work withChristopher Almost

John Lehoczky

October 4, 2012

Outline

1. What is a limit-order book?

2. “Zero-intelligence” Poisson models of limit-order books.

3. Partial history.

4. Our model.

5. Diffusion scaling of our model.

1 / 24

Outline

1. What is a limit-order book?

2. “Zero-intelligence” Poisson models of limit-order books.

3. Partial history.

4. Our model.

5. Diffusion scaling of our model.

1 / 24

Outline

1. What is a limit-order book?

2. “Zero-intelligence” Poisson models of limit-order books.

3. Partial history.

4. Our model.

5. Diffusion scaling of our model.

1 / 24

Outline

1. What is a limit-order book?

2. “Zero-intelligence” Poisson models of limit-order books.

3. Partial history.

4. Our model.

5. Diffusion scaling of our model.

1 / 24

Outline

1. What is a limit-order book?

2. “Zero-intelligence” Poisson models of limit-order books.

3. Partial history.

4. Our model.

5. Diffusion scaling of our model.

1 / 24

What is a limit-order book?Orders to buy and sell stock arrive at an exchange. Four types:

1. Market buy order. Specifies number of shares to be bought atthe best available price.

2. Market sell order. Specifies number of shares to be sold at thebest available price.

3. Limit buy order. Specifies a price and a number of shares tobe bought at that price.

4. Limit sell order. Specifies a price and a number of shares tobe sold at that price.

I Market orders are executed immediately.

I Limit orders are queued for later execution.

I Agent submitting limit order can later cancel the order.

I Limit-order book is the collection of queued limit orderswaiting for execution.

2 / 24

What is a limit-order book?Orders to buy and sell stock arrive at an exchange. Four types:

1. Market buy order. Specifies number of shares to be bought atthe best available price.

2. Market sell order. Specifies number of shares to be sold at thebest available price.

3. Limit buy order. Specifies a price and a number of shares tobe bought at that price.

4. Limit sell order. Specifies a price and a number of shares tobe sold at that price.

I Market orders are executed immediately.

I Limit orders are queued for later execution.

I Agent submitting limit order can later cancel the order.

I Limit-order book is the collection of queued limit orderswaiting for execution.

2 / 24

What is a limit-order book?Orders to buy and sell stock arrive at an exchange. Four types:

1. Market buy order. Specifies number of shares to be bought atthe best available price.

2. Market sell order. Specifies number of shares to be sold at thebest available price.

3. Limit buy order. Specifies a price and a number of shares tobe bought at that price.

4. Limit sell order. Specifies a price and a number of shares tobe sold at that price.

I Market orders are executed immediately.

I Limit orders are queued for later execution.

I Agent submitting limit order can later cancel the order.

I Limit-order book is the collection of queued limit orderswaiting for execution.

2 / 24

What is a limit-order book?Orders to buy and sell stock arrive at an exchange. Four types:

1. Market buy order. Specifies number of shares to be bought atthe best available price.

2. Market sell order. Specifies number of shares to be sold at thebest available price.

3. Limit buy order. Specifies a price and a number of shares tobe bought at that price.

4. Limit sell order. Specifies a price and a number of shares tobe sold at that price.

I Market orders are executed immediately.

I Limit orders are queued for later execution.

I Agent submitting limit order can later cancel the order.

I Limit-order book is the collection of queued limit orderswaiting for execution.

2 / 24

What is a limit-order book?Orders to buy and sell stock arrive at an exchange. Four types:

1. Market buy order. Specifies number of shares to be bought atthe best available price.

2. Market sell order. Specifies number of shares to be sold at thebest available price.

3. Limit buy order. Specifies a price and a number of shares tobe bought at that price.

4. Limit sell order. Specifies a price and a number of shares tobe sold at that price.

I Market orders are executed immediately.

I Limit orders are queued for later execution.

I Agent submitting limit order can later cancel the order.

I Limit-order book is the collection of queued limit orderswaiting for execution.

2 / 24

What is a limit-order book?Orders to buy and sell stock arrive at an exchange. Four types:

1. Market buy order. Specifies number of shares to be bought atthe best available price.

2. Market sell order. Specifies number of shares to be sold at thebest available price.

3. Limit buy order. Specifies a price and a number of shares tobe bought at that price.

4. Limit sell order. Specifies a price and a number of shares tobe sold at that price.

I Market orders are executed immediately.

I Limit orders are queued for later execution.

I Agent submitting limit order can later cancel the order.

I Limit-order book is the collection of queued limit orderswaiting for execution.

2 / 24

What is a limit-order book?Orders to buy and sell stock arrive at an exchange. Four types:

1. Market buy order. Specifies number of shares to be bought atthe best available price.

2. Market sell order. Specifies number of shares to be sold at thebest available price.

3. Limit buy order. Specifies a price and a number of shares tobe bought at that price.

4. Limit sell order. Specifies a price and a number of shares tobe sold at that price.

I Market orders are executed immediately.

I Limit orders are queued for later execution.

I Agent submitting limit order can later cancel the order.

I Limit-order book is the collection of queued limit orderswaiting for execution.

2 / 24

What is a limit-order book?Orders to buy and sell stock arrive at an exchange. Four types:

1. Market buy order. Specifies number of shares to be bought atthe best available price.

2. Market sell order. Specifies number of shares to be sold at thebest available price.

3. Limit buy order. Specifies a price and a number of shares tobe bought at that price.

4. Limit sell order. Specifies a price and a number of shares tobe sold at that price.

I Market orders are executed immediately.

I Limit orders are queued for later execution.

I Agent submitting limit order can later cancel the order.

I Limit-order book is the collection of queued limit orderswaiting for execution.

2 / 24

What is a limit-order book?Orders to buy and sell stock arrive at an exchange. Four types:

1. Market buy order. Specifies number of shares to be bought atthe best available price.

2. Market sell order. Specifies number of shares to be sold at thebest available price.

3. Limit buy order. Specifies a price and a number of shares tobe bought at that price.

4. Limit sell order. Specifies a price and a number of shares tobe sold at that price.

I Market orders are executed immediately.

I Limit orders are queued for later execution.

I Agent submitting limit order can later cancel the order.

I Limit-order book is the collection of queued limit orderswaiting for execution.

2 / 24

Bid and ask prices

Limit sell orders

Limit buy orders

Bes

tas

k

Bes

tb

id

I The (best) bid price is the highest limit buy order price in thebook. It is the best available price for a market sell.

I The (best) ask price is the lowest limit sell order price in thebook. It is the best available price for a market buy.

3 / 24

MMM on June 30, 2010, at 0.1 second intervals

Thanks to

Mark Schervish, HeadDepartment of StatisticsCarnegie Mellon University.

4 / 24

Idea of this talk

Build a “zero-intelligence” Poisson model of the limit-order bookand determine its diffusion limit.

I “Zero-intelligence” — No strategic play by the agentssubmitting orders.

I Poisson — Arrivals of buy and sell limit and market orders arePoisson processes. Cancellations are also governed by Poissonprocesses.

I Maybe a little intelligence — Arrival and cancellation ratesdepend on the state of the limit-order book.

I Diffusion limit — Scale time by n, divide volume by√

n, andpass to the limit as n→∞.

5 / 24

Idea of this talk

Build a “zero-intelligence” Poisson model of the limit-order bookand determine its diffusion limit.

I “Zero-intelligence” — No strategic play by the agentssubmitting orders.

I Poisson — Arrivals of buy and sell limit and market orders arePoisson processes. Cancellations are also governed by Poissonprocesses.

I Maybe a little intelligence — Arrival and cancellation ratesdepend on the state of the limit-order book.

I Diffusion limit — Scale time by n, divide volume by√

n, andpass to the limit as n→∞.

5 / 24

Idea of this talk

Build a “zero-intelligence” Poisson model of the limit-order bookand determine its diffusion limit.

I “Zero-intelligence” — No strategic play by the agentssubmitting orders.

I Poisson — Arrivals of buy and sell limit and market orders arePoisson processes. Cancellations are also governed by Poissonprocesses.

I Maybe a little intelligence — Arrival and cancellation ratesdepend on the state of the limit-order book.

I Diffusion limit — Scale time by n, divide volume by√

n, andpass to the limit as n→∞.

5 / 24

Idea of this talk

Build a “zero-intelligence” Poisson model of the limit-order bookand determine its diffusion limit.

I “Zero-intelligence” — No strategic play by the agentssubmitting orders.

I Poisson — Arrivals of buy and sell limit and market orders arePoisson processes. Cancellations are also governed by Poissonprocesses.

I Maybe a little intelligence — Arrival and cancellation ratesdepend on the state of the limit-order book.

I Diffusion limit — Scale time by n, divide volume by√

n, andpass to the limit as n→∞.

5 / 24

Idea of this talk

Build a “zero-intelligence” Poisson model of the limit-order bookand determine its diffusion limit.

I “Zero-intelligence” — No strategic play by the agentssubmitting orders.

I Poisson — Arrivals of buy and sell limit and market orders arePoisson processes. Cancellations are also governed by Poissonprocesses.

I Maybe a little intelligence — Arrival and cancellation ratesdepend on the state of the limit-order book.

I Diffusion limit — Scale time by n, divide volume by√

n, andpass to the limit as n→∞.

5 / 24

Partial history

1. Garman, M. (1976) “Market microstructure,” J. FinancialEconomics 3, 257–275.Poisson arrivals of buy and sell orders. Arrival rates depend onthe price, which is set by a market maker to maximize hisprofit.

2. Mendelson, H. (1982) “Market behavior in a clearinghouse,” Econometrica 50, 1505–1524.Poisson arrivals of buy and sell orders accumulate. Then amarket maker sets a price to maximize the volume of trading.

3. Cohen, K. J., Conroy, R. M. & Maier, S. F. (1985)“Order flow and the quality of the market,” in Market Makingand the Changing Structure of the Securities Industry.Poisson arrivals of limit orders at the best bid and best askand market orders. No market maker.

6 / 24

Partial history

1. Garman, M. (1976) “Market microstructure,” J. FinancialEconomics 3, 257–275.Poisson arrivals of buy and sell orders. Arrival rates depend onthe price, which is set by a market maker to maximize hisprofit.

2. Mendelson, H. (1982) “Market behavior in a clearinghouse,” Econometrica 50, 1505–1524.Poisson arrivals of buy and sell orders accumulate. Then amarket maker sets a price to maximize the volume of trading.

3. Cohen, K. J., Conroy, R. M. & Maier, S. F. (1985)“Order flow and the quality of the market,” in Market Makingand the Changing Structure of the Securities Industry.Poisson arrivals of limit orders at the best bid and best askand market orders. No market maker.

6 / 24

Partial history

1. Garman, M. (1976) “Market microstructure,” J. FinancialEconomics 3, 257–275.Poisson arrivals of buy and sell orders. Arrival rates depend onthe price, which is set by a market maker to maximize hisprofit.

2. Mendelson, H. (1982) “Market behavior in a clearinghouse,” Econometrica 50, 1505–1524.Poisson arrivals of buy and sell orders accumulate. Then amarket maker sets a price to maximize the volume of trading.

3. Cohen, K. J., Conroy, R. M. & Maier, S. F. (1985)“Order flow and the quality of the market,” in Market Makingand the Changing Structure of the Securities Industry.Poisson arrivals of limit orders at the best bid and best askand market orders. No market maker.

6 / 24

Brief history (continued)

4. Domowitz, I. & Wang, J. (1994) “Auctions asalgorithms,” J. Economic Dynamics and Control 18, 29–60.Describe the growth in electronic exchanges since 1985 andpropose a model of Poisson arrivals of buy and sell orderskeyed off the opposite best price.

5. Luckock, H. (2003) “A steady-state model of a continuousdouble auction,” Quant. Finance 3, 385–404.Points out error in 4. Arrival rates in Luckock’s model areindependent of the state of the order book.

6. Smith, E., Farmer, J.D., Gillemot, L. &Krishnamurthy, S. (2003) “Statistical theory of thecontinuous double auction,” Quant. Finance 3, 481–514.Poisson arrivals of buy and sell orders, with arrival ratesindependent on the state of the order book. Simulationstudies.

7 / 24

Brief history (continued)

4. Domowitz, I. & Wang, J. (1994) “Auctions asalgorithms,” J. Economic Dynamics and Control 18, 29–60.Describe the growth in electronic exchanges since 1985 andpropose a model of Poisson arrivals of buy and sell orderskeyed off the opposite best price.

5. Luckock, H. (2003) “A steady-state model of a continuousdouble auction,” Quant. Finance 3, 385–404.Points out error in 4. Arrival rates in Luckock’s model areindependent of the state of the order book.

6. Smith, E., Farmer, J.D., Gillemot, L. &Krishnamurthy, S. (2003) “Statistical theory of thecontinuous double auction,” Quant. Finance 3, 481–514.Poisson arrivals of buy and sell orders, with arrival ratesindependent on the state of the order book. Simulationstudies.

7 / 24

Brief history (continued)

4. Domowitz, I. & Wang, J. (1994) “Auctions asalgorithms,” J. Economic Dynamics and Control 18, 29–60.Describe the growth in electronic exchanges since 1985 andpropose a model of Poisson arrivals of buy and sell orderskeyed off the opposite best price.

5. Luckock, H. (2003) “A steady-state model of a continuousdouble auction,” Quant. Finance 3, 385–404.Points out error in 4. Arrival rates in Luckock’s model areindependent of the state of the order book.

6. Smith, E., Farmer, J.D., Gillemot, L. &Krishnamurthy, S. (2003) “Statistical theory of thecontinuous double auction,” Quant. Finance 3, 481–514.Poisson arrivals of buy and sell orders, with arrival ratesindependent on the state of the order book. Simulationstudies.

7 / 24

Brief history (continued)

7. Cont, R., Stoikov, S. & Talreja, R. (2010) “Astochastic model for order book dynamics,” OperationsResearch 58, 549–563.Poisson arrivals of buy and sell orders keyed off the oppositebest price. Compute statistics of the order-book behavior byLaplace transforms analysis.

8. Follmer, H. & Schweizer, M. (1993) “A microeconomicapproach to diffusion models for stock prices,” Math. Finance3, 1–23.Not a model of limit order books. Uses ideas of marketmicrostructure to set up a discrete-time model and then usesweak convergence to obtain continuous-time limit. First of aseries of papers using weak convergence as a tool to studymicrostructure.

8 / 24

Brief history (continued)

7. Cont, R., Stoikov, S. & Talreja, R. (2010) “Astochastic model for order book dynamics,” OperationsResearch 58, 549–563.Poisson arrivals of buy and sell orders keyed off the oppositebest price. Compute statistics of the order-book behavior byLaplace transforms analysis.

8. Follmer, H. & Schweizer, M. (1993) “A microeconomicapproach to diffusion models for stock prices,” Math. Finance3, 1–23.Not a model of limit order books. Uses ideas of marketmicrostructure to set up a discrete-time model and then usesweak convergence to obtain continuous-time limit. First of aseries of papers using weak convergence as a tool to studymicrostructure.

8 / 24

Brief history (continued)

9. Cont, R. & de Larrard, A. (2012) “Price dynamics in aMarkovian limit order market,” preprint.Poisson arrivals at best bid and best ask only. When a queuedepletes, price moves and system reinitializes. Determinediffusion limit.

9 / 24

Our “zero-intelligence” Poisson model (buy orders)

c 1 1 λ

I All orders are of size 1.

I Poisson arrivals of market buys, rate λ > 1. These execute atthe (best) ask price.

I Poisson arrivals of limit buys at one and two ticks below the(best) ask price, both at rate 1.

I Cancellations of limit buys two or more ticks below the (best)bid price, rate θ/

√n per order.

10 / 24

Our “zero-intelligence” Poisson model (buy orders)

c 1 1 λ

I All orders are of size 1.

I Poisson arrivals of market buys, rate λ > 1. These execute atthe (best) ask price.

I Poisson arrivals of limit buys at one and two ticks below the(best) ask price, both at rate 1.

I Cancellations of limit buys two or more ticks below the (best)bid price, rate θ/

√n per order.

10 / 24

Our “zero-intelligence” Poisson model (buy orders)

c 1 1 λ

I All orders are of size 1.

I Poisson arrivals of market buys, rate λ > 1. These execute atthe (best) ask price.

I Poisson arrivals of limit buys at one and two ticks below the(best) ask price, both at rate 1.

I Cancellations of limit buys two or more ticks below the (best)bid price, rate θ/

√n per order.

10 / 24

Our “zero-intelligence” Poisson model (buy orders)

c 1 1 λ

I All orders are of size 1.

I Poisson arrivals of market buys, rate λ > 1. These execute atthe (best) ask price.

I Poisson arrivals of limit buys at one and two ticks below the(best) ask price, both at rate 1.

I Cancellations of limit buys two or more ticks below the (best)bid price, rate θ/

√n per order.

10 / 24

Our “zero-intelligence” Poisson model (buy orders)

c 1 1 λ

I All orders are of size 1.

I Poisson arrivals of market buys, rate λ > 1. These execute atthe (best) ask price.

I Poisson arrivals of limit buys at one and two ticks below the(best) ask price, both at rate 1.

I Cancellations of limit buys two or more ticks below the (best)bid price, rate θ/

√n per order.

10 / 24

Our “zero-intelligence Poisson model (sell orders)

λ 1

c 1 1 λ

I All orders are of size 1.

I Poisson arrivals of market sells, rate λ > 1. These execute atthe (best) bid price.

I Poisson arrivals of limit sells at one and two ticks above the(best) bid price, both at rate 1.

I Cancellations of limit sells two or more ticks above the (best)ask price, rate θ/

√n per order.

11 / 24

Limit-order book arrivals (Poisson at rates indicated)

λ 1

c 1 1 λ

1 1 λ

λ 1 1

1 λ1

λ 1

c

λ1 1

1 1λ

1 λ

1

1

λ 1

λ 1 1 c

λ1

1 1 λ

λ 1 1

1 λ

1 c1λ

12 / 24

Limit-order book arrivals (Poisson at rates indicated)

λ 1

c 1 1 λ

1 1 λ

λ 1 1

1 λ1

λ 1

c

λ1 1

1 1λ

1 λ

1

1

λ 1

λ 1 1 c

λ1

1 1 λ

λ 1 1

1 λ

1 c1λ

12 / 24

Simulation of the model

Thanks to

Christopher AlmostPh.D. studentDepartment of Mathematical SciencesCarnegie Mellon University

13 / 24

Limit-order book arrivals (Poisson at rates indicated)

λ 1

c 1 1 λ

1 1 λ

λ 1 1

1 λ1

λ 1

c

λ1 1

1 1λ

1 λ

1

1

λ 1

λ 1 1 c

λ1

1 1 λ

λ 1 1

1 λ

1 c1λ

W X Y Z

14 / 24

Limit-order book arrivals (Poisson at rates indicated)

λ 1

c 1 1 λ

1 1 λ

λ 1 1

1 λ1

λ 1

c

λ1 1

1 1λ

1 λ

1

1

λ 1

λ 1 1 c

λ1

1 1 λ

λ 1 1

1 λ

1 c1λ

W X Y Z

14 / 24

Transitions of (X , Y ) Y

X

λ

1

1λ

1

1

11

1

λ

11

λ

λ1

1

λ

11λ1

1λ1

11

1

1

λ = 1+√

52 ⇐⇒

(X ,Y ) is “globally balanced”

G = YH = X + λY

G = YH = X + Y

G = −XH = X + Y

G = −XH = λX + Y

15 / 24

Transitions of (X , Y ) Y

X

λ

1

1λ

1

1

11

1

λ

11

λ

λ1

1

λ

11λ1

1λ1

11

1

1

λ = 1+√

52 ⇐⇒

(X ,Y ) is “globally balanced”

G = YH = X + λY

G = YH = X + Y

G = −XH = X + Y

G = −XH = λX + Y

15 / 24

Transitions of (X , Y ) Y

X

λ

1

1λ

1

1

11

1

λ

11

λ

λ1

1

λ

11λ1

1λ1

11

1

1

λ = 1+√

52 ⇐⇒

(X ,Y ) is “globally balanced”

G = YH = X + λY

G = YH = X + Y

G = −XH = X + Y

G = −XH = λX + Y

15 / 24

Transitions of (X , Y ) Y

X

λ

1

1λ

1

1

11

1

λ

11

λ

λ1

1

λ

11λ1

1λ1

11

1

1

λ = 1+√

52 ⇐⇒

(X ,Y ) is “globally balanced”

G = YH = X + λY

G = YH = X + Y

G = −XH = X + Y

G = −XH = λX + Y

15 / 24

Analysis of G

0−1−2 1 2

λ λ 1 1

1 1 λ λEssentially the dynamics of G

I Independent unit Poisson processes Ni , i = 1, 2, 3, 4.I Compensated Poisson processes Mi (t) = Ni (t)− t.I πt =

∫ t0 I{Gs>0} ds, νt =

∫ t0 I{Gs<0} ds, ζt =

∫ t0 I{Gs=0} ds.

Gt = N1(πt + ζt)− N2(λπt) + N3(λνt)− N4(νt + ζt)

= M1(πt + ζt)−M2(λπt) + M3(λνt)−M4(νt + ζt)

−(λ− 1)(πt − νt).

|Gt | = N1(πt + ζt)− N2(λπt)− N3(λνt) + N4(νt + ζt)

= M1(πt + ζt)−M2(λπt)−M3(λνt) + M4(νt + ζt)

+(λ+ 1)ζt − (λ− 1)t.

16 / 24

Analysis of G

0−1−2 1 2

λ λ 1 1

1 1 λ λEssentially the dynamics of G

I Independent unit Poisson processes Ni , i = 1, 2, 3, 4.

I Compensated Poisson processes Mi (t) = Ni (t)− t.I πt =

∫ t0 I{Gs>0} ds, νt =

∫ t0 I{Gs<0} ds, ζt =

∫ t0 I{Gs=0} ds.

Gt = N1(πt + ζt)− N2(λπt) + N3(λνt)− N4(νt + ζt)

= M1(πt + ζt)−M2(λπt) + M3(λνt)−M4(νt + ζt)

−(λ− 1)(πt − νt).

|Gt | = N1(πt + ζt)− N2(λπt)− N3(λνt) + N4(νt + ζt)

= M1(πt + ζt)−M2(λπt)−M3(λνt) + M4(νt + ζt)

+(λ+ 1)ζt − (λ− 1)t.

16 / 24

Analysis of G

0−1−2 1 2

λ λ 1 1

1 1 λ λEssentially the dynamics of G

I Independent unit Poisson processes Ni , i = 1, 2, 3, 4.I Compensated Poisson processes Mi (t) = Ni (t)− t.

I πt =∫ t0 I{Gs>0} ds, νt =

∫ t0 I{Gs<0} ds, ζt =

∫ t0 I{Gs=0} ds.

Gt = N1(πt + ζt)− N2(λπt) + N3(λνt)− N4(νt + ζt)

= M1(πt + ζt)−M2(λπt) + M3(λνt)−M4(νt + ζt)

−(λ− 1)(πt − νt).

|Gt | = N1(πt + ζt)− N2(λπt)− N3(λνt) + N4(νt + ζt)

= M1(πt + ζt)−M2(λπt)−M3(λνt) + M4(νt + ζt)

+(λ+ 1)ζt − (λ− 1)t.

16 / 24

Analysis of G

0−1−2 1 2

λ λ 1 1

1 1 λ λEssentially the dynamics of G

I Independent unit Poisson processes Ni , i = 1, 2, 3, 4.I Compensated Poisson processes Mi (t) = Ni (t)− t.I πt =

∫ t0 I{Gs>0} ds,

νt =∫ t0 I{Gs<0} ds, ζt =

∫ t0 I{Gs=0} ds.

Gt = N1(πt + ζt)− N2(λπt) + N3(λνt)− N4(νt + ζt)

= M1(πt + ζt)−M2(λπt) + M3(λνt)−M4(νt + ζt)

−(λ− 1)(πt − νt).

|Gt | = N1(πt + ζt)− N2(λπt)− N3(λνt) + N4(νt + ζt)

= M1(πt + ζt)−M2(λπt)−M3(λνt) + M4(νt + ζt)

+(λ+ 1)ζt − (λ− 1)t.

16 / 24

Analysis of G

0−1−2 1 2

λ λ 1 1

1 1 λ λEssentially the dynamics of G

I Independent unit Poisson processes Ni , i = 1, 2, 3, 4.I Compensated Poisson processes Mi (t) = Ni (t)− t.I πt =

∫ t0 I{Gs>0} ds, νt =

∫ t0 I{Gs<0} ds,

ζt =∫ t0 I{Gs=0} ds.

Gt = N1(πt + ζt)− N2(λπt) + N3(λνt)− N4(νt + ζt)

= M1(πt + ζt)−M2(λπt) + M3(λνt)−M4(νt + ζt)

−(λ− 1)(πt − νt).

|Gt | = N1(πt + ζt)− N2(λπt)− N3(λνt) + N4(νt + ζt)

= M1(πt + ζt)−M2(λπt)−M3(λνt) + M4(νt + ζt)

+(λ+ 1)ζt − (λ− 1)t.

16 / 24

Analysis of G

0−1−2 1 2

λ λ 1 1

1 1 λ λEssentially the dynamics of G

I Independent unit Poisson processes Ni , i = 1, 2, 3, 4.I Compensated Poisson processes Mi (t) = Ni (t)− t.I πt =

∫ t0 I{Gs>0} ds, νt =

∫ t0 I{Gs<0} ds, ζt =

∫ t0 I{Gs=0} ds.

Gt = N1(πt + ζt)− N2(λπt) + N3(λνt)− N4(νt + ζt)

= M1(πt + ζt)−M2(λπt) + M3(λνt)−M4(νt + ζt)

−(λ− 1)(πt − νt).

|Gt | = N1(πt + ζt)− N2(λπt)− N3(λνt) + N4(νt + ζt)

= M1(πt + ζt)−M2(λπt)−M3(λνt) + M4(νt + ζt)

+(λ+ 1)ζt − (λ− 1)t.

16 / 24

Analysis of G

0−1−2 1 2

λ λ 1 1

1 1 λ λEssentially the dynamics of G

I Independent unit Poisson processes Ni , i = 1, 2, 3, 4.I Compensated Poisson processes Mi (t) = Ni (t)− t.I πt =

∫ t0 I{Gs>0} ds, νt =

∫ t0 I{Gs<0} ds, ζt =

∫ t0 I{Gs=0} ds.

Gt = N1(πt + ζt)− N2(λπt) + N3(λνt)− N4(νt + ζt)

= M1(πt + ζt)−M2(λπt) + M3(λνt)−M4(νt + ζt)

−(λ− 1)(πt − νt).

|Gt | = N1(πt + ζt)− N2(λπt)− N3(λνt) + N4(νt + ζt)

= M1(πt + ζt)−M2(λπt)−M3(λνt) + M4(νt + ζt)

+(λ+ 1)ζt − (λ− 1)t.

16 / 24

Analysis of G

0−1−2 1 2

λ λ 1 1

1 1 λ λEssentially the dynamics of G

I Independent unit Poisson processes Ni , i = 1, 2, 3, 4.I Compensated Poisson processes Mi (t) = Ni (t)− t.I πt =

∫ t0 I{Gs>0} ds, νt =

∫ t0 I{Gs<0} ds, ζt =

∫ t0 I{Gs=0} ds.

Gt = N1(πt + ζt)− N2(λπt) + N3(λνt)− N4(νt + ζt)

= M1(πt + ζt)−M2(λπt) + M3(λνt)−M4(νt + ζt)

−(λ− 1)(πt − νt).

|Gt | = N1(πt + ζt)− N2(λπt)− N3(λνt) + N4(νt + ζt)

= M1(πt + ζt)−M2(λπt)−M3(λνt) + M4(νt + ζt)

+(λ+ 1)ζt − (λ− 1)t.

16 / 24

Analysis of G

0−1−2 1 2

λ λ 1 1

1 1 λ λEssentially the dynamics of G

I Independent unit Poisson processes Ni , i = 1, 2, 3, 4.I Compensated Poisson processes Mi (t) = Ni (t)− t.I πt =

∫ t0 I{Gs>0} ds, νt =

∫ t0 I{Gs<0} ds, ζt =

∫ t0 I{Gs=0} ds.

Gt = N1(πt + ζt)− N2(λπt) + N3(λνt)− N4(νt + ζt)

= M1(πt + ζt)−M2(λπt) + M3(λνt)−M4(νt + ζt)

−(λ− 1)(πt − νt).

|Gt | = N1(πt + ζt)− N2(λπt)− N3(λνt) + N4(νt + ζt)

= M1(πt + ζt)−M2(λπt)−M3(λνt) + M4(νt + ζt)

+(λ+ 1)ζt − (λ− 1)t.

16 / 24

Analysis of G

0−1−2 1 2

λ λ 1 1

1 1 λ λEssentially the dynamics of G

I Independent unit Poisson processes Ni , i = 1, 2, 3, 4.I Compensated Poisson processes Mi (t) = Ni (t)− t.I πt =

∫ t0 I{Gs>0} ds, νt =

∫ t0 I{Gs<0} ds, ζt =

∫ t0 I{Gs=0} ds.

Gt = N1(πt + ζt)− N2(λπt) + N3(λνt)− N4(νt + ζt)

= M1(πt + ζt)−M2(λπt) + M3(λνt)−M4(νt + ζt)

−(λ− 1)(πt − νt).

|Gt | = N1(πt + ζt)− N2(λπt)− N3(λνt) + N4(νt + ζt)

= M1(πt + ζt)−M2(λπt)−M3(λνt) + M4(νt + ζt)

+(λ+ 1)ζt − (λ− 1)t.

16 / 24

Diffusion scaling of G

G(n)t ,

1√nGnt .

Define also

M(n)i (t) ,

1√nMi (nt), i = 1, 2, 3, 4.

Functional central limit theorem implies M(n)i =⇒ B∗i , where the

B∗i are independent Brownian motions. Define

π(n)t ,

1

nπnt , ν

(n)t ,

1

nνnt , ζ

(n)t ,

1

nζnt .

G(n)t = M

(n)1 (π

(n)t + ζ

(n)t )− M

(n)2 (λπ

(n)t ) + M

(n)3 (λν

(n)t )

−M(n)4 (ν

(n)t + ζ

(n)t )−

√n(λ− 1)(π

(n)t − ν

(n)t ),∣∣G (n)

t

∣∣ = M(n)1 (π

(n)t + ζ

(n)t )− M

(n)2 (λπ

(n)t )− M

(n)3 (λν

(n)t )

+M(n)4 (ν

(n)t + ζ

(n)t ) +

√n[(λ+ 1)ζ

(n)t − (λ− 1)t

].

17 / 24

Diffusion scaling of G

G(n)t ,

1√nGnt .

Define also

M(n)i (t) ,

1√nMi (nt), i = 1, 2, 3, 4.

Functional central limit theorem implies M(n)i =⇒ B∗i , where the

B∗i are independent Brownian motions.

Define

π(n)t ,

1

nπnt , ν

(n)t ,

1

nνnt , ζ

(n)t ,

1

nζnt .

G(n)t = M

(n)1 (π

(n)t + ζ

(n)t )− M

(n)2 (λπ

(n)t ) + M

(n)3 (λν

(n)t )

−M(n)4 (ν

(n)t + ζ

(n)t )−

√n(λ− 1)(π

(n)t − ν

(n)t ),∣∣G (n)

t

∣∣ = M(n)1 (π

(n)t + ζ

(n)t )− M

(n)2 (λπ

(n)t )− M

(n)3 (λν

(n)t )

+M(n)4 (ν

(n)t + ζ

(n)t ) +

√n[(λ+ 1)ζ

(n)t − (λ− 1)t

].

17 / 24

Diffusion scaling of G

G(n)t ,

1√nGnt .

Define also

M(n)i (t) ,

1√nMi (nt), i = 1, 2, 3, 4.

Functional central limit theorem implies M(n)i =⇒ B∗i , where the

B∗i are independent Brownian motions. Define

π(n)t ,

1

nπnt , ν

(n)t ,

1

nνnt , ζ

(n)t ,

1

nζnt .

G(n)t = M

(n)1 (π

(n)t + ζ

(n)t )− M

(n)2 (λπ

(n)t ) + M

(n)3 (λν

(n)t )

−M(n)4 (ν

(n)t + ζ

(n)t )−

√n(λ− 1)(π

(n)t − ν

(n)t ),∣∣G (n)

t

∣∣ = M(n)1 (π

(n)t + ζ

(n)t )− M

(n)2 (λπ

(n)t )− M

(n)3 (λν

(n)t )

+M(n)4 (ν

(n)t + ζ

(n)t ) +

√n[(λ+ 1)ζ

(n)t − (λ− 1)t

].

17 / 24

Diffusion scaling of G

G(n)t ,

1√nGnt .

Define also

M(n)i (t) ,

1√nMi (nt), i = 1, 2, 3, 4.

Functional central limit theorem implies M(n)i =⇒ B∗i , where the

B∗i are independent Brownian motions. Define

π(n)t ,

1

nπnt , ν

(n)t ,

1

nνnt , ζ

(n)t ,

1

nζnt .

G(n)t = M

(n)1 (π

(n)t + ζ

(n)t )− M

(n)2 (λπ

(n)t ) + M

(n)3 (λν

(n)t )

−M(n)4 (ν

(n)t + ζ

(n)t )−

√n(λ− 1)(π

(n)t − ν

(n)t ),

∣∣G (n)t

∣∣ = M(n)1 (π

(n)t + ζ

(n)t )− M

(n)2 (λπ

(n)t )− M

(n)3 (λν

(n)t )

+M(n)4 (ν

(n)t + ζ

(n)t ) +

√n[(λ+ 1)ζ

(n)t − (λ− 1)t

].

17 / 24

Diffusion scaling of G

G(n)t ,

1√nGnt .

Define also

M(n)i (t) ,

1√nMi (nt), i = 1, 2, 3, 4.

Functional central limit theorem implies M(n)i =⇒ B∗i , where the

B∗i are independent Brownian motions. Define

π(n)t ,

1

nπnt , ν

(n)t ,

1

nνnt , ζ

(n)t ,

1

nζnt .

G(n)t = M

(n)1 (π

(n)t + ζ

(n)t )− M

(n)2 (λπ

(n)t ) + M

(n)3 (λν

(n)t )

−M(n)4 (ν

(n)t + ζ

(n)t )−

√n(λ− 1)(π

(n)t − ν

(n)t ),∣∣G (n)

t

∣∣ = M(n)1 (π

(n)t + ζ

(n)t )− M

(n)2 (λπ

(n)t )− M

(n)3 (λν

(n)t )

+M(n)4 (ν

(n)t + ζ

(n)t ) +

√n[(λ+ 1)ζ

(n)t − (λ− 1)t

].

17 / 24

Diffusion limit of GDrift in G

(n)t ,

−√

n(λ− 1)(π(n)t − ν

(n)t ),

forcesG (n) ⇒ 0, π(n) − ν(n) ⇒ 0.

Drift in |G (n)t |,

+√

n[(λ+ 1)ζ

(n)t − (λ− 1)t

],

forces

ζ(n) ⇒ λ− 1

λ+ 1e,

where e is the identity process e(t) = t for all t ≥ 0. But

π(n) + ζ(n)

+ ν(n) = e, so

π(n) ⇒ 1

λ+ 1e, ν(n) ⇒ 1

λ+ 1e.

18 / 24

Diffusion limit of GDrift in G

(n)t ,

−√

n(λ− 1)(π(n)t − ν

(n)t ),

forcesG (n) ⇒ 0,

π(n) − ν(n) ⇒ 0.

Drift in |G (n)t |,

+√

n[(λ+ 1)ζ

(n)t − (λ− 1)t

],

forces

ζ(n) ⇒ λ− 1

λ+ 1e,

where e is the identity process e(t) = t for all t ≥ 0. But

π(n) + ζ(n)

+ ν(n) = e, so

π(n) ⇒ 1

λ+ 1e, ν(n) ⇒ 1

λ+ 1e.

18 / 24

Diffusion limit of GDrift in G

(n)t ,

−√

n(λ− 1)(π(n)t − ν

(n)t ),

forcesG (n) ⇒ 0, π(n) − ν(n) ⇒ 0.

Drift in |G (n)t |,

+√

n[(λ+ 1)ζ

(n)t − (λ− 1)t

],

forces

ζ(n) ⇒ λ− 1

λ+ 1e,

where e is the identity process e(t) = t for all t ≥ 0. But

π(n) + ζ(n)

+ ν(n) = e, so

π(n) ⇒ 1

λ+ 1e, ν(n) ⇒ 1

λ+ 1e.

18 / 24

Diffusion limit of GDrift in G

(n)t ,

−√

n(λ− 1)(π(n)t − ν

(n)t ),

forcesG (n) ⇒ 0, π(n) − ν(n) ⇒ 0.

Drift in |G (n)t |,

+√

n[(λ+ 1)ζ

(n)t − (λ− 1)t

],

forces

ζ(n) ⇒ λ− 1

λ+ 1e,

where e is the identity process e(t) = t for all t ≥ 0. But

π(n) + ζ(n)

+ ν(n) = e, so

π(n) ⇒ 1

λ+ 1e, ν(n) ⇒ 1

λ+ 1e.

18 / 24

Diffusion limit of GDrift in G

(n)t ,

−√

n(λ− 1)(π(n)t − ν

(n)t ),

forcesG (n) ⇒ 0, π(n) − ν(n) ⇒ 0.

Drift in |G (n)t |,

+√

n[(λ+ 1)ζ

(n)t − (λ− 1)t

],

forces

ζ(n) ⇒ λ− 1

λ+ 1e,

where e is the identity process e(t) = t for all t ≥ 0.

But

π(n) + ζ(n)

+ ν(n) = e, so

π(n) ⇒ 1

λ+ 1e, ν(n) ⇒ 1

λ+ 1e.

18 / 24

Diffusion limit of GDrift in G

(n)t ,

−√

n(λ− 1)(π(n)t − ν

(n)t ),

forcesG (n) ⇒ 0, π(n) − ν(n) ⇒ 0.

Drift in |G (n)t |,

+√

n[(λ+ 1)ζ

(n)t − (λ− 1)t

],

forces

ζ(n) ⇒ λ− 1

λ+ 1e,

where e is the identity process e(t) = t for all t ≥ 0. But

π(n) + ζ(n)

+ ν(n) = e, so

π(n) ⇒ 1

λ+ 1e, ν(n) ⇒ 1

λ+ 1e.

18 / 24

Analysis of HTheorem

H is a martingale.

Proof in first quadrant:

1

λ1

In the first quadrant,

H = X + λY .

Drift in H is

(0 + λ) · 1 + (1 + 0) · 1 + (0− λ) · λ = λ+ 1− λ2,

which is zero when

λ =1 +√

5

2.

19 / 24

Rate of growth of 〈H , H〉Y

X

λ

1

1λ

1

1

11

1

λ

11

λ

λ1

1

λ

11λ1

1λ1

11

1

1

λ = 1+√

52 ⇐⇒

λ2 = λ+ 1

G = Y > 0H = X + λYd〈H,H〉t = (3λ+ 3)dt

G = −X > 0H = λX + Yd〈H,H〉t = (3λ+ 3)dt

G = 0d〈H,H〉t = (2λ+ 3)dt

G = 0

G = Y < 0H = X + Yd〈H,H〉t = (2λ+ 2)dt

G = −X < 0H = X + Yd〈H,H〉t = (2λ+ 2)dt

20 / 24

Rate of growth of 〈H , H〉Y

X

λ

1

1λ

1

1

11

1

λ

11

λ

λ1

1

λ

11λ1

1λ1

11

1

1

λ = 1+√

52 ⇐⇒

λ2 = λ+ 1

G = Y > 0H = X + λYd〈H,H〉t = (3λ+ 3)dt

G = −X > 0H = λX + Yd〈H,H〉t = (3λ+ 3)dt

G = 0d〈H,H〉t = (2λ+ 3)dt

G = 0

G = Y < 0H = X + Yd〈H,H〉t = (2λ+ 2)dt

G = −X < 0H = X + Yd〈H,H〉t = (2λ+ 2)dt

20 / 24

Rate of growth of 〈H , H〉Y

X

λ

1

1λ

1

1

11

1

λ

11

λ

λ1

1

λ

11λ1

1λ1

11

1

1

λ = 1+√

52 ⇐⇒

λ2 = λ+ 1

G = Y > 0H = X + λYd〈H,H〉t = (3λ+ 3)dt

G = −X > 0H = λX + Yd〈H,H〉t = (3λ+ 3)dt

G = 0d〈H,H〉t = (2λ+ 3)dt

G = 0

G = Y < 0H = X + Yd〈H,H〉t = (2λ+ 2)dt

G = −X < 0H = X + Yd〈H,H〉t = (2λ+ 2)dt

20 / 24

Rate of growth of 〈H , H〉Y

X

λ

1

1λ

1

1

11

1

λ

11

λ

λ1

1

λ

11λ1

1λ1

11

1

1

λ = 1+√

52 ⇐⇒

λ2 = λ+ 1

G = Y > 0H = X + λYd〈H,H〉t = (3λ+ 3)dt

G = −X > 0H = λX + Yd〈H,H〉t = (3λ+ 3)dt

G = 0d〈H,H〉t = (2λ+ 3)dt

G = 0

G = Y < 0H = X + Yd〈H,H〉t = (2λ+ 2)dt

G = −X < 0H = X + Yd〈H,H〉t = (2λ+ 2)dt

20 / 24

Rate of growth of 〈H , H〉Y

X

λ

1

1λ

1

1

11

1

λ

11

λ

λ1

1

λ

11λ1

1λ1

11

1

1

λ = 1+√

52 ⇐⇒

λ2 = λ+ 1

G = Y > 0H = X + λYd〈H,H〉t = (3λ+ 3)dt

G = −X > 0H = λX + Yd〈H,H〉t = (3λ+ 3)dt

G = 0d〈H,H〉t = (2λ+ 3)dt

G = 0

G = Y < 0H = X + Yd〈H,H〉t = (2λ+ 2)dt

G = −X < 0H = X + Yd〈H,H〉t = (2λ+ 2)dt

20 / 24

Diffusion limit of H

H(n)t ,

1√nHnt .

Then

〈H(n), H(n)〉t = (3λ+ 3)π(n)t + (2λ+ 3)ζ

(n)t + (2λ+ 2)ν

(n)t .

In particular,

〈H(n), H(n)〉 ⇒ (3λ+ 3)1

λ+ 1e + (2λ+ 3)

λ− 1

λ+ 1e + (2λ+ 2)

1

λ+ 1e

= 4λe.

Theorem 7.1.4 of Ethier and Kurtz implies

H(n) ⇒ 2√λB∗,

where B∗ is a standard Brownian motion.

21 / 24

Diffusion limit of (X (n), Y (n))Y

X

λ

1

1λ

1

1

11

1

λ

11

λ

λ1

1

λ

11λ1

1λ1

11

1

1

G (n) = Y (n) ⇒ 0H(n) = X (n) + λY (n)

⇒ 2√λB∗

G (n) = Y (n) ⇒ 0H(n) = X (n) + Y (n)

⇒ 2√λB∗

G (n) = −X (n) ⇒ 0

H(n) = X (n) + Y (n)

⇒ 2√λB∗

G (n) = −X (n) ⇒ 0

H(n) = λX (n) + Y (n)

⇒ 2√λB∗

22 / 24

Diffusion limit of limit-order book

W ∗ X ∗ Y ∗Z ∗ Z ∗

X ∗ Y ∗W ∗ W ∗ X ∗Y ∗ Z ∗

I Let B∗ be a standard Brownian motion

I X ∗ = 2√λmax{B∗, 0}

I Y ∗ = 2√λmin{B∗, 0}

This is correct only as long as the “bracketing processes” W ∗ andZ ∗ do not vanish.

23 / 24

Diffusion limit of limit-order book

W ∗ X ∗ Y ∗Z ∗ Z ∗

X ∗ Y ∗W ∗ W ∗ X ∗Y ∗ Z ∗

I Let B∗ be a standard Brownian motion

I X ∗ = 2√λmax{B∗, 0}

I Y ∗ = 2√λmin{B∗, 0}

This is correct only as long as the “bracketing processes” W ∗ andZ ∗ do not vanish.

23 / 24

Vanishing W ∗

V ∗ W ∗ X ∗Y ∗ Z ∗

Consider the case X ∗ = 0 and Y ∗ < 0:

I Z ∗ is pinned at −1

θ;

I V ∗ is pinned at1

θ;

I Y ∗ evolves as 2√λ times a Brownian motion;

I W ∗ evolves as 2√λ times a Brownian motion independent of

Y ∗;

I If W ∗ reaches zero before Y ∗ reaches zero, then V ∗ and Y ∗

become the “bracketing processes”.

The proof of the claim on this page is still under construction.

24 / 24

Vanishing W ∗

V ∗ W ∗ X ∗Y ∗ Z ∗

Consider the case X ∗ = 0 and Y ∗ < 0:

I Z ∗ is pinned at −1

θ;

I V ∗ is pinned at1

θ;

I Y ∗ evolves as 2√λ times a Brownian motion;

I W ∗ evolves as 2√λ times a Brownian motion independent of

Y ∗;

I If W ∗ reaches zero before Y ∗ reaches zero, then V ∗ and Y ∗

become the “bracketing processes”.

The proof of the claim on this page is still under construction.

24 / 24

Vanishing W ∗

V ∗ W ∗ X ∗Y ∗ Z ∗

Consider the case X ∗ = 0 and Y ∗ < 0:

I Z ∗ is pinned at −1

θ;

I V ∗ is pinned at1

θ;

I Y ∗ evolves as 2√λ times a Brownian motion;

I W ∗ evolves as 2√λ times a Brownian motion independent of

Y ∗;

I If W ∗ reaches zero before Y ∗ reaches zero, then V ∗ and Y ∗

become the “bracketing processes”.

The proof of the claim on this page is still under construction.

24 / 24

Vanishing W ∗

V ∗ W ∗ X ∗Y ∗ Z ∗

Consider the case X ∗ = 0 and Y ∗ < 0:

I Z ∗ is pinned at −1

θ;

I V ∗ is pinned at1

θ;

I Y ∗ evolves as 2√λ times a Brownian motion;

I W ∗ evolves as 2√λ times a Brownian motion independent of

Y ∗;

I If W ∗ reaches zero before Y ∗ reaches zero, then V ∗ and Y ∗

become the “bracketing processes”.

The proof of the claim on this page is still under construction.

24 / 24

Vanishing W ∗

V ∗ W ∗ X ∗Y ∗ Z ∗

Consider the case X ∗ = 0 and Y ∗ < 0:

I Z ∗ is pinned at −1

θ;

I V ∗ is pinned at1

θ;

I Y ∗ evolves as 2√λ times a Brownian motion;

I W ∗ evolves as 2√λ times a Brownian motion independent of

Y ∗;

I If W ∗ reaches zero before Y ∗ reaches zero, then V ∗ and Y ∗

become the “bracketing processes”.

The proof of the claim on this page is still under construction.

24 / 24

Vanishing W ∗

V ∗ W ∗ X ∗Y ∗ Z ∗

Consider the case X ∗ = 0 and Y ∗ < 0:

I Z ∗ is pinned at −1

θ;

I V ∗ is pinned at1

θ;

I Y ∗ evolves as 2√λ times a Brownian motion;

I W ∗ evolves as 2√λ times a Brownian motion independent of

Y ∗;

I If W ∗ reaches zero before Y ∗ reaches zero, then V ∗ and Y ∗

become the “bracketing processes”.

The proof of the claim on this page is still under construction.

24 / 24

Vanishing W ∗

V ∗ W ∗ X ∗Y ∗ Z ∗

Consider the case X ∗ = 0 and Y ∗ < 0:

I Z ∗ is pinned at −1

θ;

I V ∗ is pinned at1

θ;

I Y ∗ evolves as 2√λ times a Brownian motion;

I W ∗ evolves as 2√λ times a Brownian motion independent of

Y ∗;

I If W ∗ reaches zero before Y ∗ reaches zero, then V ∗ and Y ∗

become the “bracketing processes”.

The proof of the claim on this page is still under construction.

24 / 24