Cent. Eur. J. Math. DOI: 10.2478/s11533-012-0098-3

Central European Journal of Mathematics

Limiting distribution for a simple model of order book dynamics Research Article Łukasz Kruk1∗

1 Department of Mathematics, Maria Curie-Skłodowska University, Pl. Marii Curie-Skłodowskiej 1, 20-031 Lublin, Poland

Received 15 February 2012; accepted 25 April 2012 Abstract: A continuous-time model for the limit order book dynamics is considered. The set of outstanding limit orders is modeled as a pair of random counting measures and the limiting distribution of this pair of measure-valued processes is obtained under suitable conditions on the model parameters. The limiting behavior of the bid-ask spread and the midpoint of the bid-ask interval are also characterized. MSC:

91B26, 91B24, 91B70, 60F17, 60G57

Keywords: Order book dynamics • Bid-ask spread • Utility • Waiting costs • Measure-valued process • Weak convergence • Laplace transform

© Versita Sp. z o.o.

1.

Introduction

The advent of electronic exchanges driven purely by the order flow has revolutionized the method by which prices are formed. Examples of order-driven electronic exchanges are Archipelago, Euronext and the London International Financial Futures Exchange. The New York Stock Exchange (NYSE) has become a hybrid in which market makers exist but must compete with traders who submit limit orders. In the old system found on the NYSE and elsewhere, market makers on the floor of the exchange would match buy and sell orders, sometimes trading for their own accounts in order to “make markets”. In the new regime, orders arrive to exchanges and wait in an order book to be executed. Major traders on the exchange can see the order book, i.e., they know the size, price and type (buy or sell) of each order in the book. Moreover, the algorithm by which orders are executed is unambiguously specified and implemented by computer. This suggests that knowledge of the current limit order book should be a valuable asset in making profitable investment decisions. Indeed, there is a rising industry of algorithmic trading, wherein firms automate rapid trading of assets in large quantities based on order book information. ∗

E-mail: [email protected]

Limiting distribution for a simple model of order book dynamics

In this situation there is large and still increasing demand for mathematically rigorous, but at the same time analytically tractable models of order book dynamics. However, because of strategic play by those who place orders, capturing the behavior of limit order books is inherently complex and difficult. Early attempts to formulate and rigorously analyze mathematical models of order book dynamics may be found in, e.g., Mendelson [10], Luckock [9] and Kruk [8]. In the latter paper, tools borrowed from the area of fluid and heavy traffic analysis of queueing systems were used to characterize the limiting distributions of the outstanding limit order books in simple continuous double auctions. (See an expository paper by Bayraktar, Horst and Sircar [2] for a discussion of various queueing-theoretic approaches to market microstructure modeling and more references). More recently, Roșu [11] proposed a model of an order-driven market in which all limit traders of the same type (buyers or sellers) present at the same time have common expected utility, trading off execution price and waiting costs. Using game-theoretic tools, Roșu characterized the possible (equilibrium) states of the limit order book and the values of the corresponding expected customer utilities. He also conjectured the limiting behavior of the expected utilities of the customers as the model “granularity” (i.e., the ratio between the patience coefficient of the customers and their trading activity) gets small and heuristically justified his conjecture by analyzing the corresponding finite difference schemes. Unfortunately, a rigorous justification of these asymptotics appears to be a formidable task, due to considerable complexity of the model under consideration. In this paper we introduce a model that is similar in spirit to the one considered by Roșu [11], but much more tractable as far as asymptotic analysis is concerned. We consider a continuous-time, infinite-horizon economy with a single asset paying no dividends. There are four types of trades allowed, namely market and limit buy or sell orders for one unit of the asset, and the execution of the limit orders is subject to the usual price priority rule. A limit order may be freely cancelled and resubmitted at any time. The resulting order book is visible to all the market participants. Traders of different types arrive to the market according to independent Poisson processes. We assume that the prices are all placed in a closed, bounded interval [cB , cA ], where the endpoints cB , cA (or, equivalently, the midpoint P = (cA − cB )/2) may undergo stochastic fluctuations, but the length of the interval, i.e., the width of the price window, remains constant. The prices may take any values in [cB , cA ], so the corresponding tick size equals zero. If there are k limit sellers at the market at a given time, then the prices of their limit orders are x1 , . . . , xk , where P < xk < xk−1 < . . . < x1 = cA , and the i-th limit seller’s utility   UA (xi − P) Eh e−rT (xi ) (1) does not depend on i = 1, . . . , k, where Eh is the expectation operator (conditional on the market history) and T (xi ) is the execution time of the i-th limit sell order. Here UA is the utility function of the limit sellers and r > 0 is a given constant, which may be regarded as the patience coefficient of the customers. The utility (1) captures the tradeoff between the quoted price and the execution time of the i-th limit sell order – a trader asking for a higher price is usually required to wait more before his order clears. Common utility (1) is an equilibrium condition, analogous to the one introduced in Roșu [11]; if one seller had more utility than another, the seller with the lower utility would want to slightly undercut the price of the seller with higher utility in order to get earlier execution at nearly the same price as the high-utility seller, starting the “price war”. Similarly, if there are l limit buyers currently at the market, then the prices of their limit orders are y1 , . . . , yl , where cB = y1 < y2 < . . . < yl < P and the i-th limit buyer’s utility   UB (P − yi ) Eh e−rT (yi ) does not depend on i = 1, . . . , l, where UB is the utility function of the limit buyers and T (yi ) is the execution time of the i-th limit buy order. It is convenient to model the state of the order book at any time t as a pair of random counting measures such that the first (second) one of them has a unit atom at the current price of each limit buy (sell) order and total mass equal to the number of such orders in the book at time t. We consider a sequence of markets described above, under appropriate scaling of the model parameters, and we find the limiting distributions of the measure-valued processes describing the order book dynamics. We also characterize the limiting behavior of the bid-ask spread and the midpoint of the bid-ask interval, which may be regarded as a proxy for the fair price of the underlying asset. It is worthwhile to mention that our approach is, in some aspects, similar to the one used in the heavy-traffic analysis of some queueing systems with nonstandard service protocols, especially EDF (earliest-deadline-first), where the current state of the system is also modeled by counting measures. The interested reader is referred to [4], which is a good entry point to the relevant literature.

Ł. Kruk

A related Markovian model of the order book dynamics was recently proposed by Cont, Stoikov and Tarleja [3]. In that model, as in [8], there is a finite number of possible asset prices, so the state of the system at any time can be modeled in a finitely dimensional space. The authors prove existence and uniqueness of the stationary distribution for the underlying Markov process and evaluate some of its characteristics which are of interest to financial engineers. In contrast, our work (as well as [8]) utilizes functional central limit theorems rather than steady state analysis and our model needs an infinitely-dimensional state descriptor due to the zero tick size of our model. The paper is organized as follows. Section 2 presents the notation and gives a precise description of the model. Section 3 contains the asymptotic assumptions, two auxiliary lemmas and the statement of our main results. Finally, Section 4 is devoted to the evaluation of the Laplace transforms of the order execution times, followed by the proofs of the main results.

2.

The model

2.1.

Notation

The following notation will be used throughout the paper. Let R denote the set of real numbers and let N = {1, 2, 3, . . . }. For a, b ∈ R, we write a ∨ b for the maximum of a and b and a ∧ b for the minimum of a and b. Denote by M the set of all finite, nonnegative measures on B(R), the Borel subsets of R. Under the weak topology, M is a Polish space. Given a Polish space X , we use DX [0, ∞) to denote the space of right-continuous functions with left-hand limits (RCLL functions) from [0, ∞) to X , equipped with the Skorokhod J1 topology. See [5] for details. When dealing with DX [0, ∞), we consider X = R or Rd , with appropriate dimension d for vector-valued functions, or X = M. Given DX [0, ∞)-valued random variables Zn , n ∈ N, defined, respectively, on the probability spaces (Ωn , Fn , Pn ), n ∈ N, and a DX [0, ∞)-valued random variable Z defined on a probability space (Ω, F, P), we say Z (n) converges in distribution to Z , and write Zn ⇒ Z , if lim En [f(Zn )] = E[f(Z )], n→∞

for every bounded continuous function f on DX [0, ∞). Here En and E are expectations taken with respect to Pn and P, respectively.

2.2.

Market description

Consider a sequence of markets for an asset paying no dividends, indexed by superscript n. In each of these markets the time horizon is infinite and trading takes place in continuous time. The only types of trades allowed are market and limit buy/sell orders for one unit of the asset. The limit orders are subject to the usual price priority rule. When prices are equal, the time priority rule is applied. Limit orders may be cancelled and resubmitted for no cost at any time. There is no delay in trading, both types of orders being posted or executed simultaneously. Trading is based on a publicly observable limit order book. Traders of different types, limit sellers, limit buyers, market sellers and market buyers, (n) (n) (n) (n) arrive to the market according to independent Poisson processes with intensities λS , λB , µS and µB , respectively. We assume that for some positive constants λS , λB , βA and βB , we have (n) lim λ n→∞ S

= λS ,

λS

(n)

ρA , (n)

(n) µB

βA =1− √ , n

(n) lim λ n→∞ B

= λB ,

λB

(n)

ρB , (n)

(n) µS

βB =1− √ . n

(2)

In particular, lim µB = λS , lim µS = λB . (n)

(n)

n→∞

n→∞

Let P (n) (t), t ≥ 0, be a stochastic process taking values in (0, ∞), independent of the customer arrival processes and such that P (n) (0) = P(0) is a constant independent of n. Let cA (0), cB (0) be real constants such that cB (0) < P(0) < cA (0) and let (n) (n) cA (t) = cA (0) + P (n) (t) − P(0), cB (t) = cB (0) + P (n) (t) − P(0). (3)

Limiting distribution for a simple model of order book dynamics

Note that cA (t) − cB (t) = cA (0) − cB (0), cA (t) − P (n) (t) = cA (0) − P(0) and P (n) (t) − cB (t) = P(0) − cB (0) for every t ≥ 0. Let UA and UB be strictly increasing, concave, differentiable utility functions defined on [0, cA (0) − P(0)] and [0, P(0) − cB (0)], respectively, such that UA (0) = UB (0) = 0. We assume that at time t ≥ 0 all the prices quoted in the (n) (n) n-th market lie in the range [cB (t), cA (t)]. However, they may take any value in this range, so the tick size in the (n) markets under consideration is zero. If there are QA (t) = k limit sellers at the n-th market at time t, then the prices of (n) their limit orders are x1 , . . . , xk , where P (n) (t) < xk < xk−1 < . . . < x2 < x1 = cA (t). The price levels xi are chosen so that the i-th limit seller’s utility
  (n) UA xi − P (n) (t) E e−rTt (xi )/n |F(n) (t) , (n)

(n)

(n)

(n)

is independent of i = 1, . . . , k, where r > 0 is constant, Tt (xi ) is the time to the execution of the i-th limit sell order after time t and F(n) (t) is the σ -field generated by the events containing information about the evolution of the n-th market up to time t. In this case, xk , the lowest sell price available in the n-th market’s limit order book at time t, will (n) be called the ask price at time t and denoted by A(n) (t). Similarly, if there are QB (t) = l limit buyers at the n-th market (n) at time t, then the prices of their limit orders are y1 , . . . , yl , where cB (t) = y1 < y2 < . . . < yl < P (n) (t) and yi are chosen so that the i-th limit buyer’s utility (n)


 (n) UB P (n) (t) − yi E e−rTt (yi )/n |F(n) (t) , where Tt (yi ) is the time to the execution of the i-th limit buy order after time t, is independent of i = 1, . . . , l. In this case, yl will be called the bid price at time t and denoted by B(n) (t). Thus, the quantity A(n) (t) − B(n) (t) is the bid-ask (n) (n) spread in the n-th market at time t. We assume that QA (0) and QB (0) are constants. (n)

3.

The limiting distribution

For t ≥ 0, let b (n) (t) , √1 Q (n) (nt), Q A n A

b (n) (t) , √1 Q (n) (nt), Q B n B

b (n) (t) , A(n) (nt), A

b (n) (t) , B(n) (nt), B

b (n) (t) , P (n) (nt). P

We assume that there exists a continuous process P such that b (n) ⇒ P P

as

n → ∞.

(4)

Let γA = βA λS , γB = βB λB , σA2 = 2λS and σB2 = 2λB . Let WA (t) and WB (t) be independent Brownian motions with constant, strictly positive initial values, drifts −γA , −γB and variances σA2 , σB2 , respectively, with instantaneous reflection at 0. We assume that the pair (WA , WB ) is independent of the process P and b (n) (0) = WA (0), lim Q A

n→∞

b (n) (0) = WB (0). lim Q B

(5)

n→∞

Lemma 3.1. Under the assumptions (4)–(5), we have, as n → ∞,
b (n), Q b (n), P b (n) ⇒ (WA , WB , P). Q A B

Proof.

(6)

The number of limit sellers at the n-th market follows a birth-death process with birth intensity λS and death (n) (n) intensity µB . Thus, the distribution of the process QA is identical to that of the queue length process for a M/M/1 (n) (n) (n) queue with initial condition QA (0), arrival rate λS and service rate µB , see, e.g., [1, Section III.3a]. Consequently, the (n) b b (n) ⇒ WB fact that QA ⇒ WA follows from a classical result due to Iglehart and Whitt [6]. The proof of the convergence Q B (n) (n) (n) b (n) b (n) (n) (n) b is similar. Finally, for each n, the processes QA , QB , P (and hence QA , QB , P ) are independent, because QA (n) depends only on the arrivals of limit sellers and market buyers, while QB depends only on the arrivals of limit buyers and market sellers and P (n) is independent of the customer arrival processes, so (6) follows. (n)

Ł. Kruk

For t ≥ 0, let λt : R → R be a (random) shift operator defined by λt (x) = x + P(t) − P(0), x ∈ R. By analogy with (3), let cA (t) = λt (cA (0)) = cA (0) + P(t) − P(0),

(7)

cB (t) = λt (cB (0)) = cB (0) + P(t) − P(0). Define p

s

r β2 βA + A − , λS 4 2 s p β2 βB γB2 + 2rσB2 − γB r + B − . kB = = 2 λB 4 2 σB kA =

γA2 + 2rσA2 − γA = σA2

(8)

(9)

For t ≥ 0, let UA0 (x − P(t)) , kA UA (x − P(t)) UB0 (P(t) − x) ftB (x) = , kB UB (P(t) − x) ftA (x) =

HtA (x) =

x ∈ (P(t), cA (t)], x ∈ [cB (t), P(t)),

cA (t)

Z

ftA (ξ) dξ =

 1  log(UA (cA (t) − P(t)) − log(UA (x − P(t)) k A x  1  = log(UA (cA (0) − P(0)) − log(UA (x − P(t)) kA

(10)

for x ∈ (P(t), cA (t)], where the third equality follows from (7), and HtB (x) =

Z

x

ftB (ξ) dξ =

cB (t)

 1  log(UB (P(t) − cB (t)) − log(UB (P(t) − x) kB

for x ∈ [cB (t), P(t)). It is easy to check that H0A (x) = HtA (λt (x)),

x ∈ (P(0), cA (0)],

H0B (x) = HtB (λt (x)),

x ∈ [cB (0), P(0)).

(11)

For every t ≥ 0, the function HtA is a continuous, strictly decreasing mapping of (P(t), cA (t)] onto [0, ∞). Therefore, there exists a continuous inverse (HtA )−1 : [0, ∞) → (P(t), cA (t)]. Similarly, for each t ≥ 0, the function HtB is a continuous, strictly increasing mapping of [cB (t), P(t)) onto [0, ∞). Therefore, there exists a continuous inverse (HtB )−1 : [0, ∞) → [cB (t), P(t)). For every x ∈ (P(0), cA (0)] such that H0A (x) ≤ WA (0), let T A (x) = inf {t ≥ 0 : WA (t) = HtA (λt (x))} = inf {t ≥ 0 : WA (t) = H0A (x)}, where the second equality holds by (11). Also, for x ∈ [cB (0), P(0)) such that H0B (x) ≤ WB (0), let T B (x) = inf {t ≥ 0 : WB (t) = HtB (λt (x))} = inf {t ≥ 0 : WB (t) = H0B (x)}. We have

P[ T A (x) < ∞] = P[ T B (x) < ∞] = 1. H0A (x)

A

(12)

Indeed, WA (0) ≥ ≥ 0, so for the purpose of evaluation of P[ T (x) < ∞], we may assume that WA is just a Brownian motion with negative drift −γA and variance σA2 (without reflection at 0). Hence, by [7, (5.13), p. 197], the process WA almost surely reaches the level H0A (x) below its initial value WA (0) in finite time, i.e., P[T A (x) < ∞] = 1. By the same token, P[ T B (x) < ∞] = 1.

Limiting distribution for a simple model of order book dynamics

Lemma 3.2. For every r > 0, we have Ee−rT

A (x)

−rT B (x)

Ee

A

= ekA (H0 (x)−WA (0)) , =e

kB (H0B (x)−WB (0))

(13)

.

(14) A

As in the justification of (12), since WA (0) ≥ H0A (x) ≥ 0, for the purpose of evaluation of Ee−rT (x) we may assume that WA is just a Brownian motion with drift −γA and variance σA2 (without reflection at 0). In this case, for every λ ∈ R the process   t Mλ (t) = exp λWA (t) − λ2 σA2 + λγA t 2

Proof.

is a martingale. By (12), Wald’s identity EMλ (TA (x)) = Mλ (0) = eλWA (0) holds, see [7, Problem 5.7, p. 197]. Consequently, A

eλH0 (x) Ee−(λ

2 σ 2 /2−λγ )T (x) A A A

= eλWA (0) .

(15)

 p Since λ = −kA = − γA2 + 2rσA2 − γA /σA2 solves the equation λ2 σA2 /2 − λγA = r, (15) implies (13). The proof of (14) is similar. Define the processes A(t) and B(t) by   A(t) = P(t) + UA−1 UA (cA (0) − P(0)) e−κA WA (t) ,   B(t) = P(t) − UB−1 UB (P(0) − cB (0)) e−κB WB (t) , for t ≥ 0. Note that

HtA (A(t)) = WA (t),

HtB (B(t)) = WB (t),

(16)

(17)

so A(t) = (HtA )−1 (WA (t)), B(t) = (HtB )−1 (WB (t)) and cB (t) ≤ B(t) < P(t) < A(t) ≤ cA (t) for each t ≥ 0.

Proposition 3.3. We have, as n → ∞,
b (n) , B b (n) ⇒ (A, B) A

(18)

jointly with (6).

The proof of this proposition will be given in subsection 4.2. The following corollary characterizes the limiting distributions for the bid-ask spread and the midpoint of the bid-ask interval – a proxy of the fair price for the traded asset.

Corollary 3.4. We have, as n → ∞, b (n) − B b (n) ⇒ A − B, A jointly with (6) and (18).

1 b (n) b (n)
1 A +B ⇒ (A + B) 2 2

Ł. Kruk

We wish to characterize the asymptotic behavior of the limit order books. To this end, we shall introduce measure-valued processes B(n) and S(n) defined by B(n) (t)(B) ,



number of outstanding buy limit orders at the n-th market with prices in B ⊂ R at time t ,  S(n) (t)(B) , number of outstanding sell limit orders at the n-th market with prices in B ⊂ R at time t ,

and

for all t ≥ 0 and B ⊂ R. Their rescaled counterparts are b (n) (t) , √1 B(n) (nt), B n

1 b S(n) (t) , √ S(n) (nt). n

b (n) , b The processes B(n) , S(n) , B S(n) are RCLL, taking values in M. Let B∗ and S∗ be the M-valued processes defined by B∗ (t)(B) ,

Z

S∗ (t)(B) ,

Z

B∩[cB (t),B(t)]

B∩[A(t),cA (t)]

ftB (x) dx, ftA (x) dx.

(19)

The following theorem describes the limiting behavior of the limit order books.

Theorem 3.5. We have, as n → ∞,
b (n) , b B S(n) ⇒ (B∗ , S∗ ).

(20)

The proof of this theorem will be given in subsection 4.3.

4.

Proofs of the main results

4.1.

Laplace transforms of the order execution times

Let

    (n) (n) (n) (n) (n) ψA (z) = E e−zTt (A (t)) | F(n) (t) = E e−zTt (A (t)) ,

be the Laplace transform of the time time t. Similarly, let

(n) Tt (A(n) (t))

z ≥ 0,

to the execution of the first limit sell order in the n-th market after

    (n) (n) (n) (n) (n) ψB (z) = E e−zTt (B (t)) | F(n) (t) = E e−zTt (B (t)) ,

z ≥ 0,

be the Laplace transform of the time Tt (B(n) (t)) to the execution of the first limit buy order in the n-th market after time t. By the Markov property, (n)

 QA(n) (t)−i+1 r , n   QB(n) (t)−i+1  −rT (n) (y )/n (n)  r (n) i t E e | F (t) = ψB , n   (n) E e−rTt (xi )/n | F(n) (t) =



ψA

(n)

i = 1, . . . , QA (t), (n)

i = 1, . . . , QB (t). (n)

(21)

Limiting distribution for a simple model of order book dynamics

Lemma 4.1. For z > 0, we have z + λS + µB − (n)

ψA (z) = (n)

(n)

q (n) (n) (n) (n) (z + λS + µB )2 − 4λS µB

,

(22)

.

(23)

(n)

2λS z + λB + µS − (n)

(n) ψB (z)

(n)

q (n) (n) (n) (n) (z + λB + µS )2 − 4λB µS

=

(n)

2λB

Proof.

e (n) be the arrival time of either the first limit seller, or the first market buyer to appear at the n-th market. Let T e (n) is exponential with parameter λ(n) + µ(n) , so The distribution of T S B  e (n)  E e−zT =

Z

∞

e−zt λS + µB (n)

(n)


e−(λS

(n)

(n)

λS + µB (n)

+µB )t

dt =

0

z+

(n)

(n) λS

+ µB

.

(24)

(n)

  e (n) is a limit seller (market buyer) equals λ(n) / λ(n) + µ(n) resp. The probability that the customer arriving at time T S S B (n)  (n) (n) 
µB / λS + µB . Thus, by (24),   e (n)  (n) ψA (z) = E e−zT

λS

µB

(n)

(n)

(n) λS

+

(n) µB

+

(n) λS

+

ψA (z) (n)

(n) µB


2



µB

λS

(n)

=

z+

(n) λS

(n)

+

(n) µB

+

z+

(n) λS

+

(n) µB


2 (n) ψA (z) .

In other words, ψA (z) solves the quadratic equation (n)

(n) (n) (n)
(n) λS x 2 − z + λS + µB x + µB = 0

with roots z + λS + µB ± (n)

x=

q

(n)

(z + λS + µB )2 − 4λS µB (n)

(n)

(n) (n)

.

(n)

2λS

This, together with the fact that ψA (z) is a continuous, decreasing function of z, implies (22). The proof of (23) is similar. (n)

For future reference note that by (2) and (22),    − r T (n) (A(n) (t))  r (n) t n E e = ψA n q √ (n) √ (n) (r/n + (2 − βA / n)µB )2 − 4(1 − βA / n)(µB )2 = √ (n) 2(1 − βA / n)µB q √ (n) √   (n) (n) (2 − βA / n)µB − 4rµB + (βA µB )2 / n 1 √ + o = √ (n) n 2(1 − βA / n)µB q   (n) (n) (n) 4rµB + (βA µB )2 − βA µB 1 1 =1− √ +o √ √ (n) n n 2(1 − βA / n)µB q   (n) (n) (n) 4rµB + (βA µB )2 − βA µB 1 1 √ =1− √ + o (n) n n 2µB p     2 4rλS + (βA λS ) − βA λS 1 1 kA 1 =1− √ +o √ =1− √ +o √ , 2λS n n n n √ (n) r/n + (2 − βA / n)µB −

where kA has been defined by (8). A similar argument shows that       (n) (n) r kB 1 (n) E e−rTt (B (t))/n = ψB =1− √ +o √ , n n n where kB is as in (9).

(25)

(26)

Ł. Kruk

4.2.

Proof of Proposition 3.3

The utilities of all the limit sellers present at the n-th market at time nt are equal, so their price levels xi , (n) i = 1, . . . , QA (nt) − 1, satisfy the equation
 
  (n) (n) UA xi+1 − P (n) (nt) E e−rTnt (xi+1 )/n | F(n) (nt) = UA xi − P (n) (nt) E e−rTnt (xi )/n | F(n) (nt) . Thus, by (21), for i = 1, . . . , QA (nt) − 1 we have (n)



(n) UA xi+1 − P (n) (nt) = UA xi − P (n) (nt) ψA

  r . n

This, together with (3) and the fact that x1 = cA (nt), implies (n)

  i−1   i−1

(n) r r (n) (n) = UA (cA (0) − P(0)) ψA UA xi − P (n) (nt) = UA cA (nt) − P (n) (nt) ψA n n

(27)

for i = 1, . . . , QA (nt). Putting i = QA (nt) into (27) and recalling that xQ(n) (nt) = A(n) (nt), we get (n)

(n)

A






UA A (nt) − P (nt) = UA (cA (0) − P(0)) (n)

(n)

(n) ψA

hence A (nt) = P (nt) +

UA−1

B (nt) = P (nt) −

UB−1

(n)

(n)

 (n) ψA

UA (cA (0) − P(0))

 QA(n) (nt)−1 r , n

 QA(n) (nt)−1 ! r . n

(28)

 QB(n) (nt)−1 ! r . n

(29)

A similar argument yields

(n)

(n)

 UB (P(0) − cB (0))

(n) ψB

By (25), we have 

  QA(n) (nt)−1  √n Qb A(n) (t)−1 1 r kA (n) ψA = 1− √ +o √ = n n n

Similarly, by (26),  (n) ψB

 QB(n) (nt)−1 r = n





 √n/kA !kA (Qb A(n) (t)−1/ 1 kA 1− √ +o √ n n

 √n/kB !kB (Qb B(n) (t)−1/ kB 1 1− √ +o √ n n

Lemma 3.1, together with (28)–(31), implies the joint convergence (6) and (18).

√ n)

.

(30)

√ n)

.

(31)

Limiting distribution for a simple model of order book dynamics

4.3.

Proof of Theorem 3.5

Using Lemma 3.1 and the Skorokhod Representation Theorem, see, e.g., [5, Theorem 3.1.8], we can construct the model primitives for the whole sequence of markets under consideration on a common probability space (Ω, F, P) such that the limiting processes WA , WB , P are defined on it and
b (n) , Q b (n) , P b (n) (ω) → (WA , WB , P)(ω), Q A B

n → ∞,

(32)

in DR3 [0, ∞) for P-almost every ω ∈ Ω. Let C be the set of elementary events on which (32) holds. We have P(C ) = 1. In the remainder of the proof we assume that all the random objects under consideration are evaluated at some ω ∈ C . The equations (28)–(31), together with the convergence (32), assure that
b (n) , B b (n) → (A, B), A

n → ∞,

(33)

in DR2 [0, ∞). Let T > 0, ε > 0. Let η be large enough that P(Dη ) ≥ 1 − ε/4, where   Dη , WA (t) ≤ η : t ∈ [0, T ] . Let κ = UA−1 (UA (cA (0) − P(0))e−kA η ). Then, by (16), A(t) − P(t) ≥ κ,

0 ≤ t ≤ T,

on

Dη .

(34)

Choose ε1 > 0 such that   kA ε κ ≤ s < t ≤ cA (0) − P(0), t − s ≤ ε1 ≤ . sup | log (UA (t)) − log (UA (s))| : 2 5 For δ > 0, let

(35)

 Eδ , (n)

 n o b (n) (t) − P(t)| ∨ |A b (n) (t) − A(t)| ∨ |Q b (n) (t) − WA (t)| ≤ δ . sup |P A 5 t∈[0,T ]

Put e ε = ε ∧ κ ∧ ε1 .

(36)

By (32)–(33), there exists n0 such that P(Eeε ) ≥ 1 − ε/4 for n ≥ n0 . (n)

If cA (0) − P(0) ≥ x ≥ A(t) − P(t) + e ε , then S∗ (t)[P(t) + x, ∞) =

Z

cA (t)

ftA (x) dx =

P(t)+x

 1  log (UA (cA (0) − P(0)) − log (UA (x)) , kA

(37)

where the first line follows by (19), (7) and the second line by (10). On the other hand, in this case x ≥ A(n) (nt) − (n) P (n) (nt) + 3e ε /5 > A(n) (nt) − P (n) (nt) on Eeε and we can estimate i0 , S(n) (nt)[P (n) (nt) + x, ∞) as follows. Since UA is strictly increasing and xi0 +1 < P (n) (nt) + x ≤ xi0 , where xi is the price of the i-th limit sell order at time nt, we have

UA xi0 +1 − P (n) (nt) < UA (x) ≤ UA xi0 − P (n) (nt) . This, together with (27), implies that   i0   i0 −1 r r (n) (n) UA (cA (0) − P(0)) ψA < UA (x) ≤ UA (cA (0) − P(0)) ψA , n n

Ł. Kruk

hence

      r r (n) (n) i0 log ψA < log (UA (x)) − log (UA (cA (0) − P(0))) ≤ (i0 − 1) log ψA . n n

Using (25) and the Taylor expansion of log in a neighbourhood of 1, we get √ i0 − 1 ≤

n (log (UA (cA (0) − P(0))) − log (UA (x))) + o(1) < i0 kA

and thus, for n sufficiently large (uniformly in x), S(n) (nt) [P (n) (nt) + x, ∞) =

√

 n (log (UA (cA (0) − P(0))) − log (UA (x))) + 1. kA

If x ≤ A(t) − P(t) − e ε , then S∗ (t)[P(t) + x, ∞) =

Z

cA (t)

A(t)

ftA (x) dx = HtA (A(t)) = WA (t)

(38)

(39)

by (19), (10) and (17). In this case x ≤ A(n) (nt) − P (n) (nt) − 3e ε /5 < A(n) (nt) − P (n) (nt) on Eeε and (n)

S(n) (nt)[P (n) (nt) + x, ∞) = QA (nt). (n)

(40)

The equations (37)–(40) imply that on the set Eeε we have (n)

  2 e ε b(n) b (n) ∗ sup . S (t)[P (t) + x, ∞) − S (t)[P(t) + x, ∞) ≤ max √ , n 5 x ∈(A(t)−P(t)−e / ε ,A(t)−P(t)+e ε)

(41)

b (n) (t) + x, ∞) − S∗ (t)[P(t) + x, ∞)| for x ∈ (A(t) − P(t) − e We now want to estimate |b S(n) (t)[P ε , A(t) − P(t) + e ε ). For such x, we have  (n)
b b (n) (t) + x, ∞) − S∗ (t)[P(t) + x, ∞) = b b (t) + x, P b (n) (t) + A(t) − P(t) + e S(n) (t)[P S(n) (t) P ε  (n)
b (t) + A(t) − P(t) + e − S∗ (t)[P(t) + x, A(t) + e ε)+b S(n) (t) P ε , ∞ − S∗ (t)[A(t) + e ε , ∞).

(42)

Also, −S∗ (t)[A(t) − e ε , A(t) + e ε ) ≤ −S∗ (t)[P(t) + x, A(t) + e ε)  (n)
(n) (n) b b b ≤ S (t) P (t) + x, P (t) + A(t) − P(t) + e ε − S∗ (t)[P(t) + x, A(t) + e ε)  (n)
(n) (n) b b b ≤ S (t) P (t) + A(t) − P(t) − e ε , P (t) + A(t) − P(t) + e ε  (n)
∗ (n) b b = S (t)[A(t) − e ε , A(t) + e ε ) + S (t) P (t) + A(t) − P(t) − e ε , ∞ − S∗ (t)[A(t) − e ε , ∞)    (n)
(n) ∗ b (t) + A(t) − P(t) + e − b S (t) P ε , ∞ − S (t)[A(t) + e ε , ∞) .

(43)

On the set Dη , S∗ (t)[A(t) − e ε , A(t) + e ε) =

Z

(A(t)+e ε )∧cA (t)

A(t)

ftA (y) dy

 ε 1  = log (UA ((A(t) + e ε ) ∧ cA (t) − P(t))) − log (UA (A(t) − P(t))) ≤ , kA 5

(44)

Limiting distribution for a simple model of order book dynamics

where the first equality follows by (19), the second equality by (10) and the third one follows by (34)–(35), (7). Thus, by (n) (36) and (41)–(44), on the set C ∩ Dη ∩ Eeε for sufficiently large n we have   ε 2 e ε 4ε b(n) b (n) + ≤ , S (t)[P (t) + x, ∞) − S∗ (t)[P(t) + x, ∞) ≤ 3 max √ , 5 5 5 n x∈(A(t)−P(t)−e ε ,A(t)−P(t)+e ε) sup

which, together with (41), (36) and the fact that the above estimates are uniform over t ∈ [0, T ], implies that on C ∩Dη ∩Eeε for sufficiently large n, 4ε (n) b (n) sup sup b S (t)[P (t) + x, ∞) − S∗ (t)[P(t) + x, ∞) ≤ . (45) 5 0≤t≤T x∈R (n)

We also have b b (n) (t) + x, ∞) − S∗ (t)[P b (n) (t) + x, ∞) S(n) (t)[P b (n) (t) + x, ∞) − S∗ (t)[P(t) + x, ∞) + S∗ (t)[P(t) + x, ∞) − S∗ (t)[P b (n) (t) + x, ∞) = b S(n) (t)[P

(46)

and on the set C ∩ Dη ∩ Eeε

(n)

∗ b (n) (t) + x, ∞) S (t)[P(t) + x, ∞) − S∗ (t)[P h i Z b (n) (t) ∧ P(t)) + x, (P b (n) (t) ∨ P(t)) + x = ≤ S∗ (t) (P

b (n) (t)∨P(t))+x)∧cA (t) A(t)∨(P

b (n) (t)∧P(t))+x)∧cA (t) A(t)∨(P

ftA (y) dy





i 1 h b (n) (t) ∨ P(t)) + x ∧ cA (t) − P(t) − log UA A(t) ∨ (P b (n) (t) ∧ P(t)) + x ∧ cA (t) − P(t) log UA A(t) ∨ (P kA ε ≤ , 5

(47)

=

where the equalities follow from (10) and the last inequality follows from (7), (35)–(36), together with the definitions of (n) (n) the sets Dη , Eeε . Thus, by (45)–(47), on the set C ∩ Dη ∩ Eeε for sufficiently large n we have (n) sup sup b S (t)[x, ∞) − S∗ (t)[x, ∞) ≤ ε.

(48)

0≤t≤T x∈R

Recall that P [C ∩ Dη ∩ Eeε ] ≥ 1−ε/2. Proceeding in a similar way we can find sets F (n) , n ≥ 1, such that P(F (n) ) ≥ 1−ε/2 for each n and (n) b (t)[x, ∞) − B∗ (t)[x, ∞) ≤ ε (49) sup sup B (n)

0≤t≤T x∈R

on F (n) for n sufficiently large. In particular, there exists n1 such that for n ≥ n1 on the set C ∩ Dη ∩ Eeε ∩ F (n) both (48) (n) and (49) hold. This, together with the fact that P [C ∩ Dη ∩ Eeε ∩ F (n) ] ≥ 1 − ε, implies (20). (n)

Acknowledgements I wish to acknowledge John Lehoczky, Kavita Ramanan and Steven Shreve for helpful discussions about this work during my visit to Carnegie Mellon, which was partially supported by the NSF grant DMS-0404682.

Ł. Kruk

References

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