corresponds to the total transaction costs given by ka = 1 ε

Transcription

1 ON THE ASYMPTOTIC BEHAVIOR OF A BOLTZMANN-TYPE PRICE FORMATION MODEL MARTIN BURGER, LUIS CAFFARELLI, PETER A. MARKOWICH, AND MARIE-THERESE WOLFRAM Abstract. In this paper we study the asymptotic behavior o a Boltzmann type price ormation model, which describes the tradin dynamics in a inancial market. In many o these markets tradin happens at hih requencies and low transactions costs. This observation motivates the study o the limit as the number o transactions k tends to ininity, the transaction cost a to zero and ka = const. Furthermore we illustrate the price dynamics with numerical simulations. 1. Introduction Accordin to O Hara [11] inancial markets are characterized by two unctions: irst by providin liquidity and second by acilitatin the price. The evolution o the price emeres rom the microscopic tradin strateies o the players and the tradin system considered. Hih requency tradin (HFT) is an automated tradin stratey, which is carried out by computers that place and withdraw orders within milli- or even microseconds. In 212 HFT accounted or approximately 52% o the overall US equity tradin volume. This paper is concerned with the asymptotic behavior o t (x,t) = xx (x,t)+ 1 ε () x t (x,t) = xx (x,t) 1 ε () x. (1.1a) (1.1b) as 1 ε. System (1.1) describes the distribution o buyers = (x,t) and vendors = (x,t) tradin a particular ood at prices x. The parameter 1 ε corresponds to the total transaction costs iven by ka = 1 ε, when considerin markets with hih transactions rates k and neliible transaction costs a in the limit k, a with ka = 1 ε = const. System (1.1) can be derived rom the ollowin price ormation process: Let us consider a lare number o vendors and a lare number o buyers tradin a speciic ood. I a buyer and a vendor aree on a price p = p(t) a transaction takes place. The price o this transaction is iven by a positive constant a R +. Ater the transaction, the buyer and vendor immediately switch places. Since the actual cost or the buyer is p(t)+a, he/she will sell the ood or at least that price. The proit or the vendor is p(t) a, hence he/she will try to buy the ood or a price lower than p(t) a. Based on the situation described above Lasry & Lions [8] proposed the ollowin Institute or Computational and Applied Mathematics, Einsteinstrasse 62, Münster, Germany (martin.burer@wwu.de). University o Texas at Austin, 1 University Station, C12, Austin, Texas , USA, (caarel@math.utexas.edu) 47 Kin Abdullah University o Science and Technoloy, Thuwal , Kindom o Saudi Arabia, (p.a.markowich@damtp.cam.ac.uk). 47 Kin Abdullah University o Science and Technoloy, Thuwal , Kindom o Saudi Arabia, (marie-therese.wolram@univie.ac.at) 1

2 2 On the asymptotic behavior o a Boltzmann-type price ormation model parabolic ree boundary price ormation model: t (x,t) = σ2 2 xx(x,t)+λ(t)δ(x p(t)+a) or x < p(t) and (x,t) = or x > p(t) t (x,t) = σ2 2 xx(x,t)+λ(t)δ(x p(t) a) or x > p(t) and (x,t) = or x < p(t). (1.2a) (1.2b) The unctions = (x,t) and = (x,t) denote the density o buyers and vendors and a R + the transaction costs. The areed price p = p(t) enters as a ree boundary and λ(t) = x (p(t),t) = x (p(t),t). Tradin events take only place at the price p = p(t), since the density o buyers and vendors is zero or prices smaller or larer than p(t). The Lasry & Lions model was analyzed in a series o papers, see [1, 2, 3, 4, 6]. Lasry & Lions motivated their model usin mean ield ame theory, but did not discuss its microscopic oriin. The lack o understandin system (1.2) on the microscopic level motivated urther research in this direction. In [1] we considered a simple aent based model with standard stochastic price luctuations and discrete tradin events. This Boltzmann-type price ormation (BPF) model reads as with initial data t (x,t) = σ2 2 xx(x,t) k(x,t)(x,t)+k(x+a,t)(x+a,t) (1.3a) t (x,t) = σ2 2 xx(x,t) k(x,t)(x,t)+k(x a,t)(x a,t). (1.3b) (x,) = I (x), (x,) = I (x), (1.3c) independent o k. In system (1.3) the parameter k denotes the transaction rate and σ the diusivity. The total number o transactions at a price x is iven by µ(x,t) = k(x,t)(x,t). (1.4) Kinetic models have proposed or several applications in socio-economic sciences, we reer to [14, 12, 9] or more inormation. One o the undamental dierences between (1.2) and (1.3) is the act that tradin events in the irst take only place at the price p = p(t). In BPF (1.3) a ood can be traded at any price, with a rate µ iven by (1.4). Then the mean, median and maximum o µ ives an estimate or the price. There is however a stron connection between the BPF model (1.3) and (1.2). We showed that solutions o (1.3) convere to solutions o (1.2) as the transaction rate k tends to ininity, see [1]. This indin motivated urther research on dierent asymptotic limits, or example by considerin hih tradin requencies and little transaction costs. This market behavior corresponds to the case k, a with ka = 1 ε. For studyin this limit rewrite system (1.3) as t (x,t) = 1 ()(x+a,t) ()(x,t) ε a t (x,t) = 1 ε + σ2 2 xx(x,t) (1.5a) ()(x a,t) ()(x,t) + σ2 a 2 xx(x,t). (1.5b)

3 We showed that (1.5) converes to Burer, Caarelli, Markowich and Wolram 3 t (x,t) = 1 ε ( ) x (x,t)+ σ2 2 xx (x,t) (1.6a) t (x,t) = 1 ε ( ) x (x,t)+ σ2 2 xx(x,t), (1.6b) with solutions = (x,t) and = (x,t) as k, a and ka = 1 ε. In this note we analyze the behavior o (1.6) as 1 ε and illustrate the results with numerical simulations. The note is oranized as ollows: in Section 2 we discuss the eneral structure o the BPF model. We identiy the limitin solutions o the Boltzmann price ormation model (1.6) in Section 3. Finally we illustrate the asymptotic behavior o solutions with numerical simulations in Section Structure o the Model We start by hihlihtin some structural aspects o (1.3), which also clariy certain steps in the previous analysis in [1, 2, 3]. The eneral understandin o the structure will serve as a basis or uture eneralizations and modiications and shall be used in the analysis o the asymptotic case later on. W.l.o.. we set σ = 1 throuhout this paper. Let L denote the dierential operator Lϕ = ϕ xx, and S and T the shit-operators (Sϕ)(x) = ϕ(x+a) respectively. Then system (1.3) becomes t +L = k(s I)(), t +L = k(t I)(). (T ϕ)(x) = ϕ(x a) (2.1a) (2.1b) In the settin o kinetic equations S and T are to be interpreted as the ain terms in the collision operators. A key property, which allows to derive heat equations or transormed variables, is that L commutes with the collision operators. Hence by deinin the ormal Neumann series F := (I S) 1 = S j and G := (I T ) 1 = T j, (2.2) we ind that j= F t +LF = k G t +LG = k. j= (2.3a) (2.3b) Then F G solves the heat equation. Note that this transormation was already used or the L&L model (1.2) in [2, 3] and serve as a key eature o the perormed analysis. There the authors motivated the transormation by the structure o the Dirac-δ terms rather than by invertin the collision operator. Note also that the computations above are purely ormal. Since S and T have norm equal to one, the converence o the Neumann series is not automatically uaranteed and needs to be veriied, see [1]. Moreover, also h = (I S)G p = (I T )F, (2.4a) (2.4b)

4 4 On the asymptotic behavior o a Boltzmann-type price ormation model solve the heat equation. This transormation was used, aain without the above interpretation in [1]. In the special case o the operators above, we have T = S 1 and in the L 2 scalar product even T = S, i.e. S and T are unitary operators. Then (I S)(I T ) 1 = (I S) S 1 = S, i.e. we simply have h = +S. Note that this structure was exploited in case o the Lasry & Lions model (1.2) in [2, 3] to derive a-priori estimates. 3. Asymptotic behavior when tradin with hih requencies In this Section we study the limitin behavior o system (1.1) as 1 ε. Throuhout this paper we make the ollowin assumptions. Let the initial datum I and I satisy: (A) I, I on and I, I S(), where S() is the Schwartz space. Next we reormulate (1.1) or the new variables h = + and u =, i.e. j= h t (x,t) h xx (x,t) =. u t (x,t) u xx (x,t) = 1 2ε (h2 u 2 ) x. (3.1a) (3.1b) System (3.1) can be considered either on the whole line = R or a bounded domain = ( 1,1). Note that the bounded interval corresponds to the shited and scaled interval (,p max ), where p max denotes the maximum price consumers are willin to pay. In the later case system (3.1) is supplemented with no lux boundary conditions o the orm h x = and u x = 1 2ε (h2 u 2 ) at x = ±1, (3.1c) which are equivalent to no-lux boundary conditions or (1.1). Throuhout this note we consider system (1.1) on the bounded domain with no-lux boundary conditions (3.1c) only. The assumption o a bounded price domain is motivated by the act, that the price o a ood has to be reater than zero and that there is a maximum price consumers are willin to pay within the inite time rane considered. Note that the total mass o buyers m and vendors m is conserved in time or no lux boundary conditions (3.1c), i.e. m := I (x)dx = (x,t)dx and m := I (x)dx = (x,t)dx or all t >. Proposition 3.1. Let ε > and I and I satisy (A) and = ( 1,1). Then system (3.1) has a unique smooth solution (h,u) L (,T ;L ()) 2. Furthermore u(x,t) 2 h(x,t) 2 or all (x,t) [,T ]. Proo. System (3.1) admits a smooth and bounded solution (h,u), see [5]. Furthermore the unctions and solve transport diusion equations, which preserve non-neativity. Trivially ( +) +, and thereore the inequality u 2 (x,t) h(x,t) 2 holds.

5 Burer, Caarelli, Markowich and Wolram 5 Remark 3.2. Note that the unction h = h(x,t) satisies the heat equation (3.1a) with Neumann boundary conditions. Its equilibrated solution corresponds to h eq (x) = const, or all x. Let h eq (x) = 1, then system (3.1) reduces to the viscous Burers equation u t (x,t) = u xx (x,t) 1 ε u(x,t)u x(x,t) u(x,) = u I (x) := I (x) I (x). (3.2a) (3.2b) The analytic behavior o the classical viscous Burers equation (with viscosity µ) or small viscosity in the lon-time limit was studied in [13, 7]. The authors showed that a reversal o the limitin passaes t and µ ives dierent limitin proiles. Note however that the time scalin o (3.2) is dierent. The ollowin a-priori estimates will be used to identiy the limitin solutions o (3.1). Lemma 3.1. Let ε >, = ( 1,1), T = ( 1,1) (,T ) and let the initial datum u I (x) satisy assumption (A). Then T u h = h and (h 2 u 2 ) dxdt 4ε(1+T )max h(x,t). x Proo. The irst estimate holds because o Proposition 3.1. The second one can be deduced rom the ollowin estimate o the irst order moment: d u(x,t)x dx = x(u xx (x,t)+ 1 dt 2ε (h2 u 2 (x,t)) x ) dx [ = (u x (x,t)+ 1 ] 2ε (h2 u 2 (x,t))) dx. Thereore we conclude T [ (h 2 u 2 (x,t)) dxdt = 2ε u I (x)x dx usin that u(x,t) h(x,t). From Lemma 3.1 ives u(x,t )x dx+ T 2ε[2 max x T h(x,t)+t ( u(1,t)+u( 1,t))] 4ε(1+T ) max x T h(x,t), u x (x,t) dxdt u 2 h 2 in L 1 ( (,T )). (3.3) Next we show that the unction u x can not have a jump up rom h to h. To do so we consider the unction v = u x, which satisies ] v t (x,t) v xx (x,t) = 1 2ε (h2 (x,t) u 2 (x,t)) xx = 1 [ (h(x,t)hx (x,t)) x v 2 (x,t) ] 1 ε ε u(x,t)v x(x,t). (3.4) Then the standard maximum principle implies that v 2 (x,t) > sup x (h(x,t)h x (x,t)) x and u x (x,t) max (sup (u x (x,)), max (,sup((hh x) x ))). x x x

6 6 On the asymptotic behavior o a Boltzmann-type price ormation model Thereore u = u(x,t) cannot have a jump upward and we deduce that the limitin unction can be written as { h(x,t) or x < p(t) u(x,t) = (3.5) h(x,t) or x > p(t), where p = p(t) denotes the position o the jump, i.e. the price o the traded ood. It is uniquely determined or all t > by m = p(t) 1 h(x,t) dx or, equivalently m = 1 p(t) h(x,t) dx. (3.6) The previous calculations lead to the ollowin theorem: Theorem 3.2. Let assumption (A) be satisied. Then there exist unique limitin unctions (u,h) o system (3.1) as ε, which are iven by { h(x,t) or x < p(t) u(x,t) = h(x,t) or x > p(t), where p = p(t) is determined by (3.6) and h = h(x,t) is the solution o the heat equation (3.1a) with homoeneous Neumann boundary conditions. Remark 3.3. The behavior o the price p = p(t) is determined by the conservation o mass. This implies that p(t) 1 m m = u(x,t) dx = h(x,t) dx h(x,t) dx. 1 p(t) Dierentiation respect to time yields = 2h(p(t),t)p (t)+2h x (p(t),t)) and thereore p (t) = h x(p(t),t) h(p(t),t). The unction h = h(x,t) solves the heat equation and converes exponentially ast to its steady state, iven by h(x,t) 1 1 h I (x) dx = m +m as t. This implies exponential converence o the price p = p(t), since p (t) = (lnh(p(t),t)) x. Remark 3.4. Note that the limit o the Lasry-Lions model (1.2) as the parameter a is the same as ε in the Boltzmann-type model (1.3), see also Fiure 3.1 or an illustration. 4. Numerical simulations In this last section we illustrate the behavior o the limitin system with numerical experiments. All simulations are perormed on the interval = [ 1,1] with no-lux boundary conditions (3.1c). We split the interval into N = 4 equidistant intervals or size x = System (3.1) is discretized usin a inite dierence discretization, i.e. ḣ i = 1 x 2 (h i+1 2h i +h i 1 ) u i = 1 x 2 (u i+1 2u i +u i 1 )+ 1 4ε x (h2 i+1 u 2 i+1 h 2 i 1 +u 2 i 1). (4.1a) (4.1b)

7 Burer, Caarelli, Markowich and Wolram 7 Boltzmann type model k Lasry & Lions k, a ε = 1 ka a Limitin equations Fi Asymptotic limits o the Lasry-Lions model (1.2), the Boltzmann-type model (1.3) and the limitin system studied in the paper (1.1) The resultin system o ODEs is solved usin an explicit 4th-order Rune-Kutta method (implemented within the GSL library). We illustrate the behavior o system (3.1) or an initial data I and I which is not well prepared in the sense o Lasry and Lions. In (1.2) the initial data has to satisy the ollowin conditions: I (x) > or all x < p(t) and I (x) = or all x > p(t). I (x) > or all x > p(t) and I (x) = or all x < p(t). Hence the unction u I = I I has a unique zero. An assumption that is not satisied or the initial uess depicted in Fiure 4.1(a). We choose the ollowin set o parameters ε = and σ =.1. The evolution o both unction is illustrated in Fiure 4.1. We observe the ast separation o and and the ormation o a unique interace, which corresponds to the price p = p(t) in time. This behavior is not unexpected since system (1.1) has a similar structure as classical separation or reaction-diusion models. Furthermore we observe a ast equilibration o the price p = p(t) in time, as discussed in Remark Conclusion In this paper we study the asymptotic behavior o a Boltzmann type price ormation model, which describes the tradin dynamics in a inancial market with hih tradin requencies and low transaction costs. We identiy the limitin solutions as the number o transactions tends to ininity and observe an exponentially ast equilibration o the price in time. Numerical simulations illustrate that uneconomic situations, like tradin at dierent prices, are corrected quickly. Hence we conclude that small luctuations in the tradin requency or the transaction costs inluence the price on a very short time scale only. REFERENCES [1] M. Burer, L. Caarelli, P. A. Markowich, and M.-T. Wolram, On a Boltzmann-type price ormation model, Proceedins o the Royal Society A: Mathematical, Physical and Enineerin Science, 469 (213). [2] L. A. Caarelli, P. A. Markowich, and J.-F. Pietschmann, On a price ormation ree boundary model by Lasry and Lions, C. R. Math. Acad. Sci. Paris, 349 (211), pp

8 8 On the asymptotic behavior o a Boltzmann-type price ormation model (a) t = (b) t = (c) t = (d) t = (e) t = () t = 3 Fi Evolution o the buyer and vendor density in the case o not-well prepared initial data [3] L. A. Caarelli, P. A. Markowich, and M.-T. Wolram, On a price ormation ree boundary model by Lasry and Lions: the Neumann problem, C. R. Math. Acad. Sci. Paris, 349 (211), pp [4] L. Chayes, M. d. M. González, M. P. Gualdani, and I. Kim, Global existence and uniqueness o solutions to a model o price ormation, SIAM J. Math. Anal., 41 (29), pp [5] L. Evans, Partial Dierential Equations, Graduate studies in mathematics, American Mathematical Society, 21. [6] M. d. M. González and M. P. Gualdani, Asymptotics or a ree boundary model in price ormation, Nonlinear Anal., 74 (211), pp [7] Y. J. Kim and A. E. Tzavaras, Diusive N-waves and metastability in the Burers equation, SIAM J. Math. Anal., 33 (21), pp (electronic). [8] J.-M. Lasry and P.-L. Lions, Mean ield ames, Jpn. J. Math., 2 (27), pp [9] D. Maldarella and L. Pareschi, Kinetic models or socio-economic dynamics o speculative markets, Physica A: Statistical Mechanics and its Applications, 391 (212), pp [1] P. A. Markowich, N. Matevosyan, J.-F. Pietschmann, and M.-T. Wolram, On a parabolic ree boundary equation modelin price ormation, Math. Models Methods Appl. Sci., 19 (29), pp [11] M. O Hara, Market Microstructure Theory, Wiley, Mar [12] L. Pareschi and G. Toscani, Interactin Multiaent Systems: Kinetic equations and Monte Carlo methods, OUP Oxord, 213. [13] J. Smoller, Shock waves and reaction-diusion equations, vol. 258 o Grundlehren der Mathematischen Wissenschaten [Fundamental Principles o Mathematical Sciences], Spriner- Verla, New York, second ed., [14] V. M. Yakovenko and J. B. Rosser, Colloquium: Statistical mechanics o money, wealth, and income, Rev. Mod. Phys., 81 (29), pp Acknowledement. MTW acknowledes support rom the Austrian Science Foundation FWF via the Hertha-Firnber project T456-N23.