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A Fractional Reaction-diffusion Description of Supply and Demand

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A Fractional Reaction-diffusion Description of Supply and Demand
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  A fractional reaction-diffusion description of supply and demand Michael Benzaquen 12 and Jean-Philippe Bouchaud 2 1 Ladhyx, UMR CNRS 7646, ´Ecole Polytechnique, 91128 Palaiseau Cedex, France 2 Capital Fund Management, 23 rue de l’Universit´e, 75007, Paris, FranceReceived: date / Revised version: date Abstract.  We suggest that the broad distribution of time scales in financial markets could be a crucialingredient to reproduce realistic price dynamics in stylised Agent-Based Models. We propose a fractionalreaction-diffusion model for the dynamics of latent liquidity in financial markets, where agents are veryheterogeneous in terms of their characteristic frequencies. Several features of our model are amenable to anexact analytical treatment. We find in particular that the impact is a concave function of the transactedvolume ( aka   the “square-root impact law”), as in the normal diffusion limit. However, the impact kerneldecays as  t − β with  β   = 1 / 2 in the diffusive case, which is inconsistent with market efficiency. In the sub-diffusive case the decay exponent  β   takes any value in [0 , 1 / 2], and can be tuned to match the empiricalvalue  β   ≈ 1 / 4. Numerical simulations confirm our theoretical results. Several extensions of the model aresuggested. 1 Introduction More than 50 years have passed since Montroll & Weiss[1] introduced the continuous-time random walk (CTRW)formalism to account for a broad variety of anomalousdiffusion mechanisms. Despite countless achievements inthe past decades, non-Gaussian diffusion is still topical inmany different fields such as statistical physics, condensedmatter physics or biology – as testified by the presentspecial issue of EPJB. In the present paper, we proposean srcinal application of fractional diffusion to describethe dynamics of supply and demand in financial markets.In the past few years, the concave nature of the impactof traded volume on asset prices – coined the “square-rootimpact law” – has made its way among the most firmlyestablished stylized facts of modern finance [2,3,4,5,6,7].Several attempts have been made to build theoretical mod-els that account for non-linear market impact, see  e.g. [8,9]. Following the ideas of T´oth  et al.  [4], the notion of a locally linear “latent” order-book model (LLOB) wasintroduced in [10,11]. The latter model builds upon cou-pled continuous  reaction-diffusion equations   for the dy-namics of the bid and the ask sides of the latent orderbook [10] and allows one to compute the price trajectoryconditioned to any execution profiles. In the slow execu-tion limit, the LLOB model was shown to match the lin-ear propagator model that relates past order flow to pricechanges through a power law decaying kernel.The propagator model was initially introduced in [12]to solve the so-called diffusivity puzzle: prices are approx-imately diffusive when the order flow is highly persistent[12,13,14]. In particular, the sign of the order-flow is char-acterised by an autocorrelation function  C  ( t ) that decaysas a power law  t − γ  with an exponent  γ <  1, defining a long-memory process  . Typically  γ   is found to be  ≈  0 . 5for individual stocks. Naively, the impact of these corre-lated orders should lead to a super-diffusive price dynam-ics, with a Hurst exponent  H   = 1 − γ/ 2  >  1 / 2. In orderto compensate for order flow correlation and restore pricediffusivity, the impact of each order, described by a cer-tain kernel or “propagator”, must itself decay as a powerlaw of time, with an exponent  β   = (1 − γ  ) / 2 [12,14].One major issue of the srcinal LLOB framework isthat its kernel decays with exponent  β   = 1 / 2, which can-not fulfil the above relation since  γ   must obviously bepositive. Impact relaxation is too quick for  β   = 1 / 2, andthe price dynamics generated by the LLOB model exhibitssignificant  mean reversion   on short to medium time scales,a feature that is not observed in empirical data. However,the current version of the LLOB model postulates thatmarket participants are homogeneous, in the sense thatthe distribution of volumes, reaction times, pricing up-dates, etc. are all  thin tailed  . For example, the cancellationof orders is assumed to be a Poisson process with a singlecancellation rate  ν  .In reality, different actors in financial markets are knownto be highly heterogeneous with very widely distributedvolumes and time scales, from High Frequency Traders(HFT) to large institutional investors. This observationsuggests various generalisations of the LLOB. The paththat we follow in this paper is that of a broad distribu-tion of time scales for agents’ intentions and re-evaluation,that naturally leads to a  fractional   reaction-diffusion pro-   a  r   X   i  v  :   1   7   0   4 .   0   2   6   3   8  v   2   [  q  -   f   i  n .   M   F   ]   3   0   A  u  g   2   0   1   7  2 Michael Benzaquen, Jean-Philippe Bouchaud: A fractional reaction-diffusion description of supply and demand cess. Other possible generalisations are discussed in theconclusions.The outline of the paper is as follows. We first presentthe general fractional diffusion framework with death andsources. We then present the fractional latent order-bookmodel (FLOB) and derive its equilibrium shape, in thecase of a balanced flow of buy/sell orders. We find thatthe characteristic V-shape of the latent order book is pre-served, leading to a concave impact law. We also showthat the corresponding propagator is now described by atunable exponent  β   ∈ [0 , 1 / 2] that resolves the above men-tioned diffusivity puzzle. We finally confront these resultsto numerical simulations. 2 Fractional diffusion with death and sources We here present the CTRW framework and adapt it tothe question of interest in this paper. The CTRW modeldescribes random walkers that pause for a certain wait-ing time before resuming their motion. When the averagewaiting time is finite, the long-time, large scale descriptionof an ensemble of such walkers is the standard diffusionequation. When the average waiting time diverges, thecorresponding dynamics is described by the  fractional dif- fusion equation   [15,16].Assuming that jump lengths and waiting times are in-dependent, let  Ψ  ( t ) denote the waiting time distributionfunction and  Λ ( x ) denote the jump length distributionfunction. The evolution equation for the density of walk-ers  φ ( x,t ) at position  x  and time  t  reads [17]: φ ( x,t ) =  Φ ( t ) φ ( x, 0) +    d x    [0 ,t ] d t  Λ ( x − x  ) Ψ  ( t  ) φ ( x  ,t − t  )+   [0 ,t ] d t  Φ ( t  ) s ( x,t − t  )  ,  (1)where  Φ ( t ) = 1 −   t 0  d t  Ψ  ( t  ) denotes the often called sur-vival probability function, and where  s ( x,t ) is a generalrate-source term allowing for the injection or removal of walkers (see Appendix A). Taking the Fourier-Laplace trans-form ( FL ) of Eq. (1) yields: φ ( k,p ) =  Φ (  p ) φ 0 ( k ) + Λ ( k ) Ψ  (  p ) φ ( k,p ) + Φ (  p ) s ( k,p )  ,  (2)where  φ 0 ( k ) :=  φ ( k,t  = 0), and where  pΦ (  p ) = 1 − Ψ  (  p ). We assume the jump lengths to have a zero meanand a finite second moment. In addition, considering thatwhen its time comes a random walker can either resumeits motion or disappear with some small probability  η ,we allow the distribution of jump lengths to be be non-normalized (    d xΛ ( x ) = 1 − η ). In the diffusion limit, thisis: Λ ( k ) ≈ 1 − σ 2 k 2 − η ,  (3)where  σ  denotes the root mean square of jump lengths.We assume waiting times to be distributed according to atruncated 1 power-law function with tail exponent  α <  1and cutoff   t c  =   − 1 , such that (see Appendix B): Ψ  (  p ) ≈ 1 − τ  α [(  p +  ) α −  α ]  ,  (4)where  τ   denotes the scale of waiting times. Note thatfor short times  t    t c  (equivalently  p     ) one has: Ψ  (  p ) ≈ 1 − τ  α  p α ≈ exp[ − τ  α  p α ] consistent with fractionaldiffusion (see  e.g.  [17]), while for long times  t  t c  (equiv-alently  p     )  Ψ  (  p )  ≈  1 − ˜ τp  where ˜ τ   =  α α − 1 τ  α is theaverage time between jumps, consistent with normal dif-fusion.Injecting Eqs. (3) and (4) into Eq. (2) and rearranging terms yields: φ ( k,p ) =  G α, ( k,p )  φ 0 ( k ) + s ( k,p )   ,  (5)with: G α, ( k,p ) =   p +  ωp ( k 2 + ϕ )(  p +  ) α −  α  − 1 ,  (6)where  ω  :=  σ 2 /τ  α and  ϕ  ˆ=  η/σ 2 . Equation (5) is centralas its inverse Fourier-Laplace transform allows to com-pute the evolution of the walkers density  φ ( x,t ) for giveninitial condition and source terms. Unfortunately the in-verse Fourier-Laplace transform of the kernel  G α,  is notanalytical in the general case. In the following we thusconsider the limit cases of short and long times (see Ap-pendix C for a presentation of the problem in terms of partial-differential and integro-differential equations).For  t  t c , we may note  G − α, ( k,p ) =  p α − 1 [  p α + ω ( k 2 + ϕ )] − 1 . Taking the inverse Fourier-Laplace transform, oneobtains that the kernel in real space is given by the inverseFourier transform of the Mittag-Leffler 2 function (see  e.g. [16,18]),  G − α, ( x,t ) = F  − 1 { F  α [ ω ( k 2 + ϕ ) t α ] } . Note that inthe limit  ϕ  →  0 the kernel  G − α, ( x,t ) can be convenientlywritten as: G − α, ( x,t ) = 1 √  4 πωt α  g α   x 2 4 ωt α  ,  ( ϕ → 0) ,  (7)where we introduced the function  g α , the shape of which isdiscussed in  e.g.  [18,15]. For t  t c , one obtains G + α, ( k,p ) =[  p  + ˜ ω ( k 2 +  ϕ )] − 1 where ˜ ω  ˆ=  σ 2 / ˜ τ  . Taking the inverseFourier-Laplace transform yields the normal diffusion ker-nel (with death): G + α, ( x,t ) =  e − νt √  4 π ˜ ωt exp  −  x 2 4˜ ωt  ,  (8) 1 For seasoned readers, truncating the distribution of waitingtimes may seem unusual in the context of fractional diffusionbut is here an essential ingredient for describing a stationaryorder book. 2 The Mittag-Leffler function  E  α ( z  ) =  F  α ( − z  ) is a spe-cial function defined by the following series as:  F  α ( z  ) =  ∞ j =0( − z ) j Γ  [1+ jα ] .  Michael Benzaquen, Jean-Philippe Bouchaud: A fractional reaction-diffusion description of supply and demand 3 where  ν   = ˜ ωϕ  is the rate of death per unit time. In bothcases, the general solution in real space can thus be writtenas the sum of a convolution death/diffusion term and asource contribution: φ ± ( x,t ) = [ G ± α, ∗ φ 0 ]( x,t ) + ( FL ) − 1  G ± α, ( k,p ) s ( k,p )  . (9) 3 Fractional latent order-book model Following the assumptions of Donier  et al.  [11,19], we positthat the dynamics of the intentions of market partici-pants 3 results from order cancellation or reassessment of their reservation price. Such intentions (which are com-monly called  latent orders  ) materialise into revealed or-ders in the vicinity the transaction price. The model iszero-intelligence in the sense that agents are heterogeneousand their reservation prices are updated at random. Thecrucial difference with the LLOB model of Donier  et al. [11] is that we now assume that such events occur aftera fat-tailed waiting time. 4 The distribution of these wait-ing times is assumed to decay with a power-law exponent α <  1 some until an exponential cutoff at  t c  =   − 1 (seeAppendix B). Note that while such a cutoff is indeed quiterealistic (nobody is expected to hold a position forever),it is also indispensable to ensure that the system does notage, which would prevent dynamic stationary states of thelatent order book.An important addition to the fractional diffusion equa-tion described in the previous section is the  reaction   mech-anism, that corresponds to transactions between buy andsell orders that remove volume from the latent order bookand set the transaction price. Within this framework, thedensity of buy  φ b ( x,t ) and sell  φ s ( x,t ) intentions at price x  and time  t  solve the set of coupled evolution equationsin the reference frame of the “consensus” price 5 , as givenby Eq. (1) with: s b ( x,t ) =  λΘ ( x t − x ) − R sb ( x ) (10) s s ( x,t ) =  λΘ ( x − x t ) − R sb ( x )  ,  (11)where  Θ  denotes the Heaviside step function, and where R sb ( x ) describes a reaction rate that instantaneously re-moves buy and sell “particles” as soon as they meet (see [17,20,21] 3 Note that given the very weak revealed instantaneous liq-uidity in the order book, intentions in the sense of latent ordersare an essential (and very sound) ingredient in microstructuremodelling. Latent order book models have allowed to to ac-count for important stylised facts such as the square root im-pact law (see  e.g.  [5,11]). 4 Let us stress that price reassessments themselves are notfat-tailed distributed,  i.e.  intentions do not follow Levy flights.While such an extension would be interesting, it is not thepoint of the present paper and is left for future work. 5 We here substantially simplify the discussion given in [11]where  x t  in fact follows some additional exogenous dynamics,reflecting the evolution of the agents’ expectations about the“consensus price”. See [19] for an extended discussion. for some work on fractional reaction-diffusion). Note thattransactions remove exactly the same volume of buy andsell orders, justifying the fact that the same rate  R sb ( x )appears in the two equations above. The term propor-tional to  λ  corresponds to an incoming flux of buy/sellintentions to the left/right of the transaction price  x t .The non-linearity arising from the reaction term in theabove equations can be abstracted by defining the combi-nation  ψ ( x,t ) =  φ b ( x,t ) − φ s ( x,t ), which precisely solvesEq. (1) with: s ( x,t ) =  λ sign( x t − x ) ,  (12)and where the transaction price  x t  is fixed by the condi-tion: ψ ( x t ,t ) = 0 .  (13)The stationary order-book centred at  x ∞  = 0 can be com-puted from Eqs. (9) and (12) as  ψ eq ( x ) = lim t →∞ ψ ( x,t ).Making use of Eqs. (9), (8) and (12), one obtains (see [11]): ψ eq ( x ) = − ( λ/ν  )sign( x )[1 − exp( −√  ϕ | x | )]. In the vicinityof the transaction price, the stationary order book can beshown to be locally linear and its local shape is given by: ψ eq ( x ) = −L x + O ( x 2 ) ,  (14)where  L  :=  λ √  ϕ/ν   =  λ ˜ τ/ ( σ 2 √  ϕ ). 4 Market impact In this section we compute and analyze the impact of ameta-order with execution horizon  T   on the transactionprice Following Donier  et al. , we introduce the meta-orderof volume  Q  as an extra order flow that falls exactly atthe transaction price such that the source term becomes: s ( x,t ) =  λ sign( x t − x ) + m t δ  ( x − x t )  ,  (15)where  m t  denotes the (possibly time dependent) executionrate, with   T  0  d tm t  =  Q . We set  t  = 0 to a situation inwhich the stationary order is well established and focuson the regime where  t < T    t c . 6 The general solution asgiven by Eq. (9) with the source term of Eq. (15) reads: ψ ( x,t ) = [ G − α, ∗ ψ 0 ]( x,t ) +    t 0 d um u G − α, ( x − x u ,t − u )+  iλπ  −    d kk    ∞ 0 d uF  α  ω ( k 2 + ϕ )( t − u ) α  e ik ( x − x u ) ,  (16)where  −    denotes Cauchy’s principal value.In the following we “zoom” into the linear region of the book, close to the transaction price. More precisely,we consider the limit  ϕ,λ → 0, while keeping  L  constant. 6 Note that in the limit  T     t c , one recorvers the normaldiffusion results of Donier  et al.  [11]  4 Michael Benzaquen, Jean-Philippe Bouchaud: A fractional reaction-diffusion description of supply and demand In this limit – and starting from the equilibrium book ψ 0 ( x ) =  ψ eq ( x ) – Eq. (16) becomes: ψ ( x,t ) = −L x +    t 0 d u m u   4 πω ( t − u ) α g α   ( x − x u ) 2 4 ω ( t − u ) α  . (17)Making use of Eq. (13) yields the following self-consistentintegral equation for the transaction price: x t  = 1 L    t 0 d u m u   4 πω ( t − u ) α g α  ( x t − x u ) 2 4 ω ( t − u ) α  .  (18)Provided that impact is small (or equivalently in the limitof small execution rates) one has ( x t − x u ) 2  4 ω ( t − u ) α ,which recovers the propagator limit where the transactionprice is linearly related to the order flow through a power-law decaying kernel: 7 x t  = √  π L Γ  [1 − α/ 2]    t 0 d u m u   4 πω ( t − u ) α ,  (19)allowing us to identify the propagator decay exponent  β  with min(1 / 2 ,α/ 2). Note that for  α <  1 the equality  β   =(1  −  γ  ) / 2 can be achieved by the choice  α  = 1  −  γ   ∈ [0 , 1]. Hence, the FLOB allows the price to be diffusiveat all times in the presence of a persistent order flow. Asmentioned in the introduction, real data suggests  γ   ≈ 0 . 5which implies  α ≈ 0 . 5. For a constant execution rate  m t  = m 0  =  Q/T   and denoting  I  Q  =  x T   − x 0  the impact of ameta-order of size  Q , one obtains: I  Q  =  Q 1 − α/ 2 L m α/ 20 (2 − α ) Γ  [1 − α/ 2] √  ω .  (20)As one can see, Eq. (20) leads to  I  Q  ∼  Q 0 . 75 for  α  =0 . 5, intermediate between a square-root and a linear be-haviour. The pure square-root for small execution rates isonly recovered in the limit  α  = 1 considered by Donier  et al.  [11].In the opposite limit however of fast execution – moreprecisely when ( x t − x u ) 2   4 ω ( t − u ) α (but still in theregime  t    t c ) – one can show that the impact is againgiven by a square root law (see Appendix D): I  Q  =  h ( α )   2 Q L  ,  (21)where  h ( α ) = (2  −  α ) − 1 / 2 ≤  1. Interestingly, the re-sult is smaller than what a purely geometric argumentwould suggest, where the volume initially contained be-tween  x 0  and  x T   =  x 0 + I  Q  is executed against the incom-ing metaorder. In the latter case one can write: Q  =    I  Q 0 d x L x  ⇒  I  Q  =   2 Q L  ,  (22) 7 Here, we have made use of the Taylor expansion of thefunction  g α ( y ) when  y → 0 (see [18,15]). valid for  α ≥ 1. When  α <  1, liquidity initially outside theinterval [ x 0 ,x T  ] manages to move inside that interval andmeet the incoming metaorder, even in the fast executionlimit. This provides more resistance to the metaorder,  i.e. a slightly smaller impact.The FLOB thus provides a framework in which con-cave impact is compatible with persistent order flow, al-though one expects a cross-over from a  Q 1 − α/ 2 behaviourfor “slow” execution to a  √    Q  behaviour for “fast” execu-tion. 8 The impact decay after the meta-order execution  t c   t > T  9 can be computed by replacing the upper boundaryof the integrals in Eqs. (18) and (19) by  T  . In the limit of small execution rates, one easily obtains: I  Q ( t > T  ) I  Q =   tT   1 − α/ 2 −  t − T T   1 − α/ 2 ,  (23)which decays with an infinite slope for  t → T  + and asymp-totically equals (1 − α/ 2)( t/T  ) − α/ 2 . In the limit of highexecution rates, the impact decay for  t c    t    T   can beconveniently computed as in the slow execution limit andreads: I  Q ( t  T  ) I  Q =  m α/ 20  Q (1 − α ) / 2   2(2 − α ) ω L α 1 − α/ 2 Γ  [1 − α/ 2]   tT   − α/ 2 , (24)which only differs from the low execution result by theprefactor. For  t → T  + the calculation is more subtle andrelies on the source-free relaxation of the linearised orderbook profile in the vicinity of the price at the end of themetaorder execution (see [11]). One obtains: I  Q ( t → T  + ) I  Q = 1 −   ω ( t − T  ) α I  Q z ∗  ,  (25)where z ∗  solves: a − z − ( a + − a − )   ∞ z  d ug α ( u 2 / 4)( u − z ) / √  4 π  =0 with  a ±  = lim  → 0 ∂  x ψ ( x T   ± ,T  ). 5 Numerical simulation In order to bolster our analytical results we performed anumerical simulation of the model. Because of the pecu-liar nature of fractional diffusion, such a simulation is moretime consuming than a regular reaction-diffusion simula-tion. This is because time needs be continuous and eachparticle (order intention) must be treated independently.Each particle is labeled as  buy  / sell  , and most importantly next event time   drawn from a fat-tailed probability distri-bution function. In order to speed up the simulation we use 8 On the definition of “slow” and “fast” for real marketswhere HFT significantly contribute, see [22]. 9 For  t  t c  one eventually recovers the 1 / √  t  impact decaypredicted by Donier  et al.  [11] in the diffusive limit.

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Dec 7, 2018
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