Liquidity based modeling of asset price bubbles via random matching Francesca BiaginiAndrea MazzonThilo Meyer-Brandis

2025-05-02 0 0 4.08MB 37 页 10玖币

侵权投诉

Liquidity based modeling of asset price bubbles via random

matching

Francesca Biagini ∗Andrea Mazzon∗Thilo Meyer-Brandis∗

Katharina Oberpriller†

November 3, 2022

Abstract

In this paper we study the evolution of asset price bubbles driven by contagion eﬀects spreading

among investors via a random matching mechanism in a discrete-time version of the liquidity based

model of [25]. To this scope, we extend the Markov conditionally independent dynamic directed random

matching of [13] to a stochastic setting to include stochastic exogenous factors in the model. We derive

conditions guaranteeing that the ﬁnancial market model is arbitrage-free and present some numerical

simulation illustrating our approach.

Keywords: asset price bubbles, dynamic directed random matching with stochastic intensities, contagion,

liquidity

Mathematics Subject Classiﬁcation (2020): 60G07, 91G15, 91G30

JEL Classiﬁcation: C02, G10, G12

1 Introduction

The formation of asset price bubbles has been object of many investigations in the economic and mathe-

matical literature. Diﬀerent causes have been indicated as triggering factors for bubble birth and evolution,

such as a risk shifting problem in [2], the joint eﬀect of the individual incentive to time the market and the

inability of arbitrageurs to coordinate their selling strategies in [1], heterogenous beliefs between interacting

traders as in [17], [19], [33], [32], [40], [41], a disruption of the dynamic stability of the ﬁnancial system in

[8], [9], the diﬀusion of new investment decision rules from a few expert traders to a larger population of

amateurs in [15], the tendency of investors to adopt the behavior of other agents in [26], the presence of

short-selling constraints in [31] and of noise traders with erroneous stochastic beliefs in [11].

However, mathematical models for microﬁnancial interactions leading to the formation of asset price bubbles

are still missing. This paper aims at ﬁlling this gap by studying a random matching mechanism among

investors which impacts the trading volume of an asset and then its price via illiquidity eﬀects. To this

∗Workgroup Financial Mathematics, Department of Mathematics, Ludwig-Maximilians-Universit¨at M¨unchen, Theresienstr.

39, 80333 Munich, Germany.

†Department of Mathematics, University of Freiburg, Ernst-Zermelo-Str. 1, 79104 Freiburg im Breisgau, Germany.

arXiv:2210.13804v2 [q-fin.MF] 2 Nov 2022

purpose, we ﬁrst introduce a discrete time version of the liquidity based model of [25], where the fundamental

price of the asset is exogenously given, while the market price is inﬂuenced by the trading activities of investors

via an erosion of the limit order book. The birth of a bubble is then caused by a deviation of the market

value from the fundamental one.

Here, we model the signed volume of market orders by assuming that the investment attitudes of the traders

on the market are inﬂuenced via a random matching mechanism. To this scope, we suppose that agents on

the market can be of three types, i.e. optimistic, neutral and pessimistic regarding the future returns of the

asset, and that they trade according to their type. This means that an optimistic agent places a buy market

order while a pessimistic one places a selling market order. Neutral agents neither buy nor sell the asset.

The evolution of the signed volume of market orders is thus determined by the fraction of optimistic, neutral

and pessimistic agents, respectively.

In order to model the evolution of these quantities, we extend the Markov conditionally independent dynamic

directed random matching of [13] to a stochastic setting. More precisely, the model in [13] describes a

mechanism how a continuum of agents search in a directed way for a suitable counterparty. The word

“directed” refers to the fact that the search is not purely random, but the agents are motivated to meet

another agent that provides them with some beneﬁt. In particular, every agent is described by its type which

may change at any time step, and can randomly mutate to another type and randomly match with another

agent. This meeting may induce a further type change. Furthermore, agents can also enter some potentially

enduring partnerships with random break-up times.

These models have a broad application for example in the ﬁeld of ﬁnancial markets, monetary theory and

labor economics. The ﬁrst mathematical basis for this approach in a discrete time setting is provided in

[13] and strongly relies on techniques of non-standard analysis, as a continuum of agents is considered.

Given some deterministic functions describing the probabilities associated to the random matching and

random changes introduced above, they prove existence of a dynamical system with independent agents

types’ and deterministic cross-sectional distribution of types. We now extend this model by allowing the

probabilities driving the system to also depend on an additional state of the world to allow the random

matching mechanism to be driven by some stochastic exogenous factors. Hereby, the technical diﬃculty is

to ﬁnd a suitable setting to extend the results in [13] in a consistent way. For this purpose we construct a

probability space Ω as the product of the space ˆ

Ω of the random matching and the space ˜

Ω of the factors

which may inﬂuence the transition probabilities, and introduce a Markov kernel on Ω. After proving the

existence of such a dynamical system with input processes, we study conditional type distributions.

We then apply these results to model investment attitudes leading to bubble formation in the discretized

version of [25]. More precisely, we assume that the signed volume of market orders is described by a random

matching mechanism, where the agents can be of positive, negative or neutral type, as explained above. The

stochasticity of the transition probabilities is crucial here as it reﬂects the impact of heterogenous factors

such as socio-economic indicators, external events, public news. We are able to show that the market model

is arbitrage-free by proving the existence of an equivalent martingale measure under suitable assumptions.

Furthermore, we provide some examples for the input processes of the random matching mechanism in an

arbitrage-free market model. We illustrate these results with numerical simulations.

The paper is organized as follows. In Section 2 we introduce a discrete time version of the liquidity based

model of asset prices in [25]. In Section 3 we extend the directed random matching mechanism in [13]

to a stochastic setting. We combine these two constructions in Section 4, where we propose a model of

the signed volume of market orders inﬂuence by a random matching mechanism. In this setting we derive

some conditions guaranteeing that the ﬁnancial market model is arbitrage-free and we conclude with some

numerical simulations.

2 The formation of asset price bubbles

We consider a word-of-mouth mechanism spreading among investors who meet by random matching, giving

rise to the formation of asset price bubbles. To this scope, we introduce a discretized version of the liquidity-

based model in [25].

2.1 A liquidity-based model for asset price bubbles in discrete time

We here present a discrete time version of the continuous time model of [25], which explains the birth of

bubbles as the deviation of the market price Sfrom the fundamental price Fcaused by the impact of trading

volume and illiquidity.

Let T > 0 be a given trading horizon and consider a time discretization 0 =: t0< t1< ... < tN−1< tN=T

of the interval [0, T ]. Also introduce a ﬁltered probability space (Ω,F,(Fi)i=0,...,N , P ), where we set Fi:= Fti

for i= 1, ..., N. In Section 3 we further specify a possible construction of this space in the context of random

matching. The market model consists of the money market account B≡1 and one liquid ﬁnancial asset

(stock), which is traded through limit and market orders.

Remark 2.1. In order to be consistent with the notation of the random matching mechanism, see Section

3, we indicate the time tiwith a superscript ifor ﬁltrations or processes.

The fundamental price of the asset is given by the stochastic process F= (Fi)i=0,...,N , where Firepresents

the value at time tifor i= 0, ..., N. Such a process is exogenously given. On the other hand, the market

price of the asset is generated by the trading activity of the investors as we describe in the following.

Coherently with the construction of [25], we assume that the average price to pay per share for a transaction

of size xvia a market order at time tiis given by

Si(x) = Si+Mix, x ∈R+, i = 0, ..., N, (2.1)

where S= (Si)i=0,...,N and M= (Mi)i=0,...,N are non-negative, adapted processes on the space

(Ω,F,(Fi)i=0,...,N , P ), representing the quoted price and a measure of illiquidity, respectively. Fix a time ti

for i= 1, . . . , N. The limit order book at tiis described by the density function ρi(·), where ρi(z) is the

number of shares oﬀered at price zat time ti. As in [25], the total amount paid by a trader who wants to

buy xshares at time tiis given by

Zzx

zρi(z)dz, (2.2)

where zxis the solution of

Zzx

ρi(z)dz =x.

Due to the linear structure in (2.1) it follows that ρi(z) = 1/2Miand zx=Si+ 2Mix, see [25] for further

details.

Let X= (Xi)i=0,...,N be an adapted stochastic process representing the signed volume of aggregate market

orders (buy minus sell orders). Next, we introduce a process R= (Ri)i=0,...,N with values in [0,1] to describe

the short-term resiliency of the limit order book. In particular, if ∆Xbuy market orders are executed at time

ti,Rirepresents the proportion of new sell limit orders placed from tito ti+1, having therefore the eﬀect to

partly ﬁll the temporary gap [Si, Si+ ∆X] in the limit order book. If the gap caused by the new buy market

orders is not fully ﬁlled before other market orders are executed, the market price of the asset deviates from

the fundamental value, thus creating a bubble. However, it is observed that such a deviation decays in the

long run, see [25] for details. Such an eﬀect is quantiﬁed by the speed of decay process κ= (κi)i=0,...N .

The evolution of the market price process S= (Si)i=1,...,N is then given by

Si=Si−1+Fi−Fi−1−κi(Si−1−Fi−1)∆ti+ 2ΛiMi∆Xi, i = 1, . . . , N, (2.3)

where Λi:= 1 −Ri,i= 0, ..., N . Moreover, ∆ti:= ti−ti−1, ∆Xi:= Xi−Xi−1for i= 1, ..., N . At initial

time we have X0= 0 and S0=F0. In particular, (2.3) is a discretized version of the SDE considered in [25].

Following [25], we now provide the deﬁnition of an asset price bubble in this setting.

Deﬁnition 2.2. An asset price bubble β= (βi)i=0,...,N is deﬁned as

βi:= Si−Fi, i = 0, . . . , N.

By (2.3) we obtain that

βi=βi−1−κiβi−1∆ti+ 2ΛiMi∆Xi, i = 1, . . . , N, (2.4)

β0= 0.

The birth and the burst times of the bubble are identiﬁed by the stopping times

τ+:= t¯

lwith ¯

l:= inf{j= 0, ..., N :βj>0}∧T(2.5)

and

τ0:= t¯

kwith ¯

k:= inf{j= 0, ..., N :j≥¯

lin (2.5) such that βj= 0}∧T ,

respectively. We use here the convention inf ∅= +∞. Note that τ+is the ﬁrst time when the three process

Λ, Mand Xare diﬀerent from zero, see (2.4).

Deﬁnition 2.3. The market wealth process W= (Wi)i=0,...,N is deﬁned by

Wi=Di+Si1{ti<τ}+Fj1{tj=τ}, i = 0, . . . , N,

and the fundamental wealth process WF= (WF,i)i=0,...,N by

WF,i =Di+Fi, i = 0, . . . , N.

Note that

Wi−WF,i =Si−Fi=βi, i = 0, . . . , N.

Equation (2.4) shows that the main force driving the bubble evolution is the signed volume of market orders

X. We now focus on modeling Xby assuming that the investment attitudes of the traders on the market

are inﬂuenced via a random matching mechanism. To this scope, we suppose that agents on the market can

be of three types, i.e. optimistic, neutral and pessimistic regarding the future returns of the asset, and that

they trade according to their type. This means that an optimistic agent places a buy market order while

a pessimistic one places a selling market order. Neutral agents neither buy nor sell the asset. Based on

this characterization, from now on we refer to optimistic and pessimistic agents also as buyers and sellers,

respectively. We admit that agents may inﬂuence each other if they meet, and that they may change their

type at each time ti,i= 1, ..., N, via a random matching mechanism as we explain in Section 3. The evolution

of Xis determined by the processes pi= (pj

i)j=0,...,N ,i= 1,2,3, standing for the fraction of optimistic,

neutral and pessimistic agents, respectively. In particular, the value of Xat time tiis given by

Xi= Θi(pi

1−pi

3), i = 0, . . . , N, (2.6)

where Θ = (Θi)i=0,...,N is an adapted stochastic process modelling the average size of buy market orders as

in [6]. We assume that at time t0= 0 it holds p1

0=pi

3, i.e. that the fraction of optimistic agents is equal

to that of pessimistic ones, so that X0= 0. We now model the evolution of the fractions p1,p2and p3by

using a special case of the Markov conditionally independent dynamic directed random matching which we

introduce in the next section.

3 Markov Conditionally Independent dynamic directed random

matching

Consider a probability space (Ω,F, P ) representing all possible states of the world, on which we consider a

large economy. The space of agents is given by an atomless probability space (I, I, λ). Furthermore, it is

a common assumption that the agents also face some individual risks. The natural approach to take this

into account is to consider a random variable fon the product space (I×Ω,I ⊗ F) to a Polish space (Y, G)

which is essentially pairwise independent, see Deﬁnition 2 in [13].

Deﬁnition 3.1. Consider the random variable f: (I×Ω,I ⊗ F, λ ⊗P)→(Y, G),where Yis a Polish space

endowed with the Borelian σ-algebra G. We set fi:= f(i, ·) for all i∈I. We say that fis essentially pairwise

independent if for λ-almost all j∈I,fjis independent of fifor λ-almost all i∈I.

In Proposition 2.1 of [38] and Proposition 1.1 of [37] it is shown that an essentially pairwise independent

random variable, which is also jointly measurable, is constant for λ-almost all i∈I. This is the so called

“sample measurability problem”, which has been studied in [12], [27]. To overcome this issue, the σ-algebra

I⊗F needs to be enlarged to allow jointly measurable random variables to be essentially pairwise independent

but not constant. This measurability problem can be solved by working with an extension of the product

space (I×Ω,I ⊗ F, λ ⊗P) which still satisﬁes the Fubini property, see [38]. We here recall the deﬁnition of

a Fubini extension, see e.g. Deﬁnition 1 of [13].

Deﬁnition 3.2. A probability space (I×Ω,W, Q) is said to be a Fubini extension of the product probability

space (I×Ω,I ⊗ F, λ ⊗P) if for any real-valued Q-integrable random variable fon (I×Ω,W, Q) we have

that

1. the functions fi(·) := f(i, ·) and fω(·) := f(·, ω) are integrable on (Ω,F, P ) for λ-almost all i∈I, and

on (I, I, λ) for P-almost all ω∈Ω, respectively;

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

LiquiditybasedmodelingofassetpricebubblesviarandommatchingFrancescaBiagini*AndreaMazzon*ThiloMeyer-Brandis*KatharinaOberprillerNovember3,2022AbstractInthispaperwestudytheevolutionofassetpricebubblesdrivenbycontagioneectsspreadingamonginvestorsviaarandommatchingmechanisminadiscrete-timeversionofthe...

展开>> 收起<<

Liquidity based modeling of asset price bubbles via random matching Francesca BiaginiAndrea MazzonThilo Meyer-Brandis.pdf

共37页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Liquidity based modeling of asset price bubbles via random matching Francesca BiaginiAndrea MazzonThilo Meyer-Brandis

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: