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A dynamical stochastic coupled model for financial markets
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Physica A 381 (2007) 317–328 www.elsevier.com/locate/physa
A dynamical stochastic coupled model for financial markets T.E. Govindana, Carlos Ibarra-Valdezb,, J. Ruiz de Cha´vezb a
Depto. de Matema´ticas, Escuela Superior de Fı´sica y Matema´ticas, IPN, 07300 Me´xico D.F., Mexico Depto. de Matema´ticas, Universidad Autonoma Metropolitana Iztapalapa, Apdo., Postal 55-534, 09340 Me´xico D.F., Mexico
b
Received 5 November 2006; received in revised form 20 February 2007 Available online 18 March 2007
Abstract A model coupling a deterministic dynamical system which represents trading, with a stochastic one that represents asset prices evolution is presented. Both parts of the model have connections with well established dynamic models in mathematical economics and finance. The main objective is to represent the double feedback between trading dynamics (the demand/supply interaction) and price dynamics (assumed as largely random). We present the model, and address to some extent existence and uniqueness, continuity with respect to initial conditions and stability of solutions. The non-Lipschitz case is briefly considered as well. r 2007 Elsevier B.V. All rights reserved. Keywords: Stochastic coupled model; Trading; Itoˆ equation; Lipschitz condition; Existence; Continuous dependence; Stability
1. Introduction 1.1. Objectives We present in this paper a model which, on the one hand, has as an aim to capture some essential features of a financial trading floor, linking them with ideas and models coming from classical economic theory and recent mathematical finance, and on the other hand, attempts to engage in the recent lines of development of econophysics, where many models which combine sound foundations with rich possibilities of simulation and model improvement is now a standard [1,2]. We write out a detailed version of our model including the careful consideration of assumptions, and give to some extent, foundations for existence and uniqueness, continuous dependence and stability of solutions. Some examples are presented, including one of the non-Lipschitz class. 1.2. Determinants of price formation The main factors that could influence and determine price levels and evolution—both of commodities and financial assets—can be classified roughly into the following five types: (i) The amount of social work necessary for manufacturing commodities. This is specifically connected with the level of technological development, and it can be regarded as largely random. Corresponding author. Tel.: +52 55 5804 4655; fax: +52 55 5804 4660.
E-mail address: [email protected] (C. Ibarra-Valdez). 0378-4371/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2007.03.014
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(ii) Demand/supply interaction in commodity and services markets. (iii) Demand/supply interaction within financial markets (trading). (iv) Corporate decisions and actions. due mainly to monopolies, oligopolies and governments. These actions take usually the technical form of control actions and constraints. (v) External shocks and several other types of uncertainties. In our model, we shall take into account to some extent, (i) and (v) as part of the stochastic component of the system, while (ii) and (iii) are included in the deterministic part. Part (iv) will be considered in a future work. 1.3. A brief qualitative description of the model Roughly speaking, our model consists of two coupled dynamical systems. The first one describes the trading dynamics and is given by a matrix system of ordinary deterministic differential equations which govern the dynamics of the portfolios of the agents in the market. There is a finite number of assets of which each portfolio is composed, and a finite number of agents. The state of the market is given by the matrix (generically non-square) formed by the portfolios holdings. The basic structure of this dynamical system is motivated in part by a famous model in general economic equilibrium and stability theory [3], and partially by some simpler models already published, which concern low dimensional systems designed for making simulations (see for example [4,5]). The first part of the system is assumed deterministic since many (if not most) of trading decisions are largely deterministic as we shall argue below. The deterministic dynamical system depends nevertheless on time, the matrix state of the market, and the asset prices. Prices of the market assets can be considered as ‘largely random’ [6] and therefore, the second component of the model describes price dynamics as given by a system of Itoˆ’s SDEs whose coefficients depend again on time, the state of the market and the prices. The basic structure is a generalization of the Karatzas–Samuelson model [7]. Finally, the whole system is a coupling of the deterministic and the stochastic dynamical systems, which give rise to a double feedback dynamics. One of the consequences of such a coupling is that despite the fact that trading decisions are deterministic, however their dynamics depend on prices which are intrinsically stochastic and hence the resulting decisions seem to an observer to be random as well. 1.4. Comparison with other models As we mentioned before, the Hahn-Negishi model [3] is purely deterministic and proposes a ‘price adjustment mechanism’ which is based on the supply/demand interaction and from which a stability result can be obtained. The Samuelson–Karatzas model [7] considers only the stochastic behavior of asset prices, and assumes Lipschitz condition for the coefficients of the system. In Ref. [8], Cox, Ingersoll and Ross developed an equilibrium model for the stochastic behavior of asset prices which are considered to be endogenously given. They also use Lipschitz-type conditions. Other interesting models are Xu-Shreve and Cvitanic-Karatzas who consider a market with several assets for solving a consumption-investment problem (see [7]). These latter works follow a ‘small investor’ approach which assumes that investor’s behavior does not affect market prices. The number of market participants reduces to one ‘representative agent’. The main contrasts of such works and others in the literature with our model are that we consider any number of assets, any number of agents, non-Lipschitz conditions to some extent, and the coupling between a deterministic dynamics (trading) and a stochastic dynamics (price evolution). 1.5. Organization of the paper The organization of the rest of the paper is the following: in Section 2 we describe the model in general mathematical terms; in Section 3 we give the mathematical set-up; in Section 4 we address the Lipschitzian
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case; in Section 5 the non-Lipschitzian case is addressed. We apply the results to some examples in Section 6. In Section 7, the paper concludes, summing up our contributions and stating also some lines for further research. The proofs of some of the main results are given in the Appendix. 2. The model We shall consider a market with n assets or securities (bonds, shares, options) i.e. risky and non-risky assets and m market participants (agents, traders) who exchange inside a trading floor. Let Si ðtÞ be the spot price of security i at time t, and SðtÞ ¼ ðS 1 ðtÞ; . . . ; Sn ðtÞÞ. Now let qji ðtÞ
ð1pipn; 1pjpmÞ
be the amount of security i held by trader j at time t. Denote by qj ¼ ðqj1 ; . . . ; qjn Þ the portfolio held by participant j, and by Q ¼ ½qji i¼1;...;n
j¼1;...;m
the state of the market. Now, let us make our assumptions explicit. Assumption 1. Trading activities are modeled as deterministic since traders take decisions that follow predetermined styles depending on price levels, trends, flows of information, etc., that is to say, they take decisions that depend on several types of random magnitudes but, the decision itself is non-random. For example, most traders would begin to sell a stock that has lost more than 50% of its unitary value in a day. The falling of a stock price is of a random nature, but the rule of selling in such circumstances is non-random. Another example (analogy) for this situation are the decisions about oil extracting rates which are clearly deterministic, in spite of the stochastic character of oil prices. Therefore, we shall model the dynamics of the trading floor as a sort of Hahn–Negishi dynamical system [3]:
q ji ðtÞ ¼ f ij ðt; QðtÞ; SðtÞÞ
ð1pipn; 1pjpmÞ,
or
QðtÞ ¼ F ðt; QðtÞ; SðtÞÞ, where F ¼ ½f ij . Assumption 2. Price dynamics is ‘largely random’ since it results from the interaction between two complex forces: (a) the trading dynamics which is deterministic in principle, but its behavior could be highly complex (see [9]), and (b) a multitude of factors such as information, external shocks, rumors, etc. (see [6]). Then, we model price dynamics as a system of stochastic differential equations of the Samuelson–Karatzas type [7] in a generalized fashion: dSi ðtÞ ¼ ai ðt; QðtÞ; SðtÞÞ dt þ
K X
bik ðt; QðtÞ; SðtÞÞ dW k ðtÞ
ð1pipnÞ,
k¼1
or in the matrix/vector form as dSðtÞ ¼ Aðt; QðtÞ; SðtÞÞ dt þ Bðt; QðtÞ; SðtÞÞ dW ðtÞ. Thus, our model is the coupling of both systems, that is
q ji ðtÞ ¼ f ij ðt; QðtÞ; SðtÞÞ
ð1pipn; 1pjpmÞ,
dS i ðtÞ ¼ ai ðt; QðtÞ; SðtÞÞ dt þ
K X k¼1
bik ðt; QðtÞ; SðtÞÞ dW k ðtÞ;
ð1pipnÞ,
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or in matrix/vector notation
QðtÞ ¼ F ðt; QðtÞ; SðtÞÞ, dSðtÞ ¼ Aðt; QðtÞ; SðtÞÞ dt þ Bðt; QðtÞ; SðtÞÞ dW ðtÞ. We shall also make the following standard assumptions, as made in the Black–Scholes [10] and other wellknown models: Assumption 3. Assets can be traded continuously and divided infinitesimally. Assumption 4. Assets pay no dividends. There are also no taxes or transaction costs in the market. Comment (Interpretation of the variables): The spot prices S 1 ðtÞ; . . . ; Sn ðtÞ of the financial assets or commodities have the usual financial economics meaning. They are the (stochastic) signals that the whole economy sends to agents by means of markets. They are assumed as correlated or not depending on the combinations of noises represented by the independent Brownian motions W 1 ðtÞ; . . . ; W K ðtÞ. Obviously several types of correlations between couples or groups of assets can be studied, that could give rise to interesting specific conclusions from the model dynamics. As regarding the market state variables, qij ðtÞ represents the holdings that agent j has of asset i at time t, while q ji ðtÞ is the instantaneous trend of agent j to sell (if this derivative is negative) or buy (if it is positive) an extra unit of asset i. The state of the market is given by all the instantaneous holdings of all the agents and is denoted by the matrix QðtÞ ¼ ½qij ðtÞnm . Agent j has as portfolio at time t, q j ðtÞ ¼ ðq1j ðtÞ; . . . ; qnj ðtÞÞ which represents his/her total holdings in the market. Also q ji ðtÞ measures his/her instantaneous trend, plan or program to sell or buy one extra unit of each one of the assets. The curve fq j ðtÞ : tX0g in Rn is the portfolio trajectory of agent j; while the curve fQðtÞ : tX0g in Rnm corresponds to the evolution of the state of the market, that is to say, the dynamical description of how the market allocates resources, by means of the produced prices and as a consequence of trading actions. Our main proposal with this model is that trading behavior is ‘largely deterministic’ and prices evolution is ‘largely random’ and that both dynamics are coupled, thus influencing each other and behaving stochastically as a whole. Remark 1 (On constraints). It is apparent that several constraints for the main dynamical variables must play a role: the total balance of available shares versus the sold and bought ones; the holdings of an agent versus the number of shares that she/he can sell; and finally the amount of each agent’s wealth which must be greater or equal than the price of what she/he is going to buy. In this work, we shall set aside the above constraints for avoiding too much technical complexity, and assume that there are no bounds for agents’ wealth, companies’ number of available shares, or agents’ possibilities of selling shares. This is, of course, a strong assumption which we hope to drop in a future work, but which can match to some extent with behavior in many markets: (a) companies can have an unbounded number of shares to sell by the device of issuing new shares periodically, or as far as they need or can; (b) agents which mostly are brokers and represent other people’s interests can be supposed to have an unbounded wealth because of the flows coming from given and new people that they represent, and something similar about the number of shares they can sell. 3. Mathematical formulation Let Rn be an n-dimensional Euclidean space with the norm defined as jxj2 ¼ Sni¼1 jxi j2 , where x ¼ ðx2 ; x2 ; . . . ; xn Þ is a vector in Rn , and Rnm denote the space of n m matrices with the norm defined as kAk2 ¼ Si Sj jaij j2 ¼ tr AAT , where A ¼ ½aij is a matrix and the transpose of a matrix A is denoted by AT . By a filtered probability space with increasing family of s-algebras which is denoted as ðO; F; P; fFt gtX0 Þ we mean a probability space ðO; F; PÞ with a system fFt gtX0 of sub-s-algebras of F such that Fs Ft , if sot. It is also called a ‘stochastic basis’.
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We denote by C ¼ C½0; T the Banach space of Rn -valued continuous functions defined on [0,T], and equipped with the usual supremum norm, i.e. kvkC ¼ sup0ptpT jvðtÞj; and by Cr ¼ Cr ½0; T, for r 2 N, the vector space of r-fold continuously differentiable functions. Consider the dynamics of the trading floor modeled by the following system of deterministic differential equations:
q ji ðtÞ ¼ f ij ðt; QðtÞ; SðtÞÞ;
t40;
ð1Þ
where f ij : Rþ Rnm Rn ! R. ð1pipn; 1pjpm; Rþ ¼ ½0; 1Þ and R ¼ ð1; 1ÞÞ are continuous, or in the matrix form as
QðtÞ ¼ F ðt; QðtÞ; SðtÞÞ;
t40,
Qð0Þ ¼ Q0 ,
ð2Þ
where QðtÞ ¼ ½qij ðtÞnm and F ¼ ½f ij nm are defined as F : Rþ Rnm Rn ! Rnm ; Q : Rþ ! Rnm . Consider also the price dynamics generated by the system of stochastic differential equations dS i ðtÞ ¼ ai ðt; QðtÞ; SðtÞÞ dt þ
K X
bik ðt; QðtÞ; SðtÞÞ dW k ðtÞ;
t40,
(3)
k¼1
where ai : Rþ Rnm Rn ! R; bik : Rþ Rnm Rn ! R;
i ¼ 1; 2; . . . ; n, k ¼ 1; 2; . . . ; K
and W k ðtÞ is a standard one-dimensional Wiener process. Let Aðt; Q; sÞ ¼ ðai ðt; Q; sÞÞ, 1pipn and Bðt; Q; sÞ ¼ ðbik ðt; Q; sÞÞ; 1pipn; 1pkpK be defined on Rþ Rnm Rn and continuous in ðt; Q; sÞ. Finally W ðtÞ ¼ ðW ðtÞ; . . . ; W K ðtÞÞ is a K-dimensional Wiener process with respect to the stochastic basis. We now consider the above component-wise system (3) as the following Itoˆ’s SDE: dSðtÞ ¼ Aðt; QðtÞ; SðtÞÞ dt þ Bðt; QðtÞ; SðtÞÞ dW ðtÞ; Sð0Þ ¼ S0 ,
t40, ð4Þ
where S 0 is a random variable independent of {W ðtÞ, tX0}. (For probabilistic details see [11]). We now introduce the solution concept for the coupled system (2) and (4). Definition 1. By a solution of the coupled system (2) and (4), we mean a coupled stochastic process fSðtÞ; QðtÞ; t 2 ½0; Tg for some 0oTo1, defined on the stochastic basis ðO; F; P; fFt gtX0 Þ such that: (i) QðtÞ and SðtÞ are Ft -adapted for each t 2 ½0; T and continuous in t almost surely (a.s.). (ii) EjSðtÞj2 o1 and EkQðtÞko1, for each t 2 ½0; T. (iii) SðtÞ satisfies the stochastic integral equation: Z t Z t SðtÞ ¼ S 0 þ Aðu; QðuÞ; SðuÞÞ du þ Bðu; QðuÞ; SðuÞÞ dW ðuÞ; a:s. 0
(5)
0
for each t 2 ½0; T, where the second integral is understood in the sense of Itoˆ’s stochastic integral [11,12], and
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(iv) QðtÞ satisfies the integral equation Z t QðtÞ ¼ Q0 þ F ðu; QðuÞ; SðuÞÞ du;
t40; a:s.
(6)
0
It is apparent from this definition that QðtÞ, the state of the market, is a stochastic process, despite the (assumed) determinism of the trading decisions. This is because QðtÞ depends on the random behavior of the prices.
4. Lipschitz case To begin with, in this section, we consider the problem of our interest, namely, the existence and stability problems using the classical Lipschitz and linear growth conditions on the non-linear terms. Hypotheses: (H1) (H2)
(Lipschitz condition) For all t 2 Rþ ¼ ½0; 1Þ, Q1 ; Q2 2 Rnm and s0 ; s00 2 Rn , it holds: kF ðt; Q1 ; s0 Þ F ðt; Q2 ; s00 ÞkpC½kQ1 Q2 k þ js0 s00 j, where C40 is a constant; jAðt; Q1 ; s0 Þ Aðt; Q1 ; s00 Þj þ kBðt; Q1 ; s0 Þ Bðt; Q2 ; s00 ÞkpC½kQ1 Q2 k þ js0 s00 j. (Linear growth condition) For all t 2 Rþ , Q 2 Rnm and s 2 Rn , kF ðt; Q; sÞkpCð1 þ kQk þ jsjÞ, jAðt; Q; sÞj2 þ kBðt; Q; sÞk2 pC 2 ð1 þ kQk2 þ jsj2 Þ.
Let us define the successive approximation procedure for the coupled system (5) and (6), for t 2 ½0; T as S 0 ðtÞ ¼ S 0 ,
Z
Z
t
S k ðtÞ ¼ S 0 þ
t
Aðu; Qk1 ðuÞ; S k1 ðuÞÞ du þ 0
Bðu; Qk1 ðuÞ; S k1 ðuÞÞ dW ðuÞ, 0
and Q0 ðtÞ Q0 ,
Z
Qk ðtÞ ¼ Q0 þ
t
F ðu; Qk1 ðuÞ; S k1 ðuÞÞ du, 0
for k ¼ 1; 2; . . . : Using the successive approximation procedure just introduced, we state our first result on the existence and uniqueness of a solution. Theorem 1. Assume that Aðt; Q; sÞ, Bðt; Q; sÞ and F ðt; Q; sÞ satisfy the hypotheses (H1)–(H2), and moreover EjS0 j2 o1 and EkQ0 ko1. Then there exists a unique solution fSðtÞ; QðtÞ; t 2 ½0; Tg to the coupled model, and 2 lim E sup jS k ðtÞ SðtÞj ¼ 0 and lim E sup kQk ðtÞ QðtÞk ¼ 0 k!1
hold, where
k!1
0ptpT
fSk ðtÞg1 k¼1
and
fQk ðtÞg1 k¼1
0ptpT
are the sequences of successive approximations as defined earlier.
The following result talks of the boundedness of the so-called price dynamics/process fSðtÞ : tX0g. Note that it is also possible to obtain a similar estimate of the process fQðtÞ : tX0g. This consideration applies as well in the rest of the paper. Theorem 2. Under the hypothesis (H1) and (H2), the solution SðtÞ of (4) satisfies EjSðtÞj2 pð1 þ EjS0 j2 ÞeCt ;
tX0
where C40 is a suitable constant.
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In other words, considering the initial condition S 0 as the input and the corresponding solution SðtÞ as the output, the above inequality in Theorem 2, yields the so-called well-known BIBO (bounded input and bounded output) stability. Further, defining this relation as a map: f : S 0 !S, one has the following result about the continuous dependence on the initial condition. Theorem 3. Under the hypothesis (H1) and (H2), the map f is continuous. Let Aðt; 0; 0Þ ¼ 0; Bðt; 0; 0Þ ¼ 0 and F ðt; 0; 0Þ ¼ 0 for all tX0. Then the equilibrium position or the trivial solution SðtÞ 0; QðtÞ 0, is the unique solution of the coupled system (2) and (4) with initial values S 0 ¼ Q0 ¼ 0. Next, let us address the stability issue in the following sense. Definition 2. The trivial solution of (4) is said to be exponentially stable in the quadratic-mean if EjSðtÞj2 pC 1 EjS 0 j2 eat ;
tX0,
where C 1 ; a40 are constants. Theorem 4. Let the hypotheses of Theorem 1 be satisfied. Suppose that there exists a function v : Rn ! R of class C2 such that it satisfies: (i) jvðsÞj þ jskvs ðsÞj þ jsj2 jvss ðsÞjpZjsj2 ; 8s 2 Rn and some Z40, (ii) LvðsÞ þ avðsÞp0; 8v 2 C2 ; a40, where L is defined as LvðsÞ ¼ ðvs ðsÞ; Aðt; Q; sÞÞ þ 12 tr½Bðt; Q; sÞWBT ðt; Q; sÞ, then EvðSðtÞÞpeat EvðS0 Þ. 5. Non-Lipschitz case To relax the Lipschitz assumption for the coefficients of the system is an important issue for at least two reasons: (a) several well-known models in interest rate theory involve non-Lipschitz coefficients [13], and (b) the Cobb–Douglas type functions, widely used across economic theory, and relevant to model trading behavior in more advanced terms, are also non-Lipschitz. However, the subject seems to have not been considered in the literature as it deserves. There are few works in these lines. First a series of works by Yamada and coauthors [14–16] address the problem, but with a quite abstract approach: as it is displayed below, they consider an abstract condition for substituting the Lipschitz one, a condition which depends on a dominant function (kðxÞ described below) but without being able to give examples of SDE’s satisfying such condition. We found by ourselves an unsurmountable difficulty—for now—in applying this approach to concrete financial models. Nevertheless it is a first mathematical step directed towards the solution of the problem and we shall make a quite brief presentation of part of their results. Second, there are some more recent results, remarkably Zhang & Fang [17] more biased to construct concrete examples of SDE satisfying the new (nonLipschitz) condition, but unfortunately not completely appropriate for applying to financial models. Finally, a paper by Deelstra & Delbaen [18] addresses the non-Lipschitz case for a specific family (Ho¨lder-type functions) in dimension one. We present below a part of Yamada’s approach, and in Example 2, Section 6 we take advantage of Deelstra & Delbaen’s result. 5.1. Yamada’s approach The following assumptions are based on Yamada [13].
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Hypotheses: (H3) Aðt; Q; sÞ and Bðt; Q; sÞ satisfy jAðt; Q1 ; s0 Þ Aðt; Q2 ; s00 Þj2 þ kBðt; Q1 ; s0 Þ Bðt; Q2 ; s00 Þk2 pkðkQ1 Q2 k2 þ js0 s00 j2 Þ, 8t 2 Rþ ; Q1 ; Q2 2 Rnm s0 ; s00 2 Rn ; wheren the norms in terms of the components are given by X (i) jAðt; Q ; s0 Þ Aðt; Q ; s00 Þj2 ¼ ja ðt; Q ; s0 Þ a ðt; Q ; s00 Þj2 , 1
i
2
i
1
2
i¼1
kBðt; Q1 ; s0 Þ Bðt; Q2 ; s00 Þk2 ¼
n X m X
jbij ðt; Q1 ; s0 Þ bij ðt; Q2 ; s00 Þj2 ,
i¼1 j¼1
and (ii) kðuÞ is a function defined on ½0; 1Þ that is continuous, non-decreasing and concave such that Z du ¼ þ1. þ kðuÞ 0 (H4) F satisfies conditions (H1) and (H2) as before, that is, Lipschitz and linear growth inequalities: kF ðt; Q1 ; s0 Þ F ðt; Q2 ; s00 ÞkpC½kQ1 Q2 k þ js0 s00 j kF ðt; Q; sÞkpCð1 þ kQk þ jsjÞ, where C40 is a constant. Based upon the above hypotheses, we have the next result which guarantees the existence and uniqueness of a solution. Theorem 5. Assume that Aðt; Q; sÞ, Bðt; Q; sÞ and F ðt; Q; sÞ satisfy the hypotheses (H3) and (H4) and EjS 0 j2 o1 and EkQ0 ko1. Then 2 lim E sup jS k ðtÞ SðtÞj ¼ 0 and lim E sup kQk ðtÞ QðtÞk ¼ 0 k!1
k!1
0ptpT
0ptpT
hold. A proof of this result is sketched in the Appendix. 6. Examples We shall consider here just two examples, one Lipschitz and one of the non-Lipschitz class. Example 1 (Two traders and two assets). The first trader has as goal to get certain levels of holdings in both assets:
q 11 ðtÞ ¼ A1 ða11 q11 ðtÞÞ;
q 12 ðtÞ ¼ A2 ða12 q12 ðtÞÞ.
The second trader has as goal to stabilize the asset prices about certain ‘fundamentals’:
q 21 ðtÞ ¼ B1 ðb11 S1 ðtÞÞ;
q 22 ðtÞ ¼ B2 ðb12 S2 ðtÞÞ.
The two assets are company shares both obeying a sort of ‘variable mean reverting’ diffussion process, and coupled to trading dynamics as follows: dS 1 ðtÞ ¼ ðm1 q11 ðtÞÞS1 ðtÞ dt þ s1 S 1 ðtÞ dW 1 ðtÞ, dS 2 ðtÞ ¼ ðm2 q22 ðtÞÞS2 ðtÞ dt þ s2 S 2 ðtÞ dW 2 ðtÞ,
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with W 1 ; W 2 being independent Brownian motions. We have in this example, existence, uniqueness and continuous dependence of solutions on initial conditions, by resorting to Theorems 1–3. Example 2 (One trader, one asset). Now we consider a whole family of examples, belonging to the nonLipschitz class. It is a family because asset price is labeled by two arbitrary functions of certain type (dð Þ and gð Þ below) which can represent quite different kinds of assets. The trader wants to get a certain level (q0 ) of holdings on the asset, and to keep the asset price about certain level (S 0 ). Then, the trader’s moves depend on the asset price evolution, but there is no double feedback, since the asset price does not depend on the trading dynamics. The equations governing this behavior are the following:
qðtÞ ¼ aðq0 qðtÞÞ þ bðS 0 SðtÞÞ dSðtÞ ¼ ð2bSðtÞ þ dðtÞÞ dt þ gðSðtÞÞ dW ðtÞ, with a; b40; bp0, the function g satisfies pffiffiffiffiffiffiffiffiffiffiffiffiffiffi jgðxÞ gðyÞjpc jx yj; ðc40Þ, and d : O Rþ ! Rþ is a measurable and adapted process with
Rt 0
dðuÞ duo1, for each t 2 Rþ .
The existence and uniqueness of positive solutions for the second (non-Lipschitz) equation is guaranteed by invoking a Deelstra & Delbaen result [18]. As trader’s dynamics concern, existence and uniqueness are straightforward. In other words substitute the solution SðtÞ so obtained into the first equation. Clearly, the first equation being linear in QðtÞ and SðtÞ, its coefficients are Lipschitz. Hence a unique solution process QðtÞ exists and is also Ft -adapted as SðtÞ is. Remark 2. In commodity markets and in many financial markets as well, there is always a considerably bigger number of market participants than that of commodities or assets, that is, mbn (for example, in USA economy the number of market participants is about 200 million, while the number of commodities is about 20 million). This seems to be a sort of economic behavior law, which has surprising theoretical consequences (see [9] ). On the other hand, the numerical simulation of models with a large number of agents (just more than two or three) is highly complicated from a computational point of view. This is a difficulty that should be dealt with, since it is likely that some real financial behavior could be adequately simulated by considering many agents. 7. Conclusions In this paper, we present a dynamical model for financial markets, which is a coupling of two dynamical systems: a deterministic one, governing the dynamics of the trading floor, and a stochastic one, governing asset prices evolution. Both systems are motivated by classical dynamic models in mathematical economics and mathematical finance, respectively. Our contributions in this work are the following: (a) the coupling of both systems, deterministic and stochastic, which we believe represents adequately an important feature of financial markets; (b) to consider both the Lipschitz and the non-Lipschitz case (to some extent) for the coefficients of the system. This is important for several reasons, the main one of which is that relevant SDEs models for the term structure of interest rates belong to the second category; (c) call attention to the question of stability of the coupled system. This is a subject that has been treated extensively in commodity markets within the framework of general economic equilibrium, but, as far as we know, has been almost not considered at all in mathematical finance. We found only one reference [19] which does not belong to the mainstream of financial research; finally, (d) we gave a general set-up, which allows on one hand, to construct specific examples appropriate for numerical simulations, and on the other hand to consider deeper theoretical questions, including some of a more economical nature, since the trading dynamics can also be used to represent simple economic exchange.
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7.1. Scope for further research Several questions can be addressed for the future work. On one hand a clarification of the technical questions involved in the non-Lipschitz case, enough to cover some of the main models in interest rate dynamics has an important place in any research agenda. On the other hand, extensions of our model include the considerations of the constraints mentioned in the remark after the statement of the assumptions of the model. This could be a difficult task, since some of them are of a stochastic nature and the available tools such as those coming from stochastic control theory, only consider constraints defined by a given deterministic set. Other important question is the consideration of delays, specially for the trading decisions which are to be taken up some time after the information of spot prices is made available to the traders. Since regulatory authorities play an important role for the whole outputs of a trading floor dynamics, the introduction of controls both in the trading decisions and in the price behavior is a natural question to be explored. Another class of questions that can be posed are of a theoretical-financial nature, such as, possibilities of arbitrage and completeness of the markets. Acknowledgment The second author acknowledges fruitful conversations with J. Alvarez-Ramirez, Sergio Herna´ndez, Myriam Cisneros and Esteban Martina on the subject. Appendix. Proofs of some of the main results We shall sketch two proofs (those of Theorems 1 and 5). It could be useful even for the reader acquainted with stochastic calculus techniques, since it is shown how to deal with the coupled system within standard lines of reasoning. Proof of Theorem 1 (Sketch). Let T be an arbitrary positive number 0oTo1. Consider the Lipschitz condition jAðt; Q1 ; s0 Þ Aðt; Q2 ; s00 Þj þ kBðt; Q1 ; s0 Þ Bðt; Q2 ; s00 ÞkpC½kQ1 Q2 k þ js0 s00 j.
(7)
But, Z Q1 ðtÞ Q2 ðtÞ ¼
t
½F ðu; Q1 ðuÞ; S1 ðuÞÞ F ðu; Q2 ðuÞ; S 2 ðuÞÞ du. 0
Therefore, by exploiting the Lipschitz condition on F, we get Z t kQ1 ðtÞ Q2 ðtÞkp C½kQ1 ðuÞ Q2 ðuÞk þ jS 1 ðuÞ S 2 ðuÞj du 0 Z t Z t jS 1 ðuÞ S 2 ðuÞj du þ C kQ1 ðuÞ Q2 ðuÞk du. pC 0
0
An application of Bellman–Gronwall’s Lemma [12] then yields Z t Z t Z t CðttÞ kQ1 ðtÞ Q2 ðtÞkpC jS 1 ðuÞ S 2 ðuÞj du þ C e C jS1 ðuÞ S2 ðuÞj du dt 0 0 0 Z t pCT kS1 S 2 kC þ C 2 TkS1 S 2 kC eCðtuÞ du. 0
Hence, Z kQ1 ðtÞ Q2 ðtÞkp CT þ C 2 T
T 0
eCðtuÞ du kS1 S 2 kC .
(8)
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Using (8) in (7) we have jAðt; Q1 ; s0 Þ Aðt; Q2 ; s00 Þj þ kBðt; Q1 ; s0 Þ Bðt; Q2 ; s00 Þk Z T 2 CðtuÞ e du ks0 s00 kC þ Cks0 s00 kC pC CT þ C T 0 Z T pC CT þ C 2 T eCðtuÞ du þ 1 ks0 s00 kC .
ð9Þ
0
Next, consider the condition jAðt; Q; sÞj2 þ kBðt; Q; sÞk2 pC 2 ð1 þ kQk þ jsj2 Þ.
(10)
Using the linear growth condition on F, we have Z t kF ðt; QðuÞ; SðuÞÞk du kQðtÞkpkQ0 k þ 0 Z t Cð1 þ kQðuÞk þ kSðuÞkÞ du pkQ0 k þ 0 Z t Z t kQðuÞk du þ C kSðuÞk du. pkQ0 k þ Ct þ C 0
0
Rt
Letting hðtÞ ¼ kQ0 k þ Ct þ C 0 kSðuÞk du; we have Z t kQðuÞk du. kQðtÞkphðtÞ þ C 0
Another application of Bellman–Gronwall’s Lemma yields Z t kQðtÞkphðtÞ þ C eCðtuÞ hðuÞ du; t 2 ½0; T 0 Z t eCðtuÞ ðkQ0 k þ Cu þ CukskC Þ du pkQ0 k þ CT þ CTkskC þ C 0 Z t eCðtuÞ kQ0 k du pkQ0 k þ CT þ CTkskC þ C 0 Z t Z t CðtuÞ e Cu du þ Ckske eCðtuÞ Cu du þC 0
0
pL þ CðT þ CeCT kske pðM þ 1ÞkskC ,
ð11Þ
where M ¼ maxfL; CðT þ CeCT Þg and L ¼ kQ0 k þ CT þ eCT ð1 þ CÞ. Using (11) in (10), we get jAðt; Q; sÞj2 þ kBðt; Q; sÞk2 pC 2 ð1 þ ðM þ 1Þ2 ksk2C þ ksk2C Þ pC 2 ½1 þ ððM þ 1Þ2 þ 1Þksk2C pC 21 ½1 þ ksk2C , where C 21 ¼ maxfC 2 ; ðM þ 1Þ2 þ 1g.
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Hence, the conditions of the theorem boil down to the standard Lipschitz and linear growth conditions uniformly in Q 2 Rnm , (see [12, p. 105, or 20]). Then, the existence and uniqueness of a solution follows on standard lines. & Proof of Theorem 5 (Sketch). Consider the hypothesis (H3): jAðt; Q1 ; s0 Þ Aðt; Q2 ; s00 Þj2 þ kBðt; Q1 ; s0 Þ Bðt; Q2 ; s00 Þk2 pkðkQ1 Q2 k2 þ js0 s00 j2 Þ. Since kðuÞ is non-decreasing by assumption, it follows from (8) that: jAðt; Q1 ; s0 Þ Aðt; Q2 ; s00 Þj2 þ kBðt; Q1 ; s0 Þ Bðt; Q2 ; s00 Þk2 pkðL21 ks0 s00 k2C þ js0 s00 Þk2 pkððL21 þ 1Þks0 s00 k2C Þ, where 2
Z
L1 ¼ CT þ C T
T
eCðtuÞ du,
0
uniformly in Q1 ; Q2 2 Rnm . This verifies that the hypothesis (H3) satisfies Yamada [14] condition A, uniformly in Q1 ; Q2 2 Rnm . Hence, the existence and uniqueness of a unique solution fSðtÞ; QðtÞg to the coupled model (2) and (4) follows on similar lines as in Yamada [14]. & References [1] R. Mantegna, H.E. Stanley, An Introduction to Econophysics: Correlations and Complexity in Finance, Cambridge University Press, UK, 2000. [2] J.L. MacCauley, Dynamics of Markets. Econophysics and Finance, Cambridge University Press, UK, 2004. [3] F.H. Hahn, T. Negishi, A theorem of non-tatoˆnnement stability, Econometrica 30 (3) (1962) 463–469. [4] J. Alvarez-Ramirez, C. Ibarra-Valdez, Modeling stock market dynamics based on conservation principles, Physica A 301 (2003) 493–511. [5] T. Lux, M. Marchesi, Volatility clustering in financial markets: a microsimulation of interacting agents, Int. J. Theor. Appl. Finance 3 (4) (2000) 675–702. [6] P.A. Samuelson, Mathematics of speculative price, Mathematical Topics in Economic Theory and Computation, SIAM, Philadelphia, 1965. [7] I. Karatzas, Lectures on the mathematics of finance, CRM Monograph Series, 8, American Mathematical Society, Providence, RI, 1997. [8] J.C. Cox, J.E. Ingersoll, S.A. Ross, An intertemporal general equilibrium model of asset prices, Econometrica 53 (2) (1985) 363–384. [9] D. Saari, Mathematical complexity of simple economics, Notices Amer. Math. Soc. 42 (1995) 222–230. [10] F. Black, M. Scholes, The pricing of options and corporate liabilities, J. Political Econ. 81 (1973) 637–659. [11] D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, third ed., Springer, Berlin, 1999. [12] L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley-Interscience [John Wiley & Sons], New York-LondonSydney, 1974, (Translated from the German). [13] S.E. Shreve, Stochastic Calculus for Finance. II. Continuous-time Models, Springer Finance, Springer, New York, 2004. [14] T. Yamada, On the successive approximation of solutions of stochastic differential equations, J. Math. Kyoto Univ. 21–3 (1981) 501–515. [15] S. Watanabe, T. Yamada, On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ. 11 (1971) 155–167. [16] S. Watanabe, T. Yamada, On the uniqueness of solutions of stochastic differential equations II, J. Math. Kyoto Univ. 11 (1971) 553–563. [17] T. Zhang, S. Fang, A study of a class of stochastic differential equations with non-Lipschitzian coefficients, Probab. Theory Relat. Fields 132 (2005) 356–390. [18] G. Deelstra, F. Delbaen, Existence of solutions of stochastic differential equations related to the Bessel process, Selected papers, Homepage of Prof. Freddy Delbaen: hhttp://www.math.ethz.ch/ delbaen/ftp/preprints/existenti_rev.pdfi. [19] A.V. Svishchuk, A.V. Kalemanova, The stochastic stability of interest rates with jumps changes, Theor. Probab. Math. Stat. (61) (2000) 161–172. [20] B. Oksendal, Stochastic Differential Equations an Introduction with Applications, sixth ed., Universitext, Springer, Berlin, 2003.
Physica A 381 (2007) 317–328 www.elsevier.com/locate/physa
A dynamical stochastic coupled model for financial markets T.E. Govindana, Carlos Ibarra-Valdezb,, J. Ruiz de Cha´vezb a
Depto. de Matema´ticas, Escuela Superior de Fı´sica y Matema´ticas, IPN, 07300 Me´xico D.F., Mexico Depto. de Matema´ticas, Universidad Autonoma Metropolitana Iztapalapa, Apdo., Postal 55-534, 09340 Me´xico D.F., Mexico
b
Received 5 November 2006; received in revised form 20 February 2007 Available online 18 March 2007
Abstract A model coupling a deterministic dynamical system which represents trading, with a stochastic one that represents asset prices evolution is presented. Both parts of the model have connections with well established dynamic models in mathematical economics and finance. The main objective is to represent the double feedback between trading dynamics (the demand/supply interaction) and price dynamics (assumed as largely random). We present the model, and address to some extent existence and uniqueness, continuity with respect to initial conditions and stability of solutions. The non-Lipschitz case is briefly considered as well. r 2007 Elsevier B.V. All rights reserved. Keywords: Stochastic coupled model; Trading; Itoˆ equation; Lipschitz condition; Existence; Continuous dependence; Stability
1. Introduction 1.1. Objectives We present in this paper a model which, on the one hand, has as an aim to capture some essential features of a financial trading floor, linking them with ideas and models coming from classical economic theory and recent mathematical finance, and on the other hand, attempts to engage in the recent lines of development of econophysics, where many models which combine sound foundations with rich possibilities of simulation and model improvement is now a standard [1,2]. We write out a detailed version of our model including the careful consideration of assumptions, and give to some extent, foundations for existence and uniqueness, continuous dependence and stability of solutions. Some examples are presented, including one of the non-Lipschitz class. 1.2. Determinants of price formation The main factors that could influence and determine price levels and evolution—both of commodities and financial assets—can be classified roughly into the following five types: (i) The amount of social work necessary for manufacturing commodities. This is specifically connected with the level of technological development, and it can be regarded as largely random. Corresponding author. Tel.: +52 55 5804 4655; fax: +52 55 5804 4660.
E-mail address: [email protected] (C. Ibarra-Valdez). 0378-4371/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2007.03.014
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(ii) Demand/supply interaction in commodity and services markets. (iii) Demand/supply interaction within financial markets (trading). (iv) Corporate decisions and actions. due mainly to monopolies, oligopolies and governments. These actions take usually the technical form of control actions and constraints. (v) External shocks and several other types of uncertainties. In our model, we shall take into account to some extent, (i) and (v) as part of the stochastic component of the system, while (ii) and (iii) are included in the deterministic part. Part (iv) will be considered in a future work. 1.3. A brief qualitative description of the model Roughly speaking, our model consists of two coupled dynamical systems. The first one describes the trading dynamics and is given by a matrix system of ordinary deterministic differential equations which govern the dynamics of the portfolios of the agents in the market. There is a finite number of assets of which each portfolio is composed, and a finite number of agents. The state of the market is given by the matrix (generically non-square) formed by the portfolios holdings. The basic structure of this dynamical system is motivated in part by a famous model in general economic equilibrium and stability theory [3], and partially by some simpler models already published, which concern low dimensional systems designed for making simulations (see for example [4,5]). The first part of the system is assumed deterministic since many (if not most) of trading decisions are largely deterministic as we shall argue below. The deterministic dynamical system depends nevertheless on time, the matrix state of the market, and the asset prices. Prices of the market assets can be considered as ‘largely random’ [6] and therefore, the second component of the model describes price dynamics as given by a system of Itoˆ’s SDEs whose coefficients depend again on time, the state of the market and the prices. The basic structure is a generalization of the Karatzas–Samuelson model [7]. Finally, the whole system is a coupling of the deterministic and the stochastic dynamical systems, which give rise to a double feedback dynamics. One of the consequences of such a coupling is that despite the fact that trading decisions are deterministic, however their dynamics depend on prices which are intrinsically stochastic and hence the resulting decisions seem to an observer to be random as well. 1.4. Comparison with other models As we mentioned before, the Hahn-Negishi model [3] is purely deterministic and proposes a ‘price adjustment mechanism’ which is based on the supply/demand interaction and from which a stability result can be obtained. The Samuelson–Karatzas model [7] considers only the stochastic behavior of asset prices, and assumes Lipschitz condition for the coefficients of the system. In Ref. [8], Cox, Ingersoll and Ross developed an equilibrium model for the stochastic behavior of asset prices which are considered to be endogenously given. They also use Lipschitz-type conditions. Other interesting models are Xu-Shreve and Cvitanic-Karatzas who consider a market with several assets for solving a consumption-investment problem (see [7]). These latter works follow a ‘small investor’ approach which assumes that investor’s behavior does not affect market prices. The number of market participants reduces to one ‘representative agent’. The main contrasts of such works and others in the literature with our model are that we consider any number of assets, any number of agents, non-Lipschitz conditions to some extent, and the coupling between a deterministic dynamics (trading) and a stochastic dynamics (price evolution). 1.5. Organization of the paper The organization of the rest of the paper is the following: in Section 2 we describe the model in general mathematical terms; in Section 3 we give the mathematical set-up; in Section 4 we address the Lipschitzian
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case; in Section 5 the non-Lipschitzian case is addressed. We apply the results to some examples in Section 6. In Section 7, the paper concludes, summing up our contributions and stating also some lines for further research. The proofs of some of the main results are given in the Appendix. 2. The model We shall consider a market with n assets or securities (bonds, shares, options) i.e. risky and non-risky assets and m market participants (agents, traders) who exchange inside a trading floor. Let Si ðtÞ be the spot price of security i at time t, and SðtÞ ¼ ðS 1 ðtÞ; . . . ; Sn ðtÞÞ. Now let qji ðtÞ
ð1pipn; 1pjpmÞ
be the amount of security i held by trader j at time t. Denote by qj ¼ ðqj1 ; . . . ; qjn Þ the portfolio held by participant j, and by Q ¼ ½qji i¼1;...;n
j¼1;...;m
the state of the market. Now, let us make our assumptions explicit. Assumption 1. Trading activities are modeled as deterministic since traders take decisions that follow predetermined styles depending on price levels, trends, flows of information, etc., that is to say, they take decisions that depend on several types of random magnitudes but, the decision itself is non-random. For example, most traders would begin to sell a stock that has lost more than 50% of its unitary value in a day. The falling of a stock price is of a random nature, but the rule of selling in such circumstances is non-random. Another example (analogy) for this situation are the decisions about oil extracting rates which are clearly deterministic, in spite of the stochastic character of oil prices. Therefore, we shall model the dynamics of the trading floor as a sort of Hahn–Negishi dynamical system [3]:
q ji ðtÞ ¼ f ij ðt; QðtÞ; SðtÞÞ
ð1pipn; 1pjpmÞ,
or
QðtÞ ¼ F ðt; QðtÞ; SðtÞÞ, where F ¼ ½f ij . Assumption 2. Price dynamics is ‘largely random’ since it results from the interaction between two complex forces: (a) the trading dynamics which is deterministic in principle, but its behavior could be highly complex (see [9]), and (b) a multitude of factors such as information, external shocks, rumors, etc. (see [6]). Then, we model price dynamics as a system of stochastic differential equations of the Samuelson–Karatzas type [7] in a generalized fashion: dSi ðtÞ ¼ ai ðt; QðtÞ; SðtÞÞ dt þ
K X
bik ðt; QðtÞ; SðtÞÞ dW k ðtÞ
ð1pipnÞ,
k¼1
or in the matrix/vector form as dSðtÞ ¼ Aðt; QðtÞ; SðtÞÞ dt þ Bðt; QðtÞ; SðtÞÞ dW ðtÞ. Thus, our model is the coupling of both systems, that is
q ji ðtÞ ¼ f ij ðt; QðtÞ; SðtÞÞ
ð1pipn; 1pjpmÞ,
dS i ðtÞ ¼ ai ðt; QðtÞ; SðtÞÞ dt þ
K X k¼1
bik ðt; QðtÞ; SðtÞÞ dW k ðtÞ;
ð1pipnÞ,
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or in matrix/vector notation
QðtÞ ¼ F ðt; QðtÞ; SðtÞÞ, dSðtÞ ¼ Aðt; QðtÞ; SðtÞÞ dt þ Bðt; QðtÞ; SðtÞÞ dW ðtÞ. We shall also make the following standard assumptions, as made in the Black–Scholes [10] and other wellknown models: Assumption 3. Assets can be traded continuously and divided infinitesimally. Assumption 4. Assets pay no dividends. There are also no taxes or transaction costs in the market. Comment (Interpretation of the variables): The spot prices S 1 ðtÞ; . . . ; Sn ðtÞ of the financial assets or commodities have the usual financial economics meaning. They are the (stochastic) signals that the whole economy sends to agents by means of markets. They are assumed as correlated or not depending on the combinations of noises represented by the independent Brownian motions W 1 ðtÞ; . . . ; W K ðtÞ. Obviously several types of correlations between couples or groups of assets can be studied, that could give rise to interesting specific conclusions from the model dynamics. As regarding the market state variables, qij ðtÞ represents the holdings that agent j has of asset i at time t, while q ji ðtÞ is the instantaneous trend of agent j to sell (if this derivative is negative) or buy (if it is positive) an extra unit of asset i. The state of the market is given by all the instantaneous holdings of all the agents and is denoted by the matrix QðtÞ ¼ ½qij ðtÞnm . Agent j has as portfolio at time t, q j ðtÞ ¼ ðq1j ðtÞ; . . . ; qnj ðtÞÞ which represents his/her total holdings in the market. Also q ji ðtÞ measures his/her instantaneous trend, plan or program to sell or buy one extra unit of each one of the assets. The curve fq j ðtÞ : tX0g in Rn is the portfolio trajectory of agent j; while the curve fQðtÞ : tX0g in Rnm corresponds to the evolution of the state of the market, that is to say, the dynamical description of how the market allocates resources, by means of the produced prices and as a consequence of trading actions. Our main proposal with this model is that trading behavior is ‘largely deterministic’ and prices evolution is ‘largely random’ and that both dynamics are coupled, thus influencing each other and behaving stochastically as a whole. Remark 1 (On constraints). It is apparent that several constraints for the main dynamical variables must play a role: the total balance of available shares versus the sold and bought ones; the holdings of an agent versus the number of shares that she/he can sell; and finally the amount of each agent’s wealth which must be greater or equal than the price of what she/he is going to buy. In this work, we shall set aside the above constraints for avoiding too much technical complexity, and assume that there are no bounds for agents’ wealth, companies’ number of available shares, or agents’ possibilities of selling shares. This is, of course, a strong assumption which we hope to drop in a future work, but which can match to some extent with behavior in many markets: (a) companies can have an unbounded number of shares to sell by the device of issuing new shares periodically, or as far as they need or can; (b) agents which mostly are brokers and represent other people’s interests can be supposed to have an unbounded wealth because of the flows coming from given and new people that they represent, and something similar about the number of shares they can sell. 3. Mathematical formulation Let Rn be an n-dimensional Euclidean space with the norm defined as jxj2 ¼ Sni¼1 jxi j2 , where x ¼ ðx2 ; x2 ; . . . ; xn Þ is a vector in Rn , and Rnm denote the space of n m matrices with the norm defined as kAk2 ¼ Si Sj jaij j2 ¼ tr AAT , where A ¼ ½aij is a matrix and the transpose of a matrix A is denoted by AT . By a filtered probability space with increasing family of s-algebras which is denoted as ðO; F; P; fFt gtX0 Þ we mean a probability space ðO; F; PÞ with a system fFt gtX0 of sub-s-algebras of F such that Fs Ft , if sot. It is also called a ‘stochastic basis’.
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We denote by C ¼ C½0; T the Banach space of Rn -valued continuous functions defined on [0,T], and equipped with the usual supremum norm, i.e. kvkC ¼ sup0ptpT jvðtÞj; and by Cr ¼ Cr ½0; T, for r 2 N, the vector space of r-fold continuously differentiable functions. Consider the dynamics of the trading floor modeled by the following system of deterministic differential equations:
q ji ðtÞ ¼ f ij ðt; QðtÞ; SðtÞÞ;
t40;
ð1Þ
where f ij : Rþ Rnm Rn ! R. ð1pipn; 1pjpm; Rþ ¼ ½0; 1Þ and R ¼ ð1; 1ÞÞ are continuous, or in the matrix form as
QðtÞ ¼ F ðt; QðtÞ; SðtÞÞ;
t40,
Qð0Þ ¼ Q0 ,
ð2Þ
where QðtÞ ¼ ½qij ðtÞnm and F ¼ ½f ij nm are defined as F : Rþ Rnm Rn ! Rnm ; Q : Rþ ! Rnm . Consider also the price dynamics generated by the system of stochastic differential equations dS i ðtÞ ¼ ai ðt; QðtÞ; SðtÞÞ dt þ
K X
bik ðt; QðtÞ; SðtÞÞ dW k ðtÞ;
t40,
(3)
k¼1
where ai : Rþ Rnm Rn ! R; bik : Rþ Rnm Rn ! R;
i ¼ 1; 2; . . . ; n, k ¼ 1; 2; . . . ; K
and W k ðtÞ is a standard one-dimensional Wiener process. Let Aðt; Q; sÞ ¼ ðai ðt; Q; sÞÞ, 1pipn and Bðt; Q; sÞ ¼ ðbik ðt; Q; sÞÞ; 1pipn; 1pkpK be defined on Rþ Rnm Rn and continuous in ðt; Q; sÞ. Finally W ðtÞ ¼ ðW ðtÞ; . . . ; W K ðtÞÞ is a K-dimensional Wiener process with respect to the stochastic basis. We now consider the above component-wise system (3) as the following Itoˆ’s SDE: dSðtÞ ¼ Aðt; QðtÞ; SðtÞÞ dt þ Bðt; QðtÞ; SðtÞÞ dW ðtÞ; Sð0Þ ¼ S0 ,
t40, ð4Þ
where S 0 is a random variable independent of {W ðtÞ, tX0}. (For probabilistic details see [11]). We now introduce the solution concept for the coupled system (2) and (4). Definition 1. By a solution of the coupled system (2) and (4), we mean a coupled stochastic process fSðtÞ; QðtÞ; t 2 ½0; Tg for some 0oTo1, defined on the stochastic basis ðO; F; P; fFt gtX0 Þ such that: (i) QðtÞ and SðtÞ are Ft -adapted for each t 2 ½0; T and continuous in t almost surely (a.s.). (ii) EjSðtÞj2 o1 and EkQðtÞko1, for each t 2 ½0; T. (iii) SðtÞ satisfies the stochastic integral equation: Z t Z t SðtÞ ¼ S 0 þ Aðu; QðuÞ; SðuÞÞ du þ Bðu; QðuÞ; SðuÞÞ dW ðuÞ; a:s. 0
(5)
0
for each t 2 ½0; T, where the second integral is understood in the sense of Itoˆ’s stochastic integral [11,12], and
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(iv) QðtÞ satisfies the integral equation Z t QðtÞ ¼ Q0 þ F ðu; QðuÞ; SðuÞÞ du;
t40; a:s.
(6)
0
It is apparent from this definition that QðtÞ, the state of the market, is a stochastic process, despite the (assumed) determinism of the trading decisions. This is because QðtÞ depends on the random behavior of the prices.
4. Lipschitz case To begin with, in this section, we consider the problem of our interest, namely, the existence and stability problems using the classical Lipschitz and linear growth conditions on the non-linear terms. Hypotheses: (H1) (H2)
(Lipschitz condition) For all t 2 Rþ ¼ ½0; 1Þ, Q1 ; Q2 2 Rnm and s0 ; s00 2 Rn , it holds: kF ðt; Q1 ; s0 Þ F ðt; Q2 ; s00 ÞkpC½kQ1 Q2 k þ js0 s00 j, where C40 is a constant; jAðt; Q1 ; s0 Þ Aðt; Q1 ; s00 Þj þ kBðt; Q1 ; s0 Þ Bðt; Q2 ; s00 ÞkpC½kQ1 Q2 k þ js0 s00 j. (Linear growth condition) For all t 2 Rþ , Q 2 Rnm and s 2 Rn , kF ðt; Q; sÞkpCð1 þ kQk þ jsjÞ, jAðt; Q; sÞj2 þ kBðt; Q; sÞk2 pC 2 ð1 þ kQk2 þ jsj2 Þ.
Let us define the successive approximation procedure for the coupled system (5) and (6), for t 2 ½0; T as S 0 ðtÞ ¼ S 0 ,
Z
Z
t
S k ðtÞ ¼ S 0 þ
t
Aðu; Qk1 ðuÞ; S k1 ðuÞÞ du þ 0
Bðu; Qk1 ðuÞ; S k1 ðuÞÞ dW ðuÞ, 0
and Q0 ðtÞ Q0 ,
Z
Qk ðtÞ ¼ Q0 þ
t
F ðu; Qk1 ðuÞ; S k1 ðuÞÞ du, 0
for k ¼ 1; 2; . . . : Using the successive approximation procedure just introduced, we state our first result on the existence and uniqueness of a solution. Theorem 1. Assume that Aðt; Q; sÞ, Bðt; Q; sÞ and F ðt; Q; sÞ satisfy the hypotheses (H1)–(H2), and moreover EjS0 j2 o1 and EkQ0 ko1. Then there exists a unique solution fSðtÞ; QðtÞ; t 2 ½0; Tg to the coupled model, and 2 lim E sup jS k ðtÞ SðtÞj ¼ 0 and lim E sup kQk ðtÞ QðtÞk ¼ 0 k!1
hold, where
k!1
0ptpT
fSk ðtÞg1 k¼1
and
fQk ðtÞg1 k¼1
0ptpT
are the sequences of successive approximations as defined earlier.
The following result talks of the boundedness of the so-called price dynamics/process fSðtÞ : tX0g. Note that it is also possible to obtain a similar estimate of the process fQðtÞ : tX0g. This consideration applies as well in the rest of the paper. Theorem 2. Under the hypothesis (H1) and (H2), the solution SðtÞ of (4) satisfies EjSðtÞj2 pð1 þ EjS0 j2 ÞeCt ;
tX0
where C40 is a suitable constant.
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In other words, considering the initial condition S 0 as the input and the corresponding solution SðtÞ as the output, the above inequality in Theorem 2, yields the so-called well-known BIBO (bounded input and bounded output) stability. Further, defining this relation as a map: f : S 0 !S, one has the following result about the continuous dependence on the initial condition. Theorem 3. Under the hypothesis (H1) and (H2), the map f is continuous. Let Aðt; 0; 0Þ ¼ 0; Bðt; 0; 0Þ ¼ 0 and F ðt; 0; 0Þ ¼ 0 for all tX0. Then the equilibrium position or the trivial solution SðtÞ 0; QðtÞ 0, is the unique solution of the coupled system (2) and (4) with initial values S 0 ¼ Q0 ¼ 0. Next, let us address the stability issue in the following sense. Definition 2. The trivial solution of (4) is said to be exponentially stable in the quadratic-mean if EjSðtÞj2 pC 1 EjS 0 j2 eat ;
tX0,
where C 1 ; a40 are constants. Theorem 4. Let the hypotheses of Theorem 1 be satisfied. Suppose that there exists a function v : Rn ! R of class C2 such that it satisfies: (i) jvðsÞj þ jskvs ðsÞj þ jsj2 jvss ðsÞjpZjsj2 ; 8s 2 Rn and some Z40, (ii) LvðsÞ þ avðsÞp0; 8v 2 C2 ; a40, where L is defined as LvðsÞ ¼ ðvs ðsÞ; Aðt; Q; sÞÞ þ 12 tr½Bðt; Q; sÞWBT ðt; Q; sÞ, then EvðSðtÞÞpeat EvðS0 Þ. 5. Non-Lipschitz case To relax the Lipschitz assumption for the coefficients of the system is an important issue for at least two reasons: (a) several well-known models in interest rate theory involve non-Lipschitz coefficients [13], and (b) the Cobb–Douglas type functions, widely used across economic theory, and relevant to model trading behavior in more advanced terms, are also non-Lipschitz. However, the subject seems to have not been considered in the literature as it deserves. There are few works in these lines. First a series of works by Yamada and coauthors [14–16] address the problem, but with a quite abstract approach: as it is displayed below, they consider an abstract condition for substituting the Lipschitz one, a condition which depends on a dominant function (kðxÞ described below) but without being able to give examples of SDE’s satisfying such condition. We found by ourselves an unsurmountable difficulty—for now—in applying this approach to concrete financial models. Nevertheless it is a first mathematical step directed towards the solution of the problem and we shall make a quite brief presentation of part of their results. Second, there are some more recent results, remarkably Zhang & Fang [17] more biased to construct concrete examples of SDE satisfying the new (nonLipschitz) condition, but unfortunately not completely appropriate for applying to financial models. Finally, a paper by Deelstra & Delbaen [18] addresses the non-Lipschitz case for a specific family (Ho¨lder-type functions) in dimension one. We present below a part of Yamada’s approach, and in Example 2, Section 6 we take advantage of Deelstra & Delbaen’s result. 5.1. Yamada’s approach The following assumptions are based on Yamada [13].
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Hypotheses: (H3) Aðt; Q; sÞ and Bðt; Q; sÞ satisfy jAðt; Q1 ; s0 Þ Aðt; Q2 ; s00 Þj2 þ kBðt; Q1 ; s0 Þ Bðt; Q2 ; s00 Þk2 pkðkQ1 Q2 k2 þ js0 s00 j2 Þ, 8t 2 Rþ ; Q1 ; Q2 2 Rnm s0 ; s00 2 Rn ; wheren the norms in terms of the components are given by X (i) jAðt; Q ; s0 Þ Aðt; Q ; s00 Þj2 ¼ ja ðt; Q ; s0 Þ a ðt; Q ; s00 Þj2 , 1
i
2
i
1
2
i¼1
kBðt; Q1 ; s0 Þ Bðt; Q2 ; s00 Þk2 ¼
n X m X
jbij ðt; Q1 ; s0 Þ bij ðt; Q2 ; s00 Þj2 ,
i¼1 j¼1
and (ii) kðuÞ is a function defined on ½0; 1Þ that is continuous, non-decreasing and concave such that Z du ¼ þ1. þ kðuÞ 0 (H4) F satisfies conditions (H1) and (H2) as before, that is, Lipschitz and linear growth inequalities: kF ðt; Q1 ; s0 Þ F ðt; Q2 ; s00 ÞkpC½kQ1 Q2 k þ js0 s00 j kF ðt; Q; sÞkpCð1 þ kQk þ jsjÞ, where C40 is a constant. Based upon the above hypotheses, we have the next result which guarantees the existence and uniqueness of a solution. Theorem 5. Assume that Aðt; Q; sÞ, Bðt; Q; sÞ and F ðt; Q; sÞ satisfy the hypotheses (H3) and (H4) and EjS 0 j2 o1 and EkQ0 ko1. Then 2 lim E sup jS k ðtÞ SðtÞj ¼ 0 and lim E sup kQk ðtÞ QðtÞk ¼ 0 k!1
k!1
0ptpT
0ptpT
hold. A proof of this result is sketched in the Appendix. 6. Examples We shall consider here just two examples, one Lipschitz and one of the non-Lipschitz class. Example 1 (Two traders and two assets). The first trader has as goal to get certain levels of holdings in both assets:
q 11 ðtÞ ¼ A1 ða11 q11 ðtÞÞ;
q 12 ðtÞ ¼ A2 ða12 q12 ðtÞÞ.
The second trader has as goal to stabilize the asset prices about certain ‘fundamentals’:
q 21 ðtÞ ¼ B1 ðb11 S1 ðtÞÞ;
q 22 ðtÞ ¼ B2 ðb12 S2 ðtÞÞ.
The two assets are company shares both obeying a sort of ‘variable mean reverting’ diffussion process, and coupled to trading dynamics as follows: dS 1 ðtÞ ¼ ðm1 q11 ðtÞÞS1 ðtÞ dt þ s1 S 1 ðtÞ dW 1 ðtÞ, dS 2 ðtÞ ¼ ðm2 q22 ðtÞÞS2 ðtÞ dt þ s2 S 2 ðtÞ dW 2 ðtÞ,
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with W 1 ; W 2 being independent Brownian motions. We have in this example, existence, uniqueness and continuous dependence of solutions on initial conditions, by resorting to Theorems 1–3. Example 2 (One trader, one asset). Now we consider a whole family of examples, belonging to the nonLipschitz class. It is a family because asset price is labeled by two arbitrary functions of certain type (dð Þ and gð Þ below) which can represent quite different kinds of assets. The trader wants to get a certain level (q0 ) of holdings on the asset, and to keep the asset price about certain level (S 0 ). Then, the trader’s moves depend on the asset price evolution, but there is no double feedback, since the asset price does not depend on the trading dynamics. The equations governing this behavior are the following:
qðtÞ ¼ aðq0 qðtÞÞ þ bðS 0 SðtÞÞ dSðtÞ ¼ ð2bSðtÞ þ dðtÞÞ dt þ gðSðtÞÞ dW ðtÞ, with a; b40; bp0, the function g satisfies pffiffiffiffiffiffiffiffiffiffiffiffiffiffi jgðxÞ gðyÞjpc jx yj; ðc40Þ, and d : O Rþ ! Rþ is a measurable and adapted process with
Rt 0
dðuÞ duo1, for each t 2 Rþ .
The existence and uniqueness of positive solutions for the second (non-Lipschitz) equation is guaranteed by invoking a Deelstra & Delbaen result [18]. As trader’s dynamics concern, existence and uniqueness are straightforward. In other words substitute the solution SðtÞ so obtained into the first equation. Clearly, the first equation being linear in QðtÞ and SðtÞ, its coefficients are Lipschitz. Hence a unique solution process QðtÞ exists and is also Ft -adapted as SðtÞ is. Remark 2. In commodity markets and in many financial markets as well, there is always a considerably bigger number of market participants than that of commodities or assets, that is, mbn (for example, in USA economy the number of market participants is about 200 million, while the number of commodities is about 20 million). This seems to be a sort of economic behavior law, which has surprising theoretical consequences (see [9] ). On the other hand, the numerical simulation of models with a large number of agents (just more than two or three) is highly complicated from a computational point of view. This is a difficulty that should be dealt with, since it is likely that some real financial behavior could be adequately simulated by considering many agents. 7. Conclusions In this paper, we present a dynamical model for financial markets, which is a coupling of two dynamical systems: a deterministic one, governing the dynamics of the trading floor, and a stochastic one, governing asset prices evolution. Both systems are motivated by classical dynamic models in mathematical economics and mathematical finance, respectively. Our contributions in this work are the following: (a) the coupling of both systems, deterministic and stochastic, which we believe represents adequately an important feature of financial markets; (b) to consider both the Lipschitz and the non-Lipschitz case (to some extent) for the coefficients of the system. This is important for several reasons, the main one of which is that relevant SDEs models for the term structure of interest rates belong to the second category; (c) call attention to the question of stability of the coupled system. This is a subject that has been treated extensively in commodity markets within the framework of general economic equilibrium, but, as far as we know, has been almost not considered at all in mathematical finance. We found only one reference [19] which does not belong to the mainstream of financial research; finally, (d) we gave a general set-up, which allows on one hand, to construct specific examples appropriate for numerical simulations, and on the other hand to consider deeper theoretical questions, including some of a more economical nature, since the trading dynamics can also be used to represent simple economic exchange.
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7.1. Scope for further research Several questions can be addressed for the future work. On one hand a clarification of the technical questions involved in the non-Lipschitz case, enough to cover some of the main models in interest rate dynamics has an important place in any research agenda. On the other hand, extensions of our model include the considerations of the constraints mentioned in the remark after the statement of the assumptions of the model. This could be a difficult task, since some of them are of a stochastic nature and the available tools such as those coming from stochastic control theory, only consider constraints defined by a given deterministic set. Other important question is the consideration of delays, specially for the trading decisions which are to be taken up some time after the information of spot prices is made available to the traders. Since regulatory authorities play an important role for the whole outputs of a trading floor dynamics, the introduction of controls both in the trading decisions and in the price behavior is a natural question to be explored. Another class of questions that can be posed are of a theoretical-financial nature, such as, possibilities of arbitrage and completeness of the markets. Acknowledgment The second author acknowledges fruitful conversations with J. Alvarez-Ramirez, Sergio Herna´ndez, Myriam Cisneros and Esteban Martina on the subject. Appendix. Proofs of some of the main results We shall sketch two proofs (those of Theorems 1 and 5). It could be useful even for the reader acquainted with stochastic calculus techniques, since it is shown how to deal with the coupled system within standard lines of reasoning. Proof of Theorem 1 (Sketch). Let T be an arbitrary positive number 0oTo1. Consider the Lipschitz condition jAðt; Q1 ; s0 Þ Aðt; Q2 ; s00 Þj þ kBðt; Q1 ; s0 Þ Bðt; Q2 ; s00 ÞkpC½kQ1 Q2 k þ js0 s00 j.
(7)
But, Z Q1 ðtÞ Q2 ðtÞ ¼
t
½F ðu; Q1 ðuÞ; S1 ðuÞÞ F ðu; Q2 ðuÞ; S 2 ðuÞÞ du. 0
Therefore, by exploiting the Lipschitz condition on F, we get Z t kQ1 ðtÞ Q2 ðtÞkp C½kQ1 ðuÞ Q2 ðuÞk þ jS 1 ðuÞ S 2 ðuÞj du 0 Z t Z t jS 1 ðuÞ S 2 ðuÞj du þ C kQ1 ðuÞ Q2 ðuÞk du. pC 0
0
An application of Bellman–Gronwall’s Lemma [12] then yields Z t Z t Z t CðttÞ kQ1 ðtÞ Q2 ðtÞkpC jS 1 ðuÞ S 2 ðuÞj du þ C e C jS1 ðuÞ S2 ðuÞj du dt 0 0 0 Z t pCT kS1 S 2 kC þ C 2 TkS1 S 2 kC eCðtuÞ du. 0
Hence, Z kQ1 ðtÞ Q2 ðtÞkp CT þ C 2 T
T 0
eCðtuÞ du kS1 S 2 kC .
(8)
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Using (8) in (7) we have jAðt; Q1 ; s0 Þ Aðt; Q2 ; s00 Þj þ kBðt; Q1 ; s0 Þ Bðt; Q2 ; s00 Þk Z T 2 CðtuÞ e du ks0 s00 kC þ Cks0 s00 kC pC CT þ C T 0 Z T pC CT þ C 2 T eCðtuÞ du þ 1 ks0 s00 kC .
ð9Þ
0
Next, consider the condition jAðt; Q; sÞj2 þ kBðt; Q; sÞk2 pC 2 ð1 þ kQk þ jsj2 Þ.
(10)
Using the linear growth condition on F, we have Z t kF ðt; QðuÞ; SðuÞÞk du kQðtÞkpkQ0 k þ 0 Z t Cð1 þ kQðuÞk þ kSðuÞkÞ du pkQ0 k þ 0 Z t Z t kQðuÞk du þ C kSðuÞk du. pkQ0 k þ Ct þ C 0
0
Rt
Letting hðtÞ ¼ kQ0 k þ Ct þ C 0 kSðuÞk du; we have Z t kQðuÞk du. kQðtÞkphðtÞ þ C 0
Another application of Bellman–Gronwall’s Lemma yields Z t kQðtÞkphðtÞ þ C eCðtuÞ hðuÞ du; t 2 ½0; T 0 Z t eCðtuÞ ðkQ0 k þ Cu þ CukskC Þ du pkQ0 k þ CT þ CTkskC þ C 0 Z t eCðtuÞ kQ0 k du pkQ0 k þ CT þ CTkskC þ C 0 Z t Z t CðtuÞ e Cu du þ Ckske eCðtuÞ Cu du þC 0
0
pL þ CðT þ CeCT kske pðM þ 1ÞkskC ,
ð11Þ
where M ¼ maxfL; CðT þ CeCT Þg and L ¼ kQ0 k þ CT þ eCT ð1 þ CÞ. Using (11) in (10), we get jAðt; Q; sÞj2 þ kBðt; Q; sÞk2 pC 2 ð1 þ ðM þ 1Þ2 ksk2C þ ksk2C Þ pC 2 ½1 þ ððM þ 1Þ2 þ 1Þksk2C pC 21 ½1 þ ksk2C , where C 21 ¼ maxfC 2 ; ðM þ 1Þ2 þ 1g.
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Hence, the conditions of the theorem boil down to the standard Lipschitz and linear growth conditions uniformly in Q 2 Rnm , (see [12, p. 105, or 20]). Then, the existence and uniqueness of a solution follows on standard lines. & Proof of Theorem 5 (Sketch). Consider the hypothesis (H3): jAðt; Q1 ; s0 Þ Aðt; Q2 ; s00 Þj2 þ kBðt; Q1 ; s0 Þ Bðt; Q2 ; s00 Þk2 pkðkQ1 Q2 k2 þ js0 s00 j2 Þ. Since kðuÞ is non-decreasing by assumption, it follows from (8) that: jAðt; Q1 ; s0 Þ Aðt; Q2 ; s00 Þj2 þ kBðt; Q1 ; s0 Þ Bðt; Q2 ; s00 Þk2 pkðL21 ks0 s00 k2C þ js0 s00 Þk2 pkððL21 þ 1Þks0 s00 k2C Þ, where 2
Z
L1 ¼ CT þ C T
T
eCðtuÞ du,
0
uniformly in Q1 ; Q2 2 Rnm . This verifies that the hypothesis (H3) satisfies Yamada [14] condition A, uniformly in Q1 ; Q2 2 Rnm . Hence, the existence and uniqueness of a unique solution fSðtÞ; QðtÞg to the coupled model (2) and (4) follows on similar lines as in Yamada [14]. & References [1] R. Mantegna, H.E. Stanley, An Introduction to Econophysics: Correlations and Complexity in Finance, Cambridge University Press, UK, 2000. [2] J.L. MacCauley, Dynamics of Markets. Econophysics and Finance, Cambridge University Press, UK, 2004. [3] F.H. Hahn, T. Negishi, A theorem of non-tatoˆnnement stability, Econometrica 30 (3) (1962) 463–469. [4] J. Alvarez-Ramirez, C. Ibarra-Valdez, Modeling stock market dynamics based on conservation principles, Physica A 301 (2003) 493–511. [5] T. Lux, M. Marchesi, Volatility clustering in financial markets: a microsimulation of interacting agents, Int. J. Theor. Appl. Finance 3 (4) (2000) 675–702. [6] P.A. Samuelson, Mathematics of speculative price, Mathematical Topics in Economic Theory and Computation, SIAM, Philadelphia, 1965. [7] I. Karatzas, Lectures on the mathematics of finance, CRM Monograph Series, 8, American Mathematical Society, Providence, RI, 1997. [8] J.C. Cox, J.E. Ingersoll, S.A. Ross, An intertemporal general equilibrium model of asset prices, Econometrica 53 (2) (1985) 363–384. [9] D. Saari, Mathematical complexity of simple economics, Notices Amer. Math. Soc. 42 (1995) 222–230. [10] F. Black, M. Scholes, The pricing of options and corporate liabilities, J. Political Econ. 81 (1973) 637–659. [11] D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, third ed., Springer, Berlin, 1999. [12] L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley-Interscience [John Wiley & Sons], New York-LondonSydney, 1974, (Translated from the German). [13] S.E. Shreve, Stochastic Calculus for Finance. II. Continuous-time Models, Springer Finance, Springer, New York, 2004. [14] T. Yamada, On the successive approximation of solutions of stochastic differential equations, J. Math. Kyoto Univ. 21–3 (1981) 501–515. [15] S. Watanabe, T. Yamada, On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ. 11 (1971) 155–167. [16] S. Watanabe, T. Yamada, On the uniqueness of solutions of stochastic differential equations II, J. Math. Kyoto Univ. 11 (1971) 553–563. [17] T. Zhang, S. Fang, A study of a class of stochastic differential equations with non-Lipschitzian coefficients, Probab. Theory Relat. Fields 132 (2005) 356–390. [18] G. Deelstra, F. Delbaen, Existence of solutions of stochastic differential equations related to the Bessel process, Selected papers, Homepage of Prof. Freddy Delbaen: hhttp://www.math.ethz.ch/ delbaen/ftp/preprints/existenti_rev.pdfi. [19] A.V. Svishchuk, A.V. Kalemanova, The stochastic stability of interest rates with jumps changes, Theor. Probab. Math. Stat. (61) (2000) 161–172. [20] B. Oksendal, Stochastic Differential Equations an Introduction with Applications, sixth ed., Universitext, Springer, Berlin, 2003.