﻿ A Trading Execution Model Based on Mean Field Games and Optimal Control

Applied Mathematics
Vol.05 No.19(2014), Article ID:51277,25 pages
10.4236/am.2014.519294

A Trading Execution Model Based on Mean Field Games and Optimal Control

Lorella Fatone1, Francesca Mariani2, Maria Cristina Recchioni3, Francesco Zirilli4

1Dipartimento di Matematica e Informatica, Università di Camerino, Camerino, Italy

2Dipartimento di Scienze Economiche, Università degli Studi di Verona, Verona, Italy

3Dipartimento di Management, Università Politecnica delle Marche, Ancona, Italy

4Dipartimento di Matematica “G. Castelnuovo”, Università di Roma “La Sapienza”, Roma, Italy   Received 21 August 2014; revised 18 September 2014; accepted 8 October 2014

ABSTRACT

Keywords:

Trading Execution, Mean Field Game, Optimal Control 1. Introduction

In 1998 Bertsimas and Lo  presented a discrete time trading model describing a trader that must buy a block of asset shares in an assigned time interval minimizing the expected value of the trading cost. The asset share price is modeled as a discrete arithmetic random walk that depends linearly from the number of asset shares bought/sold by the trader. This linear term models the permanent market impact on the asset share price of the trading activity of the big trader. In  the problem of determining the order optimal trading execution strategy is modeled as a dynamic optimization problem.

Gatheral and Schied  modified the model discussed in  considering as cost functional of the control problem studied the expected value of the sum of the trading cost and of a time averaged “value at risk” associated to the trading execution strategy. The robustness of the model presented in  is established in  .

In  Ankirchner, Blanchet Scalliet and Eyraud Loisel proposed a variant of the stochastic trading execution model studied in  and  . They assumed that the Brownian motion that describes the asset share price dynamics in absence of trading has a non zero drift and that the trading rate of the big trader is a square integrable stochastic process. The drift term of the asset share price dynamics can be interpreted as the directional view of the trader about the asset share price (i.e. the bullish or bearish attitude of the trader about the asset share price). This drift term is assumed to be a linear function of the asset share price. The optimal trading execution strategy solution of the model proposed in  is a Gaussian process that becomes a deterministic function of time when the drift term of the asset share price dynamics is constant.

The interested reader is referred to the website: http://www.econ.univpm.it/recchioni/finance/w19 to find material including animations, an interactive application and an app that helps the understanding of the paper. A general reference to the work of the authors and of their coauthors in mathematical finance is the website: http://www.econ.univpm.it/recchioni/finance.

The paper is organized as follows. In Section 2 we define the trading execution model studied, that is we define the mean field game model associated to the retail traders, the asset share price dynamic equation and the optimal control problem associated to the big trader. In Section 3 we solve the mean field game model. In Section 4 we solve the optimal control problem. In Section 5 we present a numerical study of the trading execution model developed in Sections 2, 3 and 4.

Let be the set of real numbers, be a real variable that denotes time and be a positive number. We assume that is the time horizon of the mean field game model that describes the behaviour of the retail traders. This means that the mean field game model is solved for . Let us consider a state variable , , that represents the number of asset shares held by the retail traders at time , . The variable is called trading position of the retail traders at time , , and is a real stochastic process. Positive values of mean that the retail traders have a long position on the asset shares at time , negative values of mean that the retail traders have a short position on the asset shares at time ,. Recall that when we write retail traders we always mean retail traders in the “mean field approximation”. We assume that the stochastic process, , satisfies the following stochastic differential equation:

, (1)

with initial condition:

, (2)

where is a continuous function, is a real constant, , , is a standard Wiener process such that and, , is its stochastic differential. The initial condition is a known random variable whose probability density function is denoted with,. We have

, , and. We assume that the initial value problem (1), (2) has a unique solution

when. The function that appears in (1) is the trading rate of the retail traders and is the control variable of the mean field game model considered. Equation (1) is the “mean field equation” announced in the Introduction that defines the dynamics of the trading position of the “mean retail trader” that is used to describe the multitude of the retail traders in the “mean field approximation”.

For we denote with, , the probability density function of the random variable solution of (1), (2). The function, , , satisfies the forward Kolmogorov equation associated to (1):

, (3)

with the initial condition implied by (2):

. (4)

Recall that in (3) the function is not a known coefficient, is the control variable that must be determined as solution of the mean field game model. The fact that is a probability density function implies that the function solution of (3), (4) is a probability density function as a function of for that

is implies that:, , , and,.

Let denote the expected value of and be the set of the square integrable processes on

This means that the real stochastic process, , belongs to if and only if

Let, , be the trading rate of Equation (1) we assume that. Given the real parameters, and we consider the following problem:

, (5)

where:

(6)

subject to the constraints (1)-(4). The model (5), (6), (1)-(4) is the mean field game model used to describe the behaviour of the retail traders. This means that the retail traders adopt the trading execution strategy, , whose rate is the solution of problem (5), (6), (1)-(4). The parameter is called risk aversion parameter. Note that the constraints (3), (4) are consequences of the constraints (1), (2) and that they can be omitted when (1), (2) are imposed. However we prefer to mention (3), (4) explicitly in the statement of the previous problem in order to emphasize the role of, , , in the functional, , , , , defined in (6). The functional, , , , , is

called utility function or cost functional of the mean field game model and is the sum of four terms. The first one: expresses the fact that the retail traders want to behave in a similar way, the second one: expresses the fact that the retail traders try to avoid abrupt changes of their trading strategies, the third one: expresses the fact that the retail traders do not like the risk associated to the fact of having open positions on the asset shares, and the fourth one: expresses the fact that in the

long run (that is at time) the retail traders want to have a given position on the asset shares (that is the position). The choice of the term, , in (6) instead of a more general term such as, for example, , , with, , a suitable (real valued) function, is essential in Section 3 to solve elementarily the mean field game model (5), (6), (1)-(4) when, , is a Gaussian probability density function. Note that in the utility function, , , , , the

terms, , , and depend from the control variable through the

Equations (1), (3).

Now let us discuss the mathematical model that describes the behaviour of the big trader. Let be a positive number such that and be a positive integer that represents the number of asset shares held by the big trader at time. We consider the problem of determining the optimal trading execution strategy that implements the order of selling asset shares in the time interval. This order is called liquidation order since consists in the order of liquidating the position on the asset shares held by the big trader at time in the time interval. Let, , be the number of asset shares held by the big trader at time,. We assume that, , is solution of the following differential equation:

, (7)

with initial condition:

, (8)

and final condition:

, (9)

We use as asset share price dynamic equation a simple generalization of the equation introduced by Almgren in  . Let be the asset share price at time, , we assume that, is a real stochastic process defined as follows:

, (10)

, (11)

where, , and, , are positive constants.

Let, , be a trading execution strategy that satisfies (7)-(9). It is easy to see that when the big trader adopts the trading execution strategy, , and the asset share price dynamics is described by (10), (11) the expected final revenue (i.e. the expected revenue at time) for the big trader resulting from the sale of asset shares in the time interval is:

. (12)

Let us take a closer look to (12). Integrating (12) by parts and using (7) we have:

. (13)

Note that the assumption implies that when the expression

is well defined and finite. From (13) we have:

. (14)

The optimal trading execution strategy of the big trader is the strategy that maximizes the expected final revenue of the big trader (14) subject to the constraints (7)-(9). That is the problem of finding the optimal trading execution strategy of the big trader consists in solving the following optimal control problem:

, (15)

where:

, (16)

subject to the constraints (7)-(9). Note that in (16) we have dropped the term that appears in

(14). In fact this term does not depend from the control variable and can be dropped from the objective function (14) without changing the solution of the control problem considered. The optimal control problem (15), (16), (7)-(9) is a linear quadratic optimal control problem. For later convenience note that the utility function, , defined in (16) does not depend from and that the presence of, in (16) determines how the behaviour of the retail traders influences the behaviour of the big trader.

Let us consider the mean field game problem (5), (6), (1)-(4). We define:

, (17)

to be the value function of problem (5), (6), (1)-(4).

The function, , , defined in (17), satisfies the following Hamilton Jacobi Bellman equation:

, (18)

with final condition:

, (19)

where is the Hamiltonian function of the mean field games (5), (6), (1)-(4).

Therefore the optimal trading execution strategy of the retail traders can be determined solving the following system of partial differential equations, see  :

, (20)

, (21)

where the function, , , must satisfy the condition, ,. This last condition couples the Equations (20), (21). The system (20), (21) with the condition, , , is equipped with a final condition on in and an initial condition on in that is:

, (22)

. (23)

Equations (20), (21) with the conditions, , , (22), (23) are a system of nonlinear

partial differential equations with an initial and a final condition. This system expresses the first order necessary optimality condition of the mean field game model (5), (6), (1)-(4). Once known the solution of this system of partial differential equations the optimal trading rate of the retail traders solution of problem (5), (6), (1)-(4) is

determined by the condition:, where, is the solution of (1), (2) when is chosen equal to.

We look for elementary solutions of problem (20), (21) with the conditions, , ,

(22), (23). In analogy with the work of Kalman  we formulate some hypotheses that make possible to reduce

the solution of problem (20), (21) with the conditions, , , (22), (23) to the solution of

a constrained two point boundary value problem for a system of six Riccati ordinary differential equations. The choice of the logarithm function in the first addendum of (6) and as a consequence the presence of the logarithm function on the right hand side of (20) are crucial to operate this reduction.

Let be a random variable and let be the Gaussian probability distribution of mean and standard deviation. The notation means that the random variable is distributed.

We have:

Proposition 3.1

Let, and let:

, (24)

i.e. let then a solution of problem (20), (21) with the conditions, , , (22), (23) is given by:

, (25)

, (26)

, (27)

where the functions, , , , , , , are solution of the following two point boundary value problem:

, (28)

, (29)

, (30)

, (31)

, (32)

, (33)

, (34)

, (35)

, (36)

, (37)

, (38)

, (39)

and satisfy condition (47). In fact, as shown later, in order to guarantee that, , , defined in (26) is a Gaussian probability density function it is necessary that the functions, , , , satisfy condition (47). Note that (47) implies Condition (47) is a constraint imposed to the solution of (28)-(39). The equations (28)-(39) are a system of six Riccati ordinary differential equations in six unknowns defined for equipped with three initial conditions in and three final conditions in.

Proof

Let, , , where is given by (25) and is a function to be determined. We have:

, (40)

, (41)

, (42)

. (43)

Substituting (40)-(43) in (21) we have:

. (44)

In (44) we choose, , , , , , and, , , where, , , , , , , are functions to be determined. It is easy to see that with the previous choices from (20) and (44) we have (28)-(39) where, and,.

Note that when is given by (26) and, , we have:

(45)

Imposing that, , from (45) it follows that:

, (46)

that is:

. (47)

Equation (47) implies that, and guarantees that the function m given by (26) is a Gaussian probability density function in for.

Note that a solution of the two point boundary value problem (28)-(39) that satisfies (47) may or may not exist. When it does not exist the attempt of building a solution of problem (20), (21) with the conditions,

, , (22), (23), based on (25)-(27) fails. Conversely when there exist the formulae (25)-(27) and the solution of the constrained two point boundary value problem discussed above gives an elementary solution

of problem (20) (21) with the conditions, , , (22), (23) when is a Gaussian proba-

bility density function. Proposition 3.1 shows that when (28)-(39) has a solution that satisfies (47) from the hypothesis it follows that the probability density function, , , solution

of problem (20), (21) with the conditions, , , (22), (23) can be chosen as the Gaussian probability density function (26) with mean:

, (48)

and variance:

, (49)

that is in this case we have:, , with, , , given by (48), (49). In this Section and in Section 4 we assume that the system (28)-(39) has a solution that satisfies (47). In Section 5 we discuss briefly the validity of this assumption and in some test cases we determine a solution of (28)-(39) that

satisfies (47). In  Guéant studies the stationary version of problem (20), (21) with the conditions,

, (22), (23) and finds a solution given by the functions, , where, , is a polynomial of degree two in and, , is a Gaussian probability density function.

From (25) it follows that:

, (50)

where is the solution of (1), (2) when is given by (50). Therefore when (50) holds we have:

, (51)

Proposition 3.2

Let, , , and, , , , be the solution of

problem (20), (21) with the conditions, , , (22), (23) determined in Proposition 3.1

that we assume to exist. We have, , where:

(52)

and

. (53)

The functions and denote, respectively, the hyperbolic sine and the hyperbolic cosine functions of and the function, , in (53) is the solution of the two point boundary value problems (32), (33), (38), (39).

Moreover let, , be the expected value of the optimal trading execution rate of the retail traders, defined in (51), we have:

(54)

Proof

Differentiating (48) with respect to we have:

, (55)

from (55) it follows that:

. (56)

From (55), (56) we have that, , satisfies the differential equation:

. (57)

Equations (35), (39), (55), (56) imply that Equation (57) can be equipped with the following boundary conditions:

, (58)

. (59)

It is easy to see that the function (52) is the solution of (57)-(59). Finally substituting (52) in (56) we determine, , as solution of the first order ordinary differential Equation (56) with the final condition

that follows from (58), (59). An elementary computation shows that the function (54) is the solution of (56) that satisfies this last final condition.

Formula (54) shows that, , depends from but does not depend from. Note that when

we have, , so that from we have:, ,

moreover from (57)-(59) when we have:,. That is when, , , is constant, and is a linear function of,.

Let us consider the optimal control problems (15), (16), (7)-(9). The value function, , ,

of problems (15), (16), (7)-(9) is the maximum of the objective function determined solving the following constrained optimization problem:

, (60)

subject to the constraint (7), (9) and:

. (61)

Note that we have:

. (62)

The function is the solution of the following Hamilton Jacobi Bellman equation:

, (63)

with the final condition:

(64)

where is the Hamiltonian function of problem (15),

(16), (7)-(9). Note that the right hand side of (64) is not a real valued function. The final condition (64) must be

interpreted as prescribing the, A trading rate, , such that that is

a trading rate that does not satisfy (9), is a rate that at time has not achieved the goal of liquidating in the time interval the position of asset shares held by the big trader at time. To this rate the final conditions (64) attributes an infinite cost, or equivalently attributes a revenue equal to minus infinity. In this sense the final condition (64) translates to the value function the condition (9) imposed to the trading execution strategy .

Therefore in order to find the optimal trading execution strategy of the big trader we must solve the following Hamilton Jacobi Bellman equation:

, (65)

with final condition (64), where, , is given by (54). In fact from the knowledge of the value function v solution of (65), (64) we can determine the optimal control, , solution of(15), (16), (7)-(9),

using the relation:, , with, , solution of (7)-(9).

We have:

Proposition 4.1

Let, , and, , be given respectively by (52) and (54), the value function solution of problem (65), (64) can be chosen as:

, (66)

where, , , , are given by:

, (67)

, (68)

. (69)

This means that the optimal trading execution rate of the big trader is:

, (70)

where in (70), , is the solution of (7)-(9) when in (7), , is given by (70).

Proof

Let, , , be given by (66), we impose that, , satisfies (65), (64). It follows that the coefficients, , , , that define the function through (66), must satisfy the system of ordinary differential equations and the final conditions that follow:

, (71)

, (72)

, (73)

, (74)

, (75)

, (76)

where, , is given by (54). The final conditions contained in (76) must be interpreted as prescribing. It is easy to see that the functions (67)-(69) satisfy (71)-(76).

Moreover the optimal trading execution rate of the big trader is given by:

(77)

That is we have Equation (70).

Substituting (70) in (7) and imposing (8) we determine the optimal trading execution strategy of the big trader, , as solution of the following differential equation:

, (78)

with initial condition:

. (79)

Note that the fact that, , satisfies (8) (or equivalently (79)) is an hypothesis (i.e. the initial condition imposed to,) while the fact that, , satisfies (9) is a consequence of the choice of made, that is of the choice, , given by (70).

5. Numerical Experiments

We begin the numerical study of the trading execution model presented in the previous Sections discussing the problem of the existence of solutions of the mean field game problem (5), (6), (1)-(4) of the form suggested in Propositions 3.1, 3.2. The existence of this kind of solutions is equivalent to the assumption that the two point boundary value problem (28)-(39) has a solution that satisfies (47). It is easy to see that the existence of a solution of the two point boundary value problem (28)-(39) that satisfies (47) depends from the existence of a solution of the two point boundary value problem (32), (33), (38), (39) such that,. In fact when the two point boundary value problem (32), (33), (38), (39) has a solution, , such that from (52), (54) it follows that we can find:, , using Equation (48), (49);, , using Equation (51), (52), (54);

using Equation (47) and finally, using Equations

(28), (29). Conversely when the two point boundary value problem (32), (33), (38), (39) does not have a solution,

such that, , the solution of (20), (21) with the conditions, , , (22),

(23) constructed in Propositions 3.1, 3.2 does not exist. The study of the existence problem of the solution of the two point boundary value problem (32), (33), (38), (39) from the mathematical point of view is beyond our purposes here. In the numerical experiments presented in this Section we proceed as follows: first of all the two point boundary value problem (32), (33), (38), (39) is solved numerically using the shooting method (see  ) and the condition is checked. When this is done successfully from the solution of (32), (33), (38), (39) that satisfies, , found numerically we determine the solution of (28)-(31), (34)-(37). Let us study the trading execution model developed in Sections 2, 3 and 4. We present four study cases. In these study cases we use a generalized version of the mean field game problem (5), (6), (1)-(4) where the utility function defined in (6) is substituted with a new utility function that depends from some new parameters. Let us explain in detail the choices made. We recall that the utility function defined in (6) is the sum of the following terms:

, , and

with, ,. As already explained in Sections 2 and 3 the second and the third term in different ways express the fact that the retail traders are risk adverse. Maximizing (6) the retail traders pursue three goals. The first one is the desire of adopting similar strategies (i.e. the desire of the retail traders of not being alone in the

market). This goal is pursued making big the term of the utility function. The second goal is the desire of avoiding risk. This goal is pursued making big the term of the utility function. In this term the risk aversion is declined in two ways: aversion to abrupt changes of trading strategies expressed by the term and aversion to hold open positions on the asset shares expressed by the term. The third goal is the desire of having a position on the asset shares close to in the long run (i.e. is at time). This last goal is pursued making big the term. Note that to keep the formulae deduced in Sections 3 and 4 as simple as possible in the utility function (6) the four terms

, , ,

are weighted in a predetermined way. However to show the versatility of the model developed in the previous Sections in the study cases presented in this Section it is convenient to introduce a new utility function containing two parameters not present in (6), that is: the parameter w that regulates the relative weights of the terms

and and the parameter that regulates the relative weights of the terms and. That is in this Section we consider the utility function:

(80)

instead of the utility function defined in (6). Note that in (80) we choose, , and that the parameter of (8) replaces the parameter of (6). For simplicity in (80) we exclude the choices and/or from the possible choices of the parameter values. In fact when in (80) we have and/or the resulting mean field game problem is degenerate. That is the choices and/or in (80) correspond to a problem with a non convex Hamiltonian, that is correspond to a degenerate problem. For sim-

plicity we avoid degenerate problems. Note that, , where

is the utility function (6) with When the utility function (6) is substituted with the utility function (80) and instead of the problem studied in Section 3 we consider the problem of maximizing (80) subject to the constraints (1)-(4) some obvious changes must be made to the statement of Proposition 3.1, to the optimal trading execution rate (50) and to the results derived in Proposition 3.2. However it is easy to see that the analysis of the problem of maximizing (6) subject to the constraints (1)-(4) carried out in Section 3 can be extended to the problem of maximizing (80) subject to the constraints (1)-(4). To keep the exposition simple we leave to the reader the effort of working out these details.

Let us point out that the aversion to abrupt position changes and the possession of stable positions in the asset shares are typical habits of the so called buy and hold investors. Instead the so called short term investors do not like to have stable positions in the asset shares. This last kind of investors open and close their positions within a relatively short time period to exploit short term movements of the asset share price. Based on these facts we argue that the utility function (80) when is close to one describes buy and hold investors while when is close to zero describes short term investors. Moreover it is easy to see that the parameter measures the desire of the retail traders of behaving in a similar way. This desire decreases when increases.

In the study cases that follow we consider the solution of the problem:

, (81)

when is given by (80) subject to the constraints (1)-(4) when, , , , and we study how the solution of the previous problem influences the trading execution strategy adopted by the big trader to implement the liquidation order.

In the second, third and fourth study cases we study the behaviour of the big trader during the execution of a liquidation order, that is we consider the solution of the optimal control problem (15), (16), (7)-(9). In particular we study the dependence of the behaviour of the big trader from the behaviour of the retail traders determined solving problem (81), (80), (1)-(4).

Figure 1. The function, , when, , (solid line) and (dashed line).

Figure 2. The function, , when, , (solid line) and (dashed line).

Figure 3. The function, , when, , and (solid line), (dashed line) and (dotted line).

In the study cases presented we assume that the number of asset shares held by the big trader at time is and that the final time within which the sale of the asset shares must be completed is. Recall that the time horizon of the mean field game problem that describes the retail traders is chosen as. We choose, in (16). Recall that (16) does not depend from.

We solve the differential Equation (7) with the initial condition (8). Recall that in (7), , , are determined solving the mean field game problem associated to the retail traders and depend on. As consequence the functions and, determined solving the optimal control problem associated to the big trader depend from. The solution, of (7) that satisfies (8) is:

, (82)

Figure 4. The function, , when, , and (solid line), (dashed line) and (dotted line).

that corresponds to the trading rate:

. (83)

Note that when the function, , defined in (82) satisfies (9), that is we have. Moreover when that is when the behaviour of the retail traders does not contribute to the asset share price equation, the optimal trading execution strategy of the big trader resulting from (82), (83) is:

,. This strategy corresponds to the sale of the asset shares held at time

with constant trading rate, , during the time interval. That is the strategy determined using the model developed in the previous Sections when coincides with the optimal trading execution strategy of a risk neutral trader found by Almgren in  . It is easy to see that the instantaneous market impact of the trading activity of the retail traders introduced in Equation (10) (i.e. the term, of (10)) can be seen as a kind of directional view of the big trader about the asset share price dynamics. Note that

since Equation (56) implies that has the same sign of,. This means that positive values of imply that is positive, that is the “majority” of the retail traders are “buyers” of the asset shares, vice versa negative values of imply that is negative, that is the “majority” of the retail traders

are “sellers” of the asset shares. Recall that when the trading activity of the retail traders influences the asset share price dynamics through the term of the asset share price equation and, as a consequence, influences the optimal trading execution strategy of the big trader.

In the second study case we choose, and in the mean field game model that describes the retail traders and we study the corresponding optimal trading execution strategy of the big trader as

a function of and,. Recall that when we have, ,

therefore when the solution of the mean field game problem (81), (80), (1)-(4) corresponds to the solution of problems (5), (6), (1)-(4) with Recall that we have chosen. Figure 5 shows the optimal trading execution strategy of the big trader resulting from the formulae presented in Section 4 for

Figure 5. Optimal trading execution strategy of the big trader, when, , , (dotted line), (dashed line) and (solid line).

several values of. When and we have,. In this case we obtain as optimal trading execution strategy of the big trader the optimal trading execution strategy of the risk neutral trader found by Almgren in  (i.e. selling with constant rate, this is the straight line segment shown in Figure 5). When and we have, , that is the “majority” of the retail traders have long positions on

the asset shares and that is the “majority” of the retail traders are buyers of the asset shares in the time

interval and the asset share price tends to increase in the time interval. This determines the behaviour of the big trader. In fact in this case the big trader waits to sell his shares at the end of the time interval assigned to execute the liquidation order, even more the big trader buys asset shares at the beginning of the time interval assigned to execute the liquidation order (i.e. we have when is close to zero) to take advantage of the expected rise of the asset share price in the time interval induced by the

behaviour of the retail traders. That is in this case the trading execution strategy, , of the big

trader is a concave function that connects the points, of the plane as shown in Figure 5, Figure 6. When and we have, , that is the “majority” of the retail traders

have short positions on the asset shares (and that is the “majority” of the retail traders are sellers of

In the third study case in the mean field game model that describes the retail traders we assume

Figure 6. Optimal trading execution strategy of the big trader, when, , , (dotted line), (dashed line) and (solid line).

Figure 7. Optimal trading execution strategy of the big trader, when, , , (dotted line), (dashed line) and (solid line).

, and we study the corresponding optimal trading execution strategy of the big trader when varies in. The remaining parameters of the trading execution model have the same values than in the

second study case. We recall that when and we have, , , that is in

In the fourth study case we assume and in the mean field game model that describes the retail traders and we study the corresponding optimal trading execution strategy of the big trader as a function of, and,. Figure 10 shows the function, obtained as solution of problem (81),

(80), (1)-(4) when, , and. We observe that when and the

function, changes sign in the time interval in fact the function, from being positive in a neighborhood of becomes negative in a neighborhood of. Recall that Equation (56)

implies that when changes sign also changes sign. That is in the time interval the “majority”

of the retail traders from being buyers of the asset shares when is close to zero become sellers of the asset shares when is close to.

Figure 11 shows the optimal trading execution strategy of the big trader corresponding to the behaviour of the retail traders shown in Figure 10. Figure 11 shows that the big trader as a consequence of the behaviour of the retail traders initially waits to sell his shares, this is coherent with the fact that at the beginning the retail traders are buyers. However later, when the retail traders become sellers and the asset share price tends to decrease, Figure 11 shows that the big trader changes strategy and tries to anticipate the sale at the beginning of the time

Figure 8. Optimal trading execution strategy of the big trader, when, , , , (solid line) and, (dashed line).

Figure 9. Optimal trading execution strategy of the big trader, when, , , , (solid line) and, (dashed line).

Figure 10. The function, when, , , , ,.

interval that remains to conclude the execution of the liquidation order. That is the optimal trading execution strategy of the big trader, , shown in Figure 11, from being a concave function of when is close to zero becomes a convex function of when is close to and the saddle point of, , is the value of that corresponds to the zero of the function,. Finally Figure 12 shows the function, , obtained as solution of problems (81), (80), (1)-(4) when, , and. We observe that when and the function, , changes sign in the time interval, in fact the function, from being negative in a neighborhood of becomes positive in a neighborhood of. That is in the time interval the “majority” of the retail traders from being sellers of the asset shares when is close to zero become buyers of the asset shares

Figure 11. Optimal trading execution strategy of the big trader, when, , , ,.

Figure 12. The function, when, , , , ,.

Figure 13. Optimal trading execution strategy of the big trader, when, , , ,.

The interested reader is referred to the website: http://www.econ.univpm.it/recchioni/finance/w19 to find material including animations, an interactive application and an app that helps the understanding of this paper. A general reference to the work of the authors and of their coauthors in mathematical finance is the website: http://www.econ.univpm.it/recchioni/finance.

Cite this paper

LorellaFatone,FrancescaMariani,Maria CristinaRecchioni,FrancescoZirilli, (2014) A Trading Execution Model Based on Mean Field Games and Optimal Control. Applied Mathematics,05,3091-3116. doi: 10.4236/am.2014.519294

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