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Optimal Control of a Continuous-in-time Financial Model

Emmanuel Frénod, Pierre Ménard, Mohamad Safa*

University Bretagne-Sud, UMR 6205, LMBA, F-56000 Vannes, France

Abstract: The achievement of a project requires tools to monitor and adjust its evolution over time. Rather than to check at mid-term whether the objectives will be achieved or not, and adjust them, it is interesting to develop a control tool in order to effectively conduct the project’s objectives. In this paper, we improve the continuous-in-time financial model developed in previous work, that describes working of loan and repayment, in order to prepare its capability to be used in control theory approach. The aim is to determine the optimal loan schedule taking into account the objective of the project, the income and the spending. For that, we set out an optimal control method for the strategy elaboration phase to better adjust the project implementation.

Keywords: financial modeling, optimal control, optimization

Citation: Emmanuel Frénod, Pierre Ménard, Mohamad Safa, Optimal Control of a Continuous-in-time Financial Model. American Journal of Modeling and Optimization, 5 (1). 1-11. January 2017. doi:10.12691/ajmo-5-1-1

1. Introduction

A project is a temporary activity group designed to produce a product, service or result. A project is temporary in that sense it has a defined beginning and end in time and so a lifetime. When the project involves a loan, it is necessary to find the best strategy in order to achieve the goals of the project, minimizing the cost of the loan. This strategy should be able to achieve the project goals on time under constraints that may be imposed by law, taxes, banking institutions and more generally by the project environment. It gives rise to the loan scheme which is the way that the amounts concerned by the loan will be borrowed.

Once this strategy is set out, and all along the project implementation, it is needed to control the way the loan scheme is implemented. The objective is to counterbalance the gap between the reality of the environment in which the project takes place and the forecast made during the strategy elaboration.

There are several references in the literature dealing with continuous-in-time financial model. Among them we find: R. Merton [6] which provides an overview and synthesis of finance theory from the perspective of continuous-in-time analysis. In [8], S. Sundaresan surveys and assesses the development of continuous-in-time methods in finance during the period between 1970 and 2000. In addition, many studies have used control engineering methods and techniques in finance. For example Keel [5] explored and extended optimal portfolio construction techniques currently found in the literature. Grigorieva & Khailov [4] built a controlled system of differential equations modeling a firm that takes a loan in order to expand its production activities.

In this paper, we base on a continuous-in-time financial model developed in Frénod & Chakkour [3] that describes working of loan and repayment. This model is not designed for the financial market but for the public institutions. The main objective of this paper is to use this model in the framework of the control theory. For that, we set out an optimal control method for the strategy elaboration phase and an adaptive control method to better adjust the project implementation.

2. Continuous-in-time Financial Model

In this section we improve the continuous-in-time financial model of Frénod & Chakkour [3] in order to account for the project notion and to prepare its capability to be used in control theory approach. The time domain is the interval [0, Θ], where Θ > 0 is the lifetime of the project. We consider that beyond Θ the spending associated with the project are done, the loan associated with the project is completely paid off and the project is finished.

2.1. Variables of the Model

To characterize the budget of a project, we introduce the loan density and the density of repayment ρK which is connected, as explained in Frénod & Chakkour [3], to the loan density by a convolution operator:

[picture] | (1)

where γ is the repayment pattern. Since the whole amount associated with the loan has to be repaid, γ has to satisfies:

[picture] | (2)

We denoted by the current debt, given as the solution to the following differential equation:

[picture] | (3)

where is the density of repayment of the current debt at the beginning of the period. It is called initial debt repayment scheme. Initial condition for equation (3) is given by:

[picture] | (4)

We denote by the density of interest defined by:

[picture] | (5)

where is the floating rate interest. The algebraic spending density is denoted it takes into account the spending and the income and it is given by:

[picture] | (6)

where is the “financing needs density”, it is the density of spending that are intended for the project only. Function is the current spending density. We assume that because only spending are concerned.

The fact that the initial time of the project is 0 and the lifetime is Θ translates as:

[picture] | (7)

where, for any function f, supp (f) is the support of f.

2.2. Objectives of the Project

Integrating (3) over [0, t], we obtain using (1) the following relation:

[picture] | (8)

and using (4), we obtain:

[picture] | (9)

We want that the spending density balances the income density. In our model we have the following densities: which, depending on its sign, stands alternately for income or spending, which is an income density and which are spending densities. Hence the balance relation reads:

[picture] | (10)

Using (9) and (5), we deduce the following relation:

[picture] | (11)

where the operator is defined by:

[picture] | (12)

is the algebraic income density associated to the loan. In other words, it is the difference between the income density induced by the loan density on the one hand and the spending density associated with the repayment density and the interest payment density on the other hand. Using (6) we have then:

[picture] | (13)

The isolated spending density is the difference between the algebraic income density as- sociated with the loan, and the spending densities related to the following: current spending, initial debt repayment and payment of the interests of this latter.

We define an objective as a couple collection where is the amount which has to be spent for the project at the moment We suppose that and to be consistant we need that We say that the objective is reached if:

[picture] | (14)

The above equation indicating that at any the amount allocated to the project is at least the amount needed for the project.

Using this model we will establish the strategy, i.e. find the loan which allows the objectives to be reached. Furthermore, this loan is not chosen at random but has to satisfy some conditions. Typically, it must minimize the cost of the loan. This strategy can be written as an optimal control problem which is developed in the next section.

3. Strategy Elaboration: Optimize Loan Cost

Among all the variables introduced in the previous section, the only one that can be modified is the loan density . We consider that the repayment pattern γ and the interest rate α are essentially imposed by banking institutions. The density of algebraic spending σ is imposed by the project environment. Hence γ, α and σ are, at the level of the strategy elaboration, foresights of what they will be when the project will be implemented.

We introduce the loan cost which is induced by interest payments and which is defined as:

[picture] | (15a)

where is the functional that links loan density to the density of interest

[picture] | (15b)

and the total amount of the loan is defined by:

[picture] | (16)

Minimizing loan cost consists in finding loan density κE that minimizes the loan cost .

We introduce now the following two optimization problems in a intuitive and not mathematically completely rigorous way. Those problems called (O1) and (O2) will then be inserted within rigorous framework in the next section.

O1. Optimization of the spending. Given find the loan density solution to:

[picture] | (17a)

subject to:

[picture] | (17b)

In this problem we seek the optimal loan density that minimizes the cost of the loan, considering that the amount allocated to the project until is at least the amount ci needed for the project at . There is no uniqueness of the solution to problem (O1). Among its solution of problem (O1) some are subjected to perturbations which are numerical artifices that are irrelevant from financial slant. We will therefore build a method based on filtering theory to eliminate those perturbations. Several filters will be investigated in the Section 5.2 to choose a suitable one to be incorporated in our method. We introduce here the general framework of the filtering based problem which is set in terms of an optimization problem:

O2. Filtering perturbed solutions to (O1) by an optimization problem. From any solution resulting from (O1), we denote by the desired unperturbed solution obtained by eliminating the perturbations in , which is defined by:

[picture] | (18)

where represents the perturbations density. is obtained as the result of the following optimization problem (called (O2)) which can be written generically as: given find solution of:

[picture] | (19a)

where the definition of the functional F depends on the filter which is used. The problem (19a) is subject to the following constraints:

[picture] | (19b)

Moreover, since G is the perturbation to be removed, it satisfies:

[picture] | (19c)

This is then a supplementary constraint for the optimization problem (O2).

4. Optimization Problem (O1)

4.1. Mathematical Framework of Optimization Problem (O1)

The objective of this section is to build the mathematical framework of the optimal control problem. We suppose that and introduced in section 2.1 are in and are in Let be the set of admissible loan densities such that where is the maximal loan densities that can be borrowed. The admissible loan density should allow to have a positive financing needs:

[picture] | (20)

By using (13), equation (20) is equivalent to write:

[picture] | (21)

and equation (19b) can be written in the next form:

[picture] | (22)

We can rewrite the optimization problem in a rigorous form. Denoting the set of function which satisfy the conditions:

[picture] | (23a)

[picture] | (23b)

[picture] | (23c)

where

[picture] | (24)

we search a solution of the following optimization problem:

[picture] | (25)

It is recalled that functions and are considered as known and are model inputs.

From now on, we work with a fixed interest rate in this case we can show that solving the problem (25) is equivalent to solve the following problem:

[picture] | (26)

which means: minimize the cost of the loan is equivalent to minimize the total amount of the loan. The latter is the problem that we will be interested in. We have the following lemma:

Lemma 1. For a fixed each function solution of (25) is also solution of (26) and vice versa.

Proof. According to (15b), we have

[picture] | (27)

Using (7), the previous equation can be written in the form:

[picture] | (28)

where

Using the fact that

[picture] | (29)

equation (28) can be written in the next form:

[picture] | (30)

Using (2) which indicate that

we obtain

thus

Besides this, we have:

[picture] | (31)

With the change of variable: y = s − x we obtain,

[picture] | (32)

and with the change of variable: z = s − y, we obtain:

using the fact that:

[picture] | (33)

we get:

[picture] | (34)

where

[picture] | (35)

We finally obtain from (31) and (34):

[picture] | (36)

This ends the proof of the lemma.

4.2. Existence of Solutions for the Optimization Problem (O1)

In this section we show the existence of solutions for the optimization problem (26). By introducing the following notation:

[picture] | (37)

and the continuous functional defined by:

[picture] | (38)

we can then rewrite U in the next form:

[picture] | (39)

We will invoke the following theorem which proof can be found in Cohen [1].

Theorem 2. Let V be a subspace of a Hilbert space. If is a lower semi-continuous function, convexe and coercive on if V is convexe and closed, if

where

then there exists at least a solution of the optimization problem:

Using Theorem 2 we obtain the next theorem:

Theorem 3. If set U given by (39) is such that then optimization problem (26) has at least one solution.

Proof. The set U given by (39) is convex and closed because it is the intersection of closed sets. In fact is a closed set, is a continuous linear operator and is a continuous operator. The functional:

[picture] | (40)

is a continuous function, so it is a lower semi-continuous function. In addition, is a convex function because it is linear on U and coercive because U is a bounded set (constraint (23a)). Since and according to Theorem 2, if then optimization problem (26) has at least one solution.

4.3. Discretization of Optimization Problem (O1)

From now, we suppose that the interest rate α is constant and is infinite. The way to discretize problem (26) consists in introducing on points such that where We assume that for any there exists a such that At the operational level, if this condition is not verified, we substitute by the closest Then we define n intervals such that defined by:

We consider then the space made of functions that are defined on and constant on each , for We introduce the approximation operator defined by:

[picture] | (41)

where is the mean value of the function f on interval We denote and we will use these notations afterward for the quantities of the model. We have:

[picture] | (42a)

[picture] | (42b)

With this discret space on hands, we consider the discret optimization problem that consists in finding solution of (26) under constraints (23), with replaced by by and by

It is straightforward to show that solving the optimization problem (26) is equivalent to solve the next one.

[picture] | (43)

where is the set of function satisfying:

[picture] | (44a)

[picture] | (44b)

[picture] | (44c)

where u is a function that is equal to in each interval i.e:

[picture] | (45)

Because of the piecewise constante nature of we have:

[picture] | (46)

where matrix resulting from the convolution, is defined by:

[picture] | (47a)

its expression is:

[picture] | (47b)

with, by using (42a):

is the matrix resulting from the approximation of the integral in the formula (12). The constraints given by (44) may then be written as:

[picture] | (48a)

[picture] | (48b)

[picture] | (48c)

where by using (42b):

and is the discretization of (45) given by:

[picture] | (49)

The discretization of the optimization problem (26) is finally written as the following one. For

[picture] | (50a)

subject to:

[picture] | (50b)

[picture] | (50c)

[picture] | (50d)

Problem (50) is a linear optimization problem.

5. Optimization Problem (O2)

5.1. Mathematical Framework of Optimization Problem (O2)

In this section we build a mathematical framework for the optimization problem (O2) introduced in Section 3. We suppose the optimal loan density and the perturbation density are in The functional F introduced in Section 3 is a map from to . Having resulting from (O1), we search solution of the next optimization problem. Given , find solution of:

[picture] | (51)

subjected to the same constraints of (O1), where is replaced by in formulas (23), and to constraint (19c). We can rewrite this optimization problem called (O2) in a rigorous form. Denoting the set of function which satisfy the conditions:

[picture] | (52a)

[picture] | (52b)

[picture] | (52c)

where is defined by (24). For given obtained from the optimization problem (O1), we search solution of the next optimization problem:

[picture] | (53)

and we thus obtain . The type of filter that we will use depends on the choice of F. We will investigated two types of filters: the first one is based on the thresholding method and the second on the total variation denoising (TVD).

Thresholding methode. It consists in eliminating the low and very high frequency of a signal, by writing the functional F, for all Y and G in as the following form:

[picture] | (54)

where is the Fourier transform operator and for fixed if The second member of (54) collects the mid-frequency of the derivative of

Total Variation Denoising (TVD). It is a tool for noise reduction developed so as to preserve sharp edges in the underlying signal [7]. In such cases we can define the functional F as:

[picture] | (55)

In (54) and (55), and are two regularization parameters to control the degree of smoothing. Increasing _2 gives more weight to the second term which measures the total variation of In the next section we will test the efficiency of each of these filters.

Since the function F(Y,G) given by (54) or (55) is strictly convex for a fixed Y and the set defined by (52) is convex, then we have the uniqueness of the solution of the optimization problem (O2).

5.2. Numerical Algorithm for Filtering

In order to test the efficiency of the filters introduced in Section 5.1, we build a class of periodic signals perturbed by some oscillations as shown in Figure 1, and we study the ability of filters to eliminate this oscillations. The goal is to estimate an original signal X from a perturbed version of X called Y and modeled as:

where G is the oscillations. Having Y, TVD estimates unperturbed signal X by solving the next optimization problem. For a given perturbed signal Y, we search G solution to:

[picture] | (56)

and we have then the unperturbed signal X = Y − G. We solve the last optimization problem by discretizing (54) or (55) and then using a finite-dimensional optimization algorithm. For that, we use the Majorization-Minimization (MM) algorithms developed by Figueiredo, and al [2].

As we can see from Figure 2 and Figure 3, the filtering method based on the optimization problem (56) allows us to remove the perturbations. Thresholding method seems more effective for the type of signals that we have in Figure 1.

Figure 1. Type of signals used to test the filters

Figure 2. Filtering by using TVD

Figure 3. Filtering by using thresholding method

6. Numerical Simulations

We solve the optimization problem (50) by using the simplex method. The numerical simulations were performed by using the mathematical software of scientific computing Matlab. We impose the repayment pattern the current spending and we seek the optimal loan which allows the objectives of the project to be realized. We check that the constraints are verified.

6.1. Example 1

In Example 1, the repayment is done in a constant way between the first and 6th year after borrowing. We pay off nothing outside this period. The current spending density alternates between positive values, corresponding to periods where income is less than spending and negative values, corresponding to periods where income is larger than spending. The objective to be achieved are fixed at the end of the 4nd and 14th year. The optimal loan density is given in Figure 4. Figure 5 shows that the constraints are satisfied. For this example we have taken:

and

Figure 4. Example 1. Optimal loan obtained for given: repayment pattern objectives c and current spending

Figure 5 shows the isolated spending density and the isolated spending obtained from Example 1. As we can see, we get and This means that the constraints of the optimization problem (O1) are satisfied and the project objectives are reached.

Figure 5. Checking if the constraints are verified for Example 1

6.2. Example 2

In Example 2, the repayment is only made in an increasing way between the 7th and 11th year after borrowing. We pay off nothing outside this period. The optimal loan density is given in Figure 6. Figure 7 shows that the constraints are satisfied. For this example we have taken:

and

Figure 6. Optimal loan obtained for given: repayment pattern objectives c and current spending

Figure 7 shows the isolated spending density and the isolated spending obtained from Example 2. As we can see, we get and The constraints of the optimization problem are well checked.

Figure 7. Checking if the constraints are verified for Example 2

7. Conclusion

In this paper we have presented an optimal control for a continuous-in-time financial model which is designed to be used for the finances of public institutions. This model describes the loan scheme, reimbursement scheme and interest payment scheme; and using mathematical operators to describe links existing between those quantities. The aim of the optimal loan strategy is to minimize the interest payment taking into account the objective of the project, the incomes and the spending; and to better adjust the project implementation. We have built a mathematical framework for the optimal control problem which consists to solving a constraint optimization problem. We have demonstrate the existing a solution of the optimization problem which is solved using the simplex method. The numerical simulations were performed by using the mathematical software of scientific computing Matlab. The simulations results are consistent with the theoretical part and shows the performance and stability of our approach.

Fund

This work is jointly funded by MGDIS company (http://www.mgdis.fr/) and the PEPS program Labex AMIES (http://www.agence-maths-entreprises.fr/).

References

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[3]   | E. Frénod and T. Chakkour. A continuous-in-time financial model. Mathematical Finance Letters, pages 1-37, 2016.
[4]   | E.V. Grigorieva and E.N. Khailov. Optimal control of a commercial loan repayment plan. Discret and continuous dynamical systems, pages 345-354, 2005.
[5]   | S.T. Keel. Optimal Portfolio Construction and Active Portfolio Management Including Alternative Investments. Phd thesis, ETH Zurich, 2006.
[6]   | R.C. Merton. Continuous-Time Finance. Blackwell, 1st edition, 1992.
[7]   | L. Rudin, S. Osher, and E. Fatemi. Nonlinear total variation based noise removal algorithms. Physica D: Nonlinear Phenomena, 60(1-4): 259-268, 1992.
[8]   | S.M. Sundaresan. Continuous-time methods in finance: A review and an assessment. Journal of Finance, pages 1569-1622, 2000.