LINEBURG


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the problem are shown as follows.
The real optimal switching times are:




Time period [0,1] is divided into four subintervals:
Further Computational Experiments and Results 117




The control is mapped into:




The state takes values as:




The minimization of the objective function at is:




Figure 5.5. Plot of n=8, forcing function ut=1,0,1,0,1,0,1,0



Figure 5.5 also shows a better approximation than
118 OPTIMAL CONTROL MODELS IN FINANCE

The results of other cases at are shown in Table 5.2. The value
of the objective function decreases when (the number of the time intervals)
increases.




2.3 Problem of the fitting function with big slope
In this section, a special case is introduced where the fitting function™s slope
is greater than 1 at time Since the given fitting function is approximated
by a piecewise-linear function with slope 1 and 0, the fitted function does not
need any part of slope 0 near In this case, the give fitting function
has a slope 1.6 at thus a slope 1 cannot give a good fit any more. A
similar problem happened in Section 5.2.2 where the computation reached a
local minimum also occurred at case
Consider the following optimal control model of finance:




subject to:




After running the program of the algorithms 2.1-2.4, two mistakes were
found. First, the control stopped jumping at case Second, the approxi-
mation did not work well at the slope close to 1.
Further Computational Experiments and Results 119

The first problem was met while computing the optimal control problem
(5.5)-(5.9). A different start of um can fix this problem. Here, we introduce
several other methods to exam which one works well in the computation. First
(method 1), simply divide each optimal um at case by 2 as the initial
start of um for case The idea behind this method is that if the com-
putation starts from a good minimum at the smaller jumps of control, it will
also lead to a good minimum at the greater jumps of control. Second (method
2), since the non-accurate results always gave zero of um, which means no
switching times, the lower bound of the um can be modified from the original
zero to a very small number to avoid the situation of um = 0. Thus, the control
jumps. It does not guarantee the computation will reach optimum. Sometimes
these two methods can be combined together. Method 3, is the method used in
Section 5.2.2 for solving a similar problem.
Using a control that can provide a bigger slope for the fitting function can
solve the second problem mentioned earlier about the slope of the given fitting
function. Method 4 lets control take a larger value to fit a function whose slope
is great.
Figure 5.6 is the result of method 3. “*-” represents state function, “.” rep-
resents the fitting function




Figure 5.6. Plot of n=8, forcing function ut= 1,0,1,0,1,0,1,0
120 OPTIMAL CONTROL MODELS IN FINANCE

The tests of all these methods are put in Table 5.3.




J is the value of the objective function. These five cases represent four
methods described earlier. Case 1 is for method 1, Case 2 is for method 2,
Case 3 is for the combination of method 2 and method 3, Case 4 is for method
3, and Case 5 is for method 4. Method 1 and method 3 both obtain similar re-
sults. This proves that the right initialization of um0 will help the computation
searching the optimum. Method 2 did not give promising results in the test. It
can be explained that certain time intervals are not optimal, so if the control
is forced to jump in that period, the optimum will be skipped. Method 5 does
solve the problem with a big slope of the fitting function.

2.4 Conclusion
The three experiments that have been done in Section 5.2.1, 5.2.2, 5.2.3
are very important for the computer package which was developed for solv-
ing a certain class of optimal control problems. They make the program more
general, the results more accurate, and computation more stable. The optimal
control problems when controls are step functions can be solved using this
computer package. Only very small modifications are needed. The accuracy
gained from these computational algorithms is also very useful for the algo-
rithms 3.1-3.4 for oscillator problems.

3. The Financial Oscillator Model when the Control Takes
Three Values
In this section, an experiment when the control takes three values is intro-
duced to test the algorithms 3.1-3.4 in Chapter 3. The results of the problem
are also analyzed.
Further Computational Experiments and Results 121

3.1 The control takes three values in an oscillator problem
Consider the financial oscillator model as follows:




subject to:




Apply the Algorithms 3.1-3.4 to this problem. There are two changes that
need to be made in Algorithm 3.3 and Algorithm 3.4. One is re-mapping the
control from original two-value pattern to three-value pattern. Here, the con-
trol is mapped to “1,0,1 in the successive time intervals. Note the smallest
number of the subdivisions must be 3 because of the control values. Another
change is in Algorithm 3.4 for the new fitting function
Several cases are tried and the results are shown in Table 5.4. The time
horizon [0,1] is first divided by 3,6,9, thus becomes 3,6,9 time intervals.
Each time interval is further subdivided by 2,3,4,6,8 to gain the better results
of the integral in the objective function.




From the results in Table 5.4, a conclusion can be made that the subdivision
of the time intervals is necessary especially when the nb (the number of the
122 OPTIMAL CONTROL MODELS IN FINANCE

time intervals) is big. Note that in case nb = 9, ns = 1, the result of the
objective function is 1.5791, nearly double of the result of nb = 6, ns = 1.
This solution cannot be optimal. The results of states and optimal switching
times in this case are shown as follows.
The real optimal switching times are:




The states take values:




The value of the objective function is:




In this case, the control only jumps twice and then does not change. From
the computation, we notice that a local minimum is reached instead of a global
minimum. The increased number of subdivision of the subintervals will give
more time for the system to reach the global minimum.
The chosen number for the subdivisions in the computation should be con-
sidered carefully. Since the control takes three values in this problem, the num-
ber of the big time intervals must be a multiplier of 3. Theoretically whether
the number of small time intervals is a multiple of 3 does not matter for the
computation. However, from the experimental results, we know that when the
number of the subdivisions are the multiplier of 3, the computation gives a
better approximation.
Figure 5.7 and Figure 5.8 are the graphical results in the case nb = 9 and
ns = 8. In that case, the computation gives accurate results. Figure 5.7 shows
the state function and the fitting function “o-” represents
state function “+:” represents the given fitting function
Further Computational Experiments and Results 123




Figure 5.7. Plot of nb=9, ns=8, forcing function ut=-2,0,2,-2,0,2,-2,0,2



Figure 5.8 shows the relationship between two state functions during the
time period [0,1]. Since in this case, the state takes 72 values, we only give the
graphical results here.




Figure 5.8. Relationship between two state functions during the time period 1,0
124 OPTIMAL CONTROL MODELS IN FINANCE

Graphic results of this oscillator problem are shown below. There are three
sets of them representing the solutions at nb = 3,6,9.
Further Computational Experiments and Results 125
126 OPTIMAL CONTROL MODELS IN FINANCE
Further Computational Experiments and Results 127
128 OPTIMAL CONTROL MODELS IN FINANCE




Set 1 for
Further Computational Experiments and Results 129
130 OPTIMAL CONTROL MODELS IN FINANCE
Further Computational Experiments and Results 131
132 OPTIMAL CONTROL MODELS IN FINANCE
Further Computational Experiments and Results 133




Set 2 for
134 OPTIMAL CONTROL MODELS IN FINANCE
Further Computational Experiments and Results 135
136 OPTIMAL CONTROL MODELS IN FINANCE
Further Computational Experiments and Results 137
138 OPTIMAL CONTROL MODELS IN FINANCE




Set 3 for
139
Further Computational Experiments and Results

4. Conclusion
There are some financial optimal control models where the given fitting
functions are cos or sin functions; then the financial control needs to take three
values with the middle being zero, to ensure a good approximation. The exper-
iment in Section 5.3.1 verifies the accuracy of the algorithms 3.1-3.4, thus the
computer software package CSTVA (for details see Appendix A.4) based on
these algorithms can solve all these kinds of control problems. The computed
results provide insights into the dynamics of the financial system in terms of
the state and control variables.
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Chapter 6

CONCLUSION




Modeling and computation of dynamic optimization problems in finance is
an important area for research in financial modeling. The thrust of this research
has been to develop computational methods in order to solve financial optimal
control models which are difficult to solve by traditional analysis using optimal
control theories. Four computer software packages called CSTVA have been
constructed, each of them used for different optimal control problems in two
areas of finance: optimal investment planning and optimal corporate financing.

The STV approach consists of the following six major computational meth-
ods:

1 An optimization program based on the sequential quadratic programming
(SQP)

2 The switching time variable method, the switching time is made a control
variable.

3 The finite difference method for estimating gradients when gradients are
not provided.

4 The step function approach to approximate the control variable.

5 A piecewise-linear (or non linear) transformation of time (as in MATLAB™s
“constr” program similar to the Newton Method for constrained optimiza-
tion).

6 Second order differential equations represent the oscillatory dynamic finan-
cial models.
142 OPTIMAL CONTROL MODELS IN FINANCE

Financial optimal control modeling with a cost of changing control is the
main topic in this book. Normally a cost is attached to a number of switch-
ing times then added to the objective function. The cost function becomes
a new objective function to be treated. The chosen cost for such an optimal
control problem for optimal investment planning in the stock market has been
discussed in Chapter 2. The control in the problem is approximated by a step-
function. The softwares constructed in this thesis compute the optimal switch-
ing times. Basically, the time period of the problem is divided into a certain
number of subintervals to solve the differential equation and calculate the ob-
jective function as an integral. For the problem in Chapter 2, a greater number
N of subdivisions will lead to a better fit to the target function When a
cost is attached to the switching times, the integral decreases as N increases,
but the cost of switching increases with N. Hence the total cost function
reaches an optimum. There are some techniques required for this research,
such as the approximation of the control, the time scaled transformation for
using the SCOM package “nqq” function, the piece-wise linear transformation
for the calculation of the differential equation and integrals, non-linear trans-
formation for the large time period, and penalty term transformations for the
constraints of the problems. The computed results are also compared with the
theoretical results. A certain class of optimal control problems can be put into
the formula introduced in this research. The computer software packages de-
veloped in this research can then solve these problems with very little or no
change.
Financing oscillator problems can form another class of financial optimal
control models. These kinds of problems have a great number of applications
in the real world. In these problems, the dynamic system can be described by
a second-order differential equation. It is required to convert the second-order
differential equation to two equivalent pair of first-order differential equations,
to enable the software to be used. In order to get more accuracy in solving the
differential equation, a further subdivision of the time intervals is introduced.
While the control takes several discrete values, the sequences of these values
may follow more than one pattern, leading to a different computed minimum.
The computational algorithms are designed to handle these problems. The
obtained computing results give good comparisons of them.
The modeling exercises in this book show the potential for modeling dy-
namic financial systems by adopting bang-bang optimal control methods. The
results of this modeling can provide improved understanding about the behav-
ior of and the decision problems in dynamic financial systems. The roles of
switching times in financial strategies and transaction costs of controls are use-
ful for financial planning as well.
The algorithms constructed in this thesis are applied to an optimal corporate
financing model, which has two state functions and two control functions. The
Conclusion 143

computation experiments with two patterns of controls were tried. Effective
results were obtained which also agree with the analytical solution from the
original work. Another computation with the SCOM package also agreed with
the computational solution in this research and the analytical solution. Further
research of trying different parameters sets will be an interesting exercise. The
accuracy of the computational algorithms in Chapter 3 and Chapter 4 were ver-
ified by these experiments. The STV approach may be considered satisfactory
in view of its computational efficiency and time, and the plausibility of results.
In the damped oscillatory financial model, many corporate finance mod-
els are concerned with the application of optimal control. Computational ap-
proaches to the determination of the optimal financing problems were applied
in a financial model which was first introduced by Davis and Elzinga [22,
1970]. The model discusses investment allocation in order to determine the
optimal proportion of the sources of finance which can maximize the value of
the company. In particular, this model determines the proportion of its earnings
that should be retained for internal investment and what proportion should be
distributed to shareholders as dividends. The model also aims to choose the
“smart” investment program that gives the owners the most benefits.
In most of the existing optimal corporate financial structure models, the op-
timal proportion of various sources of funds is determined. In a linear dynamic
finance model, this proportion may change depending on the bang-bang char-
acter of the time path in the model. For an improved understanding of the
behavior of the dynamic path of such models, it is essential to know the op-
timal switching times for changing the optimal proportion of different funds.
No algorithm exists in the literature which can determine the optimal switch-
ing time for corporate funds. The present book has developed such a model for
optimal corporate financing and switching timing.
In this research, the computational algorithms have been improved for solv-
ing bang-bang optimal control problems. Applications of the STV algorithm
to finance have been made to show the potential and methods for developing
dynamic optimization methods in finance. Further research is necessary to im-
prove the state of the art in computing bang-bang optimal control in general
and financial optimal control models in particular.
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Appendix A
CSTVA Program List




1. Program A: Investment Model in Chapter 2
Project1.m
function J=project1(C,nn)
J=project1_1(nn)+C*nn

Project1_1.m
% Program for the project 1 in chapter2

function project1_1=project1_1(nn)
figure;

par=[1,1,nn];
% parameters: 1 state, 1 control, nn subintervals.

options(14)=2000;
% maximum number of function evaluations.

options(13)=1;
% one equality constraint

xinit=0.0;
% initialize the state function xt

um0=ones(nn,1)/nn;
% take a initial guess of starting time intervals

s=[0:0.05:1];
plot(s,1/2*s,˜:™)
ul=zeros(nn,1);
% lower bound of um, t(nn)-t(nn-1) >= 0

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