ICASTOR Journal of Mathematical Sciences Vol. 9, No. 2 (2015) 51  55
__________________________________________________________________________
Correspondence: Eghwerido Joseph Thomas,
Department of Mathematics and Computer Science, Federal University of Petroleum Resources Effurun, P.M.B 1221, Effurun, Delta State, Nigeria. Email: eghwerido.joseph@fupre.edu.ng
INVESTMENT RETURNS WITH AN NSTEP GENERALIZED UTILITY FUNCTIONS
E. J. THOMAS, E. EFEEYEFIA & E. EKUMAOKEREKE,
Department of Mathematics and Computer Science, Federal University of Petroleum Resources Effurun, Effurun, Delta State, Nigeria.
ABSTRACT
In this paper, we formulated a general way of generating optimal wealth of investment returns with Nstep utility functions through a martingale approach. The problem of maximization is solved via Lagrange method. The performance of any investment from daytoday is determined by the ratio of the probability of rising to falling and the neutral probability.
KEYWORDS AND PHRASES:
Martingale, Risk Neutral Probability, Utility Function, Optimal Wealth.
INTRODUCTION
Investors usually incur loss as a result of wrong investment options and could thus be confronted with inability to quickly determine the best allocation of wealth in their choice of investment. The amount and period of investment in achieving a particular goal in relation to probability of doubling or halving of returns can thus be of particular importance, hence, one has got a certain amount of money and tries to invest it in such a way to draw maximum or optimal returns. With payment
returns an amount
after one period. The rate of return is that value
that makes the present value of the return equal to the initial payment.
(1) where;
= amount invested in security
= investment returns
J. T. EGHWERIDO, E. EFEEYEFIA & E. EKUMAOKEREKE
52
= a nonnegative random variable
= the rate of return from investment
1. RISK NEUTRAL MEASURE:
Given a probability space
,,
Let us consider a singleperiod binomial model; then a probability measure
is called risk neutral if for all
0,1,…,
!
"#
$
Suppose we have a twostate economy. Let the initial stock price be S which can either go up to
%
&
or down to
%
'
with interest rate R > 0 and
%
'
(1)%(%
&
*
Then, the risk neutral probability of an upward stock movement is given by
+
!"!
!

"!
(2) Let the derivative with payoff be
.
&
when the stock price moves up and
.
'
when it goes down, then we can price the derivative via
.
/

2"/
3
(3) Hence, the payoff for
N
period
4
at time step
is given by
4
5
67"
with probability
58$19
67"
:0,1,…,58
for an even step or
4
5
67""
with probability
5891$19
67""
:0,…,5891
for an odd step and for initial capital
x
with probability of increase and decrease
q
and
p
respectively and risk neutral probability measure
;
. The dynamic optimization problem stated can be represented as a static optimization problem over terminal wealth:
<
=
>?@
A
BCDE
(4) subject to
BDE
F
(5) where
H =
denotes state,
U =
the utility function, and
Q =
risk neutral probability
x =
initial stake.
INVESTMENT RETURNS WITH AN NSTEP GENERALIZED UTILITY FUNCTIONS
53
2. GENERALIZED NSTEP UTILITY FUNCTIONS
Let
GH
8 G
H
utility function, such that
IJIK
Suppose
L
&
M58$
67N=
67"
G
(6) and,
L
O
L
&
9PM58$;
67"67N=
;
H
9$
(7) applying the Lagrangian method, it is easy to see that
Q
IR
S
IJ
0 8
IR
S
IJ
0
we have
G;
PT;U
67"
T;U
and for
G;
PT;U
67"
T;U
V
G;
W
PT;U
67"W
T;U
W
*
Hence,
G;
=
PT;U
67
Clearly,
G;
G;
=
X;Y
T;U
G;
=
T;;U
Also,
ZL
O
ZP[\58$
67N=
;
67"
;
G9]0
which implies that,
GB
+
);E
67
J. T. EGHWERIDO, E. EFEEYEFIA & E. EKUMAOKEREKE
54
Suppose
+
)
+
1,
then
GH
CONCLUSION
This study concluded that our initial stake is maximized by the ratio of the probability of rising to falling together with the risk neutral probability measure.
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