dc.description.abstract |
A sensitive investor seeks to diversify assets and optimal portfolio which provide
the maximum expected returns at a given level of risk. Optimal portfolio problems
of an investor with logarithmic utility have been studied. However, there is scarce
information on other utility functions, such as power utility function, which cap tures the concept of diversification of portfolios. This study was therefore designed
to consider the general expected utility of a sensitive investor in a financial market.
Two models were derived from the Itˆo’s integral with respect to power utility
function. The extension of the Itˆo’s integral by forward integral with its lofty
properties was used to diversify the investors portfolio. A filtration was built
and used as a set of information for the investor. A semimartingale was used to
enlarge the investors information. A probability function was defined to capture
the activity of an insider in the market and penalty function was established to
punish such an insider. A priority Mathematical software was used to compute the
investors varying rates of volatility.
The models derived were:
U
0
(Sβ1γ1+yφ(T)) Sβ1γ1+yφ(T)|M(y)| = S
y
β1γ1+yφ(T)|M(y)|
and n
dis
t = (1 − C1C2)(ρ
k
t + πt), respectively, where U
0
(x) = dU(x)
dx is satisfied if
supy∈(−δ,δ){E[Sβ1γ1
y+yφ(T)|M(y)|
p
] < ∞} for some p > 1
0 < E[U
0
(Sβ1γ1+yφ(T)) Sβ1γ1+yφ(T)] < ∞
Sβ1γ1+yφ(T) = Sβ1γ1+yφ(T)Nβ1γ1
(y),
where
Nβ1γ1
(y) := s0 exp Z T
0
[µ(s) − r(s) − σ
2
(s)β1(s)γ1(s)]ds +
Z t
0
σ(s)dW(s)
for all β1γ1, φ ∈ AG such that AG is the set of admissible portfolios with diversi fication and φ bounded, then there was existence of δ > 0 and y ∈ (−δ, δ), where
W(t) is the Brownian motion (representing the fluctuation of the risky asset), on
a filtered probability space (Ω, F, {Ft}t ≥ 0, P) and the coefficients r(t), µ(t), σ(t)
are G = {Gt}0≤t≤T adapted with Gt ⊃ Ft
for all [0, T], T > 0 a fixed final time.
i
The Itˆo’s integral is adapted to the filtration F = {Gt}0≤t≤T . The forward in tegral showed that when an investor buys a stochastic amount α units of this
asset at some random time τ1 and keeps all these units up to a random time
τ2 : τ1 < τ2 < T, and eventually sells them at a subsequent time, the profits re alised would be αW(τ2)−αW(τ1) expressed as forward integration of the portfolio
φ(t) = αI(τ1, τ2](t), t ∈ [0, T] with respect to the Brownian motion W(t) i.e.
Z T
0
φ(t)d
−W(t) = lim
∆j→0
X
j
φ(tj ) × ∆W(tj ) = Z τ2
τ1
dW(t) = αW(τ2) − αW(τ1)
The filtration G = {Gt}0≤t≤T outlined the information flow of the investor. The
semimartingale integral R T
0
φ(t)dW(t) = R T
0
φ(t)d
−W(t) gives a decomposition
W(t) = Wˆ (t)+A(t), 0 ≤ t ≤ T, where R T
0
φ(t)dW(t) = R T
0
φ(t)dWˆ (t)+R T
0
φ(t)dA(t);
for Gt = Ft ∨ αW(T0); 0 ≤ t ≤ T i.e. Gt
is the result created by Ft and the fi nal value W(T0), where Wˆ (t) is a Gt-Brownian motion and A(t) is a continuous
Gt-adapted finite variation process. The probability of detecting and punishing an
insider was λ1 = 1 and λ2 showed the penalty on an insider observation. The
varying rates of volatility σ = 1, 0.5, s0 = 100, µ = 1, revealed that the expected
return is more when volatility σ = 1, thereby yielding optimal portfolio.
The optimal portfolio of a sensitive investor was established using power utility
function and showed higher investors return as the investor diversified his invest ment. |
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