OPTIMAL PORTFOLIO OF A SENSITIVE INVESTOR IN A FINANCIAL MARKET

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.