اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدOpen Markets
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تأليف
Donghan Kim
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An open market is a subset of an entire equity market composed of a certain fixed number of top capitalization stocks. Though the number of stocks in the open market is fixed, the constituents of the market change over time as each companys rank by its market capitalization fluctuates. When one is allowed to invest also in the money market, the open market resembles the entire closed equity market in the sense that the equivalence of market viability (lack of arbitrage) and the existence of numeraire portfolio (portfolio which cannot be outperformed) holds. When access to the money market is prohibited, some topics such as Capital Asset Pricing Model (CAPM), construction of functionally generated portfolios, and the concept of the universal portfolio are presented in the open market setting.
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We consider the problem of optimal inside portfolio $pi(t)$ in a financial market with a corresponding wealth process $X(t)=X^{pi}(t)$ modelled by begin{align}label{eq0.1} begin{cases} dX(t)&=pi(t)X(t)[alpha(t)dt+beta(t)dB(t)]; quad tin[0, T] X(0)&=x
_0>0, end{cases} end{align} where $B(cdot)$ is a Brownian motion. We assume that the insider at time $t$ has access to market information $varepsilon_t>0$ units ahead of time, in addition to the history of the market up to time $t$. The problem is to find an insider portfolio $pi^{*}$ which maximizes the expected logarithmic utility $J(pi)$ of the terminal wealth, i.e. such that $$sup_{pi}J(pi)= J(pi^{*}), text {where } J(pi)= mathbb{E}[log(X^{pi}(T))].$$ The insider market is called emph{viable} if this value is finite. We study under what inside information flow $mathbb{H}$ the insider market is viable or not. For example, assume that for all $t<T$ the insider knows the value of $B(t+epsilon_t)$, where $t + epsilon_t geq T$ converges monotonically to $T$ from above as $t$ goes to $T$ from below. Then (assuming that the insider has a perfect memory) at time $t$ she has the inside information $mathcal{H}_t$, consisting of the history $mathcal{F}_t$ of $B(s); 0 leq s leq t$ plus all the values of Brownian motion in the interval $[t+epsilon_t, epsilon_0]$, i.e. we have the enlarged filtration begin{equation}label{eq0.2} mathbb{H}={mathcal{H}_t}_{tin[0.T]},quad mathcal{H}_t=mathcal{F}_tveesigma(B(t+epsilon_t+r),0leq r leq epsilon_0-t-epsilon_t), forall tin [0,T]. end{equation} Using forward integrals, Hida-Malliavin calculus and Donsker delta functionals we show that if $$int_0^Tfrac{1}{varepsilon_t}dt=infty,$$ then the insider market is not viable.
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