Viable Insider Markets

Abstract

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)&=π(t)X(t)[α(t)dt+β(t)dB(t)]; \quad t\in[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 ${\epsilon }_t>0$ units ahead of time, in addition to the history of the market up to time $t$. The problem is to find an insider portfolio ${\pi }^{\ast }$ which maximizes the expected logarithmic utility $J(\pi )$ of the terminal wealth, i.e. such that $$\underset{\pi }{\sup }J(\pi )=J({\pi }^{\ast }),\text{where }J(\pi )=\mathbb{E}[\log (X^{\pi }(T))].$$ 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 $t+{\epsilon }_t\ge T$ 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 $B(s);0\le s\le t$ 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\}_{t\in[0.T]},\quad \mathcal{H}_t=\mathcal{F}_t\veeσ(B(t+ε_t+r),0\leq r \leq ε_0-t-ε_t), \forall t\in [0,T]. \end{equation} Using forward integrals, Hida-Malliavin calculus and Donsker delta functionals we show that if $${\int }_0^T\frac{1}{{\epsilon }_t}dt=\infty ,$$ then the insider market is not viable.