08-NOV-2024
We study a continuous-time Kyle–Back insider trading model in which the informed agent receives the asset’s true value $V$ at a random Poisson time $\tau \sim \mathrm{Exp}(\mu)$, rather than at inception. The model decomposes into two phases: before signal arrival, the price is uninformative and the insider’s continuation value satisfies a linear ODE; after arrival, a standard Kyle–Back equilibrium operates on the residual horizon. Our main result is a closed-form formula for the insider’s expected profit, $\mathbb{E}[\Pi](\mu) = \tfrac{\sigma_z \sqrt{\Sigma_0}}{2}\!\left[\sqrt{T} – \tfrac{\sqrt{\pi}}{2\sqrt{\mu}} e^{-\mu T} \,\mathrm{erfi}(\sqrt{\mu T})\right]$, involving the imaginary error function. The formula interpolates between zero profit ($\mu \to 0$) and the classical Kyle profit $\tfrac{\sigma_z\sqrt{\Sigma_0 T}}{2}$ ($\mu \to \infty$), and is strictly increasing in $\mu$, $\Sigma_0$, $T$, and $\sigma_z$.