Portfolio optimization for an insider under partial information

Abstract

In this article, we seek to solve the problem of stochastic filtering of the unobserved drift of the stock price in the presence of privileged information. Working within a finite time investment horizon, the privileged information which is a function of the future value of the stock price, is modeled such that its quality improves as we move towards the information reveal date. The hidden/unobserved drift is modeled as a Gaussian process. Combining the techniques of progressive enlargement of filtration and stochastic filtering of linear state-space models, we obtain explicit analytic results for the insider’s estimates of the unobserved drift process. In addition, we obtain the optimal portfolio strategy for an insider having the log utility function. Our numerical results reveal that when the quality of privileged information is high, the insider would require less initial capital as compared to the regular trader who has no access to the privileged information. Further, we show how the stock price volatility influences the value of the insider’s privileged information, with period of high volatility pointing to increased value of the privileged information.

Keywords

• Kalman-Bucy filtering
• Filtration enlargement
• Portfolio optimization
• Arbitrage
• Semimartingale