## A Model for Stock Price Fluctuations Based on Information

### Authors: Larry Shepp

ABSTRACT

This paper is dedicated to the memory of Aaron Wyner, the well known information theorist. It gives a new model for stock price fluctuations based on a concept of information''. In contrast, the usual Black-Scholes-Merton-Samuelson model is based on the {\em explicit} assumption that information plays {\em no} role in stock prices. The new model is based on the non-uniformity of information in the market and the time delay until new information becomes generally known.

The new model is expected to give more accurate predictions of future prices and more accurate formulas for hedge option valuations. The new valuations have been calculated for the various standard options inside the new model in a recent PhD thesis by Xin Guo. Because the concept of information is the driving one in the new model it seemed appropriate to discuss this in the present volume even though information'' is used in a somewhat different sense here than in communication theory.

In communication theory, information is {\em intended} to be communicated. One is concerned with designing a means for the successful transfer of messages from a source to the receiver and the value of the information is in its successful transmission. In the stock market, the reverse is usually true; information is hoarded and is of value to the owner {\em only} until it becomes known to others. Some messages are avidly communicated in the market but are really meant to {\em disinform} the receivers. This may of course also be true occasionally in communication theory which usually avoids concerning itself with the {\em meaning} of the message to be transmitted. Despite these differences in the role of information in the two situations, there are similarities in the mathematics used to study them. Certainly probabilistical models play a prominent role in both theories.

In Shannon's communication theory quantitative value is assigned to a channel (its rate or its capacity). Is there anything similar in the market? We are able to assign a quantitative value to knowing an item of information {\it which is unknown to others} say that a company is in a strong (or weak) position at a given point in time, via pricing of options on the company's stock. This paper is intended as a first step towards the goal of making the use of information into a quantitative tool.

Problems with explicit solutions are of value in obtaining insights. At the end of the paper we compare several problems of mathematical interest in order to better understand which optimal stopping problems have explicit solutions.

Paper Available at: ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/2000/2000-30.ps.gz