Optimal Geometric Mean Returns of Stocks and Their Options

The optimal geometric mean return is an important property of an asset. As a derivative of the underlying asset, the option also has this property. In this paper, we show that the optimal geometric mean returns of a stock and its option are the same from Kelly criterion. It is proved by using binomial option pricing model and continuous stochastic models with self-financing assumption. A simulation study reveals the same result for the continuous option pricing model. 1. Introduction The original question of Kelly criterion [1] is how to bet the fraction of your total wealth to maximize your long-term wealth when the odds and probabilities of a gambling game are known. Latane [2] first introduced the geometric mean investment strategy into finance and economics. As an application of generalized Kelly criterion, Latane and Tuttle [3] proposed a wealth maximizing model for building portfolios using geometric mean return. Bickel [4] discovered the relationship between optimal long run growth rate and the efficient portfolios based on the minimum variance criterion. Weide et al. [5] and Maier et al. [6] developed a strategy which maximizes the geometric mean return on portfolio investment. Similar research can be found by Ziemba [7], Elton and Gruber [8], and Bernstein and Wilkinson [9]. How to optimize the geometric mean return by the Kelly criterion becomes an important question faced by many portfolio managers and researchers. In the literature, Kelly criterion is also known as growth optimal portfolio, capital growth theory of investment, geometric mean strategy, investment for the long run, and maximum expected log. Estrada [10] used it as geometric mean maximization (GMM) and compared the popular mean variance analysis and Kelly criterion from an empirical perspective. Merton [11] was the first one to address the dynamic portfolio choice problem using the idea from Kelly criterion, which becomes a well-known topic in finance. McEnally [12] provided an overview of Kelly criterion, and MacLean et al. [13] summarized desirable and undesirable properties of Kelly criterion. Stock options are popular in many financial markets. An option is a contract between a buyer and a seller that gives the buyer right to buy or to sell a particular stock at a later day with a fixed price. A call option gives buyers right to buy stock and a put option gives buyers right to sell stock. The theoretical value of an option can be evaluated according to several models. Most of the theorems and models assume that market is free of arbitrage. Arbitrage is to make a guaranteed