Option Theory Does Not Refute Time Diversification

Abstract

It is often recommended that asset allocation, in particular the proportion of stocks in a portfolio, should be influenced by the time for which the portfolio is expected to be held. Short investment horizons are seen as unsuitable for stocks. Long-term portfolios are regarded as suitable for high proportions of stocks. This is partly because long investment periods are seen as moderating the relative risk of stocks without distracting from the relatively high expected returns of stocks. Time diversification is one factor that ameliorates the long-run risk of stocks. Over the long term there will be periods of relatively good returns and periods of relatively poor returns; good periods and bad periods have a tendency to offset each other over long time spans. In consequence, risk increases less than proportionately with time, whereas returns have a compounding effect over time.

Some academics have suggested that stock market risk increases as the investment horizon lengthens, thus refuting the concept of time diversification. That argument has been based on an apparent increase in the cost of hedging stock price risk, as indicated by theoretical put option prices, which appear to increase when the investment horizon extends. This article argues that those views are based on a restrictive use of the Black-Scholes option pricing model. When the model is used less restrictively, put option prices are found to decrease as the investment horizon lengthens when investment growth exceeds a particular rate. Risk, as measured by the cost of hedging, may decline as the holding period lengthens. Depending on expected investment growth, option theory supports time diversification rather than refuting it.