Time diversification and changing volatility in an options pricing framework

Journal of the Academy of Business and Economics, March, 2004 by Ronald Best, Charles W. Hodges, Robert C. Yoder, James A. Yoder

ABSTRACT

We examine time diversification with changing asset volatility by evaluating the cost of insuring that a portfolio earns at least the risk-free rate of interest as the investment horizon lengthens. If stock returns are mean reverting, the cost of shortfall insurance will be less than if returns follow a random walk, since the annualized volatility of returns will be less under a mean-reverting process than for a random walk. Empirical evidence suggests that the degree of mean reversion is not sufficient to cause a decline in the cost of shortfall insurance. Furthermore, if bond returns are mean-averting, this implies that shortfall insurance is more costly than under a random walk. We derive theoretical conditions necessary for mean reversion to cause the cost of shortfall insurance to decrease with time.

1. INTRODUCTION

Time diversification, the idea that the risk of an investment will decrease with the length of the holding period, has recently been analyzed in an option-pricing framework. Bodie [1995] employed the Black-Scholes model to determine the cost of insuring against a stock portfolio earning less than the risk-free rate of interest. He demonstrated that the concept of time diversification does not hold, since the shortfall insurance cost may be viewed as a put option whose value increases with the length of the investment horizon. His analysis assumes that stock returns follow a random walk and are independent (zero autocorrelation) across time. One implication of this is that risk, as measured by the annualized standard deviation of stock returns, is constant over time.

Bodie further argued that mean reversion in stock returns does not affect his results. Although this statement is valid if stock price volatility is known, in reality, volatility is unknown and changing. In a world of stochastic volatility, under certain conditions, the implied volatility on a given option should equal the average volatility that is expected to prevail over the life of the option (Stein, 1989). This suggests that, if a measure of volatility is to be used to obtain an option's value, it should be estimated over a period whose length is consistent with the option's life. Since a number of previous studies have shown that stock return volatility decreases with the length of the investment horizon, it becomes an empirical question as to whether the volatility decrease is large enough to cause a decline in the cost of shortfall insurance (see Poterba and Summers [1988], Reichstein and Dorsett [1995], and Vanini and Vignola [2002]).

In this paper, we examine the issue of time diversification by specifically allowing for changes in the volatility of security returns as the length of the investment horizon changes. Our results indicate that stock return volatility decreases as the investment horizon increases, but the decrease is not large enough to cause the cost of shortfall insurance to decline. Bond return volatility actually increases with the length of the investment horizon making time diversification inapplicable.

2. THE INVESTMENT HORIZON AND THE COST OF PORTFOLIO INSURANCE

We analyze the cost of insuring against a stock or a bond portfolio not earning at least the risk-free rate as the investment horizon lengthens. This shortfall insurance can be valued as a put option in a Black-Scholes option-pricing framework. Since we wish to insure against the possibility of an investment in securities, S, earning less than the risk-free rate, the exercise price, E, is set to [Se.sup.rT]. Substituting into the put-call parity theorem yields P = C. In this special case, the put price expressed in terms of the Black-Scholes formula and stated as a fraction of the security price gives:

(1)

P/S : N(d1) - N(d2)

where: d1 - [sigma] [square root of T]/2

d2 = - [sigma] [square root of T]/2

N(*) = value of the cumulative normal density function

Equation 1 shows that the put price as a fraction of the security price depends only on the time until maturity of the option, T, and the underlying asset's volatility, [[PHI].sub.T].

Bodie shows that P/S increases with the investment horizon if the annualized standard deviation of returns, [[PHI].sub.T], is constant. Although assuming constant volatility for all time horizons is valid when the security volatility is known, in a world of stochastic volatility, the average volatility expected to prevail over the life of the option is a more appropriate measure. Thus, if stock returns are generated by a mean-reverting process and volatility decreases with the length of the investment horizon, it is possible that the cost of shortfall insurance could decline, since the put value decreases as volatility decreases.

3. MEAN AVERSION, MEAN REVERSION, AND INVESTMENT HORIZON VOLATILITY

We compare the volatility for various investment horizons using the following ratio:

(2) RT = ([sigma].sub.T]/[square root of T]/[[sigma].sub.1]

where : RT = ratio of annualized standard deviation of returns for T--year holding period