Pricing stock options under expected increasing and decreasing price cases.(Author abstract)

Introduction

Several empirical studies have provided evidence that the distributions of stock and stock index returns exhibit persistent skewness (for example, Kon, 1984; Aggarwal and Rao, 1990; and Turner and Weigel, 1992). We conduct a statistical analysis of the S&P 500 using the D'Agostino, Belanger, and D'Agostino tests of normality (1990) and find that over multiple periods of time the index often has followed patterns of persistent increases that are characterized by skewness. For example, the intermediate periods from 1982 to 1987 and 1988 to 1995 saw a trend of increasing stock prices characterized by positive average logarithmic returns and negative skewness for the S&P 500 (Figure 1). Similarly, normality tests applied to the longer period from 1970 to 2000 also show a period characterized by a positive average logarithmic return and negative skewness for the S&P 500. In contrast, for the period from 1970 to 1978, the D'Agostino, Belanger, and D'Agostino tests show a trend of stable prices characterized by a low average logarithmic return, but with skewness that is insignificant.

[FIGURE 1 OMITTED]

Empirical studies by Black (1975), MacBeth and Merville (1979), and Emanuel and MacBeth (1982) have reported pricing biases associated with the Black-Scholes (B/S) option pricing model (OPM) (1973). These biases generally are thought to be the result of the model's assumption that the option's underlying security's logarithmic return is normally distributed. Studies by Stein and Stein (1991), Wiggins (1987), and Hestin (1993) have demonstrated that when skewness exists, the Black-Scholes model consistently misprices options. To address the pricing bias resulting from the assumption of normality, Jarrow and Rudd (1982) and Corrado and Tie Su (1996) have extended the Black-Scholes model to account for cases in which there is skewness in the underlying security's return distribution. Similarly, Camara and Chung (2006) and Johnson, Pawlukiewicz, and Mehta (JPM) (1997) have extended the Cox, Ross, and Rubinstein (CRR) (1979) and Rendleman and Bartter (RB) (1979) binomial option pricing model to include skewness. In addressing skewness, Camara and Chung show how the up (u) and down (d) parameters in the binomial process are obtained from skewness and kurtosis of the distribution implied by market prices. Johnson, Pawlukiewicz, and Mehta, in turn, show that in the binomial modeling of security price patterns, the existence of skewness impacts not only the values of the up and down parameters, but also the probabilities of the underlying security increasing or decreasing each period. Specifically, a binomial process that converges to an end-of-the-period distribution of logarithmic returns that is normal will have equal probabilities of the stock increasing or decreasing each period, while one that converges to a distribution that is skewed will not.

In their paper, Johnson, Pawlukiewicz, and Mehta also show that the presence of skewness affects the relative contribution of the mean to the values of u and d. In the case of a positive mean, the mean becomes more important in determining the value of u, the greater the negative skewness. In contrast, in the case of a negative mean, the mean becomes more important to the value of d, the greater the positive skewness. The presence of skewness also changes the asymptotic properties of the u and d parameters in the Johnson, Pawlukiewicz, and Mehta skewness model. Specifically, for a large number of subperiods, n, the Cox, Ross, and Rubinstein/Rendleman and Bartter model depends only on the volatility of the underlying asset. Skewness, though, changes the order of magnitude as n becomes large, making u and d dependent on all three moments--variance, skewness, and mean. (1)

Objective

The noted observations on stock price trends suggest that in cases where stock prices are expected to increase or decrease, pricing biases may result when using the Cox, Ross, and Rubinstein/Rendleman and Bartter binomial option pricing model or the Black-Scholes option pricing model where the underlying security's logarithmic return is based on the assumption of normality. The purpose of this paper is three-fold: