INVISIBLE PARAMETERS IN OPTION PRICES

This paper characterizes contingent claim formulas that are independen t of parameters governing the probability distribution of asset return s. While these parameters may affect stock, bond, and option values, t hey are ''invisible'' because they do not appear in the option formula s. For example, the Black-Scholes (1973) formula is independent of the mean of the stock return. This paper presents a new formula based on the log-negative-binomial distribution. In analogy with Cox, Ross, and Rubinstein's (1979) log-binomial formula, the log-negative-binomial o ption price does not depend on the jump probability. This paper also p resents a new formula based on the log-gamma distribution. In this for mula, the option price does not depend on the scale of the stock retur n, but does depend on the mean of the stock return. This paper extends the log-gamma formula to continuous time by defining a gamma process. The gamma process is a jump process with independent increments that generalizes the Wiener process. Unlike the Poisson process, the gamma process can instantaneously jump to a continuum of values. Hence, it i s fundamentally ''unhedgeable.'' If the gamma process jumps upward, th en stock returns are positively skewed, and if the gamma process jumps downward, then stock returns are negatively skewed. The gamma process has one more parameter than a Wiener process; this parameter controls the jump intensity and skewness of the process. The skewness of the l og-gamma process generates strike biases in options. In contrast to th e results of diffusion models, these biases increase for short maturit y options. Thus, the log-gamma model produces a parsimonious option-pr icing formula that is consistent with empirical biases in the Black-Sc holes formula.