Option valuation with infinitely divisible distributions

Abstract

This paper develops an axiomatic framework for option valuation when option payoffs are not spanned by spot and bond prices. This framework extends the parametric ‘risk neutral valuation’ results of Rubinstein (1976) and Brennan (1979) to general distributions. The valuation relationship preserves the divisibility properties of distributions. So by using infinitely divisible distributions the theory easily extends to continuous-time processes with independent increments.

 This assumption guarantees the existence and uniqueness of the desired path-independent measure. Existence merely requires one of the tails of the (continuously compounded) stock return distribution to be exponentially bounded. Continuity at a single point guarantees uniqueness.

This paper illustrates the valuation technique with negative-binomial and inverse-binomial generalizations of Cox et al.'s (1979) binomial model. The continuous-time (gamma and inverse Gaussian) limits of these models generalize the Black–Scholes (1973) formula by incorporating an extra skewness parameter. The continuous-time examples include an infinite variance stable process of the type used by Mandelbrot (1963, 1966) and McCulloch (1987). The valuation theory extends to American options and other path-dependent claims.

Acknowledgments

The author acknowledges financial support and the Roger F Murray prize from the Institute for Quantitative Research in Finance. The author appreciates comments from Phil Dybvig, Kerry Back and Guofu Zhou. Any errors are the responsibility of the author.

Notes

 This assumption guarantees the existence and uniqueness of the desired path-independent measure. Existence merely requires one of the tails of the (continuously compounded) stock return distribution to be exponentially bounded. Continuity at a single point guarantees uniqueness.

 The path-independent marginal rate of substitution corresponds to the ‘power utility’ of Vankudre (1986), Smith (1987), or Heston (1993) and the Esscher transform of Gerber and Shiu (1993).

 The main exception to homogeneity is the constant elasticity of variance model of Cox (1975) and Cox and Ross (1976) where the stock price level affects the variance of stock returns.

 The inverse binomial distribution is a rescaled distribution of the first passage time of a binomial random walk to a fixed barrier, see Feller (1971).

 If q _n  = 1/2 and σ _n  = 2t ²/n ², the inverse binomial distribution converges to the stable distribution with density

.