Option valuation with infinitely divisible distributions

Steven Heston

Quantitative Finance, 2004, vol. 4, issue 5, 515-524

Abstract: This paper develops an axiomatic framework for option valuation when option payoffs ar 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 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 Americal options and other path-dependent claims.

Date: 2004
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DOI: 10.1080/14697680400000035

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