A note on the pricing of contingent claims with a mixture of distributions in a discrete-time general equilibrium framework

Vitiello, Luiz; Poon, Ser-Huang (2009)
Centre for International Capital Markets, London Metropolitan University
Mixtures of distributions have been applied to contingent claim pricing as a way of extending the Black and Scholes (1973) assumption of lognormally distributed assets. The pricing framework presented here delivers preference free contingent claim pricing formulae and extends the literature in two ways: First, we widen the set of distributions used in the mixture by assuming that the terminal price of the underlying security has a mixture of transformed-normal distributions. Second, we show that the components of the mixture do not need to have the same density as long as they belong to the family of transformed-normal distributions. Our framework is developed in a discrete time equilibrium economy. It is strongly related to Camara (2003) and is consistent with the su¢cient conditions of Heston (1993) and Schroder (2004). We show that by restricting the value of some distributional parameters, it is possible to obtain a risk neutral valuation relationship for the pricing of contingent claims when the terminal price of the underlying asset has a mixture of transformed normal distributions. An interesting aspect of the mixtures of distributions, and in particular of the framework developed here, is that the actual and the risk neutral distributions might not have the same shape. This fact could help to explain the non-monotonic pricing kernel obtained by Jackwerth and Rubinstein (1996), Brown and Jackwerth (2004), Ait-Sahalia and Lo (1998) among others.

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