Equilibrium preference free pricing of derivatives under the generalized beta distributions

This paper demonstrates that the risk neutral valuation relationship (RNVR) exists when the aggregate wealth and the underlying variable for derivatives follow a distribution from the family of transformed beta distributions. Specifically, the asset specific pricing kernel (ASPK) is solved for the generalized beta (GB) distribution class, which is extremely flexible to describe various shapes of underlying distributions. With the ASPK in hand, preference free call option formulas are obtained for rescaled and shifted beta distribution of the first kind (RSB1) and for the second kind (RSB2). These distributions include many well known important distributions as special cases. If the preference free formula does not exist under the GB distribution class, then the call price is shown to be numerically calculated without information of preference parameters once the spot price of the underlying is given.