Binomial pricing: a generalization

Abstract

The binomial lattice framework is generalized to include stochastic interest rates, random maturity, and compound price jumps. Valuation results for contingent claims are derived based upon the discrete-time martingale representation theory. In the case of stochastic interest rates, a unique equivalent martingale probability measure (UEMPM) is derived with respect to the probability measure governing the uncertainty of the normalized asset price. Applications considered are options on forward contracts and on the maximum or minimum of several risky assets. In the case of random maturity, a UEMPM is derived with respect to the probability measure governing the asset price uncertainties which include both stochastic price changes and random maturity. The pricing of options with default risk, and of a poison pill is considered. In the case of compound price jumps, two techniques are utilized. The first involves the assumption that the jump risk is diversifiable, then a UEMPM can be derived with respect to the probability measure governing the asset price uncertainties which include both continuous and discontinuous stochastic price changes. The second involves a stochastic time change from calendar-time to transaction-time such that the transformed asset return process becomes stationary and follows a binomial lattice. However, this lattice is shown to involve a random number of periods arising from the stochastic nature of the transformation. Therefore, the pricing problem after the transformation becomes one under stationary returns but random maturity.