Asymptotic expansion for the price of a UIP barrier option in a binomial tree model - LLemit, Dennis G. (Institute of Mathematics, College of Science, University of the Philippines Diliman, 2010)
It has been shown that the famous continuous model in pricing option contracts, the Black-Scholes model, is the limit of discrete option pricing models. However, it is not realistic to define the time steps of the discrete model to be infinite since it would imply the infinite monitoring of the asset path. Instead, the time steps of the discrete model should be defined to be large enough to approximate the Black-Scholes. We choose a knock-in type of barrier option and we will measure its order of convergence to its counterpart continuous model using asymtotic expansion of its price formula. This is done by converting all three discrete models for the the UIP barrier option into incomplete binomia sums. Their incomplete binomial sum forms will then be converted into integral. Then the symmetry method devised by Mark Joshi will be applied on the integrals by setting the exponents of the integrals to be equal. All three formulas are then shown to collapse into a single integral form. The location of the parameters K(strike price) and B (barrier level) with respect to the premium price So in the binomial tree will be determined. And lastly, the asymptotic expansion of the rice formula of a UIP barrier option will be computed fro its integral version.