A Case-Study on the Trinomial Tree

Marco Avellaneda, Arnon Levy and Antonio Paras

Given a pair of volatilities, \sigma_min and \sigma_max, and a parameter \mu, we construct a sequence of trinomial trees such that, as the time between trading periods tends to zero, the asset price becomes lognormally distributed with a drift \mu and a volatility between \sigma_min and \sigma_max. Any volatility in this range can be obtained by specifying different probabilities at each node of the tree. We study the optimal dominating strategies for pricing and hedging derivate securities in this simple model of an incomplete market. We show that, ast the time between trading periods tends to zero, the bid or ask prices of a derivative security are given by the solution of a non-linear PDE which we call the Black-Scholes-Barenblatt equation. In this equation, the input volatility is ``dynamically'' selected from the two values \sigma_min and \sigma_max according to the sign of the second derivative of the value function with respect to the price of the underlying asset.