Pricing and Hedging Derivative Securities in Markets with Uncertain Volatility:
A Case-Study on the Trinomial Tree
Marco Avellaneda, Arnon Levy and Antonio Paras
Unpublished manuscript, 1994

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.
Note: This unpublished manuscript was superseded a few months later by Avellaneda, Levy, Paras, Applied Mathematical Finance (1995). The 1994 ``case-study on the trinomial tree'' version focuses on the discrete model and was less clear (at least for one referee who rejected it :-) ). The 1995 version treats the same material in the language of diffusions and is easier to understand. (M.A. 7/2002)