A New Approach for Pricing Derivative Securities in Markets with Uncertain Volatilities: A 'Case Study' on the Trinomial Tree

Abstract

We construct a sequence of trinomial trees in which an asset's price becomes lognormally distributed with given drift mu and a volatility between given sigma_min and sigma_max as the time between trades approaches zero. In this simple model of an incomplete market, we show that, as the time between trading approaches 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, sigma_max, according to the sign of the second-order price derivative of the value function. This approach gives a new way of pricing derivative securities in markets with uncertain volatilities. It can be shown that any stochastic volatility process that stays between sigma_min and sigma_max, will give rise to a price between the bid and ask prices.

JEL Classification: G13