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

Posted: 26 Dec 1998

See all articles by Arnon Levy

Arnon Levy

affiliation not provided to SSRN

Marco Avellaneda

New York University (NYU) - Courant Institute of Mathematical Sciences; Finance Concepts LLC

Antonio Paras

New York University (NYU) - Courant Institute of Mathematical Sciences

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

Suggested Citation

Levy, Arnon and Avellaneda, Marco and Paras, Antonio, A New Approach for Pricing Derivative Securities in Markets with Uncertain Volatilities: A 'Case Study' on the Trinomial Tree. Available at SSRN: https://ssrn.com/abstract=5342

Arnon Levy (Contact Author)

affiliation not provided to SSRN

Marco Avellaneda

New York University (NYU) - Courant Institute of Mathematical Sciences ( email )

251 Mercer Street
New York, NY 10012
United States
212-998-3129 (Phone)
212-995-4121 (Fax)

Finance Concepts LLC ( email )

590 Madison Avenue
21st Floor
New York, NY 10022
United States

HOME PAGE: http://www.finance-concepts.com

Antonio Paras

New York University (NYU) - Courant Institute of Mathematical Sciences ( email )

New York University
New York, NY 10012
United States

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