Inferring Risk-Averse Probability Distributions from Option Prices Using Implied Binomial Trees: Additional Theory and Extensions

19 Pages Posted: 1 Jun 2009

See all articles by Tom Arnold

Tom Arnold

University of Richmond - E. Claiborne Robins School of Business

Timothy Falcon Crack

University of Otago - Department of Accountancy and Finance

Adam Schwartz

Washington and Lee University - Department of Business Administration

Date Written: May 30, 2009

Abstract

Arnold, Crack and Schwartz (ACS) (2010) generalize the Rubinstein (1994) risk-neutral implied binomial tree (R-IBT) model by introducing a risk premium. Their new risk-averse implied binomial tree model (RA-IBT) has both probabilistic and pricing applications. They use the RA-IBT model to estimate the pricing kernel (i.e., marginal rate of substitution) and implied relative risk aversion for a representative agent. They also use the RA-IBT to explore the differences between risk-neutral and risk-averse moments of returns. They also discuss practical applications of the RA-IBT model to Value at Risk and stochastic volatility option pricing models. This paper presents additional theoretical details not contained in ACS. We present a deeper discussion on the assumptions required for the risk-averse trees, we discuss details for extending ACS through the use of general utility functions to generate discount rates in the RA-IBT, and we present further theoretical details on the propagation of risk-averse probabilities through an RA-IBT. We also present an alternate CAPM-driven derivation of the certainty equivalent risk-adjusted discounting formula that is derived using no-arbitrage principles in ACS and an alternate direct estimation routine for the RA-IBT that is similar to Rubinstein’s “one-two-three” technique.

Keywords: Binomial Option Pricing, Implied Binomial Trees, Physical Probabilities, Risk-Neutral Probabilities, Calibration, Representative Agent, Risk-Averse Probabilities

JEL Classification: A23, G13

Suggested Citation

Arnold, Thomas M. and Crack, Timothy Falcon and Schwartz, Adam, Inferring Risk-Averse Probability Distributions from Option Prices Using Implied Binomial Trees: Additional Theory and Extensions (May 30, 2009). Available at SSRN: https://ssrn.com/abstract=1412948 or http://dx.doi.org/10.2139/ssrn.1412948

Thomas M. Arnold

University of Richmond - E. Claiborne Robins School of Business ( email )

1 Gateway Drive
Richmond, VA 23173
United States
804-287-6399 (Phone)
804-289-8878 (Fax)

Timothy Falcon Crack (Contact Author)

University of Otago - Department of Accountancy and Finance ( email )

Dunedin
New Zealand

Adam Schwartz

Washington and Lee University - Department of Business Administration ( email )

Lexington, VA 24450
United States

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