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

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