7. How Precise Are Price Distributions Predicted by Implied Binomial Trees?

Wolfgang Härdle and Jun Zheng
July 9, 2002

In recent years, especially after the 1987 market crash, it became clear that the prices of the underlying asset do not exactly follow the Geometric Brownian Motion (GBM) model of Black and Scholes. The GBM model with constant volatility leads to a log-normal price distribution at any expiration date: All options on the underlying must have the same Black-Scholes (BS) implied volatility, and the Cox-Ross-Rubinstein (CRR) binomial tree makes use of this fact via the construction of constant transition probability from one node to the corresponding node at the next level in the tree. In contrast, the implied binomial tree (IBT) method simply constructs a numerical procedure consistent with the volatility smile. The empirical fact that the market implied volatilities decrease with the strike level, and increase with the time to maturity of options is better reflected by this construction. The algorithm of the IBT is a data adaptive modification of the CRR method.

An implied tree should satisfy the following principles:

The last two conditions also eliminate arbitrage opportunities.

The basic purpose of the IBT is its use in hedging and calculations of implied probability distributions (or state price density (SPD)) and volatility surfaces. Besides these practical issues, the IBT may evaluate the future stock price distributions according to the BS implied volatility surfaces which are calculated from currently observed daily market option prices.

We describe the construction of the IBT and analyze the precision of the predicted implied price distributions. In Section 7.1, a detailed outline of the IBT algorithm for a liquid European-style option is given. We follow first the Derman and Kani (1994) algorithm, discuss its possible shortcomings, and then present the Barle and Cakici (1998) construction. This method is characterized by a normalization of the central nodes according to the forward price. Next, we study the properties of the IBT via Monte-Carlo simulations and comparison with simulated conditional density from a diffusion process with a non-constant volatility. In Section 7.3, we apply the IBT to a DAX index data set containing the underlying asset price, strike price, interest rate, time to maturity, and call or put option price from the M D *BASE database (included in XploRe ), and compare SPD estimated by historical index price data with those predicted by the IBT. Conclusions and discussions on practical issues are presented in the last section.