The Valuation of American Calls on Futures Contracts: A Comparison of Methods

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The Valuation of American Calls on Futures Contracts: A Comparison of Methods

Abstract

This study extends the analysis of approximation methods for pricing

American call options on futures contracts. Whereas previous studies

have adopted a specific method as giving the true values of American

options and then used those values as a basis for evaluating the

European model, the present study develops binomial as well as both

explicit and implicit finite difference approximations and then compares

their predictions for a range of parameter values and futures prices.

Introduction

In recent years, there has been a proliferation of traded securities whose values are derived directly from the values of other assets. Among these securities are options written on futures contracts. Although such options have been in existence in Europe for many years, they only recently have become available in North America. In 1982, the United States Commodity Futures Trading Commission allowed each commodity exchange to trade options on one of its futures contracts. Since then trading in futures options has been initiated on every major exchange. The underlying spot assets include financial instruments such as bonds, Eurodollars, stock indices, foreign currencies, precious metals, and agricultural products.

The growth in interest in futures options within the investment community has been matched by a number of recent efforts to derive models for pricing them. The first such effort was that of Black[2], who derives a model for pricing European options on forward contracts that is identical in form to the well known Black-Scholes[3] formula. Under the assumption of a nonstochastic risk free rate of interest, Cox, Ingersoll, and Ross[8] have shown that Black's model can be applied to the pricing of options on futures as well.(1) As Brenner, Courtadon, and Subrahmanyan[6] have pointed out, however, it may be optimal to exercise an American call on a futures contract before expiration. If so, the use of Black's pricing model will give inexact solutions.

The importance of the early exercise feature in pricing American options also has been studied by Courtadon[7], who analyzes Treasury bond options on both the spot and futures instruments, and Ramaswamy and Sundaresan [14], who compare options on stock indices with options on stock index futures. In the absence of an exact analytic solution to the pricing of the futures options, the latter authors employ the implicit finite difference approximation method. Whaley[16,17] similarly has calculated the prices of American calls on futur es using the Geske-Johnson[10] compound option approach and a quadratic approximation approach derived by Barone-Adesi and Whaley[1]. Finally, Shastri and Tandon[15] recently have compared the pricing performance of Black's European option model with that of a variation of Geske and Johnson's approximation formula.

Ramaswamy and Sundaresan infer, on the basis of their application of the implicit finite difference technique, that the value of the opportunity to exercise early was small. For out-of-the-money options, Whaley also finds that the prices obtained using the Black model were similar to those obtained using numerical methods, whereas for in-the-money-options, "the early exercise privilege contribute[d] meaningfully to the futures option value"[17] Similarly, Shastri and Tandon conclude that Black's model can be used to price American options on futures "if the option is not deep in the money and is near maturity" (p. 613). They also note that it performs well when the volatility of the futures price and the level of the risk free interest rate are low.

The present paper extends the analysis of approximation methods for pricing American call options on futures contracts. Whereas previous studies have adopted a specific method as giving the true values of American options and then used those values as a basis for evaluating the European model, the present study develops alternative approximation methods and then compares their respective predictions for a range of parameter values and futures prices. …