Estimating Early Exercise Premiums on Gold and Copper Options Using a Multifactor Model and Density Matched Lattices

• Author: Jimmy E. Hilliard,Jitka Hilliard
• DOI: http://doi.org/10.1111/fire.12059
• Date: 01 February 2015
• Published date: 01 February 2015

The Financial Review 50 (2015) 27–56

Estimating Early Exercise Premiums on

Gold and Copper Options Using a

Multifactor Model and Density Matched

Lattices

Jimmy E. Hilliard and Jitka Hilliard∗

The Harbert College of Business at Auburn University

Abstract

We use the standard geometric Brownian motion augmented by jumps to describe the

spot underlying and mean regressive models of interest rates and convenience yields as

state variables for gold and copper prices. Estimates of parameters of the diffusion processes

are obtained by the Kalman filter. Using these estimates, jump parameters are estimated in

the second stage by least squares. Early exercise premia on puts and calls are computed using

a lattice with probabilities assigned by the density matching technique. We find that while

deep in the money options have greater absolute early exercise premiums, the early exercise

premium is roughly constant as a percent of option price. Our findings also confirm that gold

behaves like an investment asset and copper behaveslike a commodity.

Keywords: gold, copper, options, density matching, early exercise, Kalman filter

JEL Classification:G13

∗Corresponding author: Auburn University, Raymond J. Harbert College of Business, 313 Lowder

Business Building, 405 W. Magnolia Avenue, Auburn, AL 36849; Phone: (334) 844-5520; E-mail:

jzh0023@auburn.edu.

The authors thank the editors and anonymous referees for their careful reading, insightful comments and

edits. The usual disclaimer applies.

C2015The Eastern Finance Association 27

28 J. E. Hilliard and J. Hilliard/The Financial Review 50 (2015) 27–56

1. Introduction

Pricing futures contracts on commodities is investigated by a number of scholars

including work and references in Schwartz (1997). In this extension, we estimate the

early exercise premium on American options on gold and copper futures contracts.

Ramaswamy and Sundaresan (1985) and others establish that options on futures con-

tracts have positive early exercise premiums. Essentially, the instantaneous interest

rate on a futures contract behaves like a continuous dividend.

Typical commodity pricing models progress in complexity and explanatory

power from a one-factormodel based on the underlying to a two-factor model that adds

stochastic interest rates and to a three-factor model that adds both stochastic interest

rates and convenience yield. We use standard geometric Brownian motion (gBm)

augmented by jumps to describe the spot underlying and mean regressive models of

interest rates and convenience yields as state variablesfor gold and copper prices.1An

important question with respect to pricing models is whether these metals behave like

an investment asset or a commodity. Previous work indicates that gold behaves like

an investment asset while copper behaves like a commodity. See Schwartz (1997),

Hull (2006), or Hilliard and Reis (HR, 1998) for a further review of the properties of

investment and commodity assets.

We compute early exercise premia on puts and calls using a three-dimensional

lattice with probabilities assigned by the density matching technique. The density

matching technique, developed in Hilliard (2014) and Hilliard and Hilliard (2014)

extends to multivariate applications and shows excellent convergence properties.

We contributeto the literature with our empirical findings and our methodology.

We compute the early exercise premium using a density matched lattice that prices

both European options and American options. We find, somewhat surprisingly, that

within a maturity class, the early exercise premium for these metals as a percent of

option price is relatively constant. We confirm the finding that the absolute early

exercise premium on gold and copper contracts increases with moneyness. With our

data set, we also confirm the earlier findings from Schwartz (1997) that gold behaves

like an investmentasset and copper behaves like a commodity with convenience yield.

We extend the methodology by estimating parameters of a three-factor model with

jumps using a two-stage procedure. Schwartz (1997) uses observed futures prices

and the Kalman filter to estimate parameters for an identical three-factor diffusion.

We add jumps to the underlying giving a three-factor model with jumps. Jumps,

1Models that use jump diffusions to model European option prices include those of Ball and Torous

(1983), Naik and Lee (1990), Bates (1996), Scott (1997), and Duffie, Pan and Singleton (2000). The

usual assumption is that the jump magnitude is normal. The double exponential jump model studied by

Kou (2004) and Ramezani and Zeng (2007) also has theoretical and empirical support. Huang and Wu

(2004) consider time-changed L´

evy processes and find that a high-frequency jump structure outperforms

the low-frequency compound Poisson jump specification.