A Convenience Yield Approximation Model for Mean‐Reverting Commodities

• Published date: 01 July 2015
• Author: Margarethe Rammerstorfer,Engelbert J. Dockner,Zehra Eksi
• DOI: http://doi.org/10.1002/fut.21670
• Date: 01 July 2015

ACONVENIENCE YIELD APPROXIMATION MODEL

FOR MEAN-REVERTING COMMODITIES

ENGELBERT J. DOCKNER, ZEHRA EKSI and MARGARETHE RAMMERSTORFER*

Standard option-based approximations for convenience yields make use of the assumption that

commodity spot prices follow a geometric Brownian motion. While there is some empirical

support for this assumption, prices of a wide variety of (agricultural) commodities mean revert.

Using a mean-reverting spot price process we derive a novel convenience yield approximation

analytically. It corresponds to the difference between the present values of two floating-strike

Asian options written on the spot and the futures prices, respectively. Using natural gas spot

and futures price data from four different trading locations, we compare convenience yield

estimates derived from existing approximations to those of our new measure. We find that

convenience yield estimates vary substantially across approximation methods and that differ-

ences can be attributed to the cost of carry and the moneyness of the options. ©2014 Wiley

Periodicals, Inc. Jrl Fut Mark 35:625–654, 2015

1. INTRODUCTION

Convenience yield is a standard concept in commodity price analysis that measures the

economic benefits of holding a commodity (carrying an inventory) instead of buying it when

necessary. It can be interpreted as a continuous dividend stream earned by the commodity

on stock which, together with the risk-free rate of interest and the cost of carrying the

inventory, determines the forward or futures price for a given maturity T. Convenience yield

is an integral part for the pricing of non-investment commodities that traces back to Working

(1949). While frequently used as a theoretical concept to price forward and futures contracts

of commodities, it is difficult to measure as it cannot directly be observed in either the spot

or futures markets.

There exist several different approaches to estimating (or approximating) convenience

yields from actual market prices. The traditional approach builds on a simple no-arbitrage

condition that is used to determine the forward price of a commodity. An investor with the

aim of holding a unit of the commodity at time Tis indifferent between buying the commodity

in the spot market, paying the cost of carry, and benefiting from holding it until maturity T

(receiving the convenience yield) or entering a long forward contract at the current forward

Engelbert J. Dockner is at the Vienna University of Economics and Business, Institute for Finance, Banking

and Insurance and Vienna Graduate School of Finance, Vienna, Austria. Zehra Eksi is at the Vienna Uni-

versity of Economics and Business, Institute for Statistics and Mathematics, Vienna, Austria. Margarethe

Rammerstorfer is at MODUL University Vienna, Department of International Management, Vienna,Austria.

We thank Julia Reynolds and an anonymous referee for their valuable comments.

JEL Classification: G13, O13

*Correspondence author,MODUL University Vienna, Department of International Management, Am Kahlen-

berg 1, 1190 Vienna, Austria. Tel: +43 1 3203555 650, Fax: +43 1 3203555 903, e-mail: margarethe.

rammerstorfer@modul.ac.at

Received August 2012; Accepted February 2014

The Journal of Futures Markets, Vol. 35, No.7, 625–654 (2015)

©2014 Wiley Periodicals, Inc.

Published online 8 July 2014 in Wiley Online Library (wileyonlinelibrary.com).

DOI: 10.1002/fut.21670

626 Dockner, Eksi and Rammerstorfer

price to buy the commodity at maturity. No arbitrage implies that the forward price must be

equal to the terminal value of a cash position consisting of the spot price, the cost of carry

and the convenience yield invested in the risk-free asset. As a consequence, the implied

convenience yield for a given period can be estimated from the current spot and forward

prices, the cost of carry and the risk-free rate of interest (see, e.g., Pindyck, 1993 or Pindyck,

2001 for a detailed review of the traditional approach).

While the traditional approach to estimating the convenience yield is simple and easy

to implement, it requires detailed information about cost of carry data as well as friction-

less markets, both of which are not frequently available (see, Hochradl & Rammerstorfer,

2012 for a detailed discussion of the limitations of the traditional approach to estimating

implied convenience yields). As an alternative, option-based approaches can be applied to

approximate convenience yields. The economic intuition of these approaches is rooted in

the opportunity to sell a commodity if kept in stock. Consider a period equivalent to the

maturity of a futures contract; during that period, a holder of an inventory has the option

to sell the commodity at any intermediate time if the current spot price is high enough. If,

for example, the inventory holder knows that prices will increase first and then fall again at

maturity, he will sell the commodity at the high price, invest the proceeds in the risk-free

asset and buy it back at the lower price at maturity. The value of this trading strategy can

be derived using no-arbitrage principles and the convenience yield can be related to the

difference between the values of the options written on the spot and futures prices of the

underlying commodity, respectively (see, Heaney, 2002; Heinkel, Howe, & Hughes, 1990;

Hochradl & Rammerstorfer, 2012; Milonas & Thomadakis, 1997).

The option-based approach of Heaney (2002) uses the analysis of Longstaff (1995),

who determines the present value of the marketability of an asset using risk-neutral pric-

ing. In an environment in which the inventory holder can buy and sell opportunistically, the

value of this option corresponds to the value of the marketability of an asset. This leads to the

approximation of the convenience yield as the difference between two floating-strike look-

back options written on the underlying commodity and the futures contract, respectively. In

Hochradl and Rammerstorfer (2012), the approach of Heaney (2002) is extended by relaxing

the assumption of the investors’ perfect foresight and the convenience yield is modeled as

the difference between two geometric-mean, floating-strike Asian options.

Approximating convenience yields via lookback and geometric-mean Asian options re-

quires the price process to follow a geometric Brownian motion (GBM). While this assump-

tion is in line with Hotelling’s pricing rule for exhaustible resources, there is strong empirical

evidence that prices of agricultural and other commodities follow mean-reverting processes.

Mean reversion perfectly mimics market dynamics. If the current price of a commodity is

above its long-run average, new supply will enter the market and drive down the price. If, on

the other hand, current prices are below the long-run mean, high-cost commodity producers

will exit the market, with supply decreasing and the commodity price increasing as a result

(see Schwartz, 2000 for detailed discussion of these dynamics). Bessembinder, Coughenour,

Seguin, and Smoller (1995) analyze the term structure of commodity futures prices within

the period 1982–1991 and find that mean reversion is predominantly present for agricultural

commodities and crude oil. More recently, Pindyck (2001) analyzes price series of energy

commodities for the period ranging from 1870 to 1996 and confirms that spot and futures

prices do not follow log-normal distributions and GBM is not supported by price data. Addi-

tionally, Andersson (2007) underlines the importance of using mean-reverting processes to

model commodity prices.

If commodity prices mean revert, existing convenience yield approximations such as

Heaney (2002) and Hochradl and Rammerstorfer (2012) cannot be applied. The contribu-

tion of this article is to consider a mean-reverting spot price process and use risk-neutral