Among all the metal commodities, modeling price behavior of copper poses the biggest
challenge to researchers and practitioners. In the earliest reduced form model for commodity
prices (Brennan and Schwartz, 1985), the spot commodity price follows a geometric Brownian
motion and the convenience yield is treated as a dividend yield. This specification is
inappropriate since it does not take into account the mean reversion property of spot
commodity prices and neglects the inventory-dependence property of the convenience yield.
Gibson and Schwartz (1990) introduce a two-factor, constant volatility model where the
spot price and the convenience yield follow a joint stochastic process with constant correlation.
Brennan and Schwartz (1985), Gibson and Schwartz (1990) and Schwartz (1997) found that
the one-factor models fail to capture the erratic nature of the commodity price. So, the good
parametric model describing the stochastic process of commodity prices should capture the
characteristics of the evolution of commodity prices.
As the commodity spot prices have been highly volatile in recent years, the volatility
should be examined. The Stochastic Volatility (SV) processes are flexible enough to allow us
to capture the special characteristics of copper prices. In this paper, we consider the mean
reverting process of Deng (1999) and the mean reverting square root process of Heston
(1993).
Rising price volatility has given rise to a number of financial instruments that allow
investor to minimize risk and maximize expected return. Financial derivatives are contractual
agreements with a value which changes in response to price movements in a related commodity.
So, the main reason to use future contracts is to offset the risk exposures of any fluctuations
in price. The combinations of cash and futures positions typically expressed in terms of
proportion of cash to futures positions for an asset are referred to as optimal hedge ratios.
There are various approaches that have been developed to estimate the optimal hedge
ratio. The Minimum Variance Hedge Ratio (MVHR), the Ordinary Least Squares (OLS)
model, and the Vector Autoregressive (VAR) model estimate constant hedge ratio over time.
Whereas, the Bollerslev, Engle, Kroner and Kraft (BEKK) model and the VAR model with
bivariate generalized autoregressive conditional heteroscedasticity model, VAR-MGARCH,
estimate the time-varying hedge ratios. The models estimating constant hedge ratio are
based on the assumption that the joint distribution of spot and futures prices are invariant
over time. Marmer (1986) examines the effectiveness of the MVHR, and shows that the
utility of this approach is limited. Benninga et al. (1984) derives the MVHR from an OLS
regression with the spot price changes as the dependent variable and futures price changes as
the explanatory variable. The MVHR is the slope coefficient of the OLS regression. The
optimal hedge ratio is the covariance of spot prices and futures prices and variance of futures
prices. Using OLS regression for estimating the hedge ratio has been criticized for two reasons.
Firstly, it is based on the assumption of unconditional distribution of spot and futures prices.
Secondly, the OLS regression is based on the assumption that the relationship between spot
and futures prices is time invariant.
|