Using Equity, Index and Commodity Options to Obtain Forward-Looking Betas and Conditional-CAPM Expected Crude-Oil Spot Prices

Paola Z Zerilli, Ehud Ronn

Research output: Working paper


This paper presents a parsimonious and theoretically-sound basis for extracting forward-looking measures of equity and commodity betas and the risk-premium on crude-oil futures contracts. Defining forward-looking betas as stochastic processes (perturbations of historical estimates), we use the market prices of index and commodity options under a market model to estimate the risk premia that investors require in order to be compensated for the various. This model is very general as the interaction between the market price of index and the commodity prices is characterized by two main features: • stochastic volatility in the market price of index • stochastic correlation between the market prices and the commodity prices Using crude-oil futures prices as a second factor, we extend the one-factor model to obtain a two-factor model. Applying this two-factor model to equity and commodity options, we thus obtain a second independent estimate of the forward-looking correlation between crude-oil and the S&P 500. Beginning Oct. 1, 2007, the procedure gives rise to forward-looking term structures of equity and commodity betas. Together with forward-looking (i.e., implied) volatilities on commodities and stock-market indices, we utilize these forward-looking betas and correlations to provide an ex-ante estimate of the expected future crude-oil spot price via the use of an equity ex-ante risk premium and the application of the conditional CAPM for the crude-oil risk premium. Applying the conditional CAPM to the oil markets, the 2011 Arab Spring and subsequent geopolitical crises in 2013 and 2014 provide natural experiments for the models’ abilities to discern changes in oil futures markets.
Original languageEnglish
Publication statusUnpublished - 2015

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