Spanning tests for options using principal components methods

Abstract

This article proposes a new test to evaluate whether options are a redundant asset class. The methodology permits to test between two sets of competing option pricing models: the deterministic volatility models and the stochastic models. In the deterministic volatility framework, options are redundant assets and their payoff can be replicated by dynamic trading in the underlying and the risk-free assets. A stochastic model introduces an additional risk factor and in this setting options are needed in order to complete the market. The procedure relies on principal components methods. The Monte Carlo simulations reported in the article indicate that the new procedure provides simple and useful diagnostics for detection if options are redundant. We apply the method to S&P 500 index and options with different moneyness. The empirical results suggest that options are not redundant.

Acknowledgement

The authors would like to thank Mark Taylor (the editor) and an anonymous referee for helpful suggestions. This article has benefited from discussion with Turan Bali, Victor Martinez and Liuren Wu and comments from seminar participants at Aarhus University and at the 2005 Nordic Econometrics meeting. Charlotte Hansen gratefully acknowledges the financial support provided by the Eugene Lang Junior Faculty Research Fellowship and from the PSC-CUNY Research Foundation of the City University of New York. The views expressed in this article are those of the authors and are not necessarily reflective of views at Platinum Grove Asset Management. Part of this research was performed while Charlotte Hansen was visiting Stern School of Business, New York University.

Notes

¹ See e.g. Hull (2000, p. 316).

² The stock and option returns are stationary per construction.

³ Note that we need to scale the numbers in Table 4 up by p = 2 to compare the smallest eigenvalue with the average of the eigenvalues as opposed to comparing the smallest eigenvalue with the sum of the eigenvalues.

⁴ The data are available at http://www.robertshiller.com

⁵ This is what Christensen and Prabhala (1998) denote past realized volatility.

⁶ We use the new definition of the VIX from Chicago board options exchange (CBOE).