Abstract
This paper develops a test that helps assess whether the term structure of option implied volatility is constant across different levels of moneyness. The test is based on the Hausman principle of comparing two estimators, one that is efficient but not robust to the deviation being tested, and one that is robust but not as efficient. Distribution of the proposed test statistic is investigated in a general semiparametric setting via the multivariate Delta method. Using recent S&P 500 index traded options data from September 2009 to December 2018, we find that a partially linear model permitting a flexible “volatility smile” and an additive quadratic time effect is a statistically adequate depiction of the implied volatility data for most years. The constancy of implied volatility term structure, in turn, implies that option traders shall feel confident and execute volatility-based strategies using at-the-money options for its high liquidity.
Highlights
One key assumption behind the well known Black–Scholes (B–S) formula is a constant volatility function, which has been frequently challenged after the stock market crash in October 1987
Extant studies have aimed at theoretically explaining these stylized facts using stochastic volatility (Hull and White 1987; Renault and Touzi 1996) and jump diffusion models (Bates 1996; Jorion 1988), there is an absence of prior work that formally tests whether the term structure of option implied volatility is constant across different levels of moneyness
As motivated in the introduction, we study whether an option’s moneyness and time-to-maturity affect its implied volatility in an interactive fashion, e.g., do options across different moneyness value have different term structure? ABS addresses this question by testing whether the in-sample-fit of a partially linear model is statistically the same as an unrestricted, nonparametric alternative
Summary
One key assumption behind the well known Black–Scholes (B–S) formula is a constant volatility function, which has been frequently challenged after the stock market crash in October 1987. ABS further shows that a semiparametric partially linear model permitting a flexible function of moneyness—widely known as the “volatility smile”—and a quadratic time effect is a statistically adequate depiction of the empirical option data. We note the partially linear specification above precludes the interactive effect between moneyness K/F and time-to-expiration T It restricts the term structure of volatility to be constant across moneyness. This paper attempts to fill this void by developing a specification test for the partially linear structure in ABS against a semiparametric alternative that explicitly permits interaction effects. A quadratic term structure has been widely recognized as a suitable statistical depiction of volatility data (Aıt-Sahalia et al 2003, 2001; Ahn et al 2002) To this end, we consider a semiparametric alternative in which the quadratic term structure can be imposed explicitly. While the focus of this paper is on option implied volatility, the testing framework can be applied to other contexts where a semiparametric alternative is justified
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