Diagnosing affine models of options pricing: Evidence from VIX

Abstract

a b s t r a c tAffine jump-diffusion models have been the mainstream in options pricing because of their analytical tractability. Popular affine jump-diffusion models, however, are still unsatisfactory in describing the options data and the problem is often attributed to the diffusion term of the unobserved state variables. Using prices of variance-swaps (i.e., squared VIX) implied from options prices, we provide fresh evidence regarding the misspecification of affine jump-diffusion models, as variance-swap prices are affine functions of the state variables in a broader class of models that do not restrict the diffusion term of the state variables. We apply the nonparametric methodology used by Aït-Sahalia (1996b), supplemented with bootstrap tests and other parametric tests, to the S&P 500 index options data from January 1996 to September 2008. We find that, while the affine diffusion term of the state variables may contribute to the misspecification as the literature has suggested, the affine drift of the state variables, jump intensities, and risk premiums are also sources of misspecification.

VIX term structure: The role of jump propagation risks

“…Second, it would be valuable to relax the affine restrictions of the Hawkes models. Li and Zhang (2013) provide strong evidence that affine restrictions on the drift of state variables, jump intensity, and risk premiums are an important source of model misspecification. We leave these topics for further research.…”

On the relationship between conditional jump intensity and diffusive volatility

“…Christoffersen, Jacobs and Mimouni (2010) find that a stochastic volatility model with a linear diffusion term is more consistent with the data on the underlying asset and options than a stochastic volatility model with a squareroot diffusion term is. Li and Zhang (2013) show that the affine drift of the diffusive volatility model is mis-specified because the mean reversion is particularly strong at the 1 Eraker (2004) also has estimation results without using options data in his Table 4, in which he does not report the relation between jump intensity and diffusive volatility and does not explain why not. Andersen et al (2002) attribute the insignificance result to the approximation used to calculate standard errors, and a multicollinearity type problem.…”

Market-based estimation of stochastic volatility models

“…Apart from its tractability, the affine- drift models have two additional attractive features worth mentioning. First, the diffusion functions of variance swap rates remain unrestricted except for the instantaneous variance of the underlying asset; hence, leaving room for non-affine specifications of the diffusion such as the constant elasticity of variance (CEV) model of Jones (2003), which is a relevant feature when fitting the models to the data; see, e.g., Li and Zhang (2013). Second, drift specifications richer than the ones belonging to the affine class can be obtained under the objective measure  through different specifications of market prices of risk.…”