Can Tests for Jumps be Viewed as Tests for Clusters?

Abstract

We investigate the utility in employing asymptotic results related to a clustering criterion to the problem of testing for the presence of jumps in financial models. We consider the Jump Diffusion model for option pricing and demonstrate how the testing problem can be reduced to the problem of testing for the presence of clusters in the increments data. The overarching premise behind the proposed approach is in the isolation of the increments with considerably larger mean pertaining to the jumps from the ones which arise from the diffusion component. Empirical verification is provided via simulations and the test is applied to financial datasets. component? The problem is of obvious importance when prediction is the primary concern.The ramifications, while constructing a model for asset pricing, of not incorporating a jump component when the underlying process which generates the data indeed does possess one, can be quite severe. The problem has received appreciable attention over the years based on several techniques; we refer to a few articles from an exhaustive list: Ait-Sahalia [2002], Ait-Sahalia and Jacod [2009], Carr andWu [2003], Barndoff-Nielsen and Shephard [2003], Podolskij and Ziggel [2010] and Lee and Mykland [2008].The problem can be viewed as a deconstruction problem wherein the observed series of returns are deconstructed back to their continuous and jump components. This taxonomy between the continuous and the jump components of the purported model assists us in seeking 'typical' behavior of statistics based on observations under the presence and absence of the jump components. Intuitively, by constructing a test statistic which would eventually isolate the jump component under the presence of jumps in the underlying process, a suitable asymptotic hypothesis test can be developed. To elaborate, for simplicity, suppose that jumps are all positive valued and the number of jumps are finite in [0, T ]. Based on a sampling frequency, suppose we consider the increments (difference between successive observations); we would then expect to see, primarily, two groups of data: one centered around a value which is considerably larger than the other corresponding, respectively, to the jumps and the nonjumps. Such behavior is the motivation behind the test statistics based on truncated power variations employed in Ait-Sahalia [2002] and Ait-Sahalia and Jacod [2009]. In this article, we approach the problem of constructing a suitable test statistic through a different route: we ask if the isolation of the jumps can be viewed as a model-based clustering behavior wherein the distributional properties of the model, for large samples, leads to the formation of two clusters with cluster centers far apart. Under this setup, this article ought to be viewed as a first step towards providing a general answer applicable to a broad class of semimartingale models; while the alternative hypothesis of 'jumps' encompasses a large class of models, we will focus primarily on the Merton-type model wherein the jump component i...