Functional data analysis for volatility

Abstract

We introduce a functional volatility process for modeling volatility trajectories for high frequency observations in financial markets and describe functional representations and data-based recovery of the process from repeated observations. A study of its asymptotic properties, as the frequency of observed trades increases, is complemented by simulations and an application to the analysis of intra-day volatility patterns of the S&P 500 index. The proposed volatility model is found to be useful to identify recurring patterns of volatility and for successful prediction of future volatility, through the application of functional regression and prediction techniques.

An analysis of implied volatility jump dynamics: Novel functional data representation in crude oil markets

“…Breaking the sample down into periods split by predominantly positive and predominantly negative implied volatility slopes, we see that the Merton strategy outperforms the Black-Scholes whenk values are positive and broadly matches its performance in periods of negativê k values. Kneip and Utikal (2001), Müller, Sen, andStadtmüller (2011), Ramsay, Hooker, andGraves (2009) …”

Testing for independence between functional time series

“…2.1. Several authors have studied the analysis of the shapes of price curves, see Müller et al (2011) and Kokoszka et al (2013) for a small sample of such work. In our application we use (2.1) to test for correlation between samples of curves constructed from pairs of 20 of the highest trade volume stocks listed on the New York Stock Exchange.…”

A Nonparametric Estimator for the Covariance Function of Functional Data

“…There has been some direct interest in the econometric literature, both theoretical and applied, on the use of functional data (e.g., Ramsay and Ramsey, 2002;Bugni, Hall, and Horowitz, 2009;Müller, Rituparna, and Stadtmüller, 2011;Bugni, 2012;Ferraty, Quintela-del-Río, and Vieu, 2012). Obvious examples of functional data in finance and economics are the yield curve, the implied volatility surface of derivative instruments, the volume weighted average price (VWAP), intradaily volatility processes, and many others.…”

“…Recently, the concepts of functional data analysis have been applied to estimation of intraday volatility (e.g., Müller, Stadtmüller, and Yao, 2006;Alva, Romo, and Ruiz, 2009;Müller et al, 2011). Then, for day r , X r (s) is a log volatility process that is not observable, but can be extracted from observable quantities (Müller et al, 2011, eqn.…”