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.

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We develop time series analysis of functional data observed discretely, treating the whole curve as a random realization from a distribution on functions that evolve over time. The method consists of principal components analysis of functional data and subsequently modeling the principal component scores as vector autoregressive moving averag (VARMA) process. We justify the method by showing that an underlying ARMAH structure of the curves leads to a VARMA structure on the principal component scores. We derive asymptotic properties of the estimators, fits, and forecast. For term structures of interest rates, these provide a unified framework for studying the time and maturity components of interest rates under one setup with few parametric assumptions. We apply the method to the yield curves of USA and India. We compare our forecasts to the parametric model that is based on Nelson‐Siegel curves. In another application, we study the dependence of long term interest rate on the short term interest rate using functional regression.

Diffusion processes driven by Fractional Brownian motion (FBM) have often been considered in modeling stock price dynamics in order to capture the long range dependence of stock price observed in reality. Option prices for such models had been obtained by Necula (2002) under constant drift and volatility. We obtain option prices under time varying volatility model. The expression depends on volatility and the Hurst parameter in a complicated manner. We derive a central limit theorem for the quadratic variation as an estimator for volatility for both the cases, constant as well as time varying volatility. That will help us to find estimators of the option prices and to find their asymptotic distributions.

Range value at risk (RVaR) is a quantile-based risk measure with two parameters. As special examples, the value at risk (VaR) and the expected shortfall (ES), two well-known but competing regulatory risk measures, are both members of the RVaR family. The estimation of RVaR is a critical issue in the financial sector. Several nonparametric RVaR estimators are described here. We examine these estimators’ accuracy in various scenarios using Monte Carlo simulations. Our simulations shed light on how changing p and q with respect to n affects the effectiveness of RVaR estimators that are nonparametric, with n representing the total number of samples. Finally, we perform a backtesting exercise of RVaR based on Acerbi and Szekely’s test.

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