Testing for jumps and modeling volatility in asset prices

TESTING FOR JUMPS AND MODELING VOLATILITY IN ASSET PRICES Johan Bjursell, PhD George Mason University, 2009 Dissertation Director: Dr. James E. Gentle Observers of financial markets have long noted that asset prices are very volatile and commonly exhibit jumps (price spikes). Thus, the assumption of a continuous process for asset price behavior is often violated in practice. Although empirical studies have found that the impact of such jumps is transitory, the shortterm effect in the volatility may nonetheless be considerable with important financial implications for the valuation of derivatives, asset allocation and risk management. This dissertation contributes to the literature in two areas. First, I evaluate the small sample properties of a nonparametric method for identifying jumps. I focus on the implication of adding noise to the prices and recent methods developed to contend with such market frictions. Initially, I examine the properties and convergence results of the power variations that constitute the jump statistics. Then I document the asymptotic results of these jump statistics. Finally, I estimate their size and power. I examine these properties using a stochastic volatility model incorporating alternative noise and jump processes. I find that the properties of the statistics remain close to the asymptotics when methods for managing the effects of noise are applied judiciously. Improper use leads to invalid tests or tests with low power. Empirical evidence demonstrates that the nonparametric method performs well for alternative models, noise processes, and jump distributions. In the second essay, I present a study on market data from U.S. energy futures markets. I apply a nonparametric method to identify jumps in futures prices of crude oil, heating oil and natural gas contracts traded on the New York Mercantile Exchange. The sample period of the intraday data covers January 1990 to January 2008. Alternative methods such as staggered returns and optimal sampling frequency methods are used to remove the effects of microstructure noise which biases the tests against detecting jumps. I obtain several important empirical results: (i) The realized volatility of natural gas futures exceeds that of heating oil and crude oil. (ii) In these commodities, large volatility days are often associated with large jump components and large jump components are often associated with weekly announcements of inventory levels. (iii) The realized volatility and smooth volatility components in natural gas and heating oil futures are higher in winter months than in summer months. Moreover, cold weather and inventory surprises cause the volatility in natural gas and heating oil to increase during the winter season. (iv) The jump component produces a transitory surge in total volatility, and there is a strong reversal in volatility on days following a significant jump day. (v) I find that including jump and seasonal components as explanatory variables significantly improves the modeling and forecasting of the realized volatility. Chapter 1: Introduction Observers of financial markets have long noted that asset prices are very volatile and often exhibit jumps (price spikes). Thus, the assumption of a continuous diffusion process for asset price behavior is often violated in practice. Although empirical studies often note that the impact of such jumps generally is transitory, the shortterm effect in the volatility may nonetheless be considerable with important financial implications for valuation of derivatives (Merton (1976)), asset allocation (Jarrow and Rosenfeld (1984)) and risk management (Duffie and Pan (2001)). A number of studies has shown that models including both a discontinuous jump component and a continuous component fits the data better than only a continuous process. For example, Cox and Rubinstein (1985) compare the Black-Scholes formula with Merton’s option pricing formula (Merton (1976)) and show that for large and frequent jumps in the price process of the underlying assets, Black-Scholes significantly undervalues out-of-the-money and at-the-money options. In a more recent study, Bakshi et al. (1997) compare several parametric models with and without a jump component based on model fitting, pricing, and hedging. They report that it is essential to include a jump component for pricing and internal consistency. Eraker et al. (2003) observe that jumps in the returns occur less frequently than what is reported in most literature but are nevertheless still significant. Maheu and McCurdy (2004) assume that jumps in stock market returns are generated by a nonhomogenous Poisson process and find that the addition of the jump component improves forecasts of volatility.