Pricing and Hedging Volatility Derivatives

Abstract This paper studies the pricing and hedging of variance swaps and other volatility derivatives, including volatility swaps and variance options, in the Heston stochastic volatility model. Pricing and hedging results are derived using partial differential equation techniques. We formulate an optimization problem to determine the number of options required to best hedge a variance swap. We propose a method to dynamically hedge volatility derivatives using variance swaps and a finite number of European call and put options

Modeling Financial Swaps and Geophysical data Using the Barndorff-Nielsen and Shephard Model

This dissertation uses Barndoff-Nielsen and Shephard (BN-S) models to model swap, a type of financial derivative, and analyze geophysical data for estimation of major earthquakes. From empirical observation of the stock market activity and earthquake occurrence, we observe that the distributions have high kurtosis and right skewness. Consequently, such data cannot be captured by stochastic models driven by a Wiener process. Non-Gaussian processes of Ornstein-Uhlenbeck type are one of the most significant candidates for being the building blocks of models of financial economics. These models offer the possibility of capturing important distributional deviations from Gaussianity and thus these are more practical models of dependence structures. Introduced by Barndorff-Nielsen and Shephard these processes are not only convenient to model volatility in financial market, but have an independent interest for modeling stationary time series of various kinds. For the financial data we use BN-S models to price the arbitrage-free value of volatility, variance, covariance, and correlation swap. Such swaps are used in financial markets for volatility hedging and speculation. We use the S&P500 and NASDAQ index for parameter estimation and numerical analysis. We show that with this model the error estimation in fitting the delivery price is much less than the existing models with comparable parameters. For any given time interval, the earthquake magnitude data have three main properties: (1) magnitude is a non-negative stationary stochastic process; (2) for any finite interval of time there are only finite number of jumps; (3) the sample path of the magnitude of an earthquake consists of upward jumps (significant earthquake) and a gradual decrease (aftershocks). For such geophysical data we specifically use Gamma Ornstein Uhlenbeck processes driven by a Levy process to estimate a major earthquake in a certain region in California. Rigorous regression analysis is provided, and based on that, first-passage times are computed for different sets of data. Both applications demonstrate the significance of BN-S models to phenomena that follow non-Gaussian distributions

Arithmetic variance swaps

Biases in standard variance swap rates can induce substantial deviations below market rates. Defining realised variance as the sum of squared price (not log-price) changes yields an `arithmetic' variance swap with no such biases. Its fair value has advantages over the standard variance swap rate: no discrete-monitoring or jump biases; and the same value applies for any monitoring frequency, even irregular monitoring and to any underlying, including those taking zero or negative values. We derive the fair-value for the arithmetic variance swap and compare with the standard variance swap rate by: analysing errors introduced by interpolation and integration techniques; numerical experiments for approximation accuracy; and using 23 years of FTSE 100 options data to explore the empirical properties of arithmetic variance (and higher-moment) swaps. The FTSE 100 variance risk has a strong negative correlation with the implied third moment, which can be captured using a higher-moment arithmetic swap

Discretisation-invariant swaps and higher-moment risk premia

This thesis introduces a general framework for model-free discretisation-invariant swaps. In the first main chapter a novel design for swap contracts is developed where the realised leg is modified such that the fair value is independent of the monitoring partition. An exact swap rate can then be derived from the price aportfolio of vanilla out-of-the-money options without any discrete-monitoring or jump errors. In the second main chapter the P&Ls on discretisation-invariant swaps associated with the variance, skewness and kurtosis of the log return distribution are used as estimators for the corresponding higher-moment risk premia. An empirical study on the S&P 500 investigates the factors determining these risk premia for different sampling frequencies and contract maturities. In the third main chapter the dynamics of conventional and discretisation-invariant variance swaps and variance risk premia are compared in an affine jump-diffusion setting. The ideas presented in this thesis set the ground for many interesting and practically relevant applications

Volatility forecasting

Volatility has been one of the most active and successful areas of research in time series econometrics and economic forecasting in recent decades. This chapter provides a selective survey of the most important theoretical developments and empirical insights to emerge from this burgeoning literature, with a distinct focus on forecasting applications. Volatility is inherently latent, and Section 1 begins with a brief intuitive account of various key volatility concepts. Section 2 then discusses a series of different economic situations in which volatility plays a crucial role, ranging from the use of volatility forecasts in portfolio allocation to density forecasting in risk management. Sections 3, 4 and 5 present a variety of alternative procedures for univariate volatility modeling and forecasting based on the GARCH, stochastic volatility and realized volatility paradigms, respectively. Section 6 extends the discussion to the multivariate problem of forecasting conditional covariances and correlations, and Section 7 discusses volatility forecast evaluation methods in both univariate and multivariate cases. Section 8 concludes briefly. JEL Klassifikation: C10, C53, G1

Volatility Forecasting

Volatility has been one of the most active and successful areas of research in time series econometrics and economic forecasting in recent decades. This chapter provides a selective survey of the most important theoretical developments and empirical insights to emerge from this burgeoning literature, with a distinct focus on forecasting applications. Volatility is inherently latent, and Section 1 begins with a brief intuitive account of various key volatility concepts. Section 2 then discusses a series of different economic situations in which volatility plays a crucial role, ranging from the use of volatility forecasts in portfolio allocation to density forecasting in risk management. Sections 3,4 and 5 present a variety of alternative procedures for univariate volatility modeling and forecasting based on the GARCH, stochastic volatility and realized volatility paradigms, respectively. Section 6 extends the discussion to the multivariate problem of forecasting conditional covariances and correlations, and Section 7 discusses volatility forecast evaluation methods in both univariate and multivariate cases. Section 8 concludes briefly.