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Stochastic Volatility in Financial Markets presents advanced topics in financial econometrics and theoretical finance, and is divided into three main parts. The first part aims at documenting an empirical regularity of financial price changes: the occurrence of sudden and persistent changes of financial markets volatility. This phenomenon, technically termed `stochastic volatility', or `conditional heteroskedasticity', has been well known for at least 20 years; in this part, further, useful theoretical properties of conditionally heteroskedastic models are uncovered. The second part goes beyond the statistical aspects of stochastic volatility models: it constructs and uses new fully articulated, theoretically-sounded financial asset pricing models that allow for the presence of conditional heteroskedasticity. The third part shows how the inclusion of the statistical aspects of stochastic volatility in a rigorous economic scheme can be faced from an empirical standpoint.
This paper compares the performance of three maximum likelihood estimation procedures -quasi-maximum likelihood, Monte Carlo likelihood and the particle filter to estimate stochastic volatility models of short term interest rates. The procedures are compared in an empirical study of interest rate volatility where a number of diagnostic tests in- and out-of-sample are utilized to evaluate both model specification and estimation procedure. Empirically, the results suggest interest rates follow the Cox-Ingersoll-Ross model with stochastic volatility and that volatility increases after Federal Open Market Committee meetings. Overall, the Monte Carlo likelihood procedure provided the best results.
In this article I provide new evidence on the role of nonlinear drift and stochastic volatility in interest rate modeling. I compare various model specifications for the short-term interest rate using the data from five countries. I find that modeling the stochastic volatility in the short rate is far more important than specifying the shape of the drift function. The empirical support for nonlinear drift is weak with or without the stochastic volatility factor. Although a linear drift stochastic volatility model fits the international data well, I find that the level effect differs across countries.
The Finance Division of the Faculty of Commerce and Business Administration at the University of British Columbia in Vancouver, British Columbia, Canada, presents the full text of a working paper entitled "Markov-Swtiching and Stochastic Volatility Diffusion Models of Short-Term Interest Rates," by Daniel R. Smith. The paper compares the Markov-switching and stochastic volatility diffusion models of short-term interest rates.
In this paper, we examine possible stochastic volatility and jumps in short-term interest rates for four major countries: US, UK, Germany and Japan. An econometric model with stochastic volatility and jumps in both rates and volatility is derived and fit to the daily data for futures interest rates in four major currencies and the model provides a better fit for the empirical distributions. The distributions for changes in Eurocurrency interest rate futures are leptokurtic with fat tails and an unusually large percentage of observations concentrated at zero. The implied volatilities for at-the-money options on interest rate futures reveal evidence of stochastic volatility, as well as jumps in volatility.
In this paper a three-factor model of the term structure of interest rates is developed. In the model the future short rate depends on 1) the current short rate, 2) the short-term mean of the short rate, and 3) the current volatility of the short rate. Furthermore, it is assumed that both the short term mean of the short rate and the volatility of the short rate are stochastic and follow square-root process. The model is a substantial extension the seminal Cox-Ingersoll-Ross model of interest rates. A general formula for evaluating interest rate derivatives is presented. Closed-form solutions for prices of bond, bond option, futures, futures option, swap and cap are derived. The model can fit into the Heath-Jarrow-Morton arbitrage framework. The model is also useful for other practical purposes such as managing interest rate risks and formulating fixed income arbitrage strategies.
This book on Interest Rate Derivatives has three parts. The first part is on financial products and extends the range of products considered in Interest Rate Derivatives Explained I. In particular we consider callable products such as Bermudan swaptions or exotic derivatives. The second part is on volatility modelling. The Heston and the SABR model are reviewed and analyzed in detail. Both models are widely applied in practice. Such models are necessary to account for the volatility skew/smile and form the fundament for pricing and risk management of complex interest rate structures such as Constant Maturity Swap options. Term structure models are introduced in the third part. We consider three main classes namely short rate models, instantaneous forward rate models and market models. For each class we review one representative which is heavily used in practice. We have chosen the Hull-White, the Cheyette and the Libor Market model. For all the models we consider the extensions by a stochastic basis and stochastic volatility component. Finally, we round up the exposition by giving an overview of the numerical methods that are relevant for successfully implementing the models considered in the book.