We present a set of stylized empirical facts emerging from the statistical analysis of price variations in various types of financial markets. We first discuss some general issues common to all statistical studies of financial time series. Various statistical properties of asset returns are then described: distributional properties, tail properties and extreme fluctuations, pathwise regularity, linear and nonlinear dependence of returns in time and across stocks. Our description emphasizes properties common to a wide variety of markets and instruments. We then show how these statistical properties invalidate many of the common statistical approaches used to study financial data sets and examine some of the statistical problems encountered in each case.

Introduction and motivation The problem of how to specify a correlation matrix occurs in several important areas of finance and of risk management. A few of the important applications are, for instance, the specification of a (possibly time-dependent) instantaneous correlation matrix in the context of the BGM interestrate option models, stress-testing and scenario analysis for market risk management purposes, or the specification of a correlation matrix amongst a large number of obligors for credit-derivative pricing or credit risk management. For those applications where the most important desideratum is the recovery of the real-world correlation matrix, the problem is in principle well defined and readily solvable by means of wellestablished statistical techniques. In practice, however, the estimation problems can be severe: a small number of outliers, for instance, can seriously "pollute" a sample; non-synchronous data can easily destroy or hide correlation

The prices of index options at a given date are usually represented via the corresponding implied volatility surface, presenting skew/smile features and term structure which several models have attempted to reproduce.

We present and approximation for the volatility of European swaptions in a forward rate based Brace-Gatarek-Musiela/Jamshidian framework [BGM97, Jam97] which enables us to calculate prices for swaptions without the need for Monte Carlo simulations. Also, we explain the mechanism behind the remarkable accuracy of these approximate prices. For cases where the yield curve varies noticeably as a function of maturity, a second, and even more accurate formula is derived. 1

We develop a new method for finding upper bounds for Bermudan swaptions in a swap-rate market model. By comparing with lower bounds found by exercise boundary parametrization, we find that the bounds are well within bid-o#er spread. As an application, we study the dependence of Bermudan swaption prices on the number of instantaneous factors used in the model.

We review both full-rank and reduced-rank parameterizations for correlation matrices. In particular, the full-rank parameterizations of Rebonato (1999d) and Schoenmakers and Coffey (2000), the reduced-rank angle parameterization of Rebonato (1999d) and Rebonato and Jäckel (1999), and the well known zeroed-eigenvalues reduced-rank approximation are described in detail. Numerical examples are provided for the reduced-rank parameterizations and results are compared. 1

A method is presented for the valuation of American style options as a function of an exogeneosly assigned future implied volatility surface. Subject to this specification, this is done independently of any assumptions about a stochastic process of the underlying asset.

It is well-known that the fair value of options can be determined by using the Black-Scholes model. However, for liquidly traded options, i.e. instruments for which the market price is known, there is clear evidence that the Black-Scholes model is not correct. This is reflected in the existence of the volatility smile phenomenon which is one the most challenging problems in financial economics. Recently, a rigorous analysis of the time evolution of the empirically observed volatility smile, i.e. smile dynamics, has been reported by (CdF02) and (Fen05). However, the quantification of the volatility smile dynamics as implied by smile-consistent models has not been done rigorously, so far. People have addressed this by looking at the evolution of the smile, based on asymptotic analysis and qualitative investigations. In this work, we use similar statistical techniques as employed in the empirical studies, to quantify the smile dynamics that is implied by the following smile-consistent models: Displaced Diffusion, Constant Elasticity of Variance (CEV) and SABR Stochastic Volatility Models. We find that in markets where options exhibit extreme skew, e.g. equity options markets, the displaced diffusion and CEV models should be used with care, since these models have poor fitting capabilities to market prices and impose inaccurate smile dynamics. The SABR model on the other

This thesis develops a method for market risk measurement of an interest rate derivatives portfolio. The potential value loss of the portfolio is tested on a daily basis using market scenarios for interest rates and interest rate volatilities. Test scenarios are chosen to represent plausible changes in the market factors. The portfolio being tested consists of interest rate instruments including swaps, forward rate agreements, futures contracts, futures options, interest rate caps and floors and swaptions. Valuation models for these instrument types are introduced. The market scenarios are specified from historical interest rate data using principal components analysis. The resulting principal components are interpreted as typical market movements and are therefore chosen as test scenarios. Scenarios testing is used as a supplementary market risk measurement tool in conjunction with the incumbent risk management tools. It is found that the risk measures that result from the test scenarios do help the dealers and the risk managers to evaluate the structure of the current derivatives position. However, the benefits over the current risk measures are limited, and better results could be obtained by methods like scenario simulation and Value at Risk.

Abstract. In this paper the Black–Scholes implied volatility and the GARCH volatility are compared. The implied volatility is used in the form of volatility indices. Empirical study of the performance of GARCH in forecasting the implied volatility shows that the GARCH volatility can be used to forecast the implied volatility and is better than the historical volatility. Two different volatility indices (VIX, VDAX) of two different market indices (SPX index in United States and DAX index in Germany) were considered in the empirical study.