Dynamics of state price densities

State price densities (SPDs) are an important element in applied quantitative finance. In a Black-Scholes world they are lognormal distributions but in practice volatility changes and the distribution deviates from log-normality. In order to study the degree of this deviation, we estimate SPDs using EUREX option data on the DAX index via a nonparametric estimator of the second derivative of the (European) call pricing function. The estimator is constrained so as to satisfy no-arbitrage constraints and corrects for the intraday covariance structure in option prices. In contrast to existing methods, we do not use any parametric or smoothness assumptions

Confidence Intervals for State Price Densities

The state price density is a second derivative of the discounted European options prices with respect to the strike price. We use Maximum Likelihood method to derive a simple estimator of the curve such that it is decreasing, convex and its second derivative integrates to one. Confidence intervals for this estimator can be constructed using standard Maximum Likelihood theory. The method works well in praxis as illustrated on the DAX option prices data

Constrained General Regression in Pseudo-Sobolev Spaces with Application to Option Pricing

State price density (SPD) contains important information concerning market expectations. In existing literature, a constrained estimator of the SPD is found by nonlinear least squares in a suitable Sobolev space. We improve the behavior of this estimator by implementing a covariance structure taking into account the time of the trade and by considering simultaneously both the observed Put and Call option prices

Robust Sequential Estimation

Katedra pravděpodobnosti a matematické statistikyDepartment of Probability and Mathematical StatisticsMatematicko-fyzikální fakultaFaculty of Mathematics and Physic

Spandauer Straße 1, D-10178 BerlinDynamics of State Price Densities

State price densities (SPD) are an important element in applied quantitative finance. In a Black-Scholes model they are lognormal distributions with constant volatility parameter. In practice volatility changes and the distribution deviates from log-normality. We estimate SPDs using EU-REX option data on the DAX index via a nonparametric estimator of the second derivative of the (European) call price function. The estimator is constrained so as to satisfy no-arbitrage constraints and it corrects for intraday covariance structure. Given a low dimensional representation of this SPD we study its dynamic for the years 1995–2003. We calculate a prediction corridor for the DAX for a 45 day forecast. The proposed algorithm is simple, it allows calculation of future volatility and can be applied to hedging exotic options. Key words and Phrases: option pricing, state price density estimation, nonlinear least squares, confidence intervals

On the Appropriateness of Inappropriate VaR Models

Die Berechnung des VaR führt zur Reduktion der Dimension des Raumes der Risikofaktoren. Die vorzunehmenden Vereinfachungen resultieren aus unterschiedlichen Beweggründen, z.B. technische Effizienz, Sachlogik der Ergebnisse und statistische Adäquanz des Modells. Im Kapitel 2 stellen wir drei gängige Mappingverfahren vor: das Marktindexmodell, das Hauptkomponentenmodell und das Modell mit gleichkorrelierten Risikofaktoren. Impulse für Methoden zum Vergleich dieser Modelle im Kapitel 3 kamen vor allem aus der Literatur zur Praxis der Beurteilung von Wetterprognosen (Murphy and Winkler 1992, Murphy 1997). Umfangreiche Überlegungen zu einer quantitativen Analyse werden im vierten Kapitel dieser Arbeit vorgestellt. Die empirische Analyse der DAX Daten wird abschließend mit XploRe durchgeführt.The Value-at-Risk calculation reduces the dimensionality of the risk factor space. The main reasons for such simplifications are, e.g., technical efficiency, the logic and statistical appropriateness of the model. In Chapter 2 we present three simple mappings: the mapping on the market index, the principal components model and the model with equally correlated risk factors. The comparison of these models in Chapter 3 is based on the literatere on the verification of weather forecasts (Murphy and Winkler 1992, Murphy 1997). Some considerations on the quantitative analysis are presented in the fourth chapter. In the last chapter, we present empirical analysis of the DAX data using XploRe