No arbitrage without semimartingales

We show that with suitable restrictions on allowable trading strategies, one has no arbitrage in settings where the traditional theory would admit arbitrage possibilities. In particular, price processes that are not semimartingales are possible in our setting, for example, fractional Brownian motion.Comment: Published in at http://dx.doi.org/10.1214/08-AAP554 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Interest Rate Caps Smile Too! But Can the LIBOR Market Models Capture It?

Using more than two years of daily interest rate cap price data, this paper provides a systematic documentation of a volatility smile in cap prices. We find that Black (1976) implied volatilities exhibit an asymmetric smile (sometimes called a sneer) with a stronger skew for in-the-money caps than out-of-the-money caps. The volatility smile is time varying and is more pronounced after September 11, 2001. We also study the ability of generalized LIBOR market models to capture this smile. We show that the best performing model has constant elasticity of variance combined with uncorrelated stochastic volatility or upward jumps. However, this model still has a bias for short- and medium-term caps. In addition, it appears that large negative jumps are needed after September 11, 2001. We conclude that the existing class of LIBOR market models can not fully capture the volatility smileLIBOR market models, volatility smile, interest rate caps

Restructuring Risk in Credit Default Swaps: An Empirical Analysis

This paper estimates the price for restructuring risk in the U.S. corporate bond market during 1999-2005. Comparing quotes from default swap (CDS) contracts with a restructuring event and without, we find that the average premium for restructuring risk represents 6% to 8% of the swap rate without restructuring. We show that the restructuring premium depends on firm-specific balance-sheet and macroeconomic variables. And, when default swap rates without a restructuring event increase, the increase in restructuring premia is higher for low-credit-quality firms than for high-credit-quality firms. We propose a reduced-form arbitrage-free model for pricing default swaps that explicitly incorporates the distinction between restructuring and default events. A case study illustrating the model's implementation is provided.

Filtration Reduction and Completeness in Jump-Diffusion Models

This paper studies the pricing and hedging of derivatives in frictionless and competitive, but incomplete jump-diffusion markets. A unique equivalent martingale measure (EMM) is obtained using filtration reduction to a fictitious complete market. This unique EMM in the fictitious market is uplifted to the original economy using the notion of consistency. For pedagogical purposes, we begin with simple setups and progressively extend to models of increasing generality

Specification Tests of Calibrated Option Pricing Models

In spite of the popularity of model calibration in finance, empirical researchers have put more emphasis on model estimation than on the equally important goodness-of-fit problem. This is due partly to the ignorance of modelers, and more to the ability of existing statistical tests to detect specification errors. In practice, models are often calibrated by minimizing the sum of squared difference between the modelled and actual observations. It is challenging to disentangle model error from estimation error in the residual series. To circumvent the difficulty, we study an alternative way of estimating the model by exact calibration. We argue that standard time series tests based on the exact approach can better reveal model misspecifications than the error minimizing approach. In the context of option pricing, we illustrate the usefulness of exact calibration in detecting model misspecification. Under heteroskedastic observation error structure, our simulation results shows that the Black-Scholes model calibrated by exact approach delivers more accurate hedging performance than that calibrated by error minimization

Model Error in Contingent Claim Models Dynamic Evaluation

We formally incorporate parameter uncertainty and model error in the estimation of contingent claim models and the formulation of forecasts. This allows an inference on any function of interest (option values, bias functions, hedge ratios) consistent with the uncertainty in both parameters and models. We show how to recover the exact posterior distributions of the parameters or any function of interest. It is crucial to obtain exact posterior or predictive densities because the most likely implementation, a frequent updating setup, results in small samples and requires the incorporation of specific prior information. We develop Markov Chain Monte Carlo estimators to solve the estimation problem posed. We provide both within sample and predictive model specification tests which can be used in dynamic testing or trading systems, making use of both the cross-sectional and time series information in the options data. Finally, we generalize the error distribution by allowing for the (small) probability that an observation has a larger error. For each observation, this produces the probability of its being an outlier, and may help distinguish market from model error. We apply these new techniques to equity options. When model error is taken into account, the black-Scholes appears very robust, in contrast with previous studies which at best only incorporated parameter uncertainty. We then extend the base model, e.g., Black-Scholes, by polynomial functions of parameters. This allows for intuitive specification tests. The Black-Scholes in-sample error properties can be improved by the use of these simple extended models but this does not result in major improvements in out of sample predictions. The differences between these models may be important however because, as we document it, they produce different functions of economic interest such as hedge ratios, probability of mispricing. Nous incorporons formellement l'incertitude des paramètres et l'erreur de modèle dans l'estimation des modèles d'option et la formulation de prévisions. Ceci permet l'inférence de fonctions d'intérêt (prix de l'option, biais, ratios) cohérentes avec l'incertitude des paramètres et du modèle. Nous montrons comment extraire la distribution postérieure exacte (de fonctions) des paramètres. Ceci est crucial parce que l'utilisation la plus probable, réestimation périodique des paramètres, est analogues à des échantillons de petite taille et demande l'incorporation d'informations a priori spécifiques. Nous développons des modèles Monte Carlo de chaînes markoviennes afin de résoudre les problèmes d'estimation posés. Nous fournissons des tests de spécification, à la fois pour l'échantillon et le modèle prédictif, qui peuvent être utilisés pour les tests dynamiques et les systèmes de trading en utilisant l'information en coupe transversale et temporelle des données d'option. Finalement, nous généralisons la distribution d'erreurs en tenant compte de la (faible) probabilité qu'une observation ait une plus grande probabilité d'erreur. Cela fournit pour chaque observation la probabilité d'une donnée aberrante et peut aider à différencier erreur de modèle et erreur de marché. Nous appliquons ces nouvelles techniques aux options d'équité. Quand l'erreur de modèle est prise en considération, le Black-Scholes apparaît très robuste, en contraste avec les études précédentes qui, au mieux, incluait l'erreur de paramètre. Après, nous étendons le modèle de base, i.e. Black-Schles, par des fonctions polynomiales des paramètres. Cela permet des tests intuitifs de spécification. Les erreurs en échantillon du B-S sont améliorées par l'utilisation de ces simples modèles étendus,0501s cela n'apporte pas d'amélioration majeure dans les prédictions hors-échantillon. Quoi qu'il en soit, les différences entre ces modèles peuvent être importantes parcequ'elles produisent différentes fonctions d'intérêt telles que les ratios et la probabilité d'erreur d'évaluation.