On the discretization of backward doubly stochastic differential equations

In this paper, we are dealing with the approximation of the process (Y,Z) solution to the backward doubly stochastic differential equation with the forward process X . After proving the L2-regularity of Z, we use the Euler scheme to discretize X and the Zhang approach in order to give a discretization scheme of the process (Y,Z)

Weak error in negative Sobolev spaces for the stochastic heat equation

In this paper, we make another step in the study of weak error of the stochastic heat equation by considering norms as functional

A regression Monte-Carlo method for Backward Doubly Stochastic Differential Equations

This paper extends the idea of E.Gobet, J.P.Lemor and X.Warin from the setting of Backward Stochastic Differential Equations to that of Backward Doubly Stochastic Differential equations. We propose some numerical approximation scheme of these equations introduced by E.Pardoux and S.Peng

Pricing CAC 40 Index Options under Asymmetry of Information.

This article analyses, for the first time, the financial impact on the French market of September 11th, 2001. Was there any information asymmetry around this date? How deep was the reaction of the French investors? This study measures the magnitude of the shock in the stock price process.Information costs; implied volatility; jump diffusion model;

GARCH Option Pricing Under Skew

This article is an empirical study dedicated to the GARCH Option pricing model of Duan (1995) applied to the FTSE 100 European style options for various maturities. The beauty of this model is to have used the standard GARCH theory in an option perspective and also it is its flexibility to adapt to different rich GARCH specifications. We analyze the valididy of the model given its ability to price one-day ahead out- of-sample call options and also its ability to capture the empirical dynamic of the volatility skew. We get severe mispricing for deep out- of-the-money and short term call options, which tend to decrease the global performance of the model that is relatively correct. We note that long term skews tend to be more stable across time and strikes, which explains why we had a decreasing pricing bias for longer maturity contracts. We also get that skews tend to deform into smiles as we go toward the expiry date. This model reveals a good ability to capture the change of regime in the implied volatility surface judging from the transformation observed from smiles to skews.GARCH Option models, Monte Carlo simulations, Implied Volatility,Volatility Smile.

GARCH option pricing under skew.

This article is an empirical study dedicated to the GARCH Option pricing model of Duan (1995) applied to the FTSE 100 European style options for various maturities. We analyze the validity of the model given its ability to price one-day ahead out-of-sample call options and also its ability to capture the empirical dynamic of the volatility skew. First, we get a severe mispricing for deep out-of-the-money and short term call options. Second, this model reveals a good ability to capture the change of regime in the implied volatility surface.GARCH model; Monte Carlo simulations; Implied Volatility; Volatility Smile;

Density estimates for solutions to one dimensional Backward SDE's

In this paper, we derive sufficient conditions for each component of the solution to a general backward stochastic differential equation to have a density for which upper and lower Gaussian estimates can be obtained

Systematic credit risk: CDX index correlation and extreme dependence.

Dependence is an important issue in credit risk portfolio modeling and pricing. We discuss a straightforward common factor model of credit risk dependence, which is motivated by intensity models such as Duffie and Singleton (1998), among others. In the empirical analysis, we study dependence under the risk-neutral measure using credit default swap (CDS) spread data of liquid large-cap U.S. obligors. The proxy for the commonfactor is the DJ CDX.NA.IG index. We document that (i) the CDX factor is significant but has low explanatory power, (ii) factor sensitivities show distinct time-varying nature and that (iii) systematic credit risk shows asymmetric extreme factor dependence, where extreme dependence is present for upward CDX movements only. This finding from an EVT-copula approach is what is predicted by various intensity models of joint defaults.Credit risk; Time-varying risk; Extreme dependence; Factor model;

Extreme Asymmetric Volatility, Leverage, Feedback and Asset Prices.

Asymmetric volatility in equity markets has been widely documented in finance, where two competing explanations, as considered in Bekaert and Wu (2000), are the financial leverage and the volatility feedback hypothesis. We explicitly test for the role of both hypotheses in explaining extreme daily U.S. equity market movements during the period January 1990 to September 2008. To this aim, we examine asymmetric volatility based on a novel model of market returns, conditional market volatility and volatility of volatility. We then test for extreme asymmetry and the distinct predictions of both hypotheses. Our results document significant extreme asymmetric volatility. This effect is contemporaneous, consistent with both hypotheses, and it is important for large market declines. We further point out aggregate asset pricing implications under extreme volatility feedback.Market stress; Asymmetric volatility; Leverage effect; Effet de levier; Market volatility; Volatility feedback;

The Reactive Volatility Model

We present a new volatility model, simple to implement, that includes a leverage effect whose return-volatility correlation function fits to empirical observations. This model is able to capture both the "retarded effect" induced by the specific risk, and the "panic effect", which occurs whenever systematic risk becomes the dominant factor. Consequently, in contrast to a GARCH model and a standard volatility estimate from the squared returns, this new model is as reactive as the implied volatility: the model adjusts itself in an instantaneous way to each variation of the single stock price or the stock index price and the adjustment is highly correlated to implied volatility changes. We also test the reactivity of our model using extreme events taken from the 470 most liquid European stocks over the last decade. We show that the reactive volatility model is more robust to extreme events, and it allows for the identification of precursors and replicas of extreme events