José Manuel Corcuera

Additional utility of insiders with imperfect dynamical information

In this paper we consider an insider with privileged information that is affected by an independent noise vanishing as the revelation time approaches. At this time, information is available to every trader. Our financial markets are based on Wiener space. In probabilistic terms we obtain an infinite dimensional extension of Jacod’s theorem to cover cases of progressive enlargement of filtrations. The application of this result gives the semimartingale decomposition of the original Wiener process under the progressively enlarged filtration. As an application we prove that if the rate at which the additional noise in the insider’s information vanishes is slow enough then there is no arbitrage and the additional utility of the insider is finite.

Multipower Variation for Brownian Semistationary Processes

In this paper we study the asymptotic behaviour of power and multipower variations of processes Y : Yt = Z t 1 g(t s) sW (ds) +Zt

Limit Theorems for Functionals of Higher Order Differences of Brownian Semi-Stationary Processes

We present some new asymptotic results for functionals of higher order differences of Brownian semi-stationary processes. In an earlier work [4] we have derived a similar asymptotic theory for first order differences. However, the central limit theorems were valid only for certain values of the smoothness parameter of a Brownian semistationary process, and the parameter values which appear in typical applications, e.g. in modeling turbulent flows in physics, were excluded. The main goal of the current paper is the derivation of the asymptotic theory for the whole range of the smoothness parameter by means of using second order differences. We present the law of large numbers for the multipower variation of the second order differences of Brownian semi-stationary processes and show the associated central limit theorem. Finally, we demonstrate some estimation methods for the smoothness parameter of a Brownian semi-stationary process as an application of our probabilistic results.

A generalized bayes rule for prediction

In the case of prior knowledge about the unknown parameter, the Bayesian predictive density coincides with the Bayes estimator for the true density in the sense of the Kullback-Leibler divergence, but this is no longer true if we consider another loss function. In this paper we present a generalized Bayes rule to obtain Bayes density estimators with respect to any a-divergence, including the Kullback-Leibler divergence and the Hellinger distance. For curved exponential models, we study the asymptotic behaviour of these predictive densities. We show that, whatever prior we use, the generalized Bayes rule improves (in a non-Bayesian sense) the estimative density corresponding to a bias modifica- tion of the maximum likelihood estimator. It gives rise to a correspondence between choosing a prior density for the generalized Bayes rule and fixing a bias for the maximum likelihood estimator in the classical setting. A criterion for comparing and selecting prior densities is also given.

Bipower variation for Gaussian processes with stationary increments

Convergence in probability and central limit laws of bipower variation for Gaussian processes with stationary increments and for integrals with respect to such processes are derived. The main tools of the proofs are some recent powerful techniques of Wiener/Ito/Malliavin calculus for establishing limit laws, due to Nualart, Peccati, and others.

Completion of a Lévy market by power-jump assets

Abstract.Except for the geometric Brownian model and the geometric Poissonian model, the general geometric Lévy market models are incomplete models and there are many equivalent martingale measures. In this paper we suggest to enlarge the market by a series of very special assets (power-jump assets) related to the suitably compensated power-jump processes of the underlying Lévy process. By doing this we show that the market can be completed. The very particular choice of the compensators needed to make these processes tradable is delicate. The question in general is related to the moment problem.

Pricing of contingent convertibles under smile conform models

We look at the problem of pricing CoCo bonds where the underlying risky asset dynamics are given by a smile conform model, more precisely an exponential Levy process incorporating jumps and heavy tails. A core mathematical quantity that is needed in closed form in order to produce an exact analytical expression for the price of a CoCo is the law of the infimum of the underlying equity price process at a fixed time. With the exception of Brownian motion with drift, no such closed analytical form is available within the class of Levy process that are suitable for financial modeling. Very recently however there has been some remarkable progress made with the theory of a large family of Levy processes, known as β-processes, cf. Kuznetsov [12] and Kuznetsov et al. [14]. Indeed for this class of Levy processes, the law of the infimum at an independent and exponentially distributed random time can be written down in terms of the roots and poles of its characteristic exponent; all of which are easily found within regularly spaced intervals along one of the axes of the complex plane. Combining these results together with a recently suggested Monte-Carlo technique, due to Kuznetsov et al. [13], which capitalises on the randomised law of the infimum we show the efficient and effective numerical pricing of CoCos. We perform our analysis using a special class of β-processes, known as β-VG, which have similar characteristics to the classical Variance-Gamma model. The theory is put to work by performing two case studies. After calibrating our model to market data, we price and analyze one of the Lloyds CoCos as well as the first Rabo CoCo.

Implied Levy volatility

Although several advanced asset return models have been developed these last two decades, including jumps and stochastic volatility characteristics, the Black-Scholes model has remained the standard quoting tool for many banks and financial institutions. This is partly due to the simple and widespread used concept of the Black-Scholes implied volatility and to the fact that over the years option traders have developed an intuition into this model parameter. Here a similar concept is developed but now under a Levy framework and therefore based on distributions that match more closely historical returns since they allow to introduce both skewness and excess of kurtosis into the model. In particular, we propose two models, the Levy implied space and time volatility models; the first arising when the Levy distribution is multiplied by the volatility and the second one when the time argument of the Levy distribution is multiplied by the square of the volatility. Moreover the concept of implied Levy space and time volatility is introduced and a study of the shape of implied Levy volatilities is made. Model performance is studied by analyzing delta-hedging strategies for the Normal Inverse Gaussian and the Meixner model, both qualitatively and on historical time-series of the S&P500. It is shown that under such parameter settings the model performs systematically better.

Multipower variation for Brownian semistationary processes

In this paper we study the asymptotic behaviour of power and multipower variations of processes

Efficient Pricing of Contingent Convertibles Under Smile Conform Models

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