Cette thèse porte sur l'étude mathématique du problème de projection Markovienne d'un processus aléatoire: il s'agit de construire, étant donné un processus aléatoire ξ, un processus de Markov ayant à chaque instant la même distribution que ξ. Cette construction permet ensuite de déployer les outils analytiques disponibles pour l'étude des processus de Markov (équations aux dérivées partielles ou équations integro-différentielles) dans l'étude des lois marginales de ξ, même lorsque ξ n'est pas markovien. D'abord étudié dans un contexte probabiliste, notamment par Gyöngy (1986), ce problème a connu un regain d'intêret motivé par les applications en finance, sous l'impulsion des travaux de B. Dupire. La thèse entreprend une étude systématique des aspects probabilistes (construction d'un processus de Markov mimant les lois marginales de ξ) et analytiques (dérivation d'une équation de Kolmogorov forward) de ce problème, étendant les résultats existants au cas de semimartingales discontinues. Notre approche repose sur l'utilisation de la notion de problème de martingale pour un opérateur integro-différentiel. Nous donnons en particulier un résultat d'unicité pour une équation de Kolmogorov associée à un opérateur integro-différentiel non-dégénéré. Ces résultats ont des applications en finance: nous montrons notamment comment ils peuvent servir à réduire la dimension d'un problème à travers l'exemple de l'évaluation des options sur indice en finance.

Using the dual Black-Scholes partial differential equation, Dupire derived an explicit formula, involving the ratio of partial derivatives of the evolving fair value of a European call option (ECO), for recovering information about its variable volatility. Because the prices, as a function of maturity and strike, are only available as discrete noisy observations, the evaluation of Dupire’s formula reduces to being an ill-posed numerical differentiation problem, complicated by the need to take the ratio of derivatives. In order to illustrate the nature of ill-posedness, a simple finite difference scheme is first used to approximate the partial derivatives. A new method is then proposed which reformulates the determination of the volatility, from the partial differential equation defining the fair value of the ECO, as a parameter identification activity. By using the weak formulation of this equation, the problem is localized to a subregion on which the volatility surface can be approximated by a constant or a constant multiplied by some known shape function which models the local shape of the volatility function. The essential regularization is achieved through the localization, the choice of the analytic weight function, and the application of integration-by-parts to the weak formulation to transfer the differentiation of the discrete data to the differentiation of the analytic weight function.

Using the dual Black-Scholes partial differential equation, Dupire derived an explicit formula, involving the ratio of partial derivatives of the evolving fair value of a European call option (ECO), for recovering information about its variable volatility. Because the prices, as a function of maturity and strike, are only available as discrete noisy observations, the evaluation of Dupire’s formula reduces to being an ill-posed numerical differentiation problem, complicated by the need to take the ratio of derivatives. In order to illustrate the nature of ill-posedness, a simple finite difference scheme is first used to approximate the partial derivatives. A new method is then proposed which reformulates the determination of the volatility, from the partial differential equation defining the fair value of the ECO, as a parameter identification activity. By using the weak formulation of this equation, the problem is localized to a subregion on which the volatility surface can be approximated by a constant or a constant multiplied by some known shape function which models the local shape of the volatility function. The essential regularization is achieved through the localization, the choice of the analytic weight function, and the application of integration-by-parts to the weak formulation to transfer the differentiation of the discrete data to the differentiation of the analytic weight function.

Theaim of this master-thesis is to study the impact of uncertainty in the local-and implied volatility surfaces when pricing certain structured products suchas capital protected notes and autocalls. Due to their long maturities, limitedavailability of data and liquidity issue, the uncertainty may have a crucialimpact on the choice of valuation model. The degree of sensitivity andreliability of two different valuation models are studied. The valuation models chosen for this thesis are the local volatility model of Dupire and the implied volatility model of Black&Scholes. The two models are stress tested with varying volatilities within an uncertainty interval chosen to be the volatilities obtained from Bid and Ask market prices. The volatility surface of the Mid market prices is set as the relative reference and then successively scaled up and down to measure the uncertainty.The results indicates that the uncertainty in the chosen interval for theDupire model is of higher order than in the Black&Scholes model, i.e. thelocal volatility model is more sensitive to volatility changes. Also, the pricederived in the Black&Scholes modelis closer to the market price of the issued CPN and the Dupire price is closer tothe issued Autocall. This might be an indication of uncertainty in thecalibration method, the size of the chosen uncertainty interval or the constantextrapolation assumption.A further notice is that the prices derived from the Black&Scholes model areoverall higher than the prices from the Dupire model. Another observation ofinterest is that the uncertainty between the models is significantly greaterthan within each model itself. / Syftet med dettaexamensarbete är att studera inverkan av osäkerhet, i prissättningen av struktureradeprodukter, som uppkommer på grund av förändringar i volatilitetsytan. I dennastudie värderas olika slags autocall- och kapitalskyddade struktureradeprodukter. Strukturerade produkter har typiskt långa löptider vilket medförosäkerhet i värderingen då mängden data är begränsad och man behöver ta tillextrapolations metoder för att komplettera. En annan faktor som avgörstorleksordningen på osäkerheten är illikviditeten, vilken mäts som spreadenmellan listade Bid och Ask priset. Dessa orsaker ligger bakom intresset attstudera osäkerheten för långa löptider över alla lösenpriser och dess inverkanpå två olika värderingsmodeller.Värderingsmodellerna som används i denna studie är Dupires lokala volatilitetsmodell samt Black&Scholes implicita volatilitets modell. Dessa ställs motvarandra i en jämförelse gällande stabilitet och förmåga att fånga uppvolatilitets ändringar. Man utgår från Mid volatilitetsytan som referens ochuppmäter prisändringar i intervallet från Bid upp till Ask volatilitetsytornagenom att skala Mid ytan. Resultaten indikerar på större prisskillnader inom Dupires modell i jämförelsemot Black&Scholes. Detta kan tolkas som att Dupires modell är mer känslig isammanhanget och har en starkare förmåga att fånga upp förändringar isvansarna. Vidare notering är att priserna beräknade i Dupire är relativtbilligare än motsvarande från Black&Scholes modellen. En ytterligareobservation är att osäkerheten mellan värderingsmodellerna är av högre ordningän inom var modell för sig. Ett annat resultat visar att CPN priset beräknat iBlack&Scholes modell ligger närmast marknadspriset medans marknadsprisetför Autocallen ligger närmare Dupires. Detta kan vara en indikation påosäkerheten i kalibreringsmetoden eventuellt det valda osäkerhetsintervalletoch konstanta extrapolations antagandet.

This thesis deals with regularization parameter selection methods in the context of Tikhonov-type regularization with Poisson distributed data, in particular the reconstruction of images, as well as with the identification of the volatility surface from observed option prices. In Part I we examine the choice of the regularization parameter when reconstructing an image, which is disturbed by Poisson noise, with Tikhonov-type regularization. This type of regularization is a generalization of the classical Tikhonov regularization in the Banach space setting and often called variational regularization. After a general consideration of Tikhonov-type regularization for data corrupted by Poisson noise, we examine the methods for choosing the regularization parameter numerically on the basis of two test images and real PET data. In Part II we consider the estimation of the volatility function from observed call option prices with the explicit formula which has been derived by Dupire using the Black-Scholes partial differential equation. The option prices are only available as discrete noisy observations so that the main difficulty is the ill-posedness of the numerical differentiation. Finite difference schemes, as regularization by discretization of the inverse and ill-posed problem, do not overcome these difficulties when they are used to evaluate the partial derivatives. Therefore we construct an alternative algorithm based on the weak formulation of the dual Black-Scholes partial differential equation and evaluate the performance of the finite difference schemes and the new algorithm for synthetic and real option prices.

This thesis deals with regularization parameter selection methods in the context of Tikhonov-type regularization with Poisson distributed data, in particular the reconstruction of images, as well as with the identification of the volatility surface from observed option prices. In Part I we examine the choice of the regularization parameter when reconstructing an image, which is disturbed by Poisson noise, with Tikhonov-type regularization. This type of regularization is a generalization of the classical Tikhonov regularization in the Banach space setting and often called variational regularization. After a general consideration of Tikhonov-type regularization for data corrupted by Poisson noise, we examine the methods for choosing the regularization parameter numerically on the basis of two test images and real PET data. In Part II we consider the estimation of the volatility function from observed call option prices with the explicit formula which has been derived by Dupire using the Black-Scholes partial differential equation. The option prices are only available as discrete noisy observations so that the main difficulty is the ill-posedness of the numerical differentiation. Finite difference schemes, as regularization by discretization of the inverse and ill-posed problem, do not overcome these difficulties when they are used to evaluate the partial derivatives. Therefore we construct an alternative algorithm based on the weak formulation of the dual Black-Scholes partial differential equation and evaluate the performance of the finite difference schemes and the new algorithm for synthetic and real option prices.