GRBAC Zorana

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In this thesis, we study several financial mathematics problems related to the valuation of derivatives. Through different asymptotic approaches, we develop methods to compute accurate approximations of the price of certain types of options in cases where no explicit formula exists.In the first chapter, we focus on the valuation of options whose payoff depends on the trajectory of the underlying by Monte Carlo methods, when the underlying is modeled by an affine process with stochastic volatility. We prove a principle of large trajectory deviations in long time, which we use to compute, using Varadhan's lemma, an asymptotically optimal change of measure, allowing to significantly reduce the variance of the Monte-Carlo estimator of option prices.The second chapter considers the valuation by Monte-Carlo methods of options depending on multiple underlyings, such as basket options, in Wishart's stochastic volatility model, which generalizes the Heston model. Following the same approach as in the previous chapter, we prove that the process vérifie a principle of large deviations in long time, which we use to significantly reduce the variance of the Monte Carlo estimator of option prices, through an asymptotically optimal change of measure. In parallel, we use the principle of large deviations to characterize the long-time behavior of the Black-Scholes implied volatility of basket options.In the third chapter, we study the valuation of realized variance options, when the spot volatility is modeled by a constant volatility diffusion process. We use recent asymptotic results on the densities of hypo-elliptic diffusions to compute an expansion of the realized variance density, which we integrate to obtain the expansion of the option price, and then their Black-Scholes implied volatility.The final chapter is devoted to the valuation of interest rate derivatives in the Lévy model of the Libor market, which generalizes the classical Libor market model (log-normal) by adding jumps. By writing the former as a perturbation of the latter and using the Feynman-Kac representation, we explicitly compute the asymptotic expansion of the price of interest rate derivatives, in particular, caplets and swaptions.

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The recent financial crisis has led to so-called multi-curve models for the term structure. Here we study a multi-curve extension of short rate models where, in addition to the short rate itself, we introduce short rate spreads. In particular, we consider a Gaussian factor model where the short rate and the spreads are second order polynomials of Gaussian factor processes. This leads to an exponentially quadratic model class that is less well known than the exponentially affine class. In the latter class the factors enter linearly and for positivity one considers square root factor processes. While the square root factors in the affine class have more involved distributions, in the quadratic class the factors remain Gaussian and this leads to various advantages, in particular for derivative pricing. After some preliminaries on martingale modeling in the multi-curve setup, we concentrate on pricing of linear and optional derivatives. For linear derivatives, we exhibit an adjustment factor that allows one to pass from pre-crisis single curve values to the corresponding post-crisis multi-curve values.

In this paper we consider the pricing of options on interest rates such as caplets and swaptions in the Levy Libor model developed by Eberlein and Ozkan (Financ. Stochast. 9:327-348 (2005) [9]). This model is an extension to Levy driving processes of the classical log-normal Libor market model (LMM) driven by a Brownian motion. Option pricing is significantly less tractable in this model than in the LMM due to the appearance of stochastic terms in the jump part of the driving process when performing the measure changes which are standard in pricing of interest rate derivatives. To obtain explicit approximation for option prices, we propose to treat a given Levy Libor model as a suitable perturbation of the log-normal LMM. The method is inspired by recent works by Cerný, Denkl, and Kallsen (Preprint (2013) [6]) and Menasse and Tankov (Preprint (2015) [14]). The approximate option prices in the Levy Libor model are given as the corresponding LMM prices plus correction terms which depend on the characteristics of the underlying Levy process and some additional terms obtained from the LMM model.

In this paper we consider the pricing of options on interest rates such as caplets and swaptions in the Lévy Libor model developed by Eberlein and Özkan (Financ. Stochast. 9:327-348 (2005) [9]). This model is an extension to Lévy driving processes of the classical log-normal Libor market model (LMM) driven by a Brownian motion. Option pricing is significantly less tractable in this model than in the LMM due to the appearance of stochastic terms in the jump part of the driving process when performing the measure changes which are standard in pricing of interest rate derivatives. To obtain explicit approximation for option prices, we propose to treat a given Lévy Libor model as a suitable perturbation of the log-normal LMM. The method is inspired by recent works by Černý, Denkl, and Kallsen (Preprint (2013) [6]) and Ménassé and Tankov (Preprint (2015) [14]). The approximate option prices in the Lévy Libor model are given as the corresponding LMM prices plus correction terms which depend on the characteristics of the underlying Lévy process and some additional terms obtained from the LMM model.

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We provide a unified framework for modeling LIBOR rates using general semimartingales as driving processes and generic functional forms to describe the evolution of the dynamics. We derive sufficient conditions for the model to be arbitrage-free which are easily verifiable, and for the LIBOR rates to be true martingales under the respective forward measures. We discuss when the conditions are also necessary and comment on further desirable properties such as those leading to analytical tractability and positivity of rates. This framework allows to consider several popular models in the literature, such as LIBOR market models driven by Brownian motion or jump processes, the L\'evy forward price model as well as the affine LIBOR model, under one umbrella. Moreover, we derive structural results about LIBOR models and show, in particular, that only models where the forward price is an exponentially affine function of the driving process preserve their structure under different forward measures.

Filling a gap in the literature caused by the recent financial crisis, this book provides a treatment of the techniques needed to model and evaluate interest rate derivatives according to the new paradigm for fixed income markets. Concerning this new development, there presently exist only research articles and two books, one of them an edited volume, both being written by researchers working mainly in practice. The aim of this book is to concentrate primarily on the methodological side, thereby providing an overview of the state-of-the-art and also clarifying the link between the new models and the classical literature. The book is intended to serve as a guide for graduate students and researchers as well as practitioners interested in the paradigm change for fixed income markets. A basic knowledge of fixed income markets and related stochastic methodology is assumed as a prerequisite.

The subject of this chapter are multi-curve models in the spirit of the Libor market model (LMM). The modeling is here done on a discrete tenor structure and the interest rates are discretely compounded, reflecting thus the market practice. We consider the discretely compounded forward OIS rates as reference rates, together with the forward Libor rates or, equivalently, the Libor-OIS spreads. The rates and the spreads are modeled directly under the forward martingale measures used for derivative pricing. First we describe a general framework that extends the classical BGM Libor market model to multiple curves. Then we present a multiple-curve affine Libor model based on families of exponentially affine martingales representing the forward OIS and Libor rates. Finally we also indicate modeling based on multiplicative Libor-OIS spreads. For each modeling approach, we mention corresponding methods for derivative pricing.

This chapter concerns the HJM framework for forward rate models in a multi-curve setup. As in Chap. 2, also in this chapter we shall model a basic OIS forward rate and the various risky multi-curve rates are obtained by adding a spread over the OIS rate. Since the ultimate goal is the pricing of interest rate derivatives, where the main underlying quantity are the Libor rates, one of the first objectives is to derive models for the dynamics of the Libor rates that are arbitrage-free. To this effect we shall first obtain models for OIS bond prices under a martingale measure and then choose suitable quantities connected to the FRA contracts, modeling them in the spirit of the HJM framework so that the complete model is arbitrage-free. Finally, we consider pricing of linear and optional interest rate derivatives in this HJM context.

The fixed income market is a sector of the global financial market in which various interest rate-sensitive instruments are traded, such as bonds, forward rate agreements, various forms of swaps, swaptions, caps and floors. It makes up a large portion of the global financial market. The recent financial crisis, of which the key features are counterparty and liquidity/funding risk, has heavily impacted the entire financial market and the fixed income market in particular. Inspection of quoted interest rates and derivative prices reveals that some classical relationships have broken down, which has induced the actors on the fixed income market to model as separate objects rates that correspond to different maturities (multi-curve models). In this chapter we review the basic notions and concepts in use before the crisis and describe how they have found an extension to the multi-curve setup.

Filling a gap in the literature caused by the recent financial crisis, this book provides a treatment of the techniques needed to model and evaluate interest rate derivatives according to the new paradigm for fixed income markets. Concerning this new development, there presently exist only research articles and two books, one of them an edited volume, both being written by researchers working mainly in practice. The aim of this book is to concentrate primarily on the methodological side, thereby providing an overview of the state-of-the-art and also clarifying the link between the new models and the classical literature. The book is intended to serve as a guide for graduate students and researchers as well as practitioners interested in the paradigm change for fixed income markets. A basic knowledge of fixed income markets and related stochastic methodology is assumed as a prerequisite.

In this chapter we consider mainly strict-sense short-rate models in view of constructing multiple curves. Because the pre-crisis rational pricing kernel models can be seen as short-rate models in a wider sense, we shall furthermore present some recent multi-curve extensions of these models as well. For the strict-sense short rate models we consider a basic OIS short rate and various spreads to be added on top of it, one for each of the multiple curves. The setup is mainly that of exponentially affine, but also exponentially quadratic models driven by several stochastic factors. This allows us to obtain explicit formulas for various linear and optional interest rate derivatives also in the multi-curve setting.

We introduce a multiple curve framework that combines tractable dynamics and semianalytic pricing formulas with positive interest rates and basis spreads. Negative rates and positive spreads can also be accommodated in this framework. The dynamics of overnight indexed swap and LIBOR rates are specified following the methodology of the affine LIBOR models and are driven by the wide and flexible class of affine processes. The affine property is preserved under forward measures, which allows us to derive Fourier pricing formulas for caps, swaptions, and basis swaptions. A model specification with dependent LIBOR rates is developed that allows for an efficient and accurate calibration to a system of caplet prices.

We consider the problem of valuation of interest rate derivatives in the post-crisis setup. We develop a multiple-curve model, set in the HJM framework and driven by a L ́evy process. We proceed with joint calibration to OTM swaptions and co-terminal ATM swaptions of different tenors, the calibration to OTM swaptions guaranteeing that the model correctly captures volatility smile effects and the calibration to co-terminal ATM swaptions ensuring an appropriate term structure of the volatility in the model. To account for counterparty risk and funding issues, we use the calibrated multiple- curve model as an underlying model for CVA computation. We follow a reduced-form methodology through which the problem of pricing the counterparty risk and funding costs can be reduced to a pre-default Markovian BSDE, or an equivalent semi-linear PDE. As an illustration we study the case of a basis swap and a related swaption, for which we compute the counterparty risk and funding adjustments.

We consider a general class of continuous asset price models where the drift and the volatility functions, as well as the driving Brownian motions, change at a random time \tau. Under minimal assumptions on the random time and on the driving Brownian motions, we study the behavior of the model in all the filtrations which naturally arise in this setting, establishing martingale representation results and characterizing the validity of the NA1 and NFLVR no-arbitrage conditions.

We consider the problem of valuation of interest rate derivatives in the post-crisis set-up. We develop a multiple-curve model, set in the HJM framework and driven by a Levy process. We proceed with joint calibration to OTM swaptions and co-terminal ATM swaptions of different tenors, the calibration to OTM swaptions guaranteeing that the model correctly captures volatility smile effects and the calibration to co-terminal ATM swaptions ensuring an appropriate term structure of the volatility in the model. To account for counterparty risk and funding issues, we use the calibrated multiple-curve model as an underlying model for CVA computation. We follow a reduced-form methodology through which the problem of pricing the counterparty risk and funding costs can be reduced to a pre-default Markovian BSDE, or an equivalent semi-linear PDE. As an illustration, we study the case of a basis swap and a related swaption, for which we compute the counterparty risk and funding adjustments.

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