CDS calibration under an extended JDCEV model

We propose a new methodology for the calibration of a hybrid credit-equity model to credit default swap (CDS) spreads and survival probabilities. We consider an extended Jump to Default Constant Elasticity of Variance model incorporating stochastic and possibly negative interest rates. Our approach is based on a perturbation technique that provides an explicit asymptotic expansion of the CDS spreads. The robustness and efficiency of the method is confirmed by several calibration tests on real market data

PDE models for the pricing of a defaultable coupon-bearing bond under an extended JDCEV model

open4siWe consider a two-factor model for the pricing of a non callable defaultable bond which pays coupons at certain given dates. The model under consideration is the Jump to Default Constant Elasticity of Variance (JDCEV) model. The JDCEV model is an improvement of the reduced form approach, which unifies credit and equity models into a single framework allowing for stochastic and possible negative interest rates. From the mathematical point of view, the valuation involves two partial differential equation (PDE) problems for each coupon. First, we obtain the existence of solution for these PDE problems. In order to solve them, we propose appropriate numerical schemes based on a Crank-Nicolson semi-Lagrangian method for time discretization combined with quadratic Lagrange finite elements for space discretization. Once the numerical solutions of the PDEs are obtained, a post-processing procedure is carried out in order to achieve the value of the bond. This post-processing includes the computation of an integral term which is approximated by using the composite trapezoidal rule. Finally, we present some numerical results for real market bonds issued by different firms in order to illustrate the proper behaviour of the numerical schemes. Moreover, we obtain an agreement between the numerical results from the PDE approach and those ones obtained by applying a Monte Carlo technique and an asymptotic aproximation method.openCarmen Calvo-Garrido, M.; Diop, Sidi; Pascucci, Andrea; Vázquez, CarlosCarmen Calvo-Garrido, M.; Diop, Sidi; Pascucci, Andrea; Vázquez, Carlo

PDE Models for the Pricing of a Defaultable Coupon-Bearing Bond Under an Extended JDCEV Model

Financiado para publicación en acceso aberto: Universidade da Coruña/CISUG[Abstract] We consider a two-factor model for the pricing of a non callable defaultable bond which pays coupons at certain given dates. The model under consideration is the Jump to Default Constant Elasticity of Variance (JDCEV) model. The JDCEV model is an improvement of the reduced form approach, which unifies credit and equity models into a single framework allowing for stochastic and possible negative interest rates. From the mathematical point of view, the valuation involves two partial differential equation (PDE) problems for each coupon. First, we obtain the existence of solution for these PDE problems. In order to solve them, we propose appropriate numerical schemes based on a Crank-Nicolson semi-Lagrangian method for time discretization combined with quadratic Lagrange finite elements for space discretization. Once the numerical solutions of the PDEs are obtained, a post-processing procedure is carried out in order to achieve the value of the bond. This post-processing includes the computation of an integral term which is approximated by using the composite trapezoidal rule. Finally, we present some numerical results for real market bonds issued by different firms in order to illustrate the proper behaviour of the numerical schemes. Moreover, we obtain an agreement between the numerical results from the PDE approach and those ones obtained by applying a Monte Carlo technique and an asymptotic aproximation method.Xunta de Galicia; ED431C2018/033Second, third and fourth authors have been supported by EU H2020-MSCA-ITN-2014 (WAKEUPCALL Grant Agreement 643045). First and fourth authors have partially been funded by Spanish MINECO with the grant PID2019-10858RB-I00 and by Galician Government with the grant ED431C2018/033, both including FEDER financial support. First and fourth authors also acknowledge the support received from the Centro de Investigacin de Galicia “CITIC”, funded by Xunta de Galicia and the European Union (European Regional Development Fund- Galicia 2014–2020 Program), by grant ED431G 2019/01. Funding for open access charge: Universidade da Coruña/CISUG

Exact Monte Carlo sampling of jump diffusions

The main objective of this thesis is to explore the theoretical foundations of the exact method for sampling jump diffusions proposed in [20] by Kay Giesecke and Dmitry Smelov, and implement it in order to compare the performance of the algorithm for pricing purposes against more traditional finite element methods, which generate biased samples. The method applies to a large class of models defined by a one-dimensional jump diffusion process, allowing us to generate exact simulations of a skeleton, a hitting time and other functionals of it, used for purposes like path-dependent option or interest rate derivatives pricing.O principal objetivo desta tese é explorar os fundamentos teóricos relativos ao método proposto em [20] por Kay Giesecke e Dmitry Smelov e implementá-lo de modo a comparar a sua performance face a métodos mais tradicionais de elementos finitos, que geram amostras enviesadas. O método aplica-se a uma grande parte dos modelos definidos por um processo de difusão com saltos unidimensional, permitindo gerar simulações de Monte Carlo exatas de um esqueleto, tempos de paragem e outros funcionais do mesmo, com finalidades como a avaliação de path-dependent options, derivados de taxa de juro ou outros instrumentos financeiros

Credit Risk Management and Jump Models

This doctoral thesis comprises three research papers that seek to improve and create corporate and sovereign credit risk models, to provide an approximate analytic expressions for CDS spreads and a numerical method for partial differential equation arisen from pricing defaultable coupon bond. First, an extension of Jump to Default Constant Elasticity Variance in more general and realistic framework is provided (see Chapter 3). We incorporate, in the model introduced in [9], a stochastic interest rate with possible negative values. In addition we provide an asymptotic approximation formula for CDS spreads based on perturbation theory. The robustness and efficiency of the method is conformed by several calibration tests on real market data. Next, under the model introduced in Chapter 3, we present in Chapter 4 a new numerical method for pricing non callable defaultable bond. we propose appropriate numerical schemes based on a Crank-Nicolson semi-Lagrangian method for time discretization combined with biquadratic Lagrange finite elements for space discretization. Once the numerical solutions of the PDEs are obtained, a post-processing procedure is carried out in order to achieve the value of the bond. Finally, we introduce a hybrid Sovereign credit risk model in which the intensity of default of a sovereign is based on the jump to default extended CEV model (see Chapter 5). The model captures the interrelationship between creditworthiness of a sovereign, its intensity to default and the correlation with the exchange rate between the bond's currency and the currency in which the CDS spread are quoted. We consider the Sovereign Credit Default Swaps Italy, during and after the financial crisis, as a case of study to show the effectiveness of our model