Three essays on the cutting edge of credit spread modeling

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(1)Universidad de Castilla-La Mancha. TESIS DOCTORAL. Three essays on the cutting edge of credit spread modeling. Autor: Alberto Fernández Muñoz de Morales. Director: Alfonso Novales Cinca. Facultad de Ciencias Jurı́dicas y Sociales Departamento de Análisis Económico y Finanzas. Toledo 2015.

(2) ”We choose to go to the moon in this decade and do the other things, not because they are easy, but because they are hard, because that goal will serve to organize and measure the best of our energies and skills, because that challenge is one that we are willing to accept, one we are unwilling to postpone, and one which we intend to win.” John F. Kennedy. September 12, 1962. I.

(3) Contents Acknowledgments. VI. Introducción. VII. Introduction. XI. 1 Credit spread modeling effects on counterparty risk valuation adjustments: a Spanish case study 1 1.1. 1.2. 1.3. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1. 1.1.1. Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1. 1.1.2. Literature on CVA . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3. 1.1.3. The Spanish case . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6. 1.1.4. This chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 8. Arbitrage-free valuation of counterparty risk . . . . . . . . . . . . . . . . . .. 8. 1.2.1. Symmetric CVA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. 1.2.2. Asymmetric CVA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. A dynamic model for default intensity and interest rates . . . . . . . . . . . 12 1.3.1. Interest rate model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. 1.3.2. Counterparty and Investor Credit Spread models . . . . . . . . . . . 13. 1.3.3. Spread correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14. 1.3.4. Default correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14. II.

(4) 1.3.5 1.4. 1.5. 1.6. Monte Carlo techniques . . . . . . . . . . . . . . . . . . . . . . . . . 15. Calibration of interest rates . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.4.1. Calibration procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 16. 1.4.2. Estimation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17. Calibration of default intensities . . . . . . . . . . . . . . . . . . . . . . . . 20 1.5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20. 1.5.2. Credit Default Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . 21. 1.5.3. Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22. 1.5.4. Departing from separability . . . . . . . . . . . . . . . . . . . . . . . 24. 1.5.5. Calibration procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 25. 1.5.6. Estimation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27. A Spanish case study: interest rate swap . . . . . . . . . . . . . . . . . . . . 37 1.6.1. The payoff. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37. 1.6.2. Main findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39. 1.7. Another case study: CDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39. 1.8. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41. 1.A Matching the market curve under G2++ . . . . . . . . . . . . . . . . . . . . 45 1.B Matching the market curve under CIR++ . . . . . . . . . . . . . . . . . . . 47 1.B.1 On the positivity of ψ . . . . . . . . . . . . . . . . . . . . . . . . . . 48 1.C Swaption prices under G2++ . . . . . . . . . . . . . . . . . . . . . . . . . . 49 1.D CDS Pricing under CIR++ stochastic intensity and G2++ interest rates . . 51 1.D.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 1.D.2 Gaussian mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 1.D.3 Computing expectations . . . . . . . . . . . . . . . . . . . . . . . . . 54 1.E Credit calibration results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2 Model risk in credit valuation adjustments: a Spanish case study. III. 62.

(5) 2.1. 2.2. 2.3. 2.4. 2.5. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.1.1. Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62. 2.1.2. This chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64. Default correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 2.2.1. Quantifying uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . 68. 2.2.2. Calibration results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70. Spread modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.3.1. An alternative model for credit spreads . . . . . . . . . . . . . . . . 71. 2.3.2. Calibrating the new model . . . . . . . . . . . . . . . . . . . . . . . 73. 2.3.3. Calibration results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78. A Spanish case study: interest rate swap . . . . . . . . . . . . . . . . . . . . 80 2.4.1. Default correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83. 2.4.2. Spread modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89. 2.A CDS Pricing under a fully Gaussian framework . . . . . . . . . . . . . . . . 92 2.A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 2.A.2 Computing expectations . . . . . . . . . . . . . . . . . . . . . . . . . 93 2.B Credit calibration results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 2.C Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100. 3 Credit spread modeling from a regulatory perspective. 103. 3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103. 3.2. Calculating VaR for CVA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105. 3.3. 3.2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105. 3.2.2. A parametric VaR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109. 3.2.3. Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Backtesting a risk-factor model under IMM . . . . . . . . . . . . . . . . . . 116. IV.

(6) 3.4. 3.3.1. A first approximation to backtesting . . . . . . . . . . . . . . . . . . 120. 3.3.2. Extending the backtesting method . . . . . . . . . . . . . . . . . . . 123. 3.3.3. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128. Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130. 3.A Monthly calibration results . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 3.B Extended Backtesting - Figures . . . . . . . . . . . . . . . . . . . . . . . . . 139 Conclusions. 142. Conclusiones. 145. Bibliography. 148. V.

(7) Acknowledgments The work shown below is the result of hours and hours of meditations, doubts, reviews and reflections. As in most PhD dissertations, the path was often steep, but, as in most PhD dissertations, I never walked alone. First and foremost, I would like to thank my supervisor Alfonso Novales Cinca. I knocked at his door with anything but a vague will of completing a PhD ”on CVA issues” and he welcomed me with enthusiasm. He has guided me during the whole process, helping me when needed. He has made possible this thesis and I want to thank him first for his support and encouragement. Also, my work fellows at BBVA have a lot to do with the completion of this venture. I am specially indebted to José Manuel López Pérez, who supported the project from the very beginning, accompanying me all the way; to Juan Antonio de Juan, who gave me useful suggestions; to Johan Gunnesson, a valuable person from whom I always learn something; and to Daniel Andrés and the rest of my team, who have increased my understanding of this sometimes rough field called Finance. And, of course, the support and breath that I always get from my parents, my sister and Sara, was essential in this project. Its completion is also due to them.. VI.

(8) Introducción La irrupción de los ajustes de valoración por crédito (CVA por sus siglas en inglés) en la contabilidad de derivados dio pie al inicio de una verdadera revolución en el marco valorativo de estos productos, un mundo cuyos fundamentales poco habı́an cambiado desde los trabajos de Black y Scholes (1973) y Merton (1973). La necesidad (o no) de incluir en el valor de un derivado conceptos tales como costes de financiación, de capital o de liquidez ha generado toda una familia de potenciales ajustes valorativos, los llamados XVAs, cuya misma existencia está generando multitud de opiniones a favor y en contra en cı́rculos académicos y empresariales. En este contexto, y quizá por ser los primeros, el CVA y su contraparte simétrica DVA (ajuste de valoración por débito) son los únicos ajustes ya contemplados en estándares de contabilidad internacional como IAS 39 o IFRS 13, subrayando este último la necesidad de maximizar el uso de información de mercado en el cálculo de estos ajustes. De este modo, se ha efectuado una transición de un mundo que consideraba los derivados como simples descuentos de flujos estipulados por contrato, totalmente independientes de las entidades participantes, a considerar estos productos como una terna constituida no sólo por los pagos en sı́, sino también por las dos contrapartes implicadas en la transacción. Si antes para una entidad financiera el valor contable de un swap de tipos de interés era independiente de la contraparte con quien lo cerrase, ahora dependerá también de la calidad crediticia de ésta. Sin embargo, si bien es cierto que existe una creciente unanimidad de cara a la necesidad de inclusión de estos ajustes por crédito, también se ha generado una amplia controversia respecto a los detalles. Cuestiones tales como la simetrı́a en los ajustes, las convenciones de cierre (los llamados risk-free o risky closeout) o fricciones entre un mundo regulatorio que excluye DVA y otro contable que sı́ lo incluye, han dado pie a todo tipo de artı́culos de opinión a favor y en contra de las distintas propuestas y aproximaciones. Aun abstrayéndonos de estos debates, los ajustes de crédito que ha de aplicar una entidad financiera afectan de manera transversal a toda su cartera de derivados, que tı́picamente constará de varios miles de contrapartidas con todo tipo de productos dependientes de diversos factores de riesgo como tipos de interés, equity, FX o calidades crediticias propias o de terceros. Consideraciones adicionales como acuerdos de neteo o efectos por correlación entre las probabilidades de quiebra y el valor del derivado en el. VII.

(9) momento de default (el llamado wrong-way risk) añaden un grado más de complejidad a la hora de estimar cuánto espera perder una entidad financiera por riesgo de contraparte en su operativa de derivados. Dicha dificultad lleva a menudo a la industria a realizar simplificaciones y supuestos que faciliten el cálculo de los ajustes, aun a costa de sacrificar precisión, y en ocasiones de un modo significativo. Los distintos comités reguladores, nacionales e internacionales, son conscientes de estas limitaciones y del daño potencial que puede causar la aplicación excesiva de simplificaciones o el uso de modelos mal entendidos, no sólo en el cálculo de los ajustes de crédito, sino en la valoración de productos derivados en general. Por ello, poco a poco han ido introduciendo en sus directrices la inclusión de cargas adicionales de capital que mitiguen las pérdidas potenciales que pudieran producirse por riesgo de modelo y similares, incorporando una nueva categorı́a de ajuste: los llamados AVA (Additional Valuation Adjustment). La adición de esta nueva carga de capital no es sino un ejemplo más de la creciente presión que los distintos organismos reguladores han venido ejerciendo sobre las entidades bancarias a raı́z de la crisis financiera global iniciada en 2007. A los conceptos de capital por riesgo de mercado, crédito y operacional, ya existentes bajo el marco regulatorio anterior (Basilea II), se han añadido otros tales como capital por oscilaciones en CVA, cargas por riesgo de liquidez o los ya mencionados AVAs que recogen conceptos tan diversos como riesgo de modelo o de inputs. Para aliviar la presión de tal número de nuevas cargas, las entidades financieras han de demostrar a sus respectivos organismos supervisores que son capaces de controlar y monitorizar los diferentes riesgos a que están expuestas, migrando ası́ a marcos regulatorios menos conservadores y más adaptados al perfil particular de riesgo que cada entidad bancaria esté asumiendo. Es en este contexto de transformación integral de la industria financiera en el que se encuadra la presente tesis doctoral, centrada en un tema transversal a los enunciados anteriormente: el modelado del valor de mercado de la calidad crediticia. Toda entidad financiera tiene multitud de contrapartes con las que cerrar acuerdos en su operativa diaria de derivados. Ya sean organismos públicos o privados, financieros o no financieros, de mayor o menor tamaño, la reciente normativa contable obliga a que los productos derivados cerrados con cualquier contrapartida incorporen el valor de mercado de la calidad crediticia de ésta a través del CVA. Por tanto, toda entidad bancaria ha de ser capaz de modelar la evolución de los spreads de crédito de, como mı́nimo, todas aquellas contrapartidas con las que cierre un derivado. De este modo, la incorporación de los ajustes CVA-DVA ha venido a poner en valor el correcto modelado de los spreads de crédito. Ahora bien, como se ha mencionado anteriormente, el gran número de contrapartidas y lo amplio de la casuı́stica de los diferentes derivados complican en gran medida el cálculo de los ajustes CVA-DVA, llevando a menudo a las entidades bancarias a realizar simplificaciones en el modelado de los spreads de crédito cuyos efectos son potencialmente relevantes, y como tales la reciente introducción de los AVAs obliga expresamente a la cuantificación, monitorización y reserva de capital por riesgo de modelo en el cálculo de VIII.

(10) CVA. En este contexto, el modelado de los spreads de crédito vuelve a jugar un papel protagonista. En relación al CVA, además de por riesgo de modelo en su cálculo, los organismos reguladores se mostraron especialmente preocupados por la variabilidad que este concepto podı́a llegar a incorporar a las cuentas de resultados de las entidades financieras. Es famosa la frase del Comité de Basilea refiriendo que durante la crisis financiera iniciada en 2007 dos tercios de las pérdidas atribuidas a riesgo de contraparte fueron debidas a pérdidas por CVA, y sólo un tercio por quiebras reales. Debido a esto, el nuevo marco regulador, Basilea III, incorporó cargas de capital por oscilaciones en CVA cuya métrica consiste en un VaR basado exclusivamente en movimientos extremos de los spreads de crédito. Ası́, todas aquellas entidades con un modelo de VaR paramétrico deberán contar con un modelo adecuado para describir la calidad crediticia de sus contrapartes. Por último, tradicionalmente las entidades financieras han contado con la calidad crediticia como un factor de riesgo más en su cartera de negociación. Bonos, Credit Default Swaps (CDS) y otros derivados de crédito dependen del comportamiento de la evolución de la calidad crediticia de diferentes referencias, generando exposiciones dependientes de esta componente que habrá que modelar de manera más o menos exacta. Una entidad que desee migrar a modelos regulatorios avanzados con el fin de optimizar el reparto de capital ha de ser capaz de demostrar que describe correctamente el comportamiento de sus factores de riesgo. La obligatoriedad de la inclusión de los ajustes CVA-DVA ha venido a reforzar la dependencia de la cartera de derivados de los spreads de crédito tanto propios como de las diferentes contrapartidas. Por lo tanto, el correcto modelado de la calidad crediticia de una referencia dada ha pasado a ser un hito de obligado cumplimiento para aquellas entidades que quieran migrar a modelo avanzado de exposiciones. En resumen, como se acaba de exponer, las entidades financieras están operando en un entorno altamente exigente, siendo continuamente presionadas por los reguladores, presionados a su vez por la opinión pública, demandando más cargas de capital con el objetivo de lograr una banca más segura y solvente. Sin embargo, al mismo tiempo, estas mismas entidades financieras deben competir entre sı́ en un entorno plagado de retos, donde únicamente aquéllas que mejor se acomoden a la nueva situación estarán en posición de continuar con su lı́nea de negocio. Si a todo esto sumamos la necesidad de adaptaciones a nuevos estándares de contabilidad, la situación se torna aún más complicada. Dada la relevancia de la banca para estimular e impulsar la economı́a, podrı́a afirmarse que una resolución satisfactoria de los complejos problemas a que estas entidades se enfrentan acabarı́a teniendo un impacto positivo en la economı́a. En esta tesis nos centraremos en uno de esos nuevos problemas que están afrontando las entidades financieras: el modelizado de los spreads de crédito. Abordaremos este tema desde varios puntos de vista, tratando los nuevos retos a que los bancos se están enfrentando, con la intención de aportar soluciones para una banca más sólida y a la vez competitiva. Podemos condensar nuestros principales objetivos en estos cinco:. IX.

(11) 1. Encontrar un modelo satisfactorio y sencillo para el comportamiento de los spreads de crédito, incluyendo una calibración a instrumentos de mercado en lı́nea con la reciente normativa contable. 2. Analizar el impacto de una modelización simplista de la calidad crediticia en el cálculo de ajustes CVA-DVA en derivados de tipos de interés y crédito. 3. Estudiar posibles fuentes de riesgo de modelo para los spreads de crédito y su impacto en el cálculo de ajustes CVA-DVA bajo la óptica de directivas regulatorias recientes. 4. Estudiar la capacidad del modelo de crédito propuesto para captar movimientos extremos de CVA-DVA (análisis VaR). 5. Comprobar la bondad del modelo a la hora de describir la distribución completa de la calidad crediticia de una referencia de cara a su aprobación por organismos reguladores para describir dicho factor de riesgo.. X.

(12) Introduction The emergence of Credit Valuation Adjustments (CVA) in derivatives accounting gave rise to a complete revolution in the valuation framework of these products, a world whose basic fundamentals had changed little since the seminal works of Black and Scholes (1973) and Merton (1973). The need (or not) to reflect in the value of a derivative concepts such as funding, capital or liquidity costs has generated a whole family of potential valuation adjustments, the so-called XVAs, whose very existence is generating plenty of opinions for and against in academic and business circles. In this context, and maybe because they arose first, CVA and its symmetrical counterpart DVA (Debit Valuation Adjustment) are the only adjustments to date entailed in international accounting standards such as IAS 39 or IFRS 13, with the latter one emphasizing the need to maximize the use of market information in the calculation of these adjustments. Thus, markets have witnessed a transition from a world that considered derivatives as a simple discounting of cash-flows specified by contract, with complete independence of the participating entities, to a new framework where these products are considered a trio made up not only from payoffs, but also from the two counterparties involved in the transaction. If traditionally a financial institution would price an interest rate swap regardless of the counterparty with which it closed the deal, now this value will also depend on the credit quality of such counterparty. However, although there is a growing consensus regarding the need for these credit adjustments, there also exists a widespread controversy with respect to the details. Issues such as symmetry assumptions, closeout conventions (the so-called risk-free vs risky closeout) or frictions between a regulatory world that excludes DVA and an accounting one that promotes it, have led to a wide array of proposals and approaches. Even if we set aside these debates, the kind of credit adjustments that any financial institution must apply has an effect across its whole portfolio of derivatives, which typically will involve several thousands of counterparties with all types of products depending on various risk factors such as interest rates, equity, FX or credit quality. Additional considerations such as netting agreements or correlation effects between the probability of default and the value of the derivative (so-called wrong-way risk) add another layer of complexity when estimating how much a bank is expecting to lose due to credit issues on its derivative business. This difficulty often encourages the industry to make simplistic. XI.

(13) assumptions in order to ease the burden of the calculation of adjustments, even at the cost of accuracy, and sometimes in a significant way. The various regulatory bodies, both national and international, are thoroughly aware of these limitations and the potential damage that excessive simplifications or the use of poorly understood models can cause, not only in the calculation of credit adjustments, but in the valuation of derivatives in general. Therefore, these committees have been gradually introducing in their guidelines additional capital charges to mitigate potential losses that may arise from model risk and the like, incorporating a new adjustment category: the AVAs (Additional Valuation Adjustments). The incorporation of this new capital charge is merely an example of the increasing pressure that the various regulatory agencies have put on the banks after the global financial crisis that started in 2007. On top of capital buffers for market, credit and operational risk, already existing under the previous regulatory framework (Basel II), new sources of risk have been considered such as variations in CVA, liquidity shortages or model and input risk, being these last two reflected in the aforementioned AVAs. In order to alleviate the pressure of so many new burdens, financial institutions must demonstrate to their local supervisors their ability to control and monitor the various risks which they are exposed to. By doing so, they will have earned the right to migrate to less conservative regulatory frameworks, more adapted to the particular risk profile that each bank is assuming. It is in this context of complete transformation of the financial industry that the present thesis arises, focusing on a topic that affects transversely those mentioned above: the modeling of the market value of credit quality. Let us see why. Every financial institution has many counterparties with which they arrange deals in their daily derivatives business. No matter if these are public or private, financial or not, larger or smaller: the recent financial accounting rules require derivatives closed with any kind of counterparty to incorporate the market value of the credit quality of them through CVA. Therefore, any bank must be able to model the evolution of credit spreads, at least of all those counterparties with which it has entered into derivative deals. Thus, the incorporation of CVA-DVA adjustments has added importance to the correct modeling of credit spreads. However, as has been mentioned above, the large number of counterparties and wide array of derivative specifications greatly complicate the calculation of credit and debit adjustments. This often leads banks to make simplifications when modeling credit spreads with potentially relevant effects. As such, the recently introduced AVAs expressly require quantification, monitoring and capital charges due to model risk when calculating CVA. In this context, modeling credit spreads plays again a major role. Regarding CVA, in addition to model risk, regulators were particularly concerned about the variability that this concept could introduce in the P&L reports of financial institutions. As the Basel Committee on Banking Supervision (BCBS) famously stated, during the financial crisis that began in 2007, roughly two-thirds of the losses attributed. XII.

(14) to counterparty risk were due to CVA losses and only one third to actual defaults. Because of this, in the new regulatory framework, Basel III, additional capital charges were introduced to account for large CVA oscillations based on a Value-at-Risk (VaR) calculation depending exclusively on extreme movements in credit spreads. Thus, any entity with a non-historically based VaR model would need to have an adequate way to accurately describe the creditworthiness of its counterparties. Finally, financial institutions have traditionally included credit quality as another risk factor in their trading books. Bonds, Credit Default Swaps (CDS) and other credit derivatives depend on the behavior of the evolution of the credit quality of different references, generating exposures dependent on this component that will need to be modeled more or less accurately. An entity willing to migrate to advanced regulatory models in order to optimize the allocation of capital must be able to show that it is able to correctly describe the behavior of their risk factors. Thus, the mandatory inclusion of CVA-DVA adjustments has reinforced the dependency of the derivatives portfolio of a financial institution on both the own credit spread and those of its counterparties. Therefore, the correct modeling of credit quality of a given reference has become an essential milestone for those entities intending to migrate to advanced model exposure method. To sum up, as we have just exposed, financial entities are encountering a highly demanding environment, being continuously pressed by regulators, pressed themselves by the public opinion, demanding more capital charges for the sake of safer banks. However, at the same time, these financial entities must compete with their peers in a challenging environment, where only those who better accommodate will be able to thrive. If on top of this we add adaptations to new accountancy standards, the situation gets even more complicated. Given the relevance of banks to stimulate and boost the economy, we may state that satisfactorily solving the complex problems that these entities face will end up having a positive impact in the economy. In this thesis we will focus on one of these new problems that financial entities are facing: the modeling of credit spreads. We will address this topic from several points of view, dealing with the new challenges than banks are encountering, with the intention of proposing solutions for the sake of a sound and competitive banking business. We can condense our main objectives in these five: 1. Find an appropriate and simple model for the behavior of credit spreads, including a calibration method to market instruments in line with recent accounting standards. 2. Analyze the impact of incorrect modeling of the credit quality adjustments in the calculation of CVA-DVA in plain interest rate and credit derivatives. 3. Study possible sources of model risk for credit spreads and their impact on the calculation of CVA-DVA from the perspective of recent regulatory directives. 4. Study the ability of the proposed model to capture extreme credit movements on CVA-DVA (VaR analysis). XIII.

(15) 5. Verify the accuracy of the model when describing the complete distribution of the credit quality of a reference regarding the approval by regulatory committees when describing this risk factor.. XIV.

(16) Chapter 1. Credit spread modeling effects on counterparty risk valuation adjustments: a Spanish case study 1.1 1.1.1. Introduction Context. The financial crisis that started in 2007 has caused a paradigm shift in the business of banking. Every stakeholder in the industry, from regulators to investment banks, from rating agencies to hedge funds, has been obliged to stop and reconsider the very basics of their daily tasks. In a similar fashion to the stock market crash of October 1987, when the volatility smile first appeared in equity option prices1 , this crisis is challenging traditional financial engineering in several ways. Typical non arbitrage relationships between spot and forward interest rates do not hold anymore due to the appearance of basis spreads among tenors ([86]). Interest rates are entering negative terrain for certain products ([106]). Paradoxically, at the same time traditional safe assets, such as OECD sovereign bonds, are becoming dubious if not dangerously risky ([53]). Another area in need of revision is the treatment of Counterparty Credit Risk (CCR) in market activities, that is, the risk that the counterparty defaults before the final settlement of a transaction’s cash flows. If the portfolio has a positive value for the bank at the time of default, an economic loss will occur. Notice that CCR has a bilateral nature in derivatives, since the market value of the portfolio can be positive or negative depending on time-varying market factors. In this setup, Credit Valuation Adjustment (CVA) is the 1. See, for example, [69]. 1.

(17) difference between the default-free portfolio value and the true portfolio value that takes into account the counterparty’s default. In short, CVA is the market value of CCR. The review on credit risk in derivatives has been threefold. Firstly, from an accountancy point of view, although credit issues had been previously addressed under International Accounting Standards (IAS) 39, their importance was further stressed in January 2013, when the International Financial Reporting Standard (IFRS) 13 ”Fair Value Measurement” entered into force. Largely based on usual practices applied in the United States, this accounting standard intends to harmonize the definition of fair value, which is characterized as an exit price, that is, the one that would be received or paid in an orderly transaction between market participants. In this context, CCR plays a major role in the computation of fair value. Secondly, from a regulatory perspective, the Basel Committee on Banking Supervision recognized in 2009 that capital for CCR had proved to be inadequate. The then ongoing regulatory framework, Basel II, addressed CCR as a default and credit migration risk, not fully accounting for market value losses short of default. However, as the Basel Committee pointed in [7], ”roughly two-thirds of CCR losses were due to CVA losses and only about one-third were due to actual defaults”. The identification of this and other related shortcomings led to a comprehensive reform on the calculation of capital for CCR which is being implemented by banks. The third aspect related to CCR which is being currently addressed has to do with pricing financial products. Counterparty risk has been gradually incorporated in valuation procedures2 , altering the price to be charged for specific instruments in order to account for the default risk of the counterparty. As we will see, this change in price, CVA, can be seen as an option on the residual value of the portfolio, with a random maturity given by the default time of the counterparty. Furthermore, if the investor wants to account for the possibility of him defaulting, a second change on price should be added, named Debit Valuation Adjustment (DVA). Both changes in price generate a source of risk that needs to be taken into account. The ubiquity of these concepts may lead to different definitions of CVA: an accounting one for books and records, a front office CVA for pricing new deals and a regulatory CVA for defining capital requirements. An accountant will comfortably accept the presence of DVA in defining fair value as the natural counterpart of CVA. A trader, though, will complain against considering an adjustment that takes into account the possibility of him defaulting, becoming virtually impossible to hedge (who would buy from the insured insurance insuring the insured?) The recent survey [46] about current market practices on CCR showed a rapid evolution along the past two to three years motivated by changes in regulatory and accounting guidelines. Banks are focusing on building models for advanced capital treatments, including collateral optimization and funding. Effort is also being placed on quantifying Wrong 2. In the survey [51], carried out among 19 sophisticated financial entities in 2012, all of them reported applying a credit valuation adjustment on derivative assets.. 2.

(18) Way Risk (WWR), that is, the risk that the exposure to our counterparty gets higher precisely when its credit quality worsens, that can be captured by jointly simulating credit spreads and underlying risk factors. The same survey unveiled a clear divergence in approaches and methodologies across the market. While the use of risk-neutral default probabilities via credit spreads is becoming a standard practice in the quantification of CVA3 , DVA considerations and the extent to which it should be used to reduce CVA charges are also sources of variation. Further ambiguities related to possible funding adjustments, outside the Basel III mandate but a subject of increased focus, add to the confusion, ensuring that CCR will remain a hot topic for a long time.. 1.1.2. Literature on CVA. Literature on credit valuation adjustments is extensive and relatively recent. Although some references can be traced back to the 1990s (see, for example, [48, 74, 75, 76, 102]), the review on counterparty risk triggered by the 2008 credit events has contributed to generate a large amount of works on this topic. A good (and entertaining) survey can be found in [17]. For deeper insight, one can follow the textbooks [65, 77, 35]. Let us now give an overview of the main references regarding counterparty valuation adjustments that populate the literature. We will briefly mention the main topics that surround this core area of research, jointly with some broad definitions and the most important works that addressed them. In the rest of the chapter we will revisit these topics several times, providing accurate definitions and sometimes referring to the same works that will be mentioned now. However, due to the wide variety of CVA-related terms, the interested reader might appreciate the concentration of references and concepts in a single section. As the pricing of derivatives became more and more complex at the beginning of the 2000s, theoreticians and practitioners came to realize that they were missing a fundamental piece when providing the expected value of a financial instrument. If in a given transaction the counterparty of a bank defaulted and the present value of the portfolio at default was positive for the bank, it would only be able to get a recovery fraction of the payoff. However, if such present value was negative, the bank would have to pay in full to the liquidators of the failed counterparty. Such asymmetry needed to be taken into account when pricing a derivative and it translated into a Credit Valuation Adjustment term (CVA) to be subtracted from the net present value of the equivalent derivative with no counterparty risk. First details and a discussion on this concept were provided in [24]. Following this initial perception about the need for CVA, a lot of questions began to be posed. This new term relied on default probabilities and recovery fractions that were not easy to estimate for most counterparties, triggering the appearance of proposals like 3. As we shall see below, at the end of the day, CVA is the price of a contingent option.. 3.

(19) [43] or [52] to guess the par Credit Default Swap (CDS) spread of a company when such an instrument is illiquid or does not exist. Correlations also presented problems: if upon default of the counterparty the value of the portfolio was highest for the bank, the CVA term would get significantly larger. The idea that the credit quality of my counterparty and the value of the portfolio were correlated in the worst possible way for me was named Wrong Way Risk (WWR), and generated a whole branch of literature on its own. For example, [24, 36, 31, 32] studied WWR for equity products, [29, 30] for interest rates, [19] for commodities and [21] for credit. The extent of the application of a CVA term ended up generating further complications. If two counterparties were to agree on a price, and each of them was subtracting the CVA of the other from the valuation of the equivalent default-free portfolio, no deal would be closed unless one of them could be considered unanimously as default-free. This led to the introduction of a second adjustment, the Debit Valuation Adjustment (DVA) that came to refine the valuation of a derivative where both counterparties could default, taking into account the default probabilities of each of the participants in the transaction. However, the introduction of this adjustment did not come without controversy. From a bank’s point of view, its DVA will get larger when its credit quality worsens, increasing the value of its portfolio since the chances that it will not fully pay the due amounts are greater. This gave rise to paradoxical situations when big financial institutions began reporting profits due to DVA in highly stressed environments, like the famous 2.5$ billion profit by Citigroup due to the widening of its credit spreads (see [44]). A large amount of literature has been generated around the concept of DVA and how it should be calculated. DVA was already addressed in [48] for swaps, as well as in [13], but it was more carefully defined in [20], where bilateral risk was introduced in general and applied to CDS. Moreover, [38, 34] and [37] examine other features of bilateral risk, including issues such as collateralization, margining and extreme contagion, while [67] studies the connection between DVA and the funding capabilities of the bank. A basic introduction on bilateral CVA can be found in [64]. Considering the possibility of the own default in the valuation of a derivative can end up affecting the definition of CVA. While this term used to reflect an economic loss that a bank could experience in case the counterparty defaulted, now it should be redefined as the loss that could be experienced in case that default arrived before that of the bank. Some references, like [93], ignore this ”first-to-default” nature of bilateral CVA-DVA adjustments, and compute each term as if the other party was default-free, what has been labeled as asymmetric bilateral CVA (as opposed to the symmetric approach). By doing so, scenarios where both counterparties default are clearly being double counted. The error in neglecting this first-to-default risk can be sizable, as shown in [33], but still some references, like [66], justify this approach based on closeout conventions, that is, on the amount that is recovered upon default that, according to the International Swaps and Derivatives Association (ISDA) 2009 protocol ”may take into account the creditworthiness of the Determining Party”. Therefore, when an institution negotiates with the liquidators of the defaulted company how much should be settled for a given derivative, it might consider its own DVA in the calculation of such amount. This is called the ”risky closeout” 4.

(20) paradigm, opposed to the ”risk-free closeout” one, where the value of the transactions to be settled upon default is based on risk-free valuation. According to [66], the asymmetric risk-free closeout CVA is closer to the symmetric risky closeout version than the symmetric risk-free closeout one, thus justifying the asymmetric approach. The debate on risk-free vs risky closeout has generated another branch of literature, with references like [26, 27, 28, 66] studying the implications of one or both of these approaches. And, of course, underlying all these topics we have models. CVA-DVA depend on the default probabilities of both participants in the transaction. Questions such as WWR are heavily dependent on the distributions of such default probabilities and their relationship with the corresponding portfolio. The first-to-default nature of bilateral adjustments rely on the relationship of default distributions between themselves. And not only we need to have the right model, but also be able to satisfactorily calibrate it to market data. In this thesis we will cover both topics looking for an adequate response to both questions. One of the main contributors to the literature on CVA has been Prof. Damiano Brigo, who has written several high impact papers on the topic since the early 2000s. Not only has he addressed the theoretical aspects of CVA-DVA (in fact, many of the references outlined above are authored or co-authored by him), but also has investigated the modeling of credit spreads and its impact on credit valuation adjustments. A typical ”modeling” work by Brigo is configured as follows: 1. Enunciation of a model-independent bilateral counterparty risk valuation formula based on expectations and default indicator functions. 2. Focus on a particular type of product. At this stage, a specific model is needed, for example, a Gaussian two factor (G2++) model for interest rates and Cox-IngersollRoss (CIR) without jumps for credit. 3. Numerical analysis, either computing sensitivities from a range of values for specific parameters or calibrating the model to real data. The complexity of the first step has been gradually increased when valuing financial products. Furthermore, there exists a growing trend in the banking industry on modeling collateral treatment for the sake of capital optimization rather than relying on simplifications . In this context, recent works, like [38] or [34], generalize the framework for arbitrage-free valuation of bilateral counterparty risk to the case where collateral is included, with possible re-hypotecation. Another source of variation has to do with adding jumps when modeling credit. Brigo himself has written some papers applying SSRJD (Shifted Square Root Jump Diffusion) for default intensities, like [23]. In [80], CVA is computed for credit default swaps including jumps. However, the complexity of the numerical methods required to successfully manage this type of models has prevented the literature to tackle this topic in depth.. 5.

(21) Still, there is a fundamental part of credit spread modeling that has not been satisfactorily solved by the literature. When trying to calibrate the dynamics of credit spreads, authors often complain about the scarcity of data. Quotes for CDS options, especially single name ones, are considered illiquid and not credible ([35]). Therefore, credit spread dynamics are often described using ”reasonable” distributions according to the authors (as in [29]), or base their calibration on numeric restrictions that have nothing to do with the true behavior of the reference which is being modeled, as in [21]. In this sense, this thesis contributes to the existing literature by providing a calibration method that allows practitioners to satisfactorily calibrate the dynamics of credit spreads to market data, thus being able to accurately estimate counterparty valuation adjustments.. 1.1.3. The Spanish case. This thesis is focused on the Spanish financial sector for two reasons. First, few studies, if any, have modeled the dynamics of credit spreads in this market, analyzing those of American ([81]), British ([20]) or Italian ([16]) companies instead. However, the Spanish financial sector has behaved in a rather unorthodox manner when compared to other European counterparts. As we shall see below, at the beginning of the crisis, Spanish banks were regarded as some of the most solid entities on the continent, having escaped from the subprime mortgage meltdown on the other side of the Atlantic. However, as time went by, the situation reversed. International financial markets calmed down and the focus was put on the depth of the Spanish recession and its effects on Spain-based financial entities. This quick twist, with two radically opposite views following each other in less than four years, gives us another reason for studying the Spanish case. The Spanish financial sector is highly concentrated. As pointed in [57], at the end of 2008 there were 362 credit institutions operating in Spain, with 159 banks that represented 53.53 percent of total assets and 46 savings and loans (cajas), which accumulated an additional 38.40 percent. However, among the banks, Banco Santander controlled assets of over $1.4 trillion and BBVA of around $0.75 trillion. In comparison, the then third largest bank, Banco Popular, had assets of only around $150 billion. Traditionally, Spanish banks and cajas have held long-standing relations with industry, both in terms of controlling equity positions in companies and through large credits. For individuals, Spanish banks offer their clients a wide variety of products including deposits, mortgages, credit cards or pension funds. Additionally, although there has been some timid investment banking, the local supervisory authority, the Banco de España, has prevented Spanish banks from entering into complicated structured investment vehicles. Securitization, despite increasing, involved instruments much less complicated than those in the U.S., making banks keep most of the credit risk on their own balance sheets. The apparent universal character of Spanish banks masked the excessive concentration of their lending to the real estate sector. The economic recession that hit Spain at the end. 6.

(22) of 2008, when the National Statistics Institute first published negative figures of GDP4 , revealed the ongoing collapse of a real estate bubble and the subsequent meltdown of the Spanish economy. Local unemployment rates doubling the EU average and concerns about the possibility of a financial bailout, that finally took place in June 2012, made local Treasury yields skyrocket. Despite their international character and diversification, both Santander and BBVA were not immune to the situation in their country of origin. Figure 1.1 displays the evolution of the 5 year Credit Default Swap (CDS) spreads for some of the biggest Euro area banks between 2008 and 2012. Starting at comparable levels, as time went by only Italian banks, Unicredito and Intesa San Paolo, exhibited spreads in line with Santander and BBVA, while CDS from German and French banks were perceived by the market as much less risky.. Figure 1.1: Evolution of CDS5y for main Euro area banks (2008-2012). Although all names start at comparable levels in 2008, at the end of the sample Spanish (BBVA and Santander) and Italian banks (Unicredito and Intesa San Paolo) exhibit rather high credit spreads, with German (Deutsche Bank and Commerzbank) and French (Société Générale and BNP) entities being perceived as much less risky.. This asymmetry between peripheral and core European countries will influence the pricing of financial products if counterparty risk valuation adjustments are taken into account. As an example, imagine a firm engaged in a loan linked to a floating reference who is willing to reduce interest rate risk by entering a payer swap (paying fixed and 4. See press release [71].. 7.

(23) receiving floating). Under previous accountancy rules, in this swap, a AAA-rated firm would pay the same fixed amount as another on the verge of defaulting (the par swap rate). If CVA is added, the bad firm will be charged with a prohibitively high spread over the swap rate compared with the good one. Conversely, if DVA is taken into account, the firm will be tempted to close the deal with a troubled bank, since the spread it will charge to her will be lower. Further paradoxical effects can appear if correlations are taken into account.. 1.1.4. This chapter. Our purpose here will be analyzing the effects of the financial crisis in counterparty risk valuation adjustments. Following the arbitrage-free valuation framework presented in [37], we will consider a model with stochastic Gaussian interest rates and CIR++ default intensities. Departing from previous literature, we will be able to calibrate the dynamic behavior of default intensities profiting from Gaussian mapping techniques presented in [18], and reproduce the historically observed instantaneous covariances of CDS prices. Such new calibration method, relatively simple and easy to implement, will allow us to address one of the main problems that financial entities are currently facing: the accurate pricing of CVA-DVA or, in other words, the accurate pricing of market derivatives. We will calculate adjustments involving the two major Spanish banks, BBVA and Santander, and a generic European counterpart. We will repeat the analysis before and along the Spanish recession in both interest rate and credit derivatives. We shall allow for credit spread volatility, correlation between the default times of the investor and counterparty, and for correlation of each with interest rates, and will investigate the effects of incorporating counterparty risk valuation adjustments in pricing. The chapter is structured as follows: Section 1.2 summarizes the counterparty risk valuation framework from [37], establishing the appropriate notation. Section 1.3 describes the reduced form model setup of the chapter with stochastic interest rates and intensities plus a copula on the exponential triggers. Section 1.4 presents the calibration procedure for interest rates along with estimation results. Section 1.5 deals with the calibration of default intensities, presenting the market of Credit Default Swaps and the application of the Gaussian mapping technique in our modeling framework to generate a closed-form expression for the price of a CDS, showing estimation results. Section 1.6 presents counterparty credit risk valuation adjustments in several scenarios for an interest rate swap and Section 1.7 does the same for a Credit Default Swap. Finally, Section 1.8 concludes.. 1.2. Arbitrage-free valuation of counterparty risk. As reported in the aforementioned surveys [46] and [51], there exists no consensus in the banking industry about how to calculate counterparty risk credit valuation adjustments. For a long time, the debate focused on whether an entity should account for the possi8.

(24) bility of its own default, including DVA in the valuation (symmetric CVA), or whether it should consider itself default-free (asymmetric CVA). Basel II defined counterparty credit risk as the one arising from the possibility that the counterparty to a transaction could default before the final settlement of the transaction’s cash flows. No explicit mention about considering one’s own default was done. Nevertheless, the new accountancy rules developed under the International Financial Reporting Standard (IFRS) 13, that entered into force in January 2013, explicitly mention the need for including non-performance risk (defined as the risk that the entity will not fulfill an obligation) in the fair value of a liability, thereby including an entity’s own credit risk in the valuation. As a consequence, financial entities have increasingly abandoned asymmetric CVA, including some form of DVA in the quantification of counterparty risk. However, there is still a source of divergence around bilateral counterparty risk. CVA reflects the economic loss to the investor in case the counterparty defaults. Conversely, DVA does the same for the counterparty in case the investor cannot fulfill his obligations. Once we are admitting that both participants in the transaction can default, we need to reconsider the definitions outlined above. CVA will reflect the economic loss to the investor in case the counterparty defaults before the investor. The same applies to DVA, which will become the loss to the counterparty in case the investor defaults before the counterparty. This approach is called symmetric CVA. The alternative asymmetric framework calculates CVA (resp., DVA) as if the investor (resp., counterparty) were risk-free. By doing so, we are clearly missing the first-to-default nature of CCR adjustments, double counting scenarios where both counterparties default. The error in neglecting this first-to-default risk can be sizable (see [33]) but still, the recent survey [51] showed that asymmetric CVA is calculated by part of the financial industry, often arguing that, since knowing default correlation between entities can be difficult, such first-to-default feature can be ignored. Therefore, we will explore valuation adjustments in both approaches, symmetric and asymmetric, defining a general arbitrage-free valuation framework for both approximations. Next, we will compare results obtained under both frameworks. There exists a third matter of discussion when computing these adjustments. Under the International Swaps and Derivatives Association (ISDA) 2009 protocol, in the event of default, the closeout amount ”may take into account the creditworthiness of the Determining Party”, suggesting that an institution may consider their own DVA in determining the amount to be settled. This is called the ”risky closeout” paradigm, as opposed to the ”risk-free closeout” one, where the value of the transactions to be settled in the event of default is based on risk-free valuation. For simplicity, we shall follow the latter paradigm. Both approaches have their shortcomings, outlined in, for example, [66]5 . Currently, there is a hot debate around this issue which is beyond the scope of this thesis. 5. Also in [66], CCR adjustments were calculated for simple payoffs under both risk-free and risky closeout conventions. In such situations, the asymmetric risk-free closeout CVA appeared to be closer to the symmetric risky closeout version than the symmetric risk-free closeout one. This has been cited as an indirect justification for using the asymmetric CVA approach.. 9.

(25) Following the notation of [37], we will refer to the two names involved in the transaction and subject to default risk as investor, named I, and counterparty, named C. Valuation will be seen from the point of view of the investor I, so that cash flows received by I will be positive whereas cash flows paid by I (and received by C) will be negative. We denote by τI and τC respectively the default times of the investor and counterparty. We place ourselves in a probability space (Ω; G; Gt ; Q). The filtration Gt models the flow of information of the whole market, including credit, and Q is the risk neutral measure. This space is endowed also with a right-continuous and complete sub-filtration Ft representing all the observable market quantities but the default events.. 1.2.1. Symmetric CVA. As in [37], let us call T the final maturity of the payoff which we need to evaluate and let us define the stopping time τ = min{τI , τC } If τ > T , there is neither a default of the investor, nor of his counterparty during the life of the contract and they both fulfill the agreements of the contract. On the contrary, if τ ≤ T then either the investor or his counterparty (or both) default. At τ , the Net Present Value (NPV) of the residual payoff until maturity is computed. We can distinguish two cases: • τ = τC : If the NPV is negative for the investor, it is completely paid by the investor itself. If the NPV is positive for the investor, only a recovery fraction RecC of the NPV is exchanged. • τ = τI : If the NPV is positive for the defaulted investor, it is completely received by the defaulted investor itself. If the NPV is negative for the defaulted investor, only a recovery fraction RecI of the NPV is exchanged. We can define the following (mutually exclusive and exhaustive) events ordering the default times:. A = {τI ≤ τC ≤ T }. E = {T ≤ τI ≤ τC }. B = {τI ≤ T ≤ τC }. F = {T ≤ τC ≤ τI }. C = {τC ≤ τI ≤ T } D = {τC ≤ T ≤ τI }. Notice that A to D are the default events, while E and F are the non-default ones. 10.

(26) Let us call ΠD (t; T ) the discounted payoff of a generic defaultable claim at t and Π(t; T ) the discounted payoff for an equivalent claim with a default-free counterparty. As stated in [37] (and proven in [20]), we then have that at valuation time t, and conditional on the event {τ > t}, the price of the payoff under bilateral counterparty risk is: Et [ΠD (t, T )] =. Et [Π(t, T )]+ Et [LgdI 1{A∪B} D(t, τI )(N P V (τI ))− ]− Et [LgdC 1{C∪D} D(t, τC )(N P V (τC ))+ ]. (1.1). where E is the risk-neutral expectation, D(t, T ) is the stochastic discount factor at time t for maturity T , Reci is the recovery fraction with i ∈ {I, C}, and Lgdi = 1 − Reci is the loss given default. Moreover, 1{A∪B} (respectively, 1{C∪D} ) is a default indicator process signaling when the investor (resp., the counterparty) has defaulted first. Finally, we have used the notation X + to refer to max{X, 0} (the positive part of X), and X − to refer to max{−X, 0} (the absolute value of the negative part of X). Equation 1.1, which may seem convoluted at first sight, expresses a rather simple and intuitive result: any derivative can be decomposed as the sum of its corresponding default-free equivalent, plus the value of what we would not pay to our counterparty in case we defaulted first (events A and B) and then the derivative had a negative value for us, minus what we would lose in case our counterparty defaulted first (events C and D) and then the derivative had a positive value for us. The second term and the third term are called respectively Debit Valuation Adjustment (DVA) and Credit Valuation Adjustment (CVA). There is an interesting interpretation of these adjustments in terms of option prices. For example, if we observe CVA, we notice that it can be regarded as a call option of strike zero on the remaining NPV of the derivative with random maturity (the defaulting time of the counterparty whenever it takes place before the investor’s). Similar arguments allow us to consider DVA as a put option. Under this point of view, the value of a defaultable claim can be regarded as the value of the corresponding default-free claim plus a long position in a put option plus a short position in a call option.. 1.2.2. Asymmetric CVA. In the asymmetric approach, each participant in the transaction considers itself defaultfree. Using the same notation as above, the adjustment calculated by the investor would be: Et [LgdC 1{τC <T } D(t, τC )(N P V (τC ))+ ] while the one calculated by the counterparty would be: Et [LgdI 1{τI <T } D(t, τI )(N P V (τI ))− ] One drawback of this approximation is that one adjustment is not the opposite of the other as in the symmetric case. Therefore, the two parties would not agree on the value of 11.

(27) the counterparty risk adjustment to be added to the default free price unless one of them was indisputably default-free.. 1.3. A dynamic model for default intensity and interest rates. To accurately price CVA and DVA we should consider a model with stochastic default intensities and interest rates. As exposed in [100], there are basically two types of tractable approaches when trying to model credit and interest rates simultaneously: 1. The Gaussian setup. This framework suffers from the possibility of reaching negative credit spreads and interest rates with positive probability, but a high degree of analytical tractability is retained. 2. The Cox-Ingersoll-Ross (CIR) setup. This approach gives us the required properties of non-negativity, but it loses some analytical tractability. Non-negative intensities are even more desirable than non-negative interest rates, since intensities must remain positive in order to prevent the occurrence of negative default probabilities, which do not make sense. Besides, credit traditionally shows higher levels of volatility, so a model that allows negative figures will reach more negative values when used for credit than for interest rates. Therefore, [37] goes for the hybrid approach of Gaussian setup for interest rates and CIR for intensities, that we will also follow here.. 1.3.1. Interest rate model. For interest rates, we will assume that the dynamics of the instantaneous short-rate process under the risk-neutral measure Q will be given by a G2++: rt = xt + zt + ϕ(t, α). (1.2). where α is a set of parameters and the processes x and z are Ft -adapted and satisfy dxt = −axt dt + σdW1t , x0 = 0 (1.3) dzt = −bzt dt + ηdW2t , z0 = 0 where (W1t , W2t ) is a two-dimensional Brownian motion with instantaneous correlation ρ12 , being −1 ≤ ρ12 ≤ 1, and a, b, σ and η are positive constants. These are the parameters defining α = [a, b, σ, η, ρ12 ]. The function ϕ(t, α) is set to match the initial zero coupon curve observed in the market (for insight on the appropriate choice of ϕ(t, α), see Appendix 1.A).. 12.

(28) 1.3.2. Counterparty and Investor Credit Spread models. We will deal with the credit quality of the two counterparties involved in the transaction. Thus, we will care about the likelihood of their default or, in other words, their default probability. In our modeling framework we will focus on the instantaneous default probability, that is, the default intensity, which can be defined as Q{τ ∈ [t, t + dt)|τ ≥ t} = λt dt, where τ is the defaulting time and Q is the risk-neutral probability. As stated in, for example, [25], this equation means that the probability that a company defaults in (arbitrarily small) dt years given that it has not defaulted so far is λt dt. In order to model default intensities we will set λit = yti + ψ i (t, β i ),. i ∈ {I, C}. (1.4). y is assumed to be a Cox-Ingersoll-Ross process under the risk-neutral measure: p i , (1.5) dyti = κi (µi − yti )dt + ν i yti dW3t i ∈ {I, C} where the parameter vector is β i = (κi , µi , ν i , y0i ) and each parameter is a positive constant. As usual, W3i is a standard Brownian motion process under the risk-neutral measure. The function ψ is a deterministic function and is set to match the initial survival probabilities observed in the market (for insight on the appropriate choice of ψ(t, β), see Appendix 1.B). In a CIR process, yt will always remain positive as long as 2κµ > ν 2 . Therefore, if we also impose that ψ(t, β) > 0, then we will ensure that the stochastic intensity λt will always be positive. Now, the parametric restrictions needed to make ψ(t, β) positive greatly limit the space of possible configurations of parameters, reducing the chances of attaining high levels of volatility. Moreover, if we intend to increase ν, due to the first restriction, we will need to either increase κ, which reduces the volatility of the overall process yt , or increase µ. But then µ might interfere with the positivity requirement of ψ(t, β). Thus, these restrictions limit the capabilities of the CIR++. As an example, numerical experiments carried out in [22] found that, if we stick to positive values of default intensities, this model can hardly reproduce levels of implied volatility for CDS options exceeding an order of magnitude of 30%. However, credit spreads have traditionally been regarded as highly volatile. For instance, [15] provides a few quotes of implied volatility levels in CDS options above 50%. Some authors ([35]) have suggested incorporating a jump component to the CIR++, obtaining the Jump-Diffusion CIR Model (JCIR). However, the calibration process gets more complicated when including this additional feature and, moreover, the JCIR++ version of the model, which also includes a deterministic function to match the default probabilities observed in the market, also presents difficulties when imposing positiveness 13.

(29) to this function so, at the end, adding jumps does not seem to be an appropriate solution. Therefore, some proposals ([25]) argue that the conditions making ψ(t, β) positive are excessively restrictive, and that one could relax this assumptions, allowing it to be negative, but not ”too much”, still obtaining reasonable results. As will be shown later, we will proceed following this last idea.. 1.3.3. Spread correlation. Short interest rate factors x and z are correlated with the intensity process y through their driving Brownian motions: dWj dW3i = ρji dt, j ∈ {1, 2}, i ∈ {I, C}. In order to reduce the number of free parameters, we will proceed as in [37], and consider that i ∈ {I, C}. ρ1i = ρ2i ,. Further, we also allow for correlation between default intensities of the investor and the counterparty: dW3I dW3C = ρIC dt. 1.3.4. Default correlation. There is another correlation coefficient that plays a leading role in symmetric counterparty valuation adjustments: default time correlation. Such correlation is independent to the ones that were previously shown: one can have independence on default probabilities but highly correlated defaulting times, which means that whenever one counterparty defaults, the other will follow with high probability. Let us see how we can introduce default correlation in our modeling framework. First, we define cumulated intensity as: Z t Λ(t) = λs ds 0. such that Q{τ ≥ t} = exp{−Λ(t)}. We are in a Cox process setting, which is a generalization of a Poisson process with stochastic time-dependent intensity. Thus, as recalled in [25], the first jump time of the process (the defaulting time τ ), transformed through its 14.

(30) cumulated intensity, is an standard (unit-mean) exponential random variable independent of Ft : Λ(τ ) = ξ,. with ξ a standard exponential independent of Ft . In our particular case, we have two defaulting times, one for the investor and another for the counterparty: τi = Λ−1 i (ξi ), i ∈ {I, C} with ξI and ξC standard (unit-mean) exponential random variables. We intend to connect these two random variables. To do so, we will assume that their cumulative distribution functions, given by the corresponding uniforms Ui = 1 − exp(−ξi ), i ∈ {I, C}, will be correlated through a copula function. Thus, Q{UI < uI , UC < uC } = C(uI , uC ) We choose copula C to be Gaussian with correlation parameter ρCop . Notice that this is a default correlation, connecting default times even if spreads were independent. As pointed in [37], where a Gaussian copula is used too, in general high default correlation creates more dependence between default times than a high correlation in spreads.. 1.3.5. Monte Carlo techniques. Payoffs will be valued using Monte Carlo simulation. The transition density for the G2++ model is known in closed form. As shown in, for example, [25], let us consider the stochastic process dxt = −kxt dt + ζdWt , x0 = 0 Then, for t ≥ s, xt is normally distributed with mean xs exp{−k(t − s)} and variance ζ2 2k [1 − exp{−2k(t − s)}]. Regarding default intensities, we will use the Euler Implicit positivity-preserving scheme presented in [18]. If we consider the CIR process: √ dyt = κ(µ − yt )dt + ν yt dWt , then, for t ≥ s, we set: √ yt =. r h i ν(Wt − Ws ) + ν 2 (Wt − Ws )2 + 4 ys + (κµ − 0.5ν 2 )(t − s) [1 + κ(t − s)] 2[1 + κ(t − s)]. 15.

(31) 1.4 1.4.1. Calibration of interest rates Calibration procedure. The parameters of the interest rate model under the risk-neutral measure can be calibrated to the surface of at-the-money (ATM) swaption volatilities. A swaption is an option on an interest rate swap (IRS). There are basically two types of swaptions: payer and receiver. A European payer swaption gives the right (but not the obligation) to enter a payer IRS (paying fixed, receiving floating) of a given length (tenor) at a given fixed rate (strike) and at a given future time (maturity). Conversely, a European receiver swaption gives the right to enter a receiver IRS (receiving fixed, paying floating). Consider a swaption with strike SK , maturity T = t0 and swap payment times T = {t1 , ..., tn }, t1 > T . It is a common practice to value swaptions with a Black-like formula. In this setup, the price of a swaption is6 : ES Black (0, T, T , SK , ω; σ) = ω. n X. h i τi P (0, Ti ) S(0)Φ(ωd1 ) − SK Φ(ωd2 ). i=1. where ω = 1 (ω = −1) for a payer (receiver) swaption, S(0) is the forward swap rate, P (0, T ) is the discount factor between 0 and T , τi the year fraction from ti−1 to ti , Φ is the standard normal cdf, and d1 =. ln(S(0)/SK )+σ 2 T /2 √ ; σ T. √ d2 = d1 − σ T. Swaption prices are typically displayed in a matrix, where each row is indexed by the swaption maturity Tα , whereas each column is indexed in terms of the underlying swap length, Tβ − Tα . The x×y-swaption is then the swaption in the table whose maturity is x years and whose underlying swap is y years long. Thus a 2×10 swaption is a swaption maturing in two years (Tα = 2) and giving then the right to enter a ten-year swap (Tβ = 12). It is a common practice to quote Black-volatilities instead of prices, as in Table 1.1. The expression for European swaption prices under G2++ is calculated in [25] and we show it in Appendix 1.C. Using this formula, at-the-money (ATM) swaption prices can be fitted to the market data surface by minimizing squared errors, that is, we solve. α b = argmin α. 6. X. G2++ Black M kt 2 [ESAT ) − ESAT M (0, Ti , Tj ; σ M (0, Ti , Tj ; α)]. i,j. See, for example, [69].. 16.

(32) 1y 2y 3y 4y 5y 7y 10y. 1y 48.7 55.8 48.7 41.2 36.5 31.9 26.1. 2y 50.0 48.8 43.0 37.8 35.0 30.9 26.0. 3y 48.0 44.5 39.9 36.4 33.7 30.1 26.3. 4y 46.6 42.7 38.8 35.7 33.2 29.7 26.6. 5y 45.4 41.8 38.6 35.3 32.9 29.4 27.2. 6y 43.7 40.4 37.4 34.5 32.3 29.3 27.9. 7y 42.7 39.3 36.5 33.8 31.9 29.4 28.8. 8y 42.0 38.5 35.7 33.3 31.7 29.7 29.6. 9y 41.6 38.0 35.1 33.0 31.7 30.1 30.4. 10y 40.9 37.9 35.1 33.0 31.8 30.7 31.1. Table 1.1: ATM swaption volatilities. 28-Jun-2012. Rows indicate maturities and columns, tenors. Thus, element 2y×10y (37.9) indicates the Black volatility of a swaption maturing in two years and giving then the right to enter a ten-year swap. Notice that we match swaption prices obtained from market quoted volatilities. Once we have obtained the parameters of α = [a, b, σ, η, ρ12 ], we can use ϕ(t, α) to automatically fit the initial zero coupon curve:. ϕ(t, α) = f (0, t) +. σ2 η2 B(a, 0, t)2 + B(b, 0, t)2 + ρ12 σηB(a, 0, t)B(b, 0, t) 2 2. −s(T −t). where B(s, t, T ) = 1−e s and f (0, t) is the instantaneous forward rate at time 0 for a maturity t implied by the initial zero coupon curve:. f (0, T ) = −. 1.4.2. ∂ ln P (0, T ) ∂T. Estimation results. We intend to capture counterparty valuation adjustments in two different scenarios, depending on whether the market considered Spanish financial institutions to be more or less solvent than the average European company. Figure 1.2 displays the evolution of 5-year tenor CDS of BBVA, Santander and iTraxx Europe (a CDS index) between 2008 and 2012. We will pick two dates for our analysis, less than four years apart: December 15, 2008 and June 28, 2012. Mid December 2008 can be regarded as the last moment before the situation reversed: Lehman Brothers had just defaulted a few months ago and there was a high degree of uncertainty with regard to the balance sheet of the big North- and Central European banks. Besides, it was generally agreed that the Spanish financial institutions had been 17.

(33) kept away from dubious subprime-related products. However, negative GDP figures for Spain would be published soon for the first time. Fears about the degree of exposure of the Spanish financial system to an inflated real estate sector gradually increased CDS spreads of both Spanish sovereign and financial entities, reaching its climax around June 2012, when Spain finally accepted a EUR 100 billion financial bailout from the Eurogroup. On the European Council of 28-29 June 2012, it was finally agreed7 that the European Stability Mechanism (ESM) would recapitalize troubled banks directly, thus avoiding bailouts to be counted for as additional sovereign debt in the corresponding country’s national account. Although this decision was deferred until an effective single supervisory mechanism was established for euro area banks, it was enough to alleviate the pressure on Spanish Treasury yields and corporate CDS. Between 28 and 29 of June 2012, the 5-year CDS spreads of BBVA and Santander fell by more than 50 bp (from 4.988% to 4.455% and from 4.747% to 4.242%, respectively). From then on, both CDS spreads have followed a decreasing trend, but still the market makes a clear distinction between Spanish and European institutions.. Figure 1.2: Evolution of CDS5y spreads for BBVA, Santander and iTraxx Europe (2008-2012). While in mid December 2008 the average European counterparty, represented by iTraxx, was regarded as significantly riskier than BBVA or Santander, the situation gradually reversed, with Spanish banks surpassing the barrier of 5% for CDS5y spreads in mid 2012. 7. See press release [54].. 18.

(34) June 2012 We calibrate the risk-neutral interest rate dynamics to the surface of ATM swaption volatilities on June 28, 2012. Minimization of the sum of the squares of the differences between model and market swaption prices produces the following calibrated parameters: a = 0.82682772, b = 0.01810259, σ = 0.01088233, η = 0.00934392 and ρ12 = −0.84103519. Calibration results are summarized in Tables 1.2 and 1.3. In the first, we show the fitted swaption volatilities as implied by the G2++ model, whereas in the second we report the absolute differences. Apart from some exceptions in the shortest maturities, differences are rather low given that we are fitting seventy prices with only five parameters:. 1y 2y 3y 4y 5y 7y 10y. 1y 97.4 74.8 53.9 41.0 35.4 31.5 28.3. 2y 86.2 63.9 47.5 38.7 34.8 31.2 28.1. 3y 72.9 54.8 43.4 37.2 34.1 30.7 29.0. 4y 61.5 48.8 40.8 36.0 33.3 30.3 29.1. 5y 53.8 44.9 38.8 34.8 32.5 29.7 29.0. 6y 48.7 42.0 37.1 33.8 31.7 29.8 30.2. 7y 45.1 39.7 35.7 32.8 31.0 29.6 30.9. 8y 42.3 37.9 34.4 32.0 30.8 29.4 31.4. 9y 40.0 36.3 33.4 31.6 30.5 30.0 31.6. 10y 38.2 35.0 32.8 31.2 30.1 30.4 31.6. Table 1.2: G2++ calibrated swaption volatilities. 28-Jun-2012. 1y 2y 3y 4y 5y 7y 10y. 1y 48.7 19.0 5.2 -0.3 -1.1 -0.3 2.2. 2y 36.1 15.1 4.5 0.9 -0.2 0.3 2.1. 3y 24.9 10.4 3.5 0.9 0.4 0.7 2.7. 4y 14.9 6.1 2.0 0.3 0.1 0.6 2.5. 5y 8.4 3.1 0.2 -0.4 -0.4 0.3 1.8. 6y 5.1 1.6 -0.3 -0.7 -0.6 0.5 2.2. 7y 2.4 0.4 -0.8 -1.0 -0.8 0.3 2.1. 8y 0.2 -0.6 -1.2 -1.4 -0.9 -0.3 1.8. 9y -1.6 -1.7 -1.8 -1.4 -1.2 -0.1 1.2. 10y -2.7 -2.8 -2.2 -1.8 -1.7 -0.3 0.6. Table 1.3: Swaption calibration results: absolute differences. 28-Jun-2012. December 2008 We now calibrate the risk-neutral dynamics to the surface of ATM swaption volatilities on December 15, 2008, just before the Spanish economy entered recession. Minimization of the sum of the squares of the percentage differences between model and market swaption prices produces the following calibrated parameters: a = 0.03125797, b = 0.06186271 , σ = 0.00345540, η = 0.01163478 and ρ12 = −0.30913976. Calibration results are summarized in Tables 1.4 and 1.5. In this case, there are no extreme differences as in 2012. Instead,. 19.