# Abstracts

**Positive Default Intensities**

*Jan Willemen (ING, Belgium)*

Tuesday June 3, 11:00-11:30 | session 1.4 | Credit | room K

The problem of developing tractable stochastic default intensity models that allow one to 1) reproduce the term-structure of credit default swaps, 2) calibrate to options on credit default swaps and 3) guarantee the positivity of default intensities (or at least limit the probability of the latter becoming negative) is, to the best of our knowledge, not yet fully satisfactorily solved. In this presentation we develop the Markov Functional approach as a means to achieve these three goals. Markov Functional models were first introduced by Hunt, Kennedy and Pelsser and have since been used to model e.g. interest rates, equity prices, foreign exchange rates and credit. We believe the application to default intensities presented here to be new. We then compare our Markov Functional model to the Shifted Square-Root Diffusion (SSRD or CIR++) model proposed by Brigo and Alfonsi. The advantage of the Markov Functional approach over the SSRD model is its greater flexibility combined with a better control of the (possibly) negative default intensities generated by the models. Its main disadvantage is that it is slightly less tractable in the sense that less closed-form analytical results are available. However, this is not a major drawback as Markov Functional models allow a relatively efficient numerical implementation without having to resort to e.g. Monte Carlo simulation. In addition to giving an overview of the SSRD model and presenting the theory and implementation of the Markov Functional model in quite some detail, we compare the relative performance of both models in terms of achieving the goals listed above. For the sake of completeness, we include the Hull-White model in our comparison, despite its inability to control the negativity of default intensities. Finally, we apply the three models to the pricing of callable credit-linked notes.

**Barrier Options Under Lévy Processes: An Alternative Short-Cut**

*José Fajardo (FGV, Brazil)*

Tuesday June 3, 11:30-12:00 | session 1.4 | Credit | room K

In this paper we present new pricing formulas for a barrier options of European type when the underlying process is driven by an important class of Lévy processes, that includes CGMY model, Generalized Hyperbolic Model, Mexiner Model, among others. To achieve our goal we first assume that a symmetry property, equivalent to put-call symmetry, holds and then we analyze the most relevant asymmetric case.

**Counterparty credit risk in a multivariate structural model with jumps**

*Laura Ballotta (Cass Business School, City University London, UK)*

Tuesday June 3, 12:00-12:30 | session 1.4 | Credit | room K

The aim of the paper is to provide a valuation framework for counterparty credit risk based on a structural default approach à la Merton which incorporates jumps and dependence between the assets of interest. In this model default is caused by the firm value falling below a prespecified threshold following unforeseeable shocks, which deteriorate its liquidity and ability to meet its liabilities. The presence of dependence between names captures wrong-way risk and right-way risk effects. The structural model traces back to Merton (1974), who considered only the possibility of default occurring at the maturity of the contract; first passage time models starting from the seminal contribution of Black and Cox (1976) extend the original framework to incorporate default events at any time during the lifetime of the contract. However, as the driving risk process used is the Brownian motion, all these models suffers of vanishing credit spreads over the short period - a feature not observed in reality. As a consequence, the CVA would be underestimated for short term deals as well as the so-called gap risk, i.e. the unpredictable loss due to a jump event in the market. Improvements aimed at resolving this issue include for example random default barriers and time dependent volatilities, and jumps. Hence, we adopt Lévy processes and capture dependence via a linear combination of two independent Lévy processes representing respectively the systematic risk factor and the idiosyncratic shock. We then apply this framework to the valuation of CVA and DVA related to equity contracts such as forwards and swaps. We analyse in details the case in which the driving process is a parsimonious non Gaussian process, such as the Normal Inverse Gaussian (NIG) one: this choice is motivated by the fact that the NIG process allows for skewness, excess kurtosis and a fairly rich jump dynamics although parsimonious in terms of number of parameters involved and calibration. We also compare against the results originated by the standard Gaussian assumption. The main focus is on the impact of correlation between entities on the value of CVA and DVA, with particular attention to wrong-way risk and right-way risk; this is explored via sensitivity analysis. Particular attention is also devoted to model calibration to market data; an empirical analysis is also conducted to evaluate how the cost of bilateral counterparty credit risk has varied over time.

**Markov switching affine processes and applications to credit risk**

*Misha Van Beek (University of Amsterdam, The Netherlands)*

Tuesday June 3, 12:30-13:00 | session 1.4 | Credit | room K

The hazard rate processes that drive defaults are often mutually correlated and correlated with the risk free rate. Furthermore, this correlation and the dynamics of these rates differ across different regimes. The premiums paid for these default risks are, next to the premiums associated with the default of the reference entities, categorized as credit valuation adjustment (CVA) and debt valuation adjustment (DVA). One or more of these categories matter when pricing risky bonds or credit derivatives such as credit default swaps (CDSs). In pricing such products, the hazard rates of default and interest rates are often modeled as affine processes. This paper develops methods for pricing these products when the hazard rates and interest rate, or the factors driving these rates, follow a Markov switching multivariate affine process. This is a regular multivariate affine process for which the parameters (including correlations between the rates and factors) are different for each regime. A Markov chain switches between these regimes and thus the parameters of the process. First we show that all higher mixed moments of this process can be written in closed form. Second, we derive a system of ordinary differential equations (ODEs) that can be used to price risky bonds, credit derivatives and the characteristic function. Third, we fit the term structures of interest rates and default probabilities. The process is flexible enough to fit complex term structures.