Abstracts
Tuesday June 3, 14:00-14:30 | session 2.5 | Credit | room G
The introduction of CCPs in most derivative transactions will dramatically change the landscape of derivatives pricing, hedging and risk management, and, according to the TABB group, will lead to an overall liquidity impact about 2 USD trillions. In this article we develop for the first time a comprehensive approach for pricing under CCP clearing, including variation and initial margins, gap credit risk and collateralization, showing concrete examples for interest rate swaps. Mathematically, the inclusion of asymmetric borrowing and lending rates in the hedge of a claim lead to nonlinearities showing up in claim dependent pricing measures, aggregation dependent prices, nonlinear PDEs and BSDEs. This still holds in presence of CCPs and CSA. We introduce a modeling approach that allows us to enforce rigorous separation of the interconnected nonlinear risks into different valuation adjustments where the key pricing nonlinearities are confined to a funding costs component that is analyzed through numerical schemes for BSDEs.
Tuesday June 3, 14:30-15:00 | session 2.5 | Credit | room G
It is well-known from the work of Schonbucher that the marginal laws of a loss process can be matched by a unit increasing time inhomogeneous Markov process, whose deterministic jump intensity is called local intensity. The Stochastic Local Intensity (SLI) models such as the one proposed by Arnsdorf and Halperin allow to get a stochastic jump intensity while keeping the same marginal laws. These models involve a non-linear SDE with jumps. The first contribution of this paper is to prove the existence and uniqueness of such processes. This is made by means of an interacting particle system, whose convergence rate towards the non-linear SDE is analyzed. Second, this approach provides a powerful way to compute pathwise expectations with the SLI model: we show that the computational cost is roughly the same as a crude Monte-Carlo algorithm for standard SDEs.
Tuesday June 3, 15:00-15:30 | session 2.5 | Credit | room G
In this talk we discuss changes of probability measure up to a random time $\sigma$. Working under the standing assumptions that $\sigma$ avoids stopping times and that all martingales are continuous, we extend results from Mortimer and Williams (1991) and provide new classes of examples involving honest times and pseudo-stopping times. Moreover, we study the stability of the pseudo-stopping time property with respect to certain measure changes.
While changes of measure are ubiquitous in mathematical finance due to the fundamental theorem of asset pricing, enlargements of filtrations are used to model credit risk and insider trading. We therefore investigate the following question: If we assume NFLVR with respect to the original filtration, under which conditions is the market then also arbitrage-free with respect to the progessively enlarged filtration until time $\sigma$? For an honest time $\sigma$ this question was recently studied in detail by Fontana, Jeanblanc and Song (2013). In this talk we consider the case of an arbitrary random time $\sigma$ and a continuous stock price process, and we are able to give sufficient criteria for NFLVR on the time horizon $[0,\sigma]$ in terms of the multiplicative decomposition of the Azéma supermartingale associated with $\sigma$.
Tuesday June 3, 15:30-16:00 | session 2.5 | Credit | room G
Credit ratings play a significant role in both credit risk measurement and defaultable debt valuation, whether they are assigned by public rating agencies or by some internal rating procedures. Accordingly, it is important for risk management to assess the possibility of future rating changes and defaults accurately as possible. We therefore need to consider an estimation problem of rating transition probabilities under the condition that transition probabilities are not constant in time but dependent on some current economic factor. In this paper, we adopt the intensity-based approach to construct a credit rating transition model with the hypothesis that transition intensities are represented in terms of some unobservable “macro” factor. Hence we can formulate a nonlinear filtering model to estimate the factor from observations of rating transition events and so on.
We discuss how to solve such a filtering problem numerically with the particle filter and apply the solution to analyze the rating transition of Japanese enterprises during 2009-2012. Finally, through the empirical analysis, we realize our model can capture contagion effects of credit events as well as the impacts of actual defaults on the estimated filter.