Abstracts

## MinicoursesHélyette Geman:Stochastic Time Changes, Lévy Processes and Option Pricing
The classical Black-Scholes model assumes constant volatility and continuity of trajectories. From classical observations of the financial markets however, it is clear that these assumptions are not satisfied. Fat tails, volatility smile, jumps in skewness, in option prices are examples of deviations of market prices from the Black-Scholes assumptions. A natural and parsimonious way of capturing the non constancy of volatility is to introduce a stochastic clock where time accelerates during periods of high volatility. From a financial perspective, the number of trades is exhibited to be the quantity driving the transaction time. From a mathematical perspective, it is shown that under no arbitrage, asset prices are time-changed Brownian motions. This representation allows to make a further case for jump processes to show the drawbacks of the Merton (1976) jump diffusion model, and to generate pure jump Lévy processes which include the CGMY (Carr - Geman - Madan - Yor) model. Stochastic volatility in the CGMY will finally be introduced (under the form of a time change) to allow for an excellent fit of the option volatility surface across strikes and maturities, using the characteristic function of the process and the fast Fourier transform in strike or the option prices. Lastly, the topic of incomplete markets will be discussed and the paradigm of acceptable risk introduced; electricity markets will illustrate this discussion. References[1] Hélyette Geman, Pure Jump Lévy Processes for Asset Price Modelling, published in the Journal of Banking and Finance 26 (7), 1297-1316 (2002) [2] Peter Carr, Hélyette Geman, Dilip B. Madan, Marc Yor, Stochastic Volatility for Lévy Processes
Paul Glasserman: Credit Risk: Here the challenge lies in accurate estimation of small probabilities of large losses (and associated risk measures). Importance sampling (IS) is a natural candidate for improving precision, but the application of IS is complicated by the types of dependence structures (e.g., normal copula) typically used in factor models of credit risk. We present two-part IS methods that change the distribution of the factors and increase default probabilities conditional on the factors in order to produce more scenarios with large losses. The methods are supported through asymptotics as both the portfolio size and loss threshold increase. We also discuss interactions between Monte Carlo and other computational methods. Market Risk: The problem of estimating the value at risk of a large portfolio over a relatively short horizon can also be addressed using importance sampling, in this case based on a delta-gamma (or other quadratic) approximation to portfolio value. Extensions to heavy-tailed distributions are of special importance but present particular challenges to traditional IS strategies. We present methods to address these problems and discuss their asymptotic optimality properties. We also discuss the combination of IS with other techniques.
American Options: The pricing of American options by simulation
is made difficult by the embedded optimal stopping problem.
We give an overview of methods developed in recent years to address this
problem. These methods apply weighted backward induction to simulated
paths, with weights defined through likelihood ratios, through calibration,
or implicitly through regression. We also discuss recent results on the
convergence of these methods.
## Special invited lecturesRüdiger Frey:On Dynamic Models for Portfolio Credit Risk and Credit Contagion
It is by now well known that the performance of models for portfolio credit risk is very sensitive to the modelling of dependence between defaults of different obligors. In this talk we will be concerned with dynamic models for portfolios of dependent defaults. After a survey of existing approaches, we concentrate on models for credit contagion, i.e. models where the default of one company has a direct impact on the default intensity of other firms. We introduce a Markovian model and discuss the various types of interaction. Finally we present limit results for large portfolios in a homogeneous model with mean-field interaction and analyze the impact of credit contagion on the portfolio loss distribution. References
[1] Slides of the lecture [2] Rüdiger Frey and Jochen Backhaus, Interacting Defaults and Counterparty Risk: a Markovian Approach
Wolfgang Runggaldier:
Uwe Wystup: ## Short lectures
Remco Peters:
Antoine van der Ploeg:
Raoul Pietersz:
Alessandro Sbuelz: |

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