Program of the School

The most extensively studied form of credit risk is the {\it default risk} -- that is, the risk that a counterparty in a financial contract will not fulfil a contractual commitment to meet her/his obligations stated in the contract. For this reason, the main tool in the area of credit risk modelling is a judicious specification of the random time of default. Our main goal is to present the most important mathematical tools that are used for the arbitrage valuation of defaultable claims, which are usually referred to as {\it credit derivatives}. We also examine the important issue of hedging for typical credit derivatives. We will first provide a concise summary of the main developments within the so-called ``structural approach'' to modelling and valuation of credit risk. In particular, we will study the first-passage-time approach and a random barrier case. The next part of the lectures will be devoted to the ``reduced-form approach''. This approach is purely probabilistic in nature and, technically speaking, it has a lot in common with the reliability theory. Subsequently, we will examine hedging strategies under the assumption that some defaultable assets (such as, e.g., corporate bonds or credit default swaps) are traded. Finally, we will give a brief survey of practical methods that are currently used in modelling of dependent defaults and credit migrations, and we will discuss valuation and hedging of multi-name credit derivatives, such as: credit default index swaps and synthetic collateralized debt obligations.

The Law of the Minimal Price will be presented for derivative pricing. A Diversification Theorem allows forming a proxy for the numeraire portfolio. The richer modelling framework of the benchmark approach leads to the derivation of tractable, realistic models for equity indices, exchange rates, equities and the interest rate term structure under the real world probability measure. It will be explained how the approach differs from the classical risk neutral approach. Examples on long term and extreme maturity derivatives demonstrate the important fact that a range of contracts can be less expensively priced and hedged than suggested by classical theory.

• Financial modelling using the numeraire portfolio
• Deriving the Law of the Minimal Price
• Approximating the numeraire portfolio via diversification
• Consistent utility maximization and portfolio optimization
• Pricing nonreplicable claims consistently as an investment
• Pricing and hedging long term and extreme maturity contracts
• Index, FX, equity and term structure derivatives