Tomas Björk (Stockholm School of Economics): INTEREST RATE MODELS

Abstract: The object of the lecture series is to give the students an introduction to the martingale-based theory of interest rates. Titles for the individual lectures: 1: General theory. Arbitrage. Completeness. Martingale measures, 2: The bond market. 3: Short rate models. Affine term structures. Inverting the yield curve 4: Forward rate models. Heath-Jarrow-Morton. Musiela. 5: Change of numeraire 6: Some current research topic(s).

Mark Davis (Imperial College London): MATHEMATICALS MODELS FOR DEFAULT AND CREDIT RISK

Abstract: 1. Credit risk: what is it, what data is available and what credit-related products are traded in the markets? 2. The relationship between credit spreads and default probabilities: a risk-neutral measure. 3. Stochastic models for default: ``structural form" and ``reduced-form" models. Calibration of models to term structure of market credit spreads. 4. Credit ratings and models for rating transitions. A summary of the CreditMetrics risk-management procedure. 5. Correlation effects for multiple credits. Moody's Diversity Score and infection models. Collateralized Bond Obligations. 6. Joint models for interest rates and credit spreads. Application to convertible bonds.

Paul Embrechts (Swiss Institute of Technology, Zurich): INSURANCE ANALYTICS

Abstract: Under Insurance Analytics I understand those methods from insurance mathematics which turn out to be important for handling questions in quantitative (integrated) risk management. Examples of topics to be included are: - Modelling catastrophic claims. - Ruin theory in general insurance risk models. - Measuring risk beyond VaR. - Securitisation of insurance risk: examples and (some) methodology. The aim of these lectures is to bring the students to the forefront of some of the more quantitative research in non-life insurance mathematics.

Nicole El Karoui (Ecole Polytechnique): INVERSE PROBLEMS IN FINANCE: THE EXAMPLES OF YIELD CURVES AND LOCAL VOLATILITY SURFACES

Abstract: An option pricing model establishes a relationship between the traded derivatives, the underlying asset and the market variables such that the volatility of the underlying asset.

The celebrated constant volatility Black-Scholes model is the most often used option pricing model in practice. However, recent evidence, especially since the 1987 crash shows that a constant volatility model is not adequate. The implied volatility calculated by inverting the Black-Scholes formula from the quoted price depends on the strike price and time to maturity. This dependence is often referred to as the implied volatility smile and the family as the implied volatility surface. For hedging purpose, these implied volatilities are used to calculate the "Delta" of the hedging portfolio, using different volatility values for options with different strikes and maturities. But this method is inappropriate for pricing and hedging exotic options with traded liquid standard options.

We can think of implied volatility as an expected volatility average on the life of the option, by analogy with the average interest rate on a period. As the forward spot rate, the local volatility (defined in the following equation) may be viewed as a forward volatility, meaning that is the anticipated level of the future spot volatility at expiration date if the underlying is equal to the exercise price.

As in the Black-Scholes model, in this framework, the market is complete and the ability to price and hedge exotic options is maintained.

A few different approaches have been proposed for modeling the volatility smiles: for instance, model with Jump component on the underlying (Merton); stochastic volatility models as Hull and White models, or Sterin and Stein models were very popular; more recently the two scales model with stochastic volatility from Papanicolaou, Fouque and Sircar is a new way for smile modeling. We describe these models in the last part of these lectures.

In the first part, we describe several methods to generate ``regular'' local volatility surfaces.

It is established by Dupire, and Derman-Kani (1975) that the local volatility surface can be uniquely determined from the European Call options of all strike prices and maturities under the no-arbitrage assumption of the observable European Call option prices.

Unfortunately, the European options market is typically limited to a relatively few different strikes and maturities. Therefore, the problem of determining the local volatility surface can be regarded as an function approximation non linear problem from a finite data set. This is a well-known ill-posed problem.

Smoothness of the local volatility can be important for computation or in risk management. Via penalization method, the problem can be treated as a deterministic control problem. Given a local volatility surface , let us denote by the model price for an option with parameters . The quoted prices today are . The deterministic program is to find a regular function minimizing the functional

where λ is a penalization parameter. Due to the highly non linearity, the computed solution is inaccurate.

Splines have long been used in approximating smooth curves and surfaces, as a tool for regularizing ill-posed problem of function from finite observation data: a classical example in finance is the yield curve reconstructed from levels of interest rates or swap rates with few maturities. Following Coleman, Li and Verma, 2 dimensional splines functional can be used to approximate a local volatility function. The implementation difficulties are discussed.

The drawbacks of these methods are to introduce arbitrary norm without reference to the dynamics of the underlying. By considering unknown local volatility as a control, Avellaneda and alii introduced the stochastic control program: to minimize

where η is a convex function which measures the distance to a prior . The optimal solution is Markovian i.e. with peaks in the neighbourhood of given strikes and maturities.

We can prefer to directly control the implied distribution of the underlying for a fixed maturity, or more generally for the all path. Techniques based on entropy with the above same constraints (Rubinstein, Avellaneda) may be used and reinterpreted in terms of portfolio optimization.

In conclusion, different ways to solve the ill-posed problem of calibrating volatility surfaces from a finite data set are proposed. These techniques can be used in other framework such that interest rates modelisation or risk management.

Ragnar Norberg (London School of Economics): FINANCIAL MATHEMATICS IN LIFE AND PENSION INSURANCE

Abstract: Within a theoretical framework that combines traditional actuarial models with models from mathematical finance, one can analyze life and pension insurance products with benefits and premiums linked to market indices. Theories for hedging in incomplete markets are well suited to the purpose. Particular attention will be given to interest guarantees for with-profit policies, unit linked (or variable life) insurance, and salary dependent benefits. The lectures will include an introduction to basic life insurance mathematics based on continuous time Markov chain models and also an introduction to basic concepts and methods of mathematical finance - pricing by no arbitrage, complete and incomplete markets, hedging and securitization.