Winter school on Financial Mathematics 2002
Abstracts with References


Rama Cont: Calibration of option pricing models: theory and algorithms

  • The calibration problem and its role in option pricing and hedging
  • Model calibration as an ill-posed inverse problem
  • Ill posed problems and regularization methods
  • Implied distributions and risk neutral densities
  • Implied trees and implied diffusions
  • Calibration of diffusion models: variational approach
  • Calibration of diffusion models: stochastic control approach
  • Calibration of jump-diffusion models
  • Calibration of LIBOR models
  • Calibration-free models: using implied volatility as state variable
[1] Calibration of jump-diffusion option-pricing models: a robust non-parametric approach
[2] Non-parametric calibration of jump-diffusion models (slides)

Marek Rutkowski: Credit Risk: Modelling, Valuation and Hedging

  • Introduction to Credit Risk and Credit Derivatives

    • Classification of Financial Risks
    • Overview of Credit Derivatives
    • Defaultable Claims

  • Structural (Value-of-the-Firm) Approach

    • Corporate Bonds
    • Merton's Model
    • Black and Cox Model
    • Structural Models with Random Interest Rates

  • Reduced-Form (Intensity-Based) Approach
    • Stochastic Intensity of a Random Time
    • General Valuation Formula for Defaultable Claims
    • Recovery Schemes for Corporate Bonds
    • Hybrid Models

  • Modelling of Dependent Defaults
    • Basket Credit Derivatives
    • Conditionally Independent Default Times
    • Duffie and Singleton Approach
    • Jarrow and Yu Model and its Extension

  • Credit Ratings and Credit Migrations
    • Jarrow, Lando and Turnbull Approach
    • Conditionally Markov Chains
    • Bielecki and Rutkowski Approach


    [1] Ammann, M. (2001) Credit Risk Valuation: Methods, Models, and Applications. Springer-Verlag, Berlin Heidelberg New York.
    [2] Arvanitis, A. and Gregory, J. (2001) Credit: The Complete Guide to Pricing, Hedging and Risk Management. RISK Books, London.
    [3] Bielecki, T.R. and Rutkowski, M. (2002) Credit Risk: Modeling, Valuation and Hedging. Springer-Verlag, Berlin Heidelberg New York.
    [4] Cossin, D. and Pirotte, H. (2000) Advanced Credit Risk Analysis. John Wiley & Sons, New York.
    [5] Schmid, B. (2002) Pricing Credit Linked Financial Instruments. Springer-Verlag, Berlin Heidelberg New York.
    [6] Credit Risk Modelling, Course notes
    [7] Value-of-the-firm approach (slides)
    [8] Intensity-based approach (slides)
    [9] Modelling of dependent defaults (slides)
    [10] Credit ratings and migrations (slides)

Special invited lectures

Damiano Brigo: Volatility-Smile Modeling with Density-Mixture Stochastic Differential Equations

We introduce the volatility-smile problem. Among the several approaches that are treated in the literature, we focus on the local volatility models setup, consisting in selecting particular functional forms for the diffusion coefficients in the stochastic differential equation describing the relevant dynamics of the underlying asset. We introduce a new general class of analytically tractable models for such dynamics based on the assumption that the asset-price density is given by a mixture of known basic densities. We consider the lognormal-mixture model as a fundamental example, and we derive the related explicit diffusion dynamics and show that it leads to a stochastic differential equation admitting a unique strong solution. We also provide closed form formulas for option prices and analytical approximations for the implied volatility function. We then introduce the asset-price model that is obtained by shifting the previous lognormal-mixture dynamics and investigate its analytical tractability. A seemingly paradoxical result on the correlation between the average volatility and the underlying asset is introduced, also in relation with stochastic volatility models. Further extensions of the basic model are considered. Finally, we consider specific examples of calibration to real market option data from the equity, FX and interest-rate markets.

[1] Lognormal-mixture dynamics and calibration to market volatility smiles
[2] The general mixture-diffusion dynamics for SDEswith a result on the volatility-asset covariance.
[3] Volatility-Smile Modeling with Density-Mixture Stochastic Differential Equations (slides)

Dilip Madan: Purely Discontinuous Processes in Asset Pricing

The talk will outline the case for the use of purely discontinuous price processes in modeling the statistical and risk neutral dynamics of asset prices. The theoretical and empirical case will be presented along with results emphasizing the lessons to be learned from such a perspective and the advantages attained in understanding the nature of investment. The primary example presented will be that of a Levy process but this will generalized to address issues of stochastic volatility as well. References
[1] Option Pricing, Lévy Processes, Stochastic Volatility, Stochastic Levy Volatility, VG Markov Chains and Derivative Investment (slides)
[2] Optimal Derivative Investment for Lévy Systems (slides)
[3] Purely Discontinuous Asset Price Processes

Jan Kallsen: Risk Management Based on Stochastic Volatility

Risk management approaches that do not incorporate randomly changing volatility tend to under- or overestimate the risk depending on current market conditions. We show how some popular stochastic volatility models in combination with the hyperbolic distribution can be applied quite easily for risk management purposes. Moreover, we compare their relative performance on the basis of German stock index data.

[1] Risk Management Based on Stochastic Volatility

Short lectures

Steffan Berridge: An irregular grid method for pricing high-dimensional American options

We propose and test a new method for pricing American options in a high dimensional setting. The method is centred around the approximation of the associated variational inequality on an irregular grid. We approximate the partial differential operator on this grid by appealing to the SDE representation of the stock process and computing the logarithm of the transition probability matrix of an approximating Markov chain. The results of numerical tests in five dimensions are promising.

[1] An irregular grid method for pricing high-dimensional American options

Dominique Dupont: Hedging barrier options: current methods and alternatives

Objectives of the paper: To introduce a flexible method for hedging barrier options or other exotic derivatives with regular options and to incorporate linear constraints on the hedging residual into the hedging process. Attention is limited to static hedging strategies, that is, strategies that allow trading in the hedging portfolio before maturity only at the first crossing of the barrier (The paper uses the example of an up-and-out call).

Problems with the current methods:

  • The current methods make strong assumptions on the availability of regular options with particular strikes and maturities, or on the probability distribution of the underlying asset.
  • There is no clear way on how to adapt these methods to situations where their assumptions are violated.
  • The current methods neglect model risk, that is, the risk that the prices at which one can trade the regular options when the barrier is attained are different from those implied by the pricing model.

Properties of the new method:

  • The new method is more general. It is based on a technique, mean-square hedging, designed to approximate the payoff of an asset, a barrier option for example, when exact replication may not be possible.
  • The new method is user friendly. The user chooses the strikes and the maturities of the regular options he wishes to trade, and the pricing model used to evaluate the regular and exotic options (for example, the pricing tree). The optimal hedging portfolio is then computed to minimize the mean of the square of the hedging residual.
  • Constraints on the hedging residual can be imposed, for example, to control the size of the hedging error in the tails.
  • Model risk can be taken into account within the mean-square hedging framework by introducing shocks on the local volatility of each regular option at the barrier. This creates more states-of-the-world but mean-square hedging can then be applied as before.
[1] Hedging Barrier Options: Current Methods and Alternatives

André Lucas: Extreme tails for linear portfolio credit risk models

We consider the extreme tail behavior of the CreditMetrics model for portfolio credit losses. We generalize the model to allow for alternative distributions of the risk factors. We consider two special cases and provide alternative tail approximations. The results reveal that one has to be careful in applying extreme value theory for computing extreme quantiles efficiently. The applicability of extreme value theory in characterizing the tail shape very much depends on the exact distributional assumptions for the systematic and idiosyncratic credit risk factors.

[1] Extreme tails for linear portfolio credit risk models
[2] Tail Behavior of Credit Loss Distributions for General Latent Factor Models

Berend Roorda: Dynamic aspects of coherent acceptability measures

Many problems in finance come down to deciding the acceptability of a position that generates an uncertain stream of future revenues or losses. In line with the coherent risk framework, introduced in Artzner et al. (1999), we take our starting point in a definition of acceptability in terms of worst expected value: a coherent acceptability measure is represented by a collection of probability measures, called the test set, and a position is deemed acceptable if it has nonnegative expected net value under all measures in that test set. In this talk we discuss the extension of the one-step framework in Artzner et al.(1999) to a multiperiod setting, in which acceptablity not only depends on an initial position, but evolves over time on the basis of incoming information. We first discuss some consistency problems that arise in this dynamic setting. It may occur, even in seemingly natural examples, that e.g. initially acceptable positions for sure turn into unacceptable. A notion of time-consistency for acceptability measures is proposed, in order to rule out such anomalies and to reduce mathematical and computational complexity. Secondly, we describe how the effect of hegding on acceptability is reflected at the level of test sets. In particular we show how test sets turn into martingale measures under the assumption of an ideal market for hedging instruments. This result will be exploited to indicate the close connection between the coherent risk framework and mainstream finance concepts as arbitrage, completeness, and valuation bounds.

[1] Coherent Acceptability Measures in Multiperiod Models
[2] Martingale characterizations of coherent acceptability measures

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