Minicourses

Thaleia Zariphopoulou: Portfolio Choice: Theoretical Foundations, Practice and New Directions

This minicourse will consist of three parts:
(i) A concise review of theoretical concepts of utility theory and portfolio management
(ii) An up to date survey of the investment industry best practice aiming at exposing the gap between academic and practical methodologies and approaches
(iii) Recent advances in bridging this gap spanning from abstract investment problem formulation to practical solutions in portfolio management
References
1. Thaleia Zariphopoulou, Fundamentals in Optimal Investments (slides of lecture 1)
2. Thaleia Zariphopoulou, Applied portfolio analysis (slides of lecture 2)
3. Thaleia Zariphopoulou, Expected utility models and optimal investments (slides of lecture 3)
4. Thaleia Zariphopoulou, Optimal investments under dynamic performance criteria (slides of lecture 4)
5. Thaleia Zariphopoulou, Investments, wealth and risk tolerance (slides of lecture 5)

Special invited lectures

John Schoenmakers: Policy iteration for American/Bermudan style derivatives

Effective valuation procedures for high-dimensional American/Bermudan derivatives are considered a thorny problem. In particular standard (e.g. regression) methods reveal limitations in many-dimensional and path-dependent problems. In this talk we recapitulate a recent alternative methodology based on policy iteration. By a popular example, the cancellable snowball, we show that allying this new methodology with industrial standard ones may fill the final gap. This is joint work with C. Bender and A. Kolodko. The talk is based on a couple of papers downloadable from John Schoenmakers's homepage.
Reference.
John Schoenmakers, Policy iterated lower bounds and linear MC upper bounds for Bermudan style derivatives Pricing complex structured products (slides of the lecture)

Albert Shiryaev: On the duality principle in option pricing for semimartingale models

The purpose of our talk is to develop the appropriate mathematical tools for the study of the call-put duality in option pricing for models where prices are described by general exponential semimartingales. Particular cases of these models are the ones which are driven by Brownian motions and by Lévy processes, which have been considered in many papers.
Generally speaking the duality principle states that the calculation of the price of a call option for a model with price process S = exp(H) (w.r.t. the measure P) is equivalent to the calculation of the price of a put option for a suitable dual model S' = exp(H') (w.r.t. a dual measure P').
From our talk it will clear that appealing to general exponential semimartingale models leads to a deeper insight into the essence of the duality principle.
Talk is based on the joint work of the author with Ernst Eberlein and Antonis Papapantoleon.

Short lectures

Jasper Anderluh: Double Sided Parisian Options

The talk will give an overview of Parisian option pricing from a probabilistic point of view. First, the double sided knock in call contract will be treated, which serves as a general type of Parisian contract from which the others (also the single-sided contract types) can be derived. For this contract a Fourier transform is derived in case the underlying follows the classical Geometric Brownian Motion. The second part of the talk is about hitting time simulation, a Monte-Carlo pricing method for Parisian options, different from the standard path-simulation techniques. The talk concludes with a few numerical examples giving insight into the difference between both the valuation techniques and the different types of Parisian contracts.
Reference
Jasper Anderluh, Double-Sided Parisian Options (slides of the lecture)

Vera Minina: The Cost of Risk in Option Hedging

The aim of this talk is to present an optimization model for option pricing and hedging. Our goal is to maximize the expected final payoff of a hedging portfolio while avoiding the use of utility functions. In order to make the problem bounded we introduce a punishment for the risk on the level of portfolio dynamics. The punishment is modeled by a risk function which may be interpreted as the obligatory transfer of a certain amount of money (dependent on the total portfolio risk) from the regular to a reserve bank account which has a lower interest rate. We present numerical results for a portfolio of options and a simple example of the risk function.
Reference
Vera Minina and Michel H. Vellekoop (2007), The Cost of Risk in Option Hedging

Budhi Arta Surya: On Endogeneous Default Under Levy Processes

The purpose of this talk is threefold. Firstly to revisit the previous work of Leland (1994), Leland and Toft (1996) and Hilberink and Rogers (2002) on optimal capital structure and show that the issue of choosing an optimal endogenous bankruptcy level can be dealt with both analytically and numerically when the underlying source of randomness for the value of the firm's asset is replaced by general spectrally negative Levy processes (with no positive jumps). Secondly, by working with the latter class of Levy processes we bring to light a new phenomena, namely that, depending on the nature of the small jumps, the optimal default level may be determined by a principle of continuous pasting as opposed to the usual smooth pasting. Thirdly, we are able to prove the optimality of the default level according to the appropriate choice of pasting. This improves on the results of Hilberink and Rogers (2002) who were only able to give a numerical justification for the case of smooth pasting. Our calculations are greatly eased by the recent perspective on fluctuation theory of spectrally negative Levy processes in which many new identities are expressed in terms of the so called scale functions. The talk will be concluded with a discussion over the effect of using Levy processes to the term structure of credit spreads. (This talk is based on the joint work with Andreas Kyprianou.)
References
1. A. E. Kyprianou and B. A. Surya. Principles of Smooth and Continuous Fit in the Determination of Endogenous Bankruptcy Levels. Finance and Stochastics (2007), 11, 131-152. A preprint is also available.
2. B. Hilberink and L.C.G Rogers. Optimal capital structure and endogenous default. Finance and Stochastics (2002), 6, 237-263.
3. H. E. Leland and K.B. Toft. Optimal capital structure, endogeneous bankruptcy, and the term structure of credit spreads. J. Finance (1996) 51:9877-1019.
4. Budhi Arta Surya, On Endogeneous Default Under Lévy Processes (slides of the lecture)

Martijn van der Voort: An Implied Loss Model

We present a model which is, by construction, consistent with observed market quotes for standard CDO tranches. The model is closely related to implied tree methods which can be used for valuing exotic equity derivatives consistent with observed market quotes for vanilla European call and put options. Rather than modelling default events for each name in the basket, the total basket loss is modelled directly and calibrated to CDO prices by construction. The proposed model has multiple important uses. First, the model can be used as a tool for avoiding arbitrage opportunities when pricing standard CDO tranches. This is a problem which is hard to solve when using the market standard Base Correlation approach in combination with interpolation and extrapolation rules. As a result the proposed model can be used to determine an arbitrage free distribution for portfolio losses for all maturities, which can subsequently be used as input to the more complex HJM type models which have recently become popular. Second, it provides us with a straightforward method for valuing Forward Starting CDOs, FDOs, consistently with observed market quotes on CDO tranches. A number of tests have been performed which have shown that the model performs well for pricing FDOs, when compared to a number of different factor copula models. Moreover, even under the assumption of heterogeneity of the basket in terms of recovery rates, the performance of the model is still impressive. Apart from performance tests, some additional tests will be presented, which show that the limited amount of market data still leads to a large amount of uncertainty in FDO prices. Finally forward Base Correlation skews implied by the model are considered and these are found to be rather stable.
Reference
Martijn van der Voort, An Implied Loss Model (working paper)