Cattedra Galileiana

           Cattedra Galileiana 13 - 17 settembre 2004

From September 13 to 17 Scuola Normale Superiore will host two courses of the Cattedra Galileiana series:

Prof. Nicole EL KAROUI (Ecole Polythecnique, Palaiseau Francia)
Lectures on Optimal Stopping problems and Non-linear Representations

Optimal stopping problems have received renewed interest with the theory of American Options in Finance or Real options in Economics. While the theory is well-established since the Sixties, new developments are proposed in view of more efficient computational methods of the optimal stopping time: quasi-explicit formulae, free boundary, Monte-Carlo methods...

More recently, motivated by the Bandit problems or Optimal consumption plan with habit formation problem, a non-linear representation of processes or supermartingale has been introduced. All these representations are given in terms of the conditional expectation of the running supremum of an index process that we characterize. Take the running supremum of a process may be interpreted in the Max-plus Algebra $R^{\rm max}$ (where the two algebraic operations are $\oplus={\rm max}$ in place of $+$, and $\otimes=+$ in place of $\star$), as an integral. Some applications of these new decompositions are proposed.

1. Some explicit examples of optimal stopping problems: American options with infinite horizon, Options on the maximum (Russian options) and other options from G. Peskir and A. Shyraev.

2. Optimal Stopping and Reflected Backward Stochastic Differential equations from NEK, Pardoux, Peng, Quenez.

3. General Theory of the optimal stopping problem: problems related to the jumps from Cours de Saint- Flour...

4. Monte Carlo methods in solving optimal stopping. Problems: Longstaff and Schwarz Algorithm and extensions.

5. Non linear representation in the Max-Plus Algebra from P.Bank, H.Foellmer, NEK

6. Applications to American options, American Guaranteed, and stochastic order.

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Prof. Dmitry KRAMKOV ( Carnegie Mellon University , Pittsburg )

Utility based valuation in incomplete markets

Abstract: 
As it is well known, in a complete financial market every contingent claim can be perfectly replicated by a controlled portfolio of the traded securities and therefore admits a well-defined arbitrage-free price.  In an incomplete market, to every contingent claim is associated an interval of arbitrage-free prices. In this case, in order to determine a unique price arbitrage arguments alone are not, in general, sufficient and the preferences of the economic agent under consideration should be taken into account.

The goal of the course is to present some recent results related to the theory of utility based pricing in incomplete markets, where the preferences of the investor are modeled, as it is usual in economic theory, through the expected utility of terminal wealth. The main emphasis will be on mathematical aspects of the problem such as the formulation of precise or at least sufficiently sharp conditions for the key assertions of the theory to hold true. The course will be based on joint papers with Walter Schachermayer, Julien Hogonnier and Mihai Sirbu.

1. Overview of arbitrage-free pricing.
   We introduce the general semimartingale model for financial market. We define the concept of arbitrage-free price and formulate the first and the second fundamental theorems of asset pricing. Finally, we present results related to replication and super-hedging of contingent claims and derive bounds for arbitrage-free prices of derivatives.
2. Optimal investment in incomplete markets.
   We present a careful mathematical study of the problem of expected utility maximization. The main emphasis will be on “qualitative” properties such as the existence of the optimal investment strategy, two-times differentiability of the value function and so on.  The results presented in this part will form a foundation for the future analysis of utility based prices.
3. Optimal investment with random endowments.
   We study the problem of expected utility maximization for an economic agent who, in addition to an initial capital, receives random endowments at maturity. We show that an appropriate choice of the variables of the optimization problem can significantly simplify the study of the duality relations.
4. Sensitivity analysis of utility based prices.
   We present asymptotic analysis of utility based prices with respect to “small” number of non-traded contingent claims. In particular, we show that the asymptotic analysis captures some important “qualitative” properties of these prices if and only if there is a risk-tolerance wealth process.

Moreover, some semirars will be held by Professors Rama Cont, Helyette Geman, Huyen Pham, and Wolfgang Runggaldier.

The lectures and the semirars will take place at Scuola Normale Superiore (Aula U. Dini, Palazzo Castelletto) with the following schedule:

  Practical information on travel and accommodation in Pisa can be found at http://www.crm.sns.it/info.html

  Registration:

Participation is free, and meals will be offered to registered participants by Associazione Amici della Scuola Normale Superiore.
To register, send an e-mail to (amicisns@sns.it) with the following information:

Registration Form
Name ___________________________________________

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Address _________________________________________

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Cattedra Galileiana      Novit�

Per informazioni:
Associazione Amici della Scuola Normale Superiore
Piazza dei Cavalieri, 7 - 56126 Pisa
Tel. 050/509.654 - Fax 050/509.534
E-mail: amicisns@sns.it

Cattedra Galileiana      Novit�

Per informazioni:
Associazione Amici della Scuola Normale Superiore
Piazza dei Cavalieri, 7 - 56126 Pisa
Tel. 050/509.654 - Fax 050/509.534
E-mail: amicisns@sns.it