ERCIM News No.38 - July 1999

## Financial Mathematics

by Denis Talay

Financial markets play an important economical role as everybody knows. It is not well known (except by specialists) that the traders now use not only huge communication networks but also highly sophisticated mathematical models and scientific computation algorithms.

The trading of options represents a large part of the financial activity. An option is a contract which gives the right to the buyer of the option to buy or sell a primary asset (for example, a stock or a bond) at a price and at a maturity date which are fixed at the time the contract is signed. This financial instrument can been seen as an insurance contract which protects the holder against indesirable changes of the primary asset price.

A natural and of practical importance question is: does there exist a theoretical price of any option within a coherent model for the economy? It is out of the scope of this short introduction to give a precise answer to such a difficult problem which, indeed, requires an entire book to be treated deeply (see Duffie ‘92). This introduction is limited to focusing on one element of the answer: owing to stochastic calculus and the notion of non arbitrage (one supposes that the market is such that, starting with a zero wealth, one cannot get a strictly positive future wealth with a positive probability), one can define rational prices for the options. Such a rational price is given as the initial amount of money invested in a financial portfolios which permits to exactly replicate the payoff of the option at its maturity date. The dynamic management of the portfolio is called the hedging strategy of the option.

It seems that the idea of modelling a financial asset price by a stochastic process is due to Bachelier (1900) who used Brownian motion to model a stock price, but the stochastic part of Financial Mathematics is actually born in 1973 with the celebrated Black and Scholes formula for European options and a paper by Merton; decisive milestones then are papers by Harrison and Kreps (1979), Harrison and Pliska (1981) which provide a rigorous and very general conceptual framework to the option pricing problem, particularly owing to an intensive use of the stochastic integration theory. As a result, most of the traders in trading rooms are now using stochastic processes to model the primary assets and deduce theoretical optimal hedging strategies which help to take management decisions. The related questions are various and complex, such as: is it possible to identify stochastic models precisely, can one efficiently approximate the option prices (usually given as solutions of Partial Differential Equations or as expectations of functionals of processes) and the hedging strategies, can one evaluate the risks of severe losses corresponding to given financial positions or the risks induced by the numerous mispecifications of the models?

These questions are subjects of intensive current researches, both in academic and financial institutions. They require competences in Statistics, stochastic processes, Partial Differential Equations, numerical analysis, software engineering, and so forth. Of course, in the ERCIM institutes several research groups participate to the exponentially growing scientific activity raised by financial markets and insurance companies, and motivated by at least three factors:

- this economical sector is hiring an increasing number of good students
- it is rich enough to fund research
- it is a source of fascinating new open problems which are challenging science.
The selection of papers in this special theme gives a partial activity report of the ERCIM groups, preceded by an authorized opinion developed by Björn Palmgren, Chief Actuary and member of the Data Security project at SICS, on the needs for mathematical models in Finance. One can separate the papers in three groups which correspond to three essential concerns in trading rooms:

- how to identify models and parameters in the models: papers by Arno Siebes (CWI), Kacha Dzhaparidze and Peter Spreij (University of Amsterdam), József Hornyák and László Monostori (SZTAKI)

- how to price options or to evaluate financial risks: papers by Jiri Hoogland and Dimitri Neumann (CWI), László Gerencsér (SZTAKI), Michiel Bertsch (CNR), Gerhard Paaß (GMD), Valeria Skrivankova (SRCIM), Denis Talay (INRIA)
- efficient methods of numerical resolution and softwares: papers by David Sayers (NAG Ltd), Claude Martini (INRIA) and Antonino Zanette (University of Trieste), Mireille Bossy (INRIA), Arie van Deursen (CWI), László Monostori (SZTAKI).
Several of these papers mention results obtained jointly by researchers working in different ERCIM institutes.

Please contact:Denis Talay - INRIA

Tel: +33 4 92 38 78 98

E-mail: Denis.Talay@sophia.inria.fr

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