## The Fields Institute

Seminar on Financial Mathematics

### April 29, 1998

**SCHEDULE**

4:30 - 5:30 p.m.

*Some simple Properties of High, Low, Open, Close: Simulating Financial
Time series and tracking volatility*

Donald McLeish, University of Waterloo

6:00 - 7:00 p.m.

*Maximizing the Probability of a Perfect Hedge in the Presence of Uncertainty*

Ioannis Karatzas, Columbia University

**ABSTRACTS OF THE TALKS**

*Some simple Properties of High, Low, Open, Close: Simulating
Financial Time series and tracking volatility*

Donald McLeish

Observations on security prices, currency exchange rates, interest
rates, and other financial time series usually include not only an open and
close, but also a high and low price for the period. The information on high
and low prices of considerable value, particularly for estimating volatility,
and essential in the pricing of look-back and barrier options. For pricing
more general derivatives, this information is useful to the extent that changes
in volatility is an important ingredient in the price. This available information
is used to estimate parameters, conduct tests, provide simple simulations,
and generate plots sensitive to the volatility parameter.

*Maximizing the Probability of a Perfect Hedge in the Presence
of Uncertainty*

Ioannis Karatzas

Suppose that you start with initial capital 0 < x < 1;
you have until time t=T to play, and you can invest in a market with risk-free
rate *r* and with stock-price *S*(i) satisfying d*S*(t)-*S*(t)
[Bdt=d*W*(t)]. The stock-appreciation rate B is an unobservable random
variable with known distribution m , and is independent
of the Brownian motion *W*(i). If you can only observe the stock-prices
*S*(t), 0 £ t £ T;
during the finite time horizon [0, T], if your wealth-process X ^{x,p }(T) ³ 1] of reaching the level
1$ at time T, how should you choose optimally your portfolio p (t), 0 £ t £ T of investment in the stock? We are going to provide an
answer to this question, and study some of its ramifications.

**SPEAKERS**

**Don McLeish** is a Professor (Department
of Statistics and Actuarial Science), at the University of Waterloo. He is
a founding member of the Center for Advanced Studies for Finance at the University
of Waterloo and the collaborative Master's program for finance. Also the past
editor of the Canadian Journal of Statistics, and author of two books and
and number of papers on topics including probability theory, statistical inference,
estimating equations, simulation, and models in finance.

**Ioannis Karatzas** is Professor of Mathematics
and Statistics at Columbia University. He has held visiting positions at Brown
University, MIT, Rutgers, NYU (Courant Institute) and at the Universities
of North Carolina, Paris, Pennsylvania and Montreal, where he held the Andre
Aisenstadt Chair in Spring 1996. He is coauthor of the graduate text on "Brownian
Motion and Stochastic Calculus" (Springer-Verlag 1998, now in its fifth printing),
and author of the more recent "Lectures on the Mathematics of Finance" (AMS
1996, now in its second printing). He has published numerous research papers
on Probability Theory, Random Processes, Stochastic Control, Optimization,
Mathematical Economics and Finance, and has served on the editorial boards
of "Applications of Mathematics", the "Annals of Applied Probability", "Finance
& Stochastics", "Mathematical Finance", and the "SIAM Journal on Control
and optimization".

**ORGANIZERS**

Claudio Albanese (Mathematics, University of Toronto), Phelim Boyle (Finance,
University of Waterloo), Michel Crouhy (Canadian Imperial Bank of Commerce),
Donald A. Dawson (Fields Institute), Ron Dembo (President, Algorithmics Inc.),
Thomas McCurdy (Management, University of Toronto), Gordon Roberts (Finance,
York University), and Stuart Turnbull (Economics, Queen's University)