# FINANCIAL MATHEMATICS ACTIVITIES

July 17, 2024

## The Fields Institute Seminar on Financial Mathematics

### 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 dS(t)-S(t) [Bdt=dW(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)