1st prize for young mathematicians (Institute of Mathematics, Novosibirsk, 1988).
Learning & Teaching Interests
I teach MSO4110 "Risk Measurement" and MSO4160 "Portfolios and Risk" modules for the MSc "Financial Mathematics" program at the School of Science and Technology.
MSO4110 “Risk Measurement"
Forecasting the magnitude of the next financial crisis is demanding, yet it is a task of utmost importance for every financial risk manager or data analyst.
For instance, on the “Black Monday” October 19, 1987, the markets fell by more than 20% in one day. Would it be possible to possible to forecast the magnitude of the crash using data available on the eve of the crash?
The answer is “Yes”!
At MSO4110 module students learn advanced methods of evaluating measures of financial risk from dependent heavy-tailed data. In particular, we study measures of risk called Value-at-Risk (VaR) and Expected Shortfall (ES) or CVaR.
Traditional measures of risk are static – they barely change with the inflow of new information and hence are only suitable for long-term investment decisions. MSO4110 introduces students to a dynamic measure of risk, mTA, that changes actively as data changes. Measure mTA appears more suitable for forecasting trends changes and short-term risks.
The module equips students with tools for accurate evaluation of the scale of the possible future market crashes. It provides an introduction to dynamic risk measurement and related statistical techniques (see chapter 10 of the recommended textbook S.Novak (2011) Extreme value methods with applications to finance. London: Chapman & Hall/CRC Press. ISBN 9781439835746).
Prerequisites: some mathematical skills, knowledge of undergraduate statistics.
Research Outputs & Interests
Research Track Record
Areas of Expertise:
Probabilistic and statistical problems in Extreme Value Theory (empirical point processes of exceedances, characterisation of the class of limit laws, estimation methods),
Theory of Sums of Random Variables (self-normalised sums of random variables, Student's statistic, characterisation, weak dependence, moment inequalities),
Non-parametric Statistics (methods of non-parametric statistical inference, accuracy of estimation, lower bounds),
Financial Risk Management (measures of financial risk).
Characterisation of the class of limiting distributions of empirical point processes of exceedances (EPPEs), necessary and sufficient conditions for the complete convergence of an EPPE to a particular limit law. Asymptotics of the EPPE is one of the key topics in Extreme Value Theory.
Sharp estimate of the accuracy of Poisson approximation.
An estimate of the accuracy of compound Poisson approximation. In the case of independent random variables the estimate coincides with the best available estimate of the accuracy of pure Poisson approximation.
Solution to the Lyapunov problem for Student's statistic (an estimate of the accuracy of normal approximation with explicit constants). This was one of very few long-standing problems in Probability & Statistics (see Liapunov (1901) and Chung (1946)). We have showed that the non-uniform Berry-Esseen-type inequality, in general (without extra assumptions), does not hold for Student's statistic.
Characterisation of the normal distribution in terms of a property of self-normalised random variables.
Characterisation of the property of a probability distribution to be symmetric.
A multivariate generalisation of Bradley's result concerning defining an independent copy of a random vector on the same probability space.
A version of the Borell-Cantelli lemma for dependent events.
Accurate estimators of the tail index and extreme quantiles, a procedure of choosing a tuning parameter. A new method of evaluating popular measures of financial risk (Value-at-Risk and Expected Shortfall).
Sharp minimax lower bounds to the accuracy of non-parametric tail index estimation.
Possible areas of PhD supervision:
* Inference on heavy-tailed distributions (Statistics)
* Accuracy of normal and compound Poisson approximation (Probability Theory)