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Professor Alex Novikov

Biography

Alexander Novikov is Professor of Mathematics at the Department of Mathematical Sciences, UTS.

Prior to this appointment in 1999 he was Leading Research Fellow at the Steklov Mathematical Institute (Moscow, since 1970) and Senior Lecture at the University of Newcastle (Australia, from 1996 to 1999).

He received a PhD in Mathematics in 1972 and his Doctor of Science degree in 1982, both from the Steklov Mathematical Institute. He has published more than 90 research papers in different areas of stochastic processes, statistics of random processes, sequential analysis, random fields and mathematical finance. He has also been invited to more than 80 visiting appointments at leading mathematical institutions.

Professional

Bernoulli Society

Image of Alex Novikov
Professor of Mathematics, School of Mathematical and Physical Sciences
Core Member, Quantitative Finance Research Centre
MAppM (Steklov MI), DSc (Steklov MI)
 
Phone
+61 2 9514 2242

Research Interests

My current research interests are in stochastic analysis, mathematical finance and statistics of random processes.

Particular areas of interest include option pricing, credit risk modelling, change-point analysis, boundary crossing probabilities, Monte Carlo methods.

Current Teaching:
Stochastic Processes (35361)
Stochastic Calculus in Finance (35365)
Advanced Stochastic Processes (35466)

Past Teaching:
Time Series
Regression Analysis
Mathematical Statistics
Probability Theory
Survival Analysis

Chapters

Novikov, A.A., Ling, T.G. & Kordzakhia, N. 2014, 'Pricing of volume-weighted average options: Analytical approximations and numerical results' in Inspired by Finance: The Musiela Festschrift, pp. 461-474.
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© Springer International Publishing Switzerland 2014. All rights are reserved. The volume weighted average price (VWAP) over rolling number of days in the averaging period is used as a benchmark price by market participants and can be regarded as an estimate for the price that a passive trader will pay to purchase securities in a market. The VWAP is commonly used in brokerage houses as a quantitative trading tool and also appears in Australian taxation law to specify the price of share-buybacks of publically-listed companies. Most of the existing literature on VWAP focuses on strategies and algorithms to acquire market securities at a price as close as possible to VWAP. In our setup the volume process is modeled via a shifted squared Ornstein-Uhlenbeck process and a geometric Brownian motion is used to model the asset price. We derive the analytical formulae for moments of VWAP and then use the moment matching approach to approximate a distribution of VWAP. Numerical results for moments of VWAP and call-option prices have been verified by Monte Carlo simulations.
Kordzakhia, N., Novikov, A. & Tsitsiashvili, G. 2012, 'On ruin probabilities in risk models with interest rate' in Sibillo, M. & Perna, C. (eds), Mathematical and Statistical Methods for Actuarial Sciences and Finance, Springer-Verlag Italia, Milano, pp. 245-253.
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An explicit formula for ruin probability in a discrete time risk model with interest rare is found under the assumption that claims follow a hyperexponential distribution.
Borovkov, K., Downes, A.N. & Novikov, A. 2010, 'Continuity Theorems in Boundary Crossing Problems for Diffusion Processes' in Chiarella, C. & Novikov, A. (eds), Contemporary Quantitative Finance: Essays in Honour of Eckhard Platen, Springer, Germany, pp. 335-368.
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Computing the probability for a given diffusion process to stay under a particular boundary is crucial in many important applications including pricing financial barrier options and defaultable bonds. We discuss results on the accuracy of approximations for both the Brownian motion process and general time-homogeneous diffusions and also some contiguous topics.
Kordzakhia, N. & Novikov, A. 2008, 'Pricing of Defaultable Securities under Stochastic Interest' in Sarychev, A., Shiryaev, A., Guerra, M. & Grossinho, M. (eds), Mathematical Control Theory and Finance, Springer, Berlin, pp. 251-263.
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Liptser, R. & Novikov, A., 'On tail distributions of supremum and quadratic variation of local martingales'.
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We extend some known results relating the distribution tails of a continuous local martingale supremum and its quadratic variation to the case of locally square integrable martingales with bounded jumps. The predictable and optional quadratic variations are involved in the main result.

Conferences

Novikov, A. & Chiarella, C. 2009, 'Contemporary Quantitative Finance, Essays in Honour of Eckhard Platen', Quantitative Mathematical Finance, Quantitative Mathematical Finance, Springer, Sydney, Australia, pp. 1-410.
The contributors to this volume write a series of articles outlining contemporary advances in a number of key areas of mathematical finance such as, optimal control theory applied to finance, interest rate models, credit risk and credit derivatives, use of alternative stochastic processes, numerical solution of equations of mathematical finance, estimation of stochastic processes in finance. The list of authors includes many of the researchers who have made the major contributions to these various areas of mathematical finance. This volume addresses both researchers and professionals in financial institutions, as well as regulators working in the above mentioned fields.
Novikov, A. 2007, 'Pricing of Defaultable Securities under Stochastic Interest', Workshop on Mathematical Control Theory and Finance, Workshop on Mathematical Control Theory and Finance, Instituto Superior de Economia e Gestão, Lisbon, Portugal.
This is a CD edtion of conference papers. An extended version of the paper is accepted for publication in Mathematical Control Theory and Finance, Springer, Editores: A. Sarychev, A. Shiryaev, M. Guerra e M. R. Grossinho, 2008.
Roberts, D.O. & Novikov, A. 2005, 'Pricing European and discretely monitored exotic options under the Levy process framework', International Mathematica Symposium 2005, International Mathematic Symposium, Wolfram Research, Perth, Australia, pp. 1-11.
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We shall consider both European and idscretely monitored Exotic options (Bermudan and Discrete Barrier) in a market where the underlying asset follows a Geometric Levy process. First we shall briefly introduce this extended framework, then using the Variance Gamma model we shall show how toprice European Options and then we will proceed to demonstrate the application of the recursive quadrature method to Bermudan and Discrete Barrier Options
Kordzakhia, N., Melchers, R. & Novikov, A. 2000, 'First passage analysis of a 'square wave' filtered Poisson process', Applications of Statistics and Probability, A.A. Balkeme,, Sydney, Australia, pp. 35-43.

Journal articles

Novikov, A.A. & Kordzakhia, N.E. 2014, 'Lower and upper bounds for prices of Asian-type options', Proceedings of the Steklov Institute of Mathematics, vol. 287, no. 1, pp. 225-231.
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Novikov, A., Kordzakhia, N. & Ling, T. 2014, 'On Moments of Pitman Estimators: The Case of Fractional Brownian Motion', Theory of Probability & Its Applications, vol. 58, no. 4, pp. 601-614.
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Novikov, A. & Shiryaev, A.N. 2014, 'Discussion on 'Sequential Estimation for Time Series Models' by T. N. Sriram and Ross Iaci', Sequential Analysis, vol. 33, no. 2, pp. 182-185.
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Novikov, A. & Shiryaev, A. 2013, 'Remarks on moment inequalities and identities for martingales', Statistics & Probability Letters, vol. 83, no. 4, pp. 1260-1261.
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Novikov, A. & Kordzakhia, N. 2013, 'Pitman Estimators: An Asymptotic Variance Revisited', Theory of Probability and its Applications, vol. 57, no. 3, pp. 521-529.
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We provide an analytic expression for the variance of ratio of integral functionals of fractional Brownian motion which arises as an asymptotic variance of Pitman estimators for a location parameter of independent identically distributed observations. The expression is obtained in terms of derivatives of a logarithmic moment of the integral functional of limit likelihood ratio process (LLRP). In the particular case when the LLRP is a geometric Brownian motion, we show that the established expression leads to the known representation of the asymptotic variance of Pitman estimator in terms of Riemann zeta-function.
Cetin, U., Novikov, A. & Shiryaev, A.N. 2013, 'Bayesian Sequential Estimation of a Drift of Fractional Brownian Motion', Sequential Analysis: Design Methods and Applications, vol. 32, no. 3, pp. 288-296.
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We solve explicitly a Bayesian sequential estimation problem for the drift parameter of a fractional Brownian motion under the assumptions that a prior density of is Gaussian and that a penalty function is quadratic or Dirac-delta. The optimal stopping time for this case is deterministic. Keywords: Fractional Brownian motion; Penalty function; Sequential estimation. Subject Classifications: 62L12; 62F15; 60G22.
Radchik, A., Skryabin, I., Maisano, J., Novikov, A. & Gazarian, T. 2013, 'Ensuring long term investment for large scale solar power stations: Hedging instruments for green power', Solar Energy, vol. 98, pp. 167-179.
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There is a general consensus that solar power is one of the cleanest energy technologies available. Nevertheless, investment in large-scale Solar Power Generators (SPGs) is largely impeded by the intermittent nature of solar power. Since the electricity market has a critical responsibility to maintain the reliability of energy supply, the SPG can be registered only as the market semi-scheduled generator (AEMC, 2011). This option excludes the advantages of providing baseload supply, which in turn impedes efficient market contracting for SPGs. The existing approach relies on energy storage or co-generation facilities to be built at the same connection point as the SPG to compensate for output shortages when there is insufficient sunlight. The co-located facilities require significant additional investment in infrastructure.This paper proposes a market based financial approach that does not require an additional construction effort. The approach financially links solar or other intermittent power generation with a gas-fired station through a set of tailored swap-type instruments.These swaps (based on solar energy forecasting) are designed to insure and hedge the SPG against a drop in its output. They contractually link physically separated solar and gas generators to form a single entity termed the Virtual Generator (VG). The VG arrangement requires the SPG to provide solar power when possible and the gas generator to kick in when there is any shortfall due to random clouds or at nightfall. Thus the proposed VG will have the capacity to produce reliable baseload supply. The profitability and design of the proposed financial setup for both the solar and gas generator has been tested in this research. The model has been prototyped on real market and solar data, the results demonstrate the benefits of implementation. While the paper is focused on links with gas generators, the developed financial setup is applicable to a broad range of fast-ramping power generators, su...
Christensen, S., Irle, A. & Novikov, A. 2011, 'An Elementary Approach to Optimal Stopping Problems for AR(1) Sequences', Sequential Analysis, vol. 30, no. 1, pp. 79-93.
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Hinz, J. & Novikov, A. 2010, 'On fair pricing of emission-related derivatives', Bernoulli journal, vol. 16, no. 4, pp. 1240-1261.
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Tackling climate change is at the top of many agendas. In this context, emission trading schemes are considered as promising tools. The regulatory framework for an emission trading scheme introduces a market for emission allowances and creates a need for risk management by appropriate financial contracts. In this work, we address logical principles underlying their valuation.
Liptser, R., Novikov, A. & Tartakovsky, A.G. 2010, 'Preface: Celebrating Albert Shiryaev's 75th Anniversary', Sequential Analysis, vol. 29, no. 2, pp. 107-111.
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Novikov, A. & Shiryaev, A. 2010, 'ON MARTINGALE PROOF OF THE KOLMOGOROV AND SMIRNOV DISTRIBUTIONS', Sequential Analysis, vol. 29, no. 4, pp. 439-443.
this is appendix to the Shiryaev respose
Mititelu, G., Areepong, Y., Sukparungsee, S. & Novikov, A. 2010, 'Explicit analytical solutions for the average run length of CUSUM and EWMA charts', East-West Journal of Mathematics, vol. special, no. 1, pp. 253-265.
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Novikov, A. 2009, 'Some remarks on distributions and expectation of exit times of AR(1) sequences', Teoriya Veroyatnostei i ee Primeneniya, vol. 53, no. 3, pp. 459-471.
Shiryaev, A.N. & Novikov, A. 2009, 'On a stochastic version of the trading rule 'Buy and Hold'', Statistics and Decision, vol. 26, no. 4, pp. 289-302.
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The paper deals with the problem of finding an optimal one-time rebalancing strategy assuming that in the BlackâScholes model the drift term of the stock may change its value spontaneously at some random non-observable (hidden) time. The problem is studied on a finite time interval under two criteria of optimality (logarithmic and linear). The methods of the paper are based on the results for the quickest detection of drift change for Brownian motion.
Novikov, A.A. 2009, 'On Distributions of First Passage Times and Optimal Stopping of AR(1) Sequences', Theory of Probability & Its Applications, vol. 53, no. 3, pp. 419-429.
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Novikov, A. & Kordzakhia, N. 2008, 'Martingales and first passage times of AR(1) sequences', Stochastics An International Journal of Probability and Stochastic Processes, vol. 80, no. 2-3, pp. 197-210.
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Borovkov, K. & Novikov, A. 2008, 'On exit times of Levy-driven Ornstein–Uhlenbeck processes', Statistics & Probability Letters, vol. 78, no. 12, pp. 1517-1525.
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Schmidt, T. & Novikov, A. 2008, 'A Structural Model with Unobserved Default Boundary', Applied Mathematical Finance, vol. 15, no. 2, pp. 183-203.
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Novikov, A. & Shiryaev, A. 2007, 'On a solution of the optimal stopping problem for processes with independent increments', Stochastics An International Journal of Probability and Stochastic Processes, vol. 79, no. 3-4, pp. 393-406.
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Roberts, D.O. & Novikov, A. 2007, 'Pricing European and Discretely Monitored Exotic Options under the Levy Process Framework', The Mathematica Journal, vol. 10, no. 3, pp. 489-500.
Sukparungsee, S. & Novikov, A. 2006, 'On EWMA procedure for detection of a change in observation via Martingale approach', KMITL Science Journal, vol. 6, no. 2a, pp. 373-380.
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Using martingale technique wepresent analytic approximation and exact lower bounds for the expectation of the first passage times of an Exponentially Weighted Moving Average (EWMA) procedure used for monitoring changes in distributions. Based on these results, a simple numericalprocedure for finding optimal parameters of EWMA for small changes in the means of observation processes is established.
Borovkov, K. & Novikov, A. 2005, 'Explicit bounds for approximation rates of boundary crossing probabilities for the wiener process', Journal of Applied Probability, vol. 42, no. 1, pp. 82-92.
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We give explicit upper bounds for convergence rates when approximating both one- and two-sided general curvilinear boundary crossing probabilities for the Wiener process by similar probabilities for close boundaries of simpler form, for which computation of the boundary crossing probabilities is feasible. In particular, we partially generalize and improve results obtained by Pötzelberger and Wang in the case when the approximating boundaries are piecewise linear. Applications to barrier option pricing are also discussed. © Applied Probability Trust 2005.
Novikov, A.A. & Shiryaev, A.N. 2005, 'On an Effective Solution of the Optimal Stopping Problem for Random Walks', Theory of Probability & Its Applications, vol. 49, no. 2, pp. 344-354.
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Novikov, A., Melchers, R.E., Shinjikashvili, E. & Kordzakhia, N. 2005, 'First passage time of filtered Poisson process with exponential shape function', Probabilistic Engineering Mechanics, vol. 20, no. 1, pp. 57-65.
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Solving some integro-differential equation we find the Laplace transform of the first passage time for filtered Poisson process generated by pulses with uniform or exponential distributions. Also, the martingale technique is applied for approximations of expectations and distributions of the first passage times. The approximations accuracy is verified with the help of Monte-Carlo simulations. © 2004 Elsevier Ltd. All rights reserved.
Novikov, A. 2003, 'Martingales and first-exit times for the Ornstein-Uhlenbeck process with jumps', Theory of Probability and its Applications, vol. 48, no. 2, pp. 340-358.
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Novikov, A., Frishling, V. & Kordzakhia, N. 2003, 'Time-dependent barrier options and boundary crossing probabilities', Georgian Mathematical Journal, vol. 10, no. 2, pp. 325-334.
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Borovkov, K. & Novikov, A. 2002, 'On a new approach to calculating expectations for option pricing', Journal of Applied Probability, vol. 39, no. N/A, pp. 889-895.
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Miyahara, Y. & Novikov, A. 2002, 'Geometric Levy Process Pricing Model', Proceedings of the Stekov Institute of Mathematics, vol. 237, no. 2, pp. 185-200.
Borovkov, K. & Novikov, A. 2001, 'On a piece-wise deterministic Markov process model', Statistics & Probability Letters, vol. 53, no. 4, pp. 421-428.
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Other

Hinz, J. & Novikov, A. 2010, 'On fair pricing of emission-related derivatives'.
View/Download from: UTS OPUS
Tackling climate change is at the top of many agendas. In this context, emission trading schemes are considered as promising tools. The regulatory framework for an emission trading scheme introduces a market for emission allowances and creates a need for risk management by appropriate financial contracts. In this work, we address logical principles underlying their valuation.