T. R. (Tom) Hurd
## Professor of Mathematics |

## Contact Information:Dr. T. R. Hurd, Professor of Mathematics |

- Research Interests

Contagion! The Spread of Systemic Risk in Financial Networks (book in progress) - Since 1998 my research programme has been concentrated in the
rapidly developing field of financial
mathematics. Prior to this I worked primarily in mathematical physics. Financial
crises such as the 2007/08 global meltdown lead to insolvency or
default of firms, and in turn such firm defaults create further
market distress that compounds the crisis. Understanding capital
structure of firms and the links joining them is thus a critical
area of concern to society. In accounting, a firm's balance
sheet records book values of A (assets), D (debt) and E (equity)
on a quarterly basis. Mathematical Finance takes a market
perspective that sees the firm and all the securities that trade
on it in terms of prices observed in the liquid capital markets.
Market values reflect but do not equal book values, and are
updated continuously not just 4 times a year. The two strands of
this proposal are to develop market models of capital structure
and default risk at the one-firm and system-wide levels.
Traditional firm models focus on the randomness of A while
treating D as non-random, and view securities such as credit
default swaps as derivatives on A. A new "hybrid" approach takes
A and D as joint stochastic processes and treats both credit and
equity securities as derivatives on A and D, enabling trading
and hedging between credit and equity markets. The first strand
of this proposal is to address the central question about hybrid
models: "How well can the collective of all observed market
securities written on a firm be understood as derivatives on the
unobserved processes A and D?"

Applying balance sheet modelling to the interbank network leads immediately to the problem of financial systemic risk, defined as the risk that insolvency of some banks triggers further bank defaults. Such "domino events" are transmitted through links representing interbank loans that are identifiable in banks' balance sheets. Modeling financial networks and their systemic risk is both of strategic importance for society and an important applied mathematics problem. My second strand is to investigate how capital structure and default risk for banks intertwine with random network theory and to address the primary question "Which structural aspects of a financial network most affect systemic risk?" In a nutshell, my research aims to extend the range of Mathematical Finance to give a practical market-based understanding of firms and their links. I am Principal Investigator of a major research project entitled Financial Systemic Risk: a Network Science Approach sponsored by the Global Risk Institute (GRI). In the past couple of years, have given minicourses on Systemic Risk at a number of research institutions: IMPA in Rio, the 11th Winter school on Mathematical Finance in the Netherlands, and MACSI at the University of Limerick. Most recently, I gave the Nachdiplom lecture series at ETH in Zurich. This was a 24 lecture PhD level course entitled "Mathematics of Financial Systemic Risk":

The economic crisis of 2007-08 was first and foremost a crisis of the financial system, a particularly complex example of a complex adaptive system. This course will take a three-stranded approach to understanding the scientific and economic foundations for the transmission of dangerous shocks between financial institutions, and determining conditions under which these shocks can amplify into a network wide disruption. The first strand will consider the structure and dynamics of banks, their balance sheets, and their interconnections. The second strand will review and extend the general probability theory of information cascades in random networks, and to determine the different ways a financial crisis can be considered as a network cascade. The final strand will develop analytical and simulation-based algorithms for large scale computation of such idealized cascades. By the end of the course, we will have the means to model and test financial networks to determine their susceptibility to systemic collapse.

- Contagion!
The Spread of Systemic Risk in Financial Networks (book
in progress)

The title of this book, to be published in 2015, suggests that financial contagion is analogous to the spread of disease, and that damaging financial crises may be better understood by bringing to bear ideas that have been developed to understand the breakdown of other complex systems in our world. It also suggests that the aim of systemic risk management is similar to a primary aim of epidemiology, namely to identify situations when contagion danger is high, and then make targetted interventions to damp out the risk.

This book is intended to be two things, a timely summary of a growing body of systemic risk research as well as a unified mathematical framework for the primary channels that can transmit damaging shocks through financial systems. Much of its contents are new, not having appeared previously in published journals. It aims to serve as a coherent guide of equal interest to quantitative finance practitioners, financial regulators and a broad range of academics including economists, physicists, applied mathematicians and computer scientists.

- Teaching Interests

In recent years, my primary interest is in teaching specialized courses in financial mathematics. In recent years, I often teach Math 771 Mathematics of Finance, a course which serves as an introduction to modern financial security analysis and is a core course of our M-Phimac coursework MSc in Financial Mathematics. Since January 2015, I have been teaching Math 774 The Mathematics of Credit Risk. - PhiMAC: the financial math lab at McMaster

I supervise PhiMAC, a group of researchers in the Mathematics Department at McMaster who share a common interest in computational finance. We encourage interest from prospective graduate students and researchers from around the world. Have a look at our website! - M-PhiMAC: Master's in Financial Mathematics at
McMaster

M-Phimac, the coursework M.Sc. program offered by the Mathematics and Statistics Department at McMaster, is for the student who wants a fast track to a finance industry career in the areas of risk management, derivative securities analysis and portfolio design. After completing eight specialized grad courses in eight months, plus a number of optional training activities, you will be well prepared to go after one of many opportunities available in banking, insurance and the investment business. Have a look at our brochure. - Mprime

Mprime, the only Network of Centres of Excellence for the mathematical sciences, brings together academia, industry and the public sector to develop cutting edge mathematical tools vital to our knowledge-based economy. I am a core investigator of an Mprime initiative titled "Modelling Trading and Risk in the Market", involving researchers at University of Calgary, University of Toronto, University of Western Ontario, University of British Columbia and McMaster University. The general theme of our work is to develop mathematical tools for measurement and management of financial risk. For a detailed description of our Mprime project and others see the Mprime website. - Published Research Papers

- T. R. Hurd, James Gleeson
*"On Watts' Cascade Model with Random Link Weights",*to appear in Journal of Complex Networks, 2013.

Download pdf file - James Gleeson, T. R. Hurd, Sergey Melnik, Adam Hackett
*"Systemic risk in banking networks without Monte Carlo simulation"*, to appear in "Advances in Network Analysis and its Applications", Mathematics in Industry series, ed. E. Kranakis, Springer Verlag, Berlin Heidelberg New York, 2011

Download pdf file - M. Grasselli, T. R. Hurd
*“The Fields Institute Thematic Program on Quantitative Finance: Foundations and Applications January to June, 2010”*Quantitative Finance, 11, 21 - 29, 2011

Download pdf file - Chuang Yi, A. Tchernitser, T. R. Hurd
*"Randomized structural models of credit spreads"*, Quantitative Finance, 2010, http://www.informaworld.com/10.1080/14697688.2010.507213. DOI: 10.1080/14697688.2010.507213

Download pdf file - Ienkaran Arasaratnam, Simon Haykin, T. R. Hurd
*"Cubature Filtering for Continuous-Discrete Systems: Theory with an Application to Tracking"*, IEEE Transactions on Signal Processing, 2010, 29 pages.

Download pdf file - T. R. Hurd, Zhuowei Zhou
*"A Fourier transform method for spread option pricing”*, SIAM Journal of Financial Mathematics,**1**, 142-157, 2009.

Download pdf file - T. R. Hurd
*“Credit Risk Modelling using time-changed Brownian motion”*, Int. J. Theor. App. Fin.**12**, 1213–1230, 2009.

Download pdf file - T. R. Hurd, A. Kuznetsov
*"On the first passage time for Brownian motion subordinated by a Levy process"*, Journal of Applied Probability**46.1**, 181-198, 2009.

Download pdf file - T. R. Hurd
*"Saddlepoint approximations in portfolio credit risk"*, in Encyclopedia of Quantitative Finance, ed. R. Cont, Wiley-UK, 7 pages, 2008.

Download pdf file - Y. Ait-Sahalia, J. Cacho-Diaz, T. Hurd
*"Portfolio Choice with Jumps: A Closed Form Solution"*, Annals of Applied Probability,**19.2**, 277-290, 2008.

Download pdf file - T. R. Hurd, A. Kuznetsov
*"Explicit formulas for Laplace transforms of stochastic integrals"*, Markov Processes and Related Fields,**14**, 277-290, 2008.

Download pdf file - T. R. Hurd, A. Kuznetsov
*"Affine Markov chain models of multifirm credit migration"*, Journal of Credit Risk**3**, 3-29, 2007.

Download pdf file - M. R. Grasselli, T. R. Hurd
*"Indifference pricing and hedging for volatility derivatives"*, Applied Mathematical Finance**14**, 303-317, 2007.

Download pdf file - Jingping Yang, T. R. Hurd, Xuping Zhang
*"Saddlepoint approximation method for pricing CDOs"*, Journal of Computational Finance**10**, 1-20, 2006.

Download pdf file - T. R. Hurd
*"A note on log-optimal portfolios in exponential Levy markets"*, Statistics and Decisions,**22**, pp. 225-236, 2004

Download pdf file - M. R. Grasselli, T. R. Hurd
*"Weiner chaos and the Cox-Ingersoll-Ross model"*, Proc. R. Soc. A,**461**, pp. 459-479, 2004

Download pdf file - T. Choulli, T. R. Hurd
*"The role of Hellinger processes in mathematical finance"*, Entropy**3**, pp. 152-163, 2001

Download pdf file - J. Boland, T.R. Hurd, M. Pivato, L. Seco
*"Measures of dependence for multivariate Levy distributions"*, Proceedings of the Conference on Disordered and Complex Systems, edited by P. Sollich et al, American Institute of Physics, 2001

Download ps file ; pdf file

- T. R. Hurd, James Gleeson
- Research Working Papers
- T. R. Hurd, Davide Cellai, Huibin Cheng, Sergey Melnik,
Quentin Shao, "Illiquidity
and Insolvency: a Double Cascade Model of Financial
Crises", working paper, October 2013.

Download pdf file - Yacine Aït-Sahalia, T. R. Hurd, Portfolio Choice in Markets with Contagion",
working paper, September 2012.

Download pdf file - T. R. Hurd, Zhuowei Zhou
*"Two-factor capital structure models for equity and credit”*, working paper, October 2011

Download pdf file - T. R. Hurd, James Gleeson
*"A framework for analyzing contagion in banking networks"*, working paper, October 2011

Download pdf file - T. R. Hurd, Zhuowei Zhou
*"Structural credit risk using time-changed Brownian motions: a tale of two models”*, working paper, September 2011

Download pdf file - J. Abad, T. R. Hurd
*"Error bounds for Monte Carlo based portfolio optimization"*, working paper, February 2004

Download pdf file - M. R. Grasselli, T. R. Hurd
*"A Monte Carlo method for exponential hedging of contingent claims"*, working paper, November 2002

Download ps file; pdf file - T. Choulli, T. R. Hurd
*"The portfolio selection problem via Hellinger processes"*, working paper, October 2001

Download pdf file - T. R. Hurd
*"Pricing formulas, model error and hedging derivative portfolios"*, working paper, August 2001

Download pdf file - Applied Math at McMaster

Applied math is thriving at McMaster. Come and visit the website of the McMaster Applied Math Research Group - Directory of Personal Home Pages for the Department of Mathematics and Statistics
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last updated 17/10/2011