Professor’s New Book Relates Math To Finance

SOCORRO, N.M. December 15, 2015 – Dr. Ivan Avramidi, professor in the Mathematics Department at Tech, has published a new book that delves into the math of high finance.


Dr. Ivan Avramidi


Avramidi-productFlyer 978-3-319-26265-9

The new book now available online. 


Heat Kernel Methods and Its Applications is intended for advanced undergraduate or beginning graduate students in physics and mathematics. However, the book also provides a useful reference for professional physicists and applied mathematicians, as well as for quantitative analysts and financial industry practitioners.

The primary focus of his research, until recently, had been developing advanced methods of geometric analysis and applying them to quantum theory. He previously published a text titled Heat Kernel and Quantum Gravity in 2000.

“I have been working in the field of mathematics pretty long, including 60 papers on the subject,” he said. “But it was always tied to physics and math. As it turns out, in finance, they are using very similar techniques mathematically to solve certain equations that I am an expert in.”

“Quantitative analysts – or quants – look at the prices of stocks, options and prices and they seem to behave randomly or stochastically,” Avramidi said. “But the mathematic equation that describes this is not random. It’s deterministic and you can predict it. You can predict how much an option will cost, but you need to solve that equation. It turned out to be the same equation that describes the propagation of heat.”

The heart of the book is the development of a short-time asymptotic expansion for the heat kernel. This is explained in detail and explicit examples of some advanced calculations are given. In addition some advanced methods and extensions, including path integrals, jump diffusion and others are presented.


The book consists of four parts: Analysis, Geometry, Perturbations and Applications. The first part shortly reviews of some background material and gives an introduction to partial differential equations. The second part is devoted to a short introduction to various aspects of differential geometry. The third part and heart of the book presents a systematic development of effective methods for various approximation schemes for parabolic differential equations. The last part is devoted to applications in financial mathematics, in particular, stochastic differential equations

Avramidi was first invited to present a series of lectures to a banking group in 2007 – but he didn’t initially think the invitation was legitimate.

“I got an email from a banker saying that he’d like me to give a talk, but I thought it was spam,” he said. “Then they called me and said they’re serious.”

The head of Research and Development at NATIXIS Corporate and Investment Bank in Paris invited Avramidi to present a series of lectures over a full week for the members of the quantitative finance group. Avramidi’s lectures were well received and he was asked repeatedly to publish them.

The field of quantitative analysis is relatively young, having arisen in the 1990s. Avramidi said that many physicists were left looking for work when the Superconducting Super Collider was de-fund in 1993.

“Many Ph.D.s were finishing grad school and preparing to work on the Superconducting Super Collider and then its funding was cut,” he said. “So this crowd – some found places in academia, but some went to Wall Street. There’s many mathematicians and physicists working in finance. And I met many of those people who work as quants at the Paris conference.”

Avramidi joined the New Mexico Tech faculty in 1999, achieving full professorship in 2005. He earned his Ph.D. at Moscow State University in 1987.

– NMT –

By Thomas Guengerich/New Mexico Tech