# Demystifying The Mathematics of Financial Markets: Making Complex Concepts Accessible to All - In Conversation with Dr.Alberto Bueno Guerrero

Alberto Bueno Guerrero was born in 1969 in Granada (Spain). From a young age he showed a strange attraction for mathematical formulas, which led him to study Physics at the University of Granada. After obtaining a Bachelor's degree in Physics, with a specialty in Theoretical Physics, given the few prospects that a research career offered him, he decided to prepare for the examinations for secondary education Mathematics teacher in public education.

Once he got the position and, already practicing as a teacher, he realized that he was going to lose all the fruit of the intense mental training that he obtained studying Physics. Aware that mathematical models are frequently used in Economics, he made the decision to pursue these studies. To make it compatible with work and family life, he enrolled in the National Distance Education University (UNED). After obtaining a degree in Economics, he re-examined to obtain accreditation to practice as an Economics professor, obtaining it in 2006.

The same desire not to let go of all the learning led him to pursue a PhD in Finance, which he achieved in 2014. During his thesis development, he discovered his true passion: research. He began a career as an independent researcher that has led him to publish more than ten works, solo or in collaboration, including articles and book chapters. His papers have been published in magazines such as: Physica A; Chaos, Solitons and Fractals; Journal of Derivatives or Quantitative Finance. Among the topics addressed in his research are the term structure of interest rates, the valuation and hedging of derivatives, the immunization of bond portfolios, or a quantum-mechanical model for interest rate derivatives. His article "Bond Market Completeness Under Stochastic Strings with Distribution-Valued Strategies" has been considered a feature article in Quantitative Finance.

In 2020 he was interviewed for the article "Interviews with Researchers Who Started Their Career in Physics but Moved to Finance", for the special issue of the Journal of Derivatives entitled "Physics and Financial Derivatives", along with Emanuel Derman, Alexander Lipton, David Gershon, Matt Lorig, Peter Tankov and Alexandre Antonov.

The lack of popular science books in the field of Mathematical Finance led him to write the book "The Mathematics of Financial Markets Within Everyone's Reach" which he also translated into Spanish.

He currently makes weekly posts in Linkedin related to Mathematical Finance formulas, which has led him to more than 6,800 followers and more than 1,700,000 impressions over the last year.

**Excerpts from the Interview : **

**Excerpts from the Interview :**

**Q1) Can you tell us about your academic background and how you developed a passion for mathematical modelling and its application in financial markets?**

**Alberto : **I have a degree in Physics and another in Economics. I also have a PhD in Finance. From a very young age I have shown a strange interest in mathematical formulas. I remember that during the summer before senior year in high school I asked a friend for his older brother's Physics book to have a look at before starting the course. Many times, in bookstores or libraries, I spent time looking at Physics or Mathematics books only to see their pages full of formulas, even if I didn't understand anything. This interest has guided my studies, and during the PhD courses I oriented myself towards the mathematical modelling of financial markets.

**Q2) Your book aims to make Mathematical Finance accessible to a broader audience without advanced mathematical knowledge. How did you approach the challenge of presenting rigorous concepts in a more understandable and intuitive manner?**

**Alberto : **As I say in my book “The Mathematics of Financial Markets Within Everyone’s Reach”, I am an avid reader of popular science literature, especially that related to Theoretical Physics. In this area you can find great researchers, including Nobel Prize winners, who try to get scientific results to the general population, without using sophisticated mathematics. In the field of Quantitative Finance there are excellent textbooks but, as far as I know, there are no books dedicated to bringing this knowledge to an interested public, but lacking the necessary mathematical level. So, I thought that I could do the type of popular science of my favorite authors in Physics, but applied to the world of financial markets. From this approach the idea of the book arose.

The way I approached the challenge was to use the usual resources of popular science: simple numerical examples, graphs, diagrams, analogies, biographical anecdotes and a little sense of humor. In addition, in the footnotes I have included the formulas, which together with the references will allow those readers with greater mathematical skill to initiate or extend a study in more depth.

**Q3) In the context of financial markets, what are some of the key mathematical laws and principles that you believe every investor should be aware of?**

**Alberto : **I do not belong to the financial industry and, therefore, I cannot offer principles based on my practical knowledge. However, in the mathematical modelling of the markets, certain principles are used that are considered valid and that, in certain circumstances, will be accepted as approximately valid by any market operator.

The first principle would be the absence of arbitrage opportunities. According to this principle, in the markets it is not possible to find out a strategy with zero net cost, without the possibility of losses, but with a positive probability of profit. In other words, there are no free lunches. In a real context such opportunities may exist, but market operators will quickly find them and, by exploiting them, will make them disappear.

The second principle that I would highlight is that diversification reduces risk. The construction of portfolios with assets with low correlations in their returns will cause unfavorable movements in certain assets to be offset by favorable movements in other assets, thus reducing risk.

**Q4) You mentioned that the book includes bibliographic references for readers with a stronger mathematical background. Could you highlight some of the recommended resources that can help readers delve deeper into the concepts covered in the book?**

**Alberto : **Each of the references cited in the book refer to a particular field of Financial Mathematics. However, there are some textbooks that cover more general content, which are the first ones that I would recommend to a reader who wants to delve into these concepts. Among them, my favorite is "Arbitrage Theory in Continuous Time" by Tomas Björk. It treats almost all the aspects covered in my book with all the rigor, but with intuitive explanations and different degrees of difficulty. In addition, it incorporates the necessary mathematical requirements.

For those readers who want to dedicate themselves to the study of Stochastic Calculus, which is the main tool of Quantitative Finance, I would recommend that they start their study with "Introduction to Stochastic Calculus with Applications", by Fima Klebaner. Two very advanced level texts would be "Brownian Motion and Stochastic Calculus" by Ioannis Karatzas and Steven Shreve and "Stochastic Integration and Differential Equations" by Phillip Protter.

**Q5) The inclusion of footnotes with mathematical formulas adds an interesting dimension to the book. How do these formulas enhance readers' understanding and appreciation of the underlying mathematics in financial markets?**

**Alberto : **The book is intended to be an introduction to Mathematical Finance suitable for curious people without specialized knowledge. On the one hand, the visualization of the formulas after having known the associated intuition can help to concretize that knowledge in quantitative terms, and in turn can stimulate in the readers a taste for a more formal study of the concepts.

On the other hand, today there are many graduates in technical studies such as Physics or Mathematics, who decide to continue their career in Quantitative Finance, knowing that a large part of the jobs in this field are covered with profiles of this type. I wanted my book to be useful for these people as well, so that with the necessary mathematical knowledge they could take advantage of the basic introduction of the book and use it as a springboard towards postgraduate studies in Finance. The inclusion of formulas in the footnotes and references to academic literature serves this purpose.

**Q6) Financial markets are constantly evolving, and new investment possibilities emerge. How do you ensure that the concepts presented in the book remain relevant and adaptable to changing market dynamics?**

**Alberto : **In mathematical modelling, one always has to be open to the possibility of new situations appearing that cannot be explained with current models. However, in the case of the aspects that I deal with in my book, this possibility is remote. Firstly, because it is based on very general principles, such as the principle of absence of arbitrage opportunities, which must be valid, at least approximately, in any sufficiently efficient market. Secondly, mathematical modelling, on which the entire book is based, always seeks maximum generality, so that any mathematical model tries to express every result in such a way that it has the broadest possible conditions of validity, covering not only the known situations, but any other possibility compatible with the assumptions.

An example of this can be found in the derivatives market, in which new types have been emerging for decades, and for which valuation formulas and methods have been developed, usually based on the Option Valuation Theory that emerged in the second half of the last century.

**Q7) Given the increased accessibility of financial markets through mobile applications, how important do you think it is for individual investors to have a basic understanding of Mathematical Finance?**

**Alberto : **Today there are many individual investors who access the markets through their mobile applications, and who base their trades on supposed information that they find through those same devices. Sometimes this information is the result of the application of mathematical models. The problem is that these results are valid under certain assumptions that are not usually made explicit. If the investor does not know these assumptions, he will be applying the results incorrectly.

On the other hand, the increase in the use of mobile devices has also increased the dissemination of incomplete or outright false information. A basic knowledge of Mathematical Finance would help investors to complete the information, or to discern between true and false information.

In my book there are several examples in which mathematical methods allow to determine the falsity of some statements related to the markets that appear on the Internet.

**Q8) In your view, how can a solid understanding of probability and statistical concepts help investors make more informed decisions in financial markets?**

**Alberto : **Financial markets are inherently random. Knowing the price of an asset at an instant, it is not possible to know what the price of that asset will be at a later instant. The most that can be done is to assign a certain probability (which is always unknown, and which in most cases is obtained from historical data) to each possible value.

Given this probabilistic nature of the markets, for someone who regularly trades in them, it may be even more important to have a deep understanding of Probability and Statistics than Mathematical Finance, since while the latter discipline relies heavily on mathematical models (which are always simplified approximations to reality), the first can be based directly on real data, without the need for previous models.

**Q9) The book serves as a gentle introduction to Mathematical Finance, but for readers seeking to apply these concepts practically, what are some key takeaways or real-world applications they can benefit from?**

**Alberto : **The book cannot be understood as a manual for investors, that is, ideas cannot be directly obtained from it to carry out investment strategies. There are already many other books on the market that are dedicated to that with varying degrees of success. I consider that my book falls within the genre of popular science, so that the main benefit that any reader can obtain is to increase their knowledge and satisfy their curiosity.

That being said, as I have commented in a previous answer, an introduction to the mathematics of the markets can help any investor to distinguish the true information from the false and to apply investment techniques knowing their limitations. In addition, it can be the base on which to build deeper knowledge, which can lead to the discovery of investment opportunities.

**Q10) Mathematical Finance is a specialized field that often requires advanced training. For readers who aspire to pursue a career in this area, what additional steps or academic paths would you recommend?**

**Alberto :**A few years ago, it was possible to find a quantitative analyst position at an investment bank with a degree in Physics or Mathematics and a knack for programming. Later, postgraduate degrees in Quantitative Finance emerged, which became an essential requirement in many of these positions. Today the profile has been expanded to include knowledge of Data Science and Machine Learning. To the best of my knowledge, these might be the most appropriate credentials to pursue a career in this field.

**Q11) How do you address the challenge of bridging the gap between theory and practice in financial markets within the context of your book?**

**Alberto : **My book is purely theoretical because that is the facet that I know and that interests me the most. To close this gap between theory and practice there are basically three ways. The first would be self-study. Today there are many audiovisual materials through which it is possible to learn to program and do numerical calculations with the models that appear in the book. A second route would be to take one of the many Masters in Quantitative Finance that are available. These studies tend to put a lot of emphasis on the practical implementation of the models. The third track would be working in the financial industry, which would obviously lead to the most significant learning by doing.

**Q12) The book also features an alphabetical index, which makes it a useful reference for basic concepts. Could you highlight some of the essential terms that readers will find in the index and their significance in understanding financial markets?**

**Alberto : **The alphabetical index is a common feature of academic texts that makes it easy to locate terms that appear in the text and that the reader may have forgotten. Among the many terms that appear in the alphabetical index, I would highlight the following:

Arbitrage: As I have already commented, it is the fundamental concept that underlies Mathematical Finance. In a simplified way, an arbitrage opportunity is a strategy with no net cost and in which it is possible to have profits but no losses. As I have commented before, it is an accepted principle in Mathematical Finance that these opportunities do not exist in the markets.

Martingale: Basically, it is understood as the result of a fair play, so that the best average that we can estimate of its future value, with the information available today, is its current value. It is intimately related to the concept of arbitrage and its role is key in Mathematical Finance.

**Q13) As a passionate advocate of mathematical modelling, how do you think quantitative analysis and mathematical approaches are shaping the future of financial markets?**

**Alberto : **As far as I know, until recently, quantitative techniques were more typical of the valuation and hedging of derivatives, but they are increasingly being incorporated into portfolio management. In addition, algorithmic trading is growing in all markets, which after all is another type of quantitative technique.

In my opinion, quantitative analysis will eventually dominate all areas of financial markets. What is not clear to me is whether this will happen through the classical techniques of Mathematical Finance or through the development of Artificial Intelligence algorithms that learn directly from the data and are not based on mathematical models.

**Q14) In your research and writing process, were there any surprising findings or insights that you discovered about the relationship between mathematics and financial markets?**

**Alberto : **One of the facts that has fascinated me the most, and that stimulates me in my research work, is the ubiquitous presence of mathematical concepts. The same concept or mathematical tool that may have been conceived by a pure mathematician can be found in applications in different branches of knowledge. Thus, for example, I myself have used mathematical concepts such as Dirac deltas, Hilbert spaces or Mercer kernels in my research papers in Mathematical Finance, which can also be found in different applications in Physics or Machine Learning.

The great physicist Eugene Wigner wrote in 1960 a famous paper entitled "The Unreasonable Effectiveness of the Mathematics in the Natural Sciences". I believe that this "unreasonable effectiveness" can be perfectly applied to other fields of science, such as Quantitative Finance.

**Q15) With the ever-increasing complexity of financial products and instruments, how do you envision the role of mathematical finance evolving in the future to meet the demands of the financial industry?**

**Alberto : **As I have said before, one of the facts that guarantee the solidity of the building of Financial Mathematics is that it is based on very general principles that must be true in any market circumstance. All the new developments that have already happened and will happen should have a place in this framework.

That being said, what is not clear to me is the future role of this discipline. The way I see it, there are two long-term possibilities. It is possible that all the new challenges that arise will be addressed within this mathematical scheme, developing new models and concepts that explain them.

The other possibility is that Artificial Intelligence plays that role, through Machine Learning algorithms that respond to new challenges without the need for new mathematical developments. In this case, Financial Mathematics would stop being developed and would remain as one more subject in Quantitative Finance programs, but without practical relevance.

A fact that corroborates the increasing presence of this last approach is that many professors and researchers who dedicated themselves a few years ago to developing advanced mathematical models within this field are beginning to dedicate their efforts to the development of algorithms and models within the field of Artificial Intelligence, with application to Finance. This can be seen in the growing number of papers focused on Machine Learning techniques published in journals such as Mathematical Finance.

**Thank you all for reading and a big thanks to Dr. Alberto for collaborating in today’s post!**

**It’s a pleasure!**