Theoretical Foundations for Quantitative Finance

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Luca Spadafora, Gennady P Berman;;;

• Publisher: WSPC
• Year: 2017
• eBook ISBN: 9789813202498
• Topic: Negocios y empresa
• Subtopic: Ingeniería financiera

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This book provides simple introduction to quantitative finance for students and junior quants who want to approach the typical industry problems with practical but rigorous ambition. It shows a simple link between theoretical technicalities and practical solutions. Mathematical aspects are discussed from a practitioner perspective, with a deep focus on practical implications, favoring the intuition and the imagination. In addition, the new post-crisis paradigms, like multi-curves, x-value adjustments (xVA) and Counterparty Credit Risk are also discussed in a very simple framework. Finally, real world data and numerical simulations are compared in order to provide a reader with a simple and handy insight on the actual model performances.

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Chapter 1

Introduction

Quantitative Finance (QF) is a discipline that lies between hard sciences, from which a quantitative approach is taken, and finance that refers to the field of economic modeling. In other words, QF is a field that aims to develop mathematical tools that contribute to a rational description of financial markets, for example, for the estimation of the risk of an investment or of the estimated price to be attributed to a financial instrument. This rational approach is somehow in contrast with the usual perception of financial-related activities that may look completely chaotic and non-scientific. From this perspective, a rational approach to this field could be understood, more than a utopia, as a regular subject with practical implications because of the multitude of hypotheses to be introduced. On the other side, we think that this rational approach is somehow needed as a way to facilitate the communication between financial stakeholders, starting from the commonly accepted rational cornerstone.

In the last decades of the 20th century and in the initial years of the 21st century, many new models for different objectives were developed, with increasing complexity from the mathematical point of view, that contributed to the coherent framework that we call Classical Quantitative Finance framework. Unfortunately, the situation dramatically changed in August 2007, when Lehman Brothers’s firm closed its subprime lender (BNC Mortgage), several crises (e.g. Sovereign crisis in 2011) came in succession, and the financial world understood that its paradigms had to be modified. Nowadays, after about 9 years since that financial crisis, some classical paradigms of QF were reformulated, and a new Modern Quantitative Finance framework has been established.

This book introduces the classical and modern QF topics in a unified framework, discussing which of the old paradigms are still valid in this new financial world and which ones need to be reformulated. The book is organized as follows.

In Chapter 2, we introduce the main mathematical tools used in the following chapters to develop financial models. In particular, it focuses on Probability Theory and Stochastic Calculus. As these topics are quite advanced, a formal description for them would require extensive mathematical review; on the other side, a very high-level introduction (as done in many books on QF) would not give the readers the opportunity to control the mathematical tools and to develop models on their own. For this reason, we preferred to present a formal treatment of the topic, with only an intuitive description of the mathematical tools, explaining why each tool is necessary and what can be done with it. In this respect, we observe that the typical questions asked for Quantitative Analyst positions are usually based on these theoretical tools; for example, interview questions are often raised concerning Itô’s integral definition.

In Chapter 3, we introduce the pricing problem and the main objectives of our models. This aspect is of fundamental importance in our opinion as we have observed that, in many cases, students and also practitioners have a wrong perception of what a pricing model should be able to do and what is not required. In addition, we present a very simple and robust method to price financial instruments, starting from a few hypotheses about the financial markets. In spite of its simplicity, this method is used in practice in many situations and, when applicable, it is usually preferred to more sophisticated models. Finally, in this chapter, we introduce the main financial instruments we consider in the rest of the book.

Chapter 4 contains the description of a very general algorithm to price a financial derivative that represents an alternative approach to the one presented in Chapter 3. Applications of these algorithms are presented in Chapter 5 where Black–Scholes (BS) model and its generalization are discussed. In this chapter, two demonstrations of the same model are presented: one based on a formal approach and one on a more intuitive description. The resulting BS formula is then exploited to obtain a more general framework that is closely related to the models used in practice by professionals. By simple pricing problems, different modeling techniques, such as the change-of-measure and the use of stochastic volatility are presented.

In Chapter 6, we consider the other fundamental topic of QF, i.e. the risk modeling. In particular, we focus on the market risk, and we show different methodologies to estimate the risk of an investment in a rational way. In the same chapter, we also discuss different statistical effects that characterize extreme events and how it is possible to test the performance of a risk model in practice.

Finally, in Chapter 7, we discuss how the classical QF framework can be extended to take into consideration the new context of the modern QF, giving different examples on how pricing formulas have to be changed and risk models have to be adjusted.

The details of derivations of some expressions are presented in Appendix.

Chapter 2

All the Financial Math You Need to Survive with Interesting Applications

2.1. Introduction

In this chapter, we are going to discuss the main mathematical tools that are used in the Quantitative Finance (QF). As the reader could figure out, this field can be potentially huge, and entire books have been written on this [1–3], and the interested reader can find therein all the mathematical details to develop a deep knowledge about the topic.

In this chapter, we would like to mention only the key aspects that can help the reader survive throughout the book, avoiding technical demonstrations or excessive formalism, and following the lines of Ref. [1]. The price to pay for this approach is a knowledge that is mainly based on intuition rather than on a formal basis that could be developed after a general understanding of the topic. On the other hand, we think that the intuition will help the readers develop an understanding of the general framework, allowing them to easily make connections between topics that could seem unrelated at first sight.

2.2. Probability Space

The first mathematical concept we need to introduce is the idea of probability and probability space. This aspect is of fundamental importance for the models we are going to introduce in the following chapters as we are implicitly assuming that the system we want to model, i.e. the financial market, can be efficiently described by the events subject to the probability law. Even if this choice could seem quite natural for practitioners, it should be mentioned from the very beginning that this is an arbitrary choice that will condition the development of the whole mathematical framework for the system representation. In particular, we are going to assume that our knowledge about the system is somehow influenced by uncertain events that cannot be known exactly; as a result, the models we are going to deal with implicitly incorporate this uncertainty in their mathematical formulation.

In general, people have quite an intuitive understanding that the probability concept is strongly related to the idea of frequency. From a classical point of view, probability was defined as the ratio of favorable outcomes to the total number of possible outcomes, assuming that the latter have all the same probability. This definition is quite simple, but unfortunately has some drawbacks:

• It is a circular definition, i.e. we used the concept of the “same probability” to define the probability itself.

• It does not tell anything when events do not have the same probability.

• It works only if the total number of possible outcomes is finite.

In order to avoid these issues, some generalization is needed, requiring the formulation of a theoretical framework based on the measure theory. Actually, it is quite natural to define the probability as a measure of how likely an event is to occur. From this perspective, we observe that, in order to obtain a formal probability framework, we need to specify three fundamental elements:

• what we consider to be our whole set of events,

• all the combinations of events we consider of interest and we want to measure,

• how we want to measure the probability of an event.

The first element is quite natural. If we want to talk about the probability that an event occurs, we should be able, at least, to specify the events among which we want to move, the ones we want to describe and measure. We define this set as Ω, and we will call it a set of scenarios, meaning that events are properly defined subsets of Ω.

It is quite natural to expect that if one is able to measure two elements separately, one should also be able to measure the union of these two elements. In a probabilistic sense, if one is able to measure how likely two events are separately, it is quite natural to expect that one should be able to say how likely they are together. This requirement could hide some difficulties fr...

Table of contents

• Cover page
• Title page
• Copyright
• Preface
• Contents
• Chapter 1. Introduction
• Chapter 2. All the Financial Math You Need to Survive with Interesting Applications
• Chapter 3. The Pricing of Financial Derivatives — The Replica Approach
• Chapter 4. Risk-Neutral Pricing
• Chapter 5. The Black and Scholes Framework and Its Extensions
• Chapter 6. Risk Modeling
• Chapter 7. The New Post-Crisis Paradigms
• Appendix
• Bibliography
• Index