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graduate or M.B.A. course such as “Quantitative Methods for Finance,” . book's primary focus is the mathematics and quantitative technique required to cre-. PDF | Quantitative analysts or “Quants” are a source of competitive advantage for financial institutions. They occupy the relatively powerful but often. Regression-Based Hedge Ratios. I Trading on Regression Models. I Summary and Conclusions. I.5 Numerical Methods in Finance.

History[ edit ] Robert C. Merton , one of the pioneers of quantitative analysis, promoted stochastic calculus into the study of finance. Quantitative finance started in with Louis Bachelier 's doctoral thesis Theory of Speculation, which provided a model to price options under a Normal Distribution. Harry Markowitz 's doctoral thesis "Portfolio Selection" and its published version was one of the first efforts in economics journals to formally adapt mathematical concepts to finance mathematics was until then confined to mathematics, statistics or specialized economics journals. He showed how to compute the mean return and variance for a given portfolio and argued that investors should hold only those portfolios whose variance is minimal among all portfolios with a given mean return.

Historically this was a distinct activity from trading but the boundary between a desk quantitative analyst and a quantitative trader is increasingly blurred, and it is now difficult to enter trading as a profession without at least some quantitative analysis education.

In the field of algorithmic trading it has reached the point where there is little meaningful difference. Front office work favours a higher speed to quality ratio, with a greater emphasis on solutions to specific problems than detailed modeling.

FOQs typically are significantly better paid than those in back office, risk, and model validation. Although highly skilled analysts, FOQs frequently lack software engineering experience or formal training, and bound by time constraints and business pressures, tactical solutions are often adopted.

Quantitative investment management[ edit ] Quantitative analysis is used extensively by asset managers. Some, such as FQ, AQR or Barclays, rely almost exclusively on quantitative strategies while others, such as Pimco, Blackrock or Citadel use a mix of quantitative and fundamental methods. Library quantitative analysis[ edit ] Major firms invest large sums in an attempt to produce standard methods of evaluating prices and risk. LQs spend more time modeling ensuring the analytics are both efficient and correct, though there is tension between LQs and FOQs on the validity of their results.

LQs are required to understand techniques such as Monte Carlo methods and finite difference methods , as well as the nature of the products being modeled.

Algorithmic trading quantitative analyst[ edit ] Often the highest paid form of Quant, ATQs make use of methods taken from signal processing , game theory , gambling Kelly criterion , market microstructure , econometrics , and time series analysis. Algorithmic trading includes statistical arbitrage , but includes techniques largely based upon speed of response, to the extent that some ATQs modify hardware and Linux kernels to achieve ultra low latency.

Risk management[ edit ] This has grown in importance in recent years, as the credit crisis exposed holes in the mechanisms used to ensure that positions were correctly hedged, though in no bank does the pay in risk approach that in front office.

A core technique is value at risk , and this is backed up with various forms of stress test financial , economic capital analysis and direct analysis of the positions and models used by various bank's divisions. Innovation[ edit ] In the aftermath of the financial crisis, there surfaced the recognition that quantitative valuation methods were generally too narrow in their approach.

An agreed upon fix adopted by numerous financial institutions has been to improve collaboration. Model validation[ edit ] Model validation MV takes the models and methods developed by front office, library, and modeling quantitative analysts and determines their validity and correctness. The MV group might well be seen as a superset of the quantitative operations in a financial institution, since it must deal with new and advanced models and trading techniques from across the firm.

Before the crisis however, the pay structure in all firms was such that MV groups struggle to attract and retain adequate staff, often with talented quantitative analysts leaving at the first opportunity. This gravely impacted corporate ability to manage model risk, or to ensure that the positions being held were correctly valued. An MV quantitative analyst would typically earn a fraction of quantitative analysts in other groups with similar length of experience.

In the years following the crisis, this has changed. Regulators now typically talk directly to the quants in the middle office such as the model validators, and since profits highly depend on the regulatory infrastructure, model validation has gained in weight and importance with respect to the quants in the front office.

More recently, banks have started to manage credit risks actively by transferring them to the capital markets instead of warehousing them. Bonds are replacing loans, mortgages and other loans are securitized, and many of the remaining credit risks can now be covered with credit default swaps. Thus credit risks are being converted into market risks. The rapid development of capital markets and, in particular, of derivative products bears witness to these changes.

These derivative markets are zero-sum games; they are all about market risk management hedging, arbitrage and speculation. This does not mean, however, that all market risk management problems have been resolved. We may have developed the means and the techniques, but we do not necessarily 23 Foreword xxi understand how to address the problems. Regulators and other experts setting standards and policies are particularly concerned with several fundamental issues. To name a few: 1.

How do we decide what market risks should be assessed and over what time horizons? For example, should the loan books of banks or long-term liabilities of pension funds be marked to market, or should we not be concerned with pricing things that will not be traded in the near future?

We think there is no general answer to this question about the most appropriate description of risks.

The descriptions must be adapted to specific management problems. In what contexts should market risks be assessed? Thus, what is more risky, fixed or floating rate financing? Answers to such questions are often dictated by accounting standards or other conventions that must be followed and therefore take on economic significance.

But the adequacy of standards must be regularly reassessed. To wit, the development of International Accounting Standards favouring mark-to-market and hedge accounting where possible whereby offsetting risks can be reported together. To what extent should risk assessments be objective? Modern regulations of financial firms Basel II Amendment, have been a major driver in the development of risk assessment methods.

Regulators naturally want a level playing field and objective rules. This reinforces a natural tendency to assess risks purely on the basis of statistical evidence and to neglect personal, forward-looking views. Thus one speaks too often about risk measurements as if risks were physical objects instead of risk assessments indicating that risks are potentialities that can only be guessed by making a number of assumptions i.

Regulators try to compensate for this tendency by asking risk managers to draw scenarios and to stress-test their models. There are many other fundamental issues to be debated, such as the natural tendency to focus on micro risk management because it is easy rather than to integrate all significant risks and to consider their global effect because that is more difficult.

In particular, the assessment and control of systemic risks by supervisory authorities is still in its infancy. But I would like to conclude by calling attention to a particular danger faced by a nascent market risk management profession, that of separating risks from returns and focusing on downside-risk limits. It is central to the ethics of risk managers to be independent and to act with integrity. Thus risk managers should not be under the direct control of line managers of profit centres and they should be well remunerated independently of company results.

But in some firms this is also understood as denying risk managers access to profit information. I remember a risk commission that had to approve or reject projects but, for internal political reasons, could not have any information about their expected profitability. For decades, credit officers in most banks operated under such constraints: they were supposed to accept or reject deals a priori, without knowledge of their pricing.

Times have changed. We understand now, at least in principle, that the essence of risk management is not simply to reduce or control risks but to achieve an optimal balance between risks and returns.

Yet, whether for organizational reasons or out of ignorance, risk management is often confined to setting and enforcing risk limits.

Most firms, especially financial firms, claim to have well-thought-out risk management policies, but few actually state trade-offs between risks and returns. Attention to risk limits may be unwittingly reinforced by regulators. Of course it is not the role of the supervisory authorities to suggest risk return trade-offs; so supervisors impose risk limits, such as value at risk relative to capital, to ensure safety and 24 xxii Foreword fair competition in the financial industry.

But a regulatory limit implies severe penalties if breached, and thus a probabilistic constraint acquires an economic value. Banks must therefore pay attention to the uncertainty in their value-at-risk estimates.

The effect would be rather perverse if banks ended up paying more attention to the probability of a probability than to their entire return distribution. With Market Risk Analysis readers will learn to understand these long-term problems in a realistic context. Carol is an academic with a strong applied interest. She has helped to design the curriculum for the Professional Risk Managers International Association PRMIA qualifications, to set the standards for their professional qualifications, and she maintains numerous contacts with the financial industry through consulting and seminars.

In Market Risk Analysis theoretical developments may be more rigorous and reach a more advanced level than in many other books, but they always lead to practical applications with numerous examples in interactive Excel spreadsheets. In summary, if there is any good reason for not treating market risk management as a separate discipline, it is that market risk management should be the business of all decision makers involved in finance, with primary responsibilities on the shoulders of the most senior managers and board members.

However, there is so much to be learnt and so much to be further researched on this subject that it is proper for professional people to specialize in it. These four volumes will fulfil most of their needs. They only have to remember that, to be effective, they have to be good communicators and ensure that their assessments are properly integrated in their firm s decision-making process.

Its development began during the s, spurred on by the first Basel Accord, between the G10 countries, which covered the regulation of banking risk. Over the past 30 years banks have begun to understand the risks they take, and substantial progress has been made, particularly in the area of market risks. Here the availability of market data and the incentive to reduce regulatory capital charges through proper assessment of risks has provided a catalyst to the development of market risk management software.

Nowadays this software is used not only by banks, but also by asset managers, hedge funds, insurance firms and corporate treasurers. Understanding market risk is the first step towards managing market risk. Yet, despite the progress that has been made over the last 30 years, there is still a long way to go before even the major banks and other large financial institutions will really know their risks.

At the time of writing there is a substantial barrier to progress in the profession, which is the refusal by many to acknowledge just how mathematical a subject risk management really is. Asset management is an older discipline than financial risk management, yet it remains at a less advanced stage of quantitative development.

Unfortunately the terms equity analyst, bond analyst and more generally financial analyst are something of a misnomer, since little analysis in the mathematical sense is required for these roles. I discovered this to my cost when I took a position as a bond analyst after completing a postdoctoral fellowship in algebraic number theory.

One reason for the lack of rigorous quantitative analysis amongst asset managers is that, traditionally, managers were restricted to investing in cash equities or bonds, which are relatively simple to analyse compared with swaps, options and other derivatives.

Also regulators have set few barriers to entry. Almost anyone can set up an asset management company or hedge fund, irrespective of their quantitative background, and risk-based capital requirements are not imposed. Instead the risks are borne by the investors, not the asset manager or hedge fund.

The duty of the fund manager is to be able to describe the risks to their investors accurately. Fund managers have been sued for not doing this properly. But a legal threat has less impact on good practice than the global regulatory rules that are imposed on banks, and this is why risk management in banking has developed faster than it has in asset management. Still, there is a very long way to go in both professions before a firm could claim that it has achieved best practice in market risk assessment, despite the claims that are currently made.

At the time of writing there is a huge demand for properly qualified financial risk managers and asset managers, and this book represents the first step towards such qualification. With this book readers will master the basics of the mathematical subjects that lay the foundations 26 xxiv Preface for financial risk management and asset management.

Readers will fall into two categories. The first category contains those who have been working in the financial profession, during which time they will have gained some knowledge of markets and instruments.

But they will not progress to risk management, except at a very superficial level, unless they understand the topics in this book. The second category contains those readers with a grounding in mathematics, such as a university degree in a quantitative discipline. Readers will be introduced to financial concepts through mathematical applications, so they will be able to identify which parts of mathematics are relevant to solving problems in finance, as well as learning the basics of financial analysis in the mathematical sense and how to apply their skills to particular problems in financial risk management and asset management.

The level should be accessible to anyone with a moderate understanding of mathematics at the high school level, and no prior knowledge of finance is necessary. For ease of exposition the emphasis is on understanding ideas rather than on mathematical rigour, although the latter has not been sacrificed as it is in some other introductory level texts.

Illustrative examples are provided immediately after the introduction of each new concept in order to make the exposition accessible to a wide audience.

Some other books with similar titles are available.

These tend to fall into one of two main categories: Those aimed at quants whose job it is to price and hedge derivative products. These books, which include the collection by Paul Wilmott , , focus on continuous time finance, and on stochastic calculus and partial differential equations in particular. They are usually written at a higher mathematical level than the present text but have fewer numerical and empirical examples.

Those which focus on discrete time mathematics, including statistics, linear algebra and linear regression. Among these books are Watsham and Parramore and Teall and Hasan , which are written at a lower mathematical level and are less comprehensive than the present text.

Continuous time finance and discrete time finance are subjects that have evolved separately, even though they approach similar problems. As a result two different types of notation are used for the same object and the same model is expressed in two different ways.

One of the features that makes this book so different from many others is that I focus on both continuous and discrete time finance, and explain how the two areas meet.

Although the four volumes of Market Risk Analysis are very much interlinked, each book is self-contained. This book could easily be adopted as a stand-alone course text in quantitative finance or quantitative risk management, leaving more advanced students to follow up cross references to later volumes only if they wish. Because finance is the study of the behaviour of agents operating in financial markets, it has a lot in common with economics.

This is a so-called soft science because it attempts to model the behaviour of human beings. Human behaviour is relatively unpredictable compared with repetitive physical phenomena. Hence the mathematical foundations of economic and econometric models, such as utility theory and regression analysis, form part of the essential mathematical toolkit for the financial analyst or market risk manager.

Also, since the prices of liquid financial instruments are determined by demand and supply, they do not obey precise rules of behaviour with established analytic solutions. As a result we must often have recourse to numerical methods to resolve financial problems. Of course, to understand these subjects fully we must first introduce readers to the elementary concepts in the four core mathematics subjects of calculus, linear algebra, probability and statistics.

Besides, these subjects have far-reaching applications to finance in their own right, as we shall see. The introduction to Chapter 1, Basic Calculus for Finance, defines some fundamental financial terminology. Then the chapter describes the mathematics of graphs and equations, functions of one and of several variables, differentiation, optimization and integration. We use these concepts to define the return on a portfolio, in both discrete and continuous time, discrete and continuous compounding of the return on an investment, geometric Brownian motion and the Greeks of an option.

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