The Role of Mathematics in Finance: A Historical Perspective and Current Developments, Study notes of Mathematics

Download The Role of Mathematics in Finance: A Historical Perspective and Current Developments and more Study notes Mathematics in PDF only on Docsity! Mathematics in Finance Riaz Ahmad These days it is hard to escape financial news; whether we watch the news reports on TV, breeze through the newspapers, or look at our hand-held devices. Terms such as derivatives, LIBOR, short selling, quantitative easing or FTSE 100 are constantly reminding us that finance is the most global of industries. The financial crisis has been the chief headline across worldwide news bulletins and remains a fierce topic of discussion, with its effects still conspicuous today. There has never been a more crucial time for understanding the underlying mechanics of the complex products traded in the markets, together with a responsible approach to managing the associated risk. Undeniably, the way finance has developed in recent years can be attributed in large part to mathematics, which has played a central role and remains the chief driving force allowing the financial markets to become increasingly sophisticated. There is little doubt that mathematics has ‘hijacked’ most disciplines and its appeal and influence is noticeable in most branches of knowledge. In addition to the traditional areas of scholarship that depend on maths for its framework; less obvious academic themes are also enjoying the tangible advances being made due to the reliance on maths, such as Political Science, Medical Research and Sociology. ‘Quantitative Finance’ as a branch of modern finance continues to be one of the fastest-growing areas within the corporate world. The sophistication and complexity of modern financial products, has acted as the motivating factor for new mathematical models and the subsequent development of associated computational schemes. Natural and Behavioural Sciences Social Sciences, Arts and Humanities Physical and Engineering Sciences Medical and Clinical Sciences MATHS Physics; Civil, Electronic & Aeronautical Engineering Biology, Chemistry, Geology, Psychology Materials Science, Oceanography, Astronomy Radiology, Nuclear Medicine Physiology, Oncology, Cardiology, Neurology Economics, Sociology Human Geography History, Music History of Science Genetics, Medical Ethics independent format (or pseudo code), followed by writing a computer program(s). 3. Analysis and interpretation of the result: The most important part is studying the results obtained and understanding them. In addition being able to explain the output to both a technical and non-scientific audience, so a solid understanding of the finance based principles is equally important in the concluding stages. This requires a solid foundation and confidence in the use of the relevant branches of mathematics. Quantitative Finance embraces the complete range of pure and applied mathematical subjects, which include probability and statistics, partial differential equations, numerical analysis, computation and operational research. Consequently an extraordinary number of quantitative-based scientists from a wide variety of backgrounds have moved into this area of research. In addition, the interdisciplinary nature of this subject matter has meant successful collaborative work being conducted by mathematicians, economists, finance professionals, theoretical physicists, and computer scientists. Even the psychologists are now playing a role through behavioural finance. Unless a mathematical modeling problem is ideal/simple, it is unlikely that an analytical/closed form solution can be obtained. As with all industrial advances, which rely on information technology, the field of finance (in particular Financial Engineering) has benefited from the availability of computing power and programming design; enabling mathematicians to study increasingly difficult problems. There is no doubt that since the nineties Object Oriented Programming (OOP) has created much excitement. As a branch of software development OOP has been a major innovative theme, and C++ is the most widely used language, which supports the object-oriented paradigm. OOP is a design philosophy that superseded languages such as Pascal and C; defined broadly as procedural/structured programming this was the principle of reducing a code into smaller and independently functioning parts. It supported efficient programming when applied to moderately complex systems. With the requirement of larger and further complex programs, this mode of programming was not so effective. Whilst OOP has inherited the best ideas from structured programming, what makes it vastly different is that it encourages the decomposition of the problem into related subgroups or self-sustainable ‘objects’. Each object contains its own related data and instructions. This functionality promotes reusability and maintainability thus reducing the overall complexity of the problem. In the financial markets C++ continues to retain its status as a ‘sexy’ language and arguable the most popular language in the financial markets. However what makes the field of Quantitative Finance so exciting and dynamic is the fast pace at which it detects and adopts new technologies. The programming language Python is rapidly becoming the standard in scientific computing, receiving much excitement about the application to mathematical finance; its appeal continues to grow in both academia and industry. It is simple to use; available on multiple platforms; easy to maintain; free to download and has a growing amount of add-on modules. It is particularly easy to interface with C++. In the last two decades, there has been great interest in acquiring knowledge in financial mathematics, ranging from one-term university modules to lengthier taught course programmes. Programmes that are aimed at leading graduates towards technical careers which include quantitative analysts (quants), quant developers, and quant traders, in investment banks, hedge funds and other financial institutions. Advanced instruction that is both demanding in mathematics and related to practice, concurrently, has become a joint concern and a success factor for both educational bodies and the capital markets. This was the motivating factor behind the decision by the mathematics department at UCL to create a MSc Financial Mathematics degree; the first cohort admitted in autumn 2012. The emphasis in the new UCL Financial Mathematics programme is to develop mathematical skills, programming proficiency and confidence in exploring financial data. The department also offers two financial maths modules at the 3rd year BSc. and 4th year MSci levels. Those familiar with the Michael Moore documentary, Capitalism A Love story will recall the scene where Moore stands outside the New York Stock Exchange asking for an explanation of what a derivative is. There are three broad classes of financial derivative: 1. Futures and Forwards 2. Options 3. Swaps Undoubtedly it is derivatives that gives the field of Quant Finance its attraction, mystique and fear in equal measures – invoking many emotions. The right to buy or sell a financial asset in the future for a predefined amount is not always intuitive. So here’s an example. Imagine you'd like to sell your house in, say, a year's time. Of course you don't know what the market will be doing during that period, and you want to be sure that you'll achieve at least the current value of the house, let's say $1,000,000. Now suppose I come along and am interested in buying your house in a year's time. I want to be sure that I don't pay much more than its current value. I can enter into a contract with you, called an option, which stipulates that on a specified date in a year's time – in this case a year from now (called the expiry), I can buy the house for $1,000,000 (called the strike price ) if I choose to. For this contract, which derives its value from the price of the house (the house is the underlying), I pay you a fee, called the option price (or premium), say $104505. I am essentially paying you for allowing me the privilege of locking in at today's price. I thus have the option of buying your house if I so choose, in which case you then have an obligation to sell me the house. If I decide not to exercise this right, that is perfectly fine, as the choice is mine - you will still keep the $104505 fee which is paid at the time we enter into the contract. Each party in this contract has a name. You are the writer (and have obligations) and I am the holder (and have choices). At the end of the one year period, we will be faced by one of two possible scenarios; the house value will have decreased or increased. If in a year's time the value of the house has gone down to $950,000, I certainly won't pay £1M for something which is worth £950,000 in the market, so you will have the house and the contract fee.