Mathematical finance

Mathematical finance is a field of applied mathematics, concerned with financial markets. The subject has a close relationship with the discipline of financial economics, which is concerned with much of the underlying theory. Generally, mathematical finance will derive, and extend, the mathematical or numerical models suggested by financial economics. Thus, for example, while a financial economist might study the structural reasons why a company may have a certain share price, a financial mathematician may take the share price as a given, and attempt to use stochastic calculus to obtain the fair value of derivatives of the stock.

In terms of practice, mathematical finance also overlaps heavily with the field of computational finance (also known as financial engineering). The fundamental theorem of arbitrage-free pricing is one of the key theorems in mathematical finance. Many universities around the world now offer degree and research programs in mathematical finance.

The history of mathematical finance starts with The Theory of Speculation (published 1900) by Louis Bachelier, which discussed the use of Brownian motion to evaluate stock options. However, it hardly caught any attention outside academia. The first influential work of mathematical finance is the theory of portfolio optimization by Harry Markowitz on using mean-variance estimates of portfolios to judge investment strategies, causing a shift away from the concept of trying to identify the best individual stock for investment. Using a linear regression strategy to understand and quantify the risk (i.e. variance) and return (i.e. mean) of an entire portfolio of stocks and bonds, an optimization strategy was used to choose a portfolio with largest mean return subject to acceptable levels of variance in the return. Simultaneously, William Sharpe developed the mathematics of determining the correlation between each stock and the market. For their pioneering work, Markowitz and Sharpe, along with Merton Miller, shared the 1990 Nobel Memorial Prize in Economic Sciences, for the first time ever awarded for a work in finance.

The next major revolution in mathematical finance came with the work of Fischer Black and Myron Scholes along with fundamental contributions by Robert C. Merton, by modeling financial markets with stochastic models. For this M. Scholes and R. Merton were awarded the 1997 Nobel Memorial Prize in Economic Sciences. Black was ineligible for the prize because of his death in 1995.

More sophisticated mathematical models and derivative pricing strategies were then developed but their credibility was damaged by the financial crisis of 2007-2010. Bodies such as the Institute for New Economic Thinking are now attempting to establish more effective theories and methods.

Financial modeling

Financial modeling is the task of building an abstract representation (a model) of a financial decision making situation. This is a mathematical model designed to represent the performance of a financial asset or a portfolio, of a business, a project, or any other investment. Financial modeling is a general term that means different things to different users; the reference usually relates either to accounting applications, or to quantitative finance applications. While there has been some debate in the industry as to the nature of financial modeling – whether it is a tradecraft, such as welding, or a science - the task of financial modeling has been gaining acceptance and rigor over the years. Several scholarly books have been written on the topic, in addition to numerous scientific articles.

In corporate finance, investment banking and the accounting profession financial modeling is synonymous with cash flow forecasting. This usually involves the preparation of large, detailed company specific models used for decision making purposes. Applications include:

Business valuation, especially discounted cash flow, but including other valuation problems;

Scenario planning and management decision making «what is»; «what if»; «what has to be done»;

Capital budgeting;

Cost of capital calculations;

Financial analysis and / or Financial statement analysis;

Project finance.

These models are generally built around financial statements, and therefore outputs, and calculations, are typically monthly or quarterly. This means that accounting based financial models are mathematically (at least implicitly) in discrete time. Model inputs often take the form of “assumptions,” where the analyst specifies the values that will apply in each period for external / global variables (exchange rates, tax percentage, etc.) and internal / company specific variables (wages, unit costs, etc.). Given that these input values are specified, the models are mathematically (implicitly) deterministic.

In quantitative finance, financial modeling entails the development of a sophisticated mathematical model. Models here deal with asset prices, market movements, portfolio returns and the like. Applications include:

 Option pricing;

 Modeling the term structure of interest rates (short rate modeling) and credit spreads;

 credit scoring and provisioning;

 Portfolio problems;

 Real options;

 Risk modeling.

These problems are often stochastic and continuous in nature, and models here thus require complex algorithms, entailing computer simulation, advanced numerical methods such as numerical differential equations, and / or the development of optimization models.

Modelers are generally referred to as «quants» (quantitative analysts), and typically have strong (Ph.D. level) backgrounds in quantitative disciplines such as physics, computer science, mathematics or engineering. Alternatively, they have completed a finance masters with a quantitative orientation – such as the Master of Quantitative Finance or Master of Finance – or the more specialized Master of Computational Finance or Master of Financial Engineering.

Although spreadsheets (e.g. Microsoft Excel) are widely used here, custom C++ or numerical analysis software such as MATLAB is often preferred, particularly where stability or speed is a concern. Additionally, for many derivative and portfolio applications, commercial software is available, and the choice as to whether the model is to be developed in-house, or whether existing products are to be deployed, will depend on the problem in question.

Criticism of financial modeling emphasizes the differences between the mathematical and physical sciences and finance. Some experts go further and question whether mathematical and statistical modeling may be applied to finance at all, at least with the assumptions usually made (for options, portfolios, etc.).

Task 1. Fill in the table with the words and phrases from the text and give their Russian equivalents:

General Science	Mathematics	Finance
master – магистр	variable – переменная	interest rate – процентная ставка

Text 6. Make the written translation into Russian (time 90 minutes)