3. The classical theory of option pricing

‘The classical theory of option pricing’ explains the theory of arbitrage pricing, which is closely related to the Dutch Book Arguments, but which brings in a new factor: prices in financial markets evolve over time and participants are able to trade at any time, instead of just taking bets and awaiting the result. In addition to the general theory, pricing models and methods have been developed for specific markets—foreign exchange, interest rates, and credit. The binomial and continuous-time mathematical models for stock prices are introduced along with the Black–Scholes formula, the volatility surface, the difference between European and American options, and the Fundamental Theorem of Asset Pricing.

1. Money, banking, and financial markets

‘Money, banking, and financial markets’ describes the playing field of mathematical finance: the world of money, banking, and financial markets. It considers the history of money as a medium of exchange and as a store of value, before discussing the activities of banks and financial markets. The financial markets, as a whole, provide a myriad of mechanisms for raising funds and managing and trading the attendant risks, providing opportunities for investors, fund managers and speculators. The main mechanisms of financial mechanisms are outlined: equities; bonds; credit risk; foreign exchange; and forwards, futures, and options.

8. The banking crisis and its aftermath

What happened to the banks in 2008 and what has been the fallout for mathematical finance? ‘The banking crisis and its aftermath’ explains that for mathematical finance it has meant a complete reorientation. The traditional topics of pricing and arbitrage theory live on, and are still important, but are no longer the sole focus of research since fundamentally the theory is complete, and also because most of the more ‘exotic’ contracts have become unviable in the new regulatory environment. On the other hand, a whole raft of new questions concerning the behaviour and interaction of markets has opened up, and the implications of new technology—FinTech—must be taken on board.

7. Risk management

The risk management function of a financial company monitors a whole range of risks that the company faces: market risk, credit risk, liquidity risk, operational risk, reputational risk, and legal risk. Some of these are connected to regulatory requirements, while others are internal procedures designed to assist the management of the company’s assets and liabilities. ‘Risk management’ focuses on market risk, which is concerned with assessing how sensitive the value of the company’s trading book is to anticipated movements in the market prices of the assets it contains. Evaluations are carried out at various levels of aggregation from individual trading desk to the company as a whole.

2. Quantifying risk

A large part of the purpose of the financial markets is to manage, and indeed profit from, risk—the uncertainty surrounding future events, any event, in fact, that affects the views of investors as to the value of securities. ‘Quantifying risk’ reviews the main ideas of probability theory, explaining random variables, normal distribution, standard deviation, strong law of large numbers, and the central limit theorem. It also discusses ‘subjective probability’ or probability as degree of belief, which are based around so-called Dutch Book Arguments that concern the existence or otherwise of arbitrage strategies. It concludes with an explanation of stochastic modelling.

6. Fund management

‘Fund management’ discusses the objective to form portfolios of assets so as to maximize the investment return. A mathematical finance-oriented approach to optimal investment, in the context of the Black–Scholes price model, was proposed by Robert Merton in 1969. Fund management is a huge industry, and has become much more technical with the emergence of hedge funds deploying sophisticated strategies. There have been many attempts at constructing mathematical models for asset allocation that match real market behaviour more closely. The basic problem is that markets appear so erratic. Is there anything about them that is more invariant? The scenario tree model for long-term asset liability management is explained.

5. Credit risk

Credit risk is the risk that your counterparty might default on future obligations. There are a small number of credit rating agencies operating globally that assign a credit rating to each company under consideration. ‘Credit risk’ explains credit risk modelling and analysis, including credit default swaps, multi-asset credit risk, and collateralized debt obligations. Credit risk models are divided into two main categories: ‘structural form’ and ‘reduced form’. A pervasive problem in credit risk modelling is that while some parameters can be backed out by the calibration process, there are usually others about which the available data is insufficient for us to do anything more than take an educated guess.

4. Interest rates

In ordinary life, there are many different interest rates—mortgage rates, rates on credit cards, and rates paid by savings accounts—which are mirrored in the commercial world by loans for short-term funding and longer term loans for business development and project finance. The reasons for so many different rates are market conditions and credit risk. ‘Interest rates’ discusses the trading of interest rates, focusing on the interbank market, which is the area where most trading takes place, most quant effort is deployed and in whose development mathematical finance has played a key role. The basic terminology—including zero coupon bonds, swap rate, yield curve generator, LIBOR rates, and whole yield models—are explained.