Problems and solutions in mathematical finance volume 2: equity derivatives.

Abstract

Mathematical finance is a highly challenging and technical discipline. Its fundamentals and applications are best understood by combining a theoretically solid approach with extensive exercises in solving practical problems. That is the philosophy behind all four volumes in this series on mathematical finance. This second of four volumes in the series Problems and Solutions in Mathematical Finance is devoted to the discussion of equity derivatives. In the first volume we developed the probabilistic and stochastic methods required for the successful study of advanced mathematical finance, in particular different types of pricing models. The techniques applied in this volume assume good knowledge of the topics covered in Volume 1. As we believe that good working knowledge of mathematical finance is best acquired through the solution of practical problems, all the volumes in this series are built up in a way that allows readers to continuously test their knowledge as they work through the texts. This second volume starts with the analysis of basic derivatives, such as forwards and futures, swaps and options. The approach is bottom up, starting with the analysis of simple contracts and then moving on to more advanced instruments. All the major classes of options are introduced and extensively studied, starting with plain European and American options. The text then moves on to cover more complex contracts such as barrier, Asian and exotic options. In each option class, different types of options are considered, including time-independent and time-dependent options, or non-path-dependent and path-dependent options. Stochastic financial models frequently require the fixing of different parameters. Some can be extracted directly from market data, others need to be fixed by means of numerical methods or optimisation techniques. Depending on the context, this is done in differentways. In the riskneutral world, the drift parameter for the geometric Brownian motion (Black–Scholes model) is extracted from the bond market (i.e., the returns on risk-free debt). The volatility parameter, in contrast, is generally determined from market prices, as the so-called implied volatility. However, if a stochastic process is to be fitted to known price data, other methods need to be consulted, such as maximum-likelihood estimation. This method is applied to a number of stochastic processes in the chapter on volatility models. In all option models, volatility presents one of the most important quantities that determine the price and the risk of derivatives contracts. For this reason, considerable effort is put into their discussion in terms of concepts, such as implied, local and stochastic volatilities, as well as the important volatility surfaces. At the end of this volume, readers will be equipped with all the major tools required for the modelling and the pricing of a whole range of different derivatives contracts. They will therefore be ready to tackle new techniques and challenges discussed in the next two volumes, including interest-rate modelling in Volume 3 and foreign exchange/commodity derivatives in Volume 4. As in the first volume, we have the following note to the student/reader: Please try hard to solve the problems on your own before you look at the solutions!