In the preface, the author explains that the book aims to be accessible to—among others—advanced undergraduate or graduate mathematics students. The book serves it purpose perfectly as it balances nicely between introducing the main concepts around financial markets and covering a wide range of relevant mathematical and probabilistic methods.
Starting from the Introduction, this part is very useful for readers not familiar with financial terms. Such readers are mathematics students with an interest, but no strong background in finance. In particular, the definitions of various types of financial contracts such as European put and call options and concepts such as pricing and hedging are essential for the reader to understand the theory and applications in the following chapters.
Chapter 1 focuses on the single-step model. The portfolio problem is introduced along with relevant theory. This theory includes the definition of a risk-neutral probability measure (section 1.3) and the concepts of arbitrage opportunities, contingent claims and market completeness. The last section of the chapter (1.6) provides an application for a simple portfolio consisting of one risky and one risk-free assets.
In Chapters 2 and 3 the portfolio problem is extended to the discrete-time model and the concepts studied in Chapter 1 are studied for this new multi-step model (along with other concepts such as self-financing portfolio). For the purpose of studying market completeness (section 2.5), martingales are introduced in section 2.4. An application used for this model is the Cox-Ross-Rubinstein (CRR) model (Cox, Ross, and Rubinstein, 1973). Pricing and hedging for contingent claims and vanilla options are covered in Chapter 3.
In Chapter 4, the author introduces the Brownian motion (BM) and stochastic calculus which are fundamental for continuous-time models. The portfolio problem for continuous-time models is then defined in Chapter 5, with concepts such as self-financing portfolios, arbitrage etc., being adjusted as to formulate similar theory as for discrete models.
In Chapter 6, under the assumptions of absence of arbitrage and self-financing portfolio, the Black-Scholes (1973) Partial Differential Equation (PDE) is derived and applied for pricing and hedging of European call and put options. The martingale approach to pricing and hedging is studied in Chapter 7.
One of the assumptions of the Black-Scholes model was the constant volatility. In Chapter 8, more general (realistic) models are introduced that allow non-constant volatility. In such models volatility is modeled by time-dependent processes. Methods of how to estimate volatility are studied in Chapter 9.
Chapter 10 derives further results of the BM, such as the reflection principle (section 10.2) and the distributions of the maximum of the BM (section 10.3) and maximum of the geometric BM (section 10.4) over a specific time interval. These results are used in Chapters 11 and 12 for the pricing and hedging of Barrier and Lookback options, respectively.
Chapters 13 and 15 focus on two other types of options, the Asian and American options, respectively. For the latter, stopping times and the Optional Stopping Theorem play an essential role and they are covered in Chapter 14.
In Chapter 16, two important forms of stochastic processes are introduced: the numéraire and forward (deflated) prices. The change of numéraire method (section 16.2) provides a method for the pricing of options under specific assumptions. An application of this method is studied in Chapter 19 for the pricing and hedging of interest rates derivatives.
Chapter 17 focuses in modeling short-term interest rates, while Chapter 18 focused in modeling forward (future) interests rates.
In Chapter 20 the Poisson and compound Poisson processes are defined, aiming to capture the feature of discontinuities (jumps) at random times. Stochastic calculus is then introduced for the jump processes. Jump processes are often used to model asset prices; the pricing and hedging of such models is studied in Chapter 21.
Finally, Chapter 22 provides some numerical methods to solve partial and stochastic differential equations.
The rigorous proofs of all results throughout the book, is very important for a textbook aiming at undergraduates or graduates students. Moreover, at the end of each chapter there is a significant number of exercises, which consist an excellent tool for the reader to evaluate their understanding.
Most mathematics students with an interest in probability, will be familiar with stochastic processes such as Poisson, BM and martingales, by the second or third year of their undergraduate studies. They would most likely be less familiar with financial markets, for example, the different types of financial contracts and their special features. Combining the relevant theories from mathematics and finance is a challenging task. This book manages to do this brilliantly. The order of the chapters, the summary at the start of each chapter, the real-life applications and the continuous linkage between the sections, are some of the factors which make the book reader-friendly for its target population.
To conclude, Introduction to Stochastic Finance with Market Examples (2nd ed.), by N. Privault (2023), would be an excellent choice for a textbook to support units such as Financial Mathematics.
University of Bristol, Bristol, UK
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References
- Black, F., and Scholes, M. (1973), “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, 81, 637–654. DOI: 10.1086/260062.
- Cox, J., Ross, S., and Rubinstein, M. (1979), “Option Pricing: A Simplified Approach,” Journal of Financial Economics, 7, 87–106. DOI: 10.1016/0304-405X(79)90015-1.