Introduction to stochastic finance
Stochastic finance, as an important discipline, aims to investigate theoretical and applied problems, which are often motivated from practices in the finance industry, using advanced theories and techniques in probability theory and stochastic analysis. It provides an elegant and beautiful mathematical language to describe and understand randomness attributable to activities in financial markets and the economy. It throws light on deepening the understanding of randomness in financial markets. The landmark contributions by \textit{J. M. Harrison} and \textit{D. M. Kreps} [J. Econ. Theory 20, 381--408 (1979; Zbl 0431.90019)], \textit{J. M. Harrison} and \textit{S. R. Pliska} [Stochastic Processes Appl. 11, 215--260 (1981; Zbl 0482.60097); Stochastic Processes Appl. 15, 313--316 (1983; Zbl 0511.60094)] laid the foundations of stochastic finance, where the relationship between the economic notion of no-arbitrage and the martingale theory was explicitly and rigorously established. This is coined as the fundamental theorem of asset pricing which is at the heart of stochastic finance. The general theory of processes and the semimartingale calculus pioneered by the ``Strasbourg school'' led by Paul-André Meyer have been widely employed in stochastic finance. See [\textit{J. M. Harrison} and \textit{S. R. Pliska}, Stochastic Processes Appl. 11, 215--260 (1981; Zbl 0482.60097)] and this monograph by Yan (2018). It was also noted by some other authors, for example, \textit{C.F. Huang} [``Continuous-time stochastic processes'' in: The new palgrave: finance. London: Palgrave Macmillan. 110--118 (1989)] and \textit{F. Delbaen} and \textit{W. Schachermayer} [Math. Ann. 300, No. 3, 463--520 (1994; Zbl 0865.90014)], that the importance for the modern finance theory of the general theory of processes may give an impression that the general theory of processes is specially designed for the modern finance theory. Though the treatments of problems in stochastic finance may sometimes lean on the style of pure mathematics, say the developments of statements, theses and knowledge from axioms via rigorous deductive systems, stochastic finance may not be purely mathematical for mathematical objects studied in the discipline are often given meanings and interpretations from the perspectives of economics and finance. The situation might not be very unlike geometry, some of whose mathematical objects, as pointed out by \textit{B. Russell} [The analysis of matter. London; Kegan, Trench, Trübner \& Co (1927; JFM 53.0052.05)], may have intimate links with concepts in physical sciences. It is a pleasant and enjoyable task to write a review on this monograph on stochastic finance. The monograph, authored by a leading authority in the field, provides a fine, elegant and comprehensive treatment to the discipline. It covers a good range of fundamental and important topics such as asset pricing, portfolio selection, derivative pricing and hedging, modelling term structures of interest rate and risk measures. The uses of martingale theory and Itô's stochastic calculus to discuss these topics are emphasized. The author strikes a good balance between the rigorousness and accessibility of the materials presented in the monograph. This makes the monograph not only an excellent reference for researchers, but also a fantastic text from the pedagogical perspective. To make the monograph self-contained, the authors present some essential materials in probability theory, martingale theory and Itô's stochastic calculus. The monograph is a wonderful text for graduate courses in mathematical finance and related fields. It appears that, from the pedagogical perspective, it would be helpful if some end-of-chapter exercises are provided. The materials presented in the monograph are organised in a thoughtful way. It appears that the topics presented may be roughly grouped into three parts. The first part may consist of Chapters 1--3 where discrete-time models are considered. Chapters 4--9 focuses on continuous-time models. Some advanced topics are discussed in Chapters 10--14. A brief chapter-by-chapter review now follows. Chapter 1 discusses some basic concepts in probability theory and discrete-time martingale theory. This chapter provides some preliminary mathematical background for the discrete-time financial models discussed in Chapters 2-3. Chapter 2 discusses some important portfolio selection models and asset pricing models which lay the foundations of the modern finance theory. Specifically, the mean-variance portfolio selection model and the utility theory in discrete time are discussed. Two major asset pricing models, namely the capital asset pricing model (CAPM) and the arbitrage pricing theory (APT), are presented. Some variants of the mean-variance portfolio selection model and the asset pricing models, such as the mean-semi-variance and multistage mean-variance models, the consumption-based asset pricing model, are briefly discussed. Chapter 3 introduces some fundamental concepts in financial markets. Some key topics in stochastic finance, such as the risk-neutral valuation, pricing and hedging European and American contingent claims, the expected utility maximization, the utility-based pricing and market equilibrium pricing, are discussed in general discrete-time models. Binomial-tree pricing models, in both static and multi-period cases, are adopted to convey important intuition behind the economic notions of arbitrage-free pricing and risk-neutral valuation. Then a rigorous mathematical treatment for arbitrage-free pricing in a general discrete-time model based on the fundamental theorem of asset pricing is presented. This provides the theoretical foundations for pricing and hedging contingent claims, the expected utility maximization, the utility-based pricing and market equilibrium pricing, where the martingale theory is the main mathematical tool that is adopted to discuss these problems. Chapter 4 introduces the mathematics, such as continuous-time stochastic processes, the martingale theory and Itô's stochastic calculus, which are essential for continuous-time stochastic finance. Specifically, the three pillars for the mathematics that have been used in continuous-time stochastic finance, namely Itô's formula, Girsanov's theorem and the martingale representation, are discussed. The pricing and hedging of contingent claims in the Black-Scholes model and other important option pricing models, such as the constant elasticity volatility model, the local volatility model, the stochastic volatility model, the stochastic alpha-beta-rho model, the variance-gamma model and the GARCH model, are considered in Chapter 5. Chapter 6 studies the pricing and hedging of exotic options, such as barrier options, Asian options, lookback options and reset options. Chapter 7 first focuses on the martingale methods for pricing and hedging European contingent claims. Then, under a modelling framework based on a diffusion process, the use of the partial differential equation approach for pricing and hedging is presented. Finally, the pricing of American contingent claims based on partial differential variational inequalities is briefly discussed. Chapter 8 discusses the term structure of interest rates. Various stochastic interest models are considered. Some short rate models, such as the Vasicek model, the Ho-Lee model, the Hull-White model and the Cox-Ross-Ingersoll (CIR) model, are discussed. The Heath-Jarrow-Morton (HJM) model for forward rates is considered. The uses of the partial differential equation approach, the forward measure method and the change of numeraire method for pricing interest rate derivatives, such as bonds, swaps, caps and floors, are discussed. The potential approach to the term structure of interest rates, namely the Flesaker-Hughston positive interest model, is presented. Finally, the Brace-Gaterek-Musiela (BGM) model, which may be thought of as a ``cousin'' of the HJM model, is discussed. Chapter 9 introduces optimal consumption-investment decision makings in continuous-time diffusion models. Specifically, the martingale method is adopted to discuss the expected utility maximization problems, the mean-risk portfolio selection problems with the Markowitz's mean-variance and weighted mean-variance criteria. Chapter 10 focuses on static risk measures. Two important classes of measures for risks, namely coherent risk measures and convex risk measures, are introduced, where the latter may be considered as a generalization of the former. An attention is given to the representations for these two classes of risk measures and their variants. Finally, the notion of law-invariant risk measures and their representations are presented. Chapter 11 discusses the general theory of processes and the semimartingale calculus. Particularly, stochastic integration with respect to semimartingales, Itô's differentiation rule and the Doléans-Dade stochastic exponential for semimartingales are presented. The optimal decomposition theorem, superhedging and attainability of contingent claims are also discussed. Chapter 12 focuses on optimal investment decision makings in incomplete markets. The use of the convex duality approach to the expected utility maximization is discussed in both the cases of complete and incomplete markets. A numeraire-free modelling framework for financial markets and the utility-based approaches for option valuation are also presented. Chapter 13 considers the use of the martingale method for the expected utility maximization and the valuation of contingent claims in a general semimartingale model, where the approach based on the completion of an incomplete market via an introduction to fictitious assets does not work well. The utility-based valuation approach is emphasized, and its links with the minimum relative entropy and the maximum Hellinger integral are highlighted. An attention is given to a market driven by random shocks modelled by Lévy processes and the class of HARA utility functions. The final chapter considers optimal growth portfolios and their application to option valuation in semimartingale markets. Specifically, optimal growth portfolio strategies are derived under a geometric Lévy process and a jump-diffusion-type process for asset price dynamics. Some important approaches for option valuation under incomplete markets, namely the Föllmer-Schweizer approach, the Davis's utility approach and the Esscher transform approach, are also discussed.
scientific article; zbMATH DE number 6928408