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Mathematical finance. (English) Zbl 1452.91001
Springer Finance. Cham: Springer (ISBN 978-3-030-26105-4/hbk; 978-3-030-26106-1/ebook). xvii, 772 p. (2019).
Mathematical finance studies practical problems and theoretical problems which may have their origins from practices in the finance, banking, insurance and related industries by drawing on concepts, theories and methods from disciplines such as mathematics, statistics, economics, finance and computing. Specifically, the interaction between mathematical finance and actuarial science has been fruitful. Recently, it seems that there may be some interests in exploring the interplay between mathematical finance and machine learning. Over the past several decades or so, mathematical finance has attracted the attention of academics from different fields and market practitioners from the industries. Probability theory and stochastic analysis, particularly the general theory of processes pioneered by the Strasbourg school of probability and the semimartingale calculus, play a significant role in the developments of some fundamental concepts, theories and techniques in mathematical finance.
This masterpiece on mathematical finance is written by two leading authorities in the field. It provides an excellent treatment of important topics in mathematical finance. The monograph discusses some fundamental issues including arbitrage theory, valuation, hedging, optimal portfolio selection and interest rate models. It focuses on continuous-time models with jumps as well as semimartingales, their calculus and control. The monograph consists of two major parts. The first part of the monograph discusses important concepts and results in stochastic processes and analysis which are relevant to mathematical finance. Topics covered in the first part include Levy processes, Markov processes, affine and polynomial processes, semimartingales, stochastic calculus and control. The second part of the monograph focuses on fundamental principles, theories and important models in mathematical finance. Topics covered in the second part include equity models based on geometric Levy processes and stochastic volatility models, fundamental theorems of asset pricing, optimal portfolio selection, valuation and hedging of derivatives, mean-variance hedging, utility-based valuation and hedging, risk assessment as well as term structure models. A nice feature of the monograph is that the intuitions and practical motivations of theories, methods and models are well explained. The mathematics behind the theories, methods and models are presented at a rigorous, but accessible, level. Another nice feature of the monograph is that problems or exercises are provided in each of the chapters. A chapter-by-chapter review of the monograph is presented in the sequel.
The first part of the monograph consists of Chapter 1–Chapter 7. Chapter 1 focuses on stochastic calculus in discrete time. It starts with presenting some basic concepts in stochastic processes, stopping times and martingales in Section 1.1. Stochastic integration, Ito’s formula, stochastic exponential and logarithm, Yor’s formula, martingale representation and optimal decomposition are discussed in Section 1.2. The jump characteristics of stochastic processes, which provide local descriptions for the processes, are considered in Section 1.3. The effects of operations such as stopping, change of measure and stochastic integration on jump characteristics are discussed. Markov processes, their transition semigroups, backward and forward equations are presented in Section 1.4. Stochastic control in discrete time and deterministic calculus are considered in Section 1.5 and Section 1.6, respectively. Chapter 2 considers Levy processes. The characterisation and properties of Levy processes are discussed in Section 2.2. The constructions of Levy processes based on subordination, convolution and the Esscher transform are presented in Section 2.3. Some examples of Levy processes, such as Compound Poisson Processes, the Merton model, the Kou model, the Variance Gamma process, Generalized Hyperbolic Levy Processes and the Meixner process, are considered in Section 2.4. Finally, the Levy-Ito decomposition for a Levy process is provided in Section 2.5. Chapter 3 concerns stochastic processes and calculus in continuous time, where the calculus is discussed from the perspective of integration. It starts with presenting some results on general semimartingale theory in Section 3.1. Stochastic integrals for processes and random measures in continuous time are discussed in Section 3.2 and Section 3.3, respectively. Ito semimartingales, stochastic differential equations and the stochastic exponential are considered in Section 3.4, Section 3.5 and Section 3.6, respectively. Exponential Levy processes and time-inhomogeneous Levy processes are presented in Section 3.7 and Section 3.8, respectively. A Levy-driven Ornstein-Uhlenbeck process is considered in Section 3.9. Martingale representation, backward stochastic differential equations and change of measures are discussed in Section 3.10, Section 3.11 and Section 3.12, respectively. Chapter 4 focuses on the descriptions for local behaviours of Levy processes and general semimartingales using semimartingale characteristics. Some representations for semimartingales such as the canonical representation, the Cerny and Cerny-Ruf representations are provided in Section 4.3 and Section 4.8, respectively. Martingale problems and limit theorems are discussed in Section 4.6 and Section 4.7, respectively. Chapter 5 considers Markov processes and the descriptions for their local behaviours using infinitesimal generators. The backward and forward equations for a Markov process are presented in Section 5.4 and Section 5.5, respectively. The relationships between Markov processes and semimartingales are explored in Section 5.6. The existence and uniqueness issues are discussed in Section 5.7. Finally, the relation between a solution to the martingale problem and a Feller process is presented in Section 5.8. Chapter 6 considers affine processes and polynomial processes. As noted in the introduction of Chapter 6, Page 337, affine processes provide an analytically tractable way to handle some important modelling issues in mathematical finance such as stochastic volatility models, term structure models and intensity-based default risk models. Moments of affine processes and exponentially affine martingales are presented in Section 6.2. Some structure-preserving operations for affine processes are discussed in Section 6.3. A change of measure for semimartingales with affine local characteristics is presented in Section 6.4. Time-inhomogeneous affine Markov processes and their characteristic functions are provided in Section 6.5. Finally, polynomial processes are considered in Section 6.6. The existence and uniqueness of a polynomial process are discussed. Some examples of polynomial processes, such as affine processes, generalized Ornstein-Uhlenbeck processes and Jacobi processes, are presented in Example 6.30, Example 6.31 and Example 6.33, respectively. Chapter 7 discusses stochastic control in continuous time. Dynamic programming, optimal stopping and the stochastic maximum principle are considered in Section 7.1, Section 7.2 and Section 7.4, respectively.
The second part of the monograph consists of Chapter 8–Chapter 14. Chapter 8 considers some classes of equity models which are tractable from the mathematical perspective and incorporate some empirical features of financial data. Specifically, two classes of models, namely geometric Levy processes and stochastic volatility models, are considered in Section 8.1 and Section 8.2, respectively. The estimation of Levy processes using the maximum likelihood estimation method and the moment estimators is discussed in Section 8.1.1. Multivariate geometric Levy processes, whose dependent structures are described Levy copulas, are presented in Section 8.1.2. Some stochastic volatility models considered in Section 8.2 include, for example, the Ornstein-Uhlenbeck-type stochastic volatility model, stochastic volatility models with jumps, the CGMY model and the affine ARCH-like model. Chapter 9 begins with discussing the mathematics for some basic concepts in Mathematical Finance, namely price processes, trading strategies, discounting and dividends in Section 9.1. Trading strategies allowing consumptions are considered in Section 9.2. The fundamental theorem of asset pricing is presented in Section 9.3. Specifically, different versions of the fundamental theorem of asset pricing, which incorporate dividends, trading constraints and bid-ask spreads, are provided in Section 9.3.1, Section 9.3.2 and Section 9.3.3, respectively. Chapter 10 discusses optimal portfolio selection. Firstly, optimal portfolio selection under the expected utility of terminal wealth is considered in Section 10.1. A version of the stochastic maximum principle is used to characterise optimality. The relation between an optimal investment strategy and an equivalent martingale measure is provided, where the latter depends on the marginal utility of optimal terminal wealth. The dual minimization problems for determining the equivalent martingale measures under the logarithmic utility, the power utility and the exponential utility are presented. The solutions to optimal portfolio selection for the logarithmic utility, the power utility and the exponential utility under a multivariate semimartingale asset price model are determined in Section 10.1.1, Section 10.1.2 and Section 10.1.3, respectively. Optimal investment and consumption maximizing the expected utility of consumption is considered in Section 10.2. The relation between an optimal-trading-consumption pair and an equivalent martingale measure is provided. The solutions to optimal investment and consumption for the logarithmic utility and the power utility are determined in Section 10.2.1 and Section 10.2.2, respectively. Optimal trading under the utility of running profits and losses is discussed in Section 10.3. A characterization for the optimal portfolio strategy in terms of an equivalent martingale measures is provided. Two utility functions, namely the exponential utility and the hyperbolic utility, are considered in Example 10.38 and Example 10.39, respectively. The mean-variance efficient portfolios are discussed in a multivariate semimartingale asset price model in Section 10.4. A characterisation for a mean-variance efficient portfolio in terms of an opportunity process and an adjustment process is provided. Finally, optimal portfolio selection in the presence of bid-ask spreads attributed to proportional transaction costs is discussed using the concept of shadow price in Section 10.5. The situations of maximizing the expected utility of terminal wealth or consumption and the expected utility of profits and losses are considered in Section 10.5.1 and Section 10.5.2, respectively. Chapter 11 discusses the valuation and hedging of derivatives. The valuation of liquidly traded derivatives is considered in Section 11.2. Specifically, the valuation of European options using the fundamental theorem of asset pricing is discussed in Section 11.2.1. Section 11.2.2 considers the valuation of American options and future contracts using a constrained version of the fundamental theorem of asset pricing and a dividend version of the fundamental theorem of asset pricing, respectively. Section 11.2.3 discusses making inference on the risk-neutral pricing measure from observed market option prices. Section 11.3 considers the valuation of over-the-counter derivatives using a superreplication theorem which is based on an upper price and a lower price. Some examples based on the geometric Levy model and the Heston stochastic volatility model are provided. The upper and lower price bounds for American options are presented. Section 11.4 discusses the computation of hedging strategies based on sensitivities under multivariate Markov models. In Section 11.5 and Section 11.6, the computation of option prices is discussed using approaches based on integration, partial integro-differential equations and integral transforms. Section 11.7 provides an in-depth analysis for arbitrage theory in continuous time. Some fundamental principles and theorems such as the law of one price, the fundamental theorem of asset pricing, the superreplication theorem are studied rigorously. Chapter 12 concerns mean-variance hedging which gives rise to a variance-optimal hedging strategy minimizing the mean-squared hedging error. Section 12.1 considers the martingale case, where the liquidly traded underlying securities are martingales. In Section 12.1.1, the Galtchouk-Kunita-Watanabe decomposition of the payoff of a contingent claim with respect to the underlying securities is used to characterise a variance-optimal hedging strategy for the claim. Section 12.1.2 uses the solution of a partial integro-differential equation to compute the value process, the variance-optimal hedging strategy and the minimal quadratic risk for the claim under a multivariate Markov asset price model. Section 12.1.3 adopts the Laplace transform approach to compute the value process, the variance-optimal hedging strategy and the minimal quadratic risk for the claim under a multivariate affine semimartingale. Section 12.2 considers the semimartingale case, where the variance-optimal logarithm process is introduced to represent the density process of the variance-optimal martingale measure and the opportunity-neutral process is introduced to represent the optimal hedging strategy. There are two parts for the variance-optimal hedging strategy, namely a pure hedge and an investment term. A formula for the minimal quadratic risk for a given initial endowment is provided. The utility-based valuation and hedging of derivatives is considered in Chapter 13. In Section 13.1, the situation of an exponential utility of terminal wealth is discussed, from the perspective of an option’s writer, under a general modelling framework with multiple liquidly traded securities. The concepts of utility indifference price and utility-based hedging strategy are defined. The exact solution to utility indifference pricing and hedging is provided in Section 13.1.1. The case of large number of contingent claims is studied in Section 13.1.2. It is shown that when the number of claims becomes large, the utility indifference price per claim approaches the upper price and the utility-based hedging strategy per claim is asymptotically the cheapest super-replicating strategy. The case of small number of claims is considered in Section 13.1.3, where the concepts of approximate utility indifference price of a claim and the approximate utility-based hedge per claim are introduced. It is shown that the approximate utility indifference price and the approximate utility-based hedge per claim are obtained from the solution to the mean-variance hedging problem with respect to the minimal entropy martingale measure. In Section 13.2, the utility of profits and losses is considered to deal with the situation where over-the-counter derivative contracts with different maturities are traded. The utility indifference price and the utility-based hedge are defined using the concept of shadow price. An optimal trading strategy maximizing the expected utility of profits and losses and the respective shadow price are characterized by an equation depending on the local characteristics of the shadow price process. When an Ito’s process is considered, the shadow price process is given by the solution of a backward stochastic differential equation in Example 13.10. The situation of small number of claims for the utility of profits and losses is considered in Section 13.2.1. Section 13.3 considers convex risk measures and discusses the utility indifference pricing from the perspective of convex risk measures. Using a dual representation, convex risk measure with hedging is represented in terms of a penalty function on the space of equivalent martingale measures. Section 13.4 compares different valuation and hedging approaches for liquidly traded derivatives and over-the-counter derivatives discussed in the preceding chapters. Some numerical illustrations are presented. Chapter 14 considers interest rate models. In Section 14.1, some basic concepts and aspects of interest rates are discussed. Concepts such as short rates and forward rates are reviewed in Section 14.1.1. Some interest rate products such as swaps, caps, floors and swaptions are considered in Section 14.1.2. Some aspects of term structure modelling such as modelling real-world and risk-neutral measures and change of numeraire are discussed in Section 14.1.3. Short rate models are considered in Section 14.2. Specifically, a short rate model described by an affine semimartingale is discussed under the spot martingale measure and the respective bond price and forward rate processes are provided in Section 14.2.1. In Section 14.2.2, the modelling starts from the specification for the short rate process under the real-world measure. The valuation of interest rate derivatives is considered using the Laplace transform in Section 14.2.3. Some examples of short rate models such as a Levy-driven Ornstein-Uhlenbeck model, the Vasicek model, the Ho-Lee model, the Gamma Ornstein-Uhlenbeck model, the Inverse-Gaussian Ornstein-Uhlenbeck model, the Hull-White model, the Cox-Ingersoll-Ross model, Levy-driven two-factor models are presented in Section 14.2.4. The Heath-Jarrow-Morton approach is considered in Section 14.3. A general modelling framework, where the forward rate process is driven by a semimartingale, is described in Section 14.3.1. A representation of bond prices, the Heath-Jarrow-Morton drift condition and the characteristic exponent of the driving semimartingale under a forward measure are provided. The valuation of interest-rate derivatives under Levy-driven term structure models using the Laplace transform is considered in Section 14.3.2. The market completeness is discussed in Section 14.3.3. In Section 14.3.4, some examples of interest rate models under the Heath-Jarrow-Morton framework are considered, where the Vasicek model and the Ho-Lee model with an exponential-type volatility structure are presented. Some short rate models such as the Hull-White model are re-considered under the Heath-Jarrow-Morton framework. In Section 14.3.5, affine term structure models are considered with a view to valuing interest rate derivatives. Section 14.4 discusses the forward process approach, where LIBOR market models driven by a semimartingale are considered. In Section 14.4.1, forward process drift conditions and the characteristic exponent of the driving semimartingale under a forward measure are provided. Using the Laplace transform, the valuation of interest rate derivatives under Levy-driven forward process models is discussed in Section 14.4.2. Some examples such as the Heath-Jarrow-Morton models and the lognormal LIBOR market models are presented in Section 14.4.3. Semi-explicit formulas for bond options under a forward process model driven by an affine semimartingale are provided in Section 14.4.4. The Flesaker-Hughston approach to term structure modelling is considered in Section 14.5. The Flesaker-Hughston approach starts with a state price density process which is a positive semimartingale such that the product process of this semimartingale and any price process is a martingale. This is discussed in Section 14.5.1. A rational model is described in Section 14.5.2, where the state price process is assumed to be a specific form. The bond price process is represented as a ratio under the rational model. Pricing formulas for call and put options on a bond portfolio are provided in Section 14.5.3. Two examples, namely a Levy-driven rational model and a lognormal rational model, are provided in Section 14.5.4. The linear-rational model approach, which mitigates the non-stationarity of the Flesaker-Hughston model, is considered in Section 14.6. The state price density of the linear-rational model is described in Section 14.6.1. Pricing formulas for call and put options on a bond portfolio are provided under the assumption that the driving semimartingale is affine in Section 14.6.2. Two examples, namely the square-root process and the Levy-driven Ornstein-Uhlenbeck process, are provided in Section 14.6.3.

MSC:
91-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to game theory, economics, and finance
91G20 Derivative securities (option pricing, hedging, etc.)
91G10 Portfolio theory
91G30 Interest rates, asset pricing, etc. (stochastic models)
60G51 Processes with independent increments; Lévy processes
60G40 Stopping times; optimal stopping problems; gambling theory
60G44 Martingales with continuous parameter
60H30 Applications of stochastic analysis (to PDEs, etc.)
91G80 Financial applications of other theories
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
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