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**Stochastic calculus for finance. II: Continuous-time models.**
*(English)*
Zbl 1068.91041

Springer Finance. New York, NY: Springer (ISBN 0-387-40101-6/hbk). xix, 550 p. (2004).

This is the latter of the two-volume series evolving from the author’s mathematics courses in M.Sc. Computational Finance program at Carnegie Mellon University (USA). The contents of this book is organized such as to give the reader precise statements of results, plausibility arguments, mathematical proofs and, more importantly, the intuitive explanations of the financial and economic phenomena. Each chapter concludes with summary of the discussed matter, bibliographic notes, and a set of really useful exercises.

Here it is a short characterization of the former book of the two-volume series. Volume I introduces the (discrete-time) binomial asset-pricing model as a paradigm of practice and a prologue to the much more complex concepts and results needed in the continuous-time theory of stochastic calculus exposed in the present Volume II. Volume I represents a prerequisite for Computational Finance courses based on Volume II, and prepares the reader for the more general setting in this latter volume by treating several different concepts in a simpler, discrete-time, and also less technical style, e.g. martingales, Markov processes, change of measure, risk-neutral pricing etc. A brief review on the contents of the six (6) chapters in Volume I: Chapter 1 (The Binomial No-Arbitrage Pricing Model) presents the no-arbitrage method of option pricing in a binomial model, and the fundamental concept of risk-neutral pricing in a first mathematical setting. Chapter 2 (Probability Theory on Coin Toss Space) introduces martingales, Markov processes, and the risk-neutral pricing formula for European derivative securities. Chapter 3 (State Prices) discusses the change of measure associated with risk-neutral pricing of European derivative securities, and its application to solving the problem of optimal investment in a binomial model. Chapter 4 (American Derivative Securities) investigates derivative securities whose owner can choose the exercise time. Chapter 5 (Random Walk) explains the reflection principle for random walk, while Chapter 6 (Interest-Rate-Dependent Assets) considers models with random interest rates, examines the difference between forward and futures prices, and introduces the concept of a forward measure. The Appendix of Volume I provides the Proof of Fundamental Properties of Conditional Expectations.

Volume II comprises 11 (eleven) chapters and 3 (three) appendixes. Chapter 1 (General Probability Theory) presents probability spaces, Lebesgue integrals, and change of measure. Chapter 2 (Information and Conditioning) introduces independence, conditional expectations, and their properties. These two first chapters of Volume II provide the measure-theoretic foundation for probability theory required for a treatment of continuous-time models. Chapter 3 (Brownian Motion) studies Brownian motion and its properties, including quadratic variation that is important for stochastic calculus. The core of Volume II is Chapter 4 (Stochastic Calculus), which contains the Itō integral, the Itō-Doeblin formula and its consequences, the characterization of Brownian motion in terms of its quadratic variation (Lévy’s theorem), the Black-Scholes-Merton equation for European call price, and the Brownian Bridge. Chapter 5 (Risk-Neutral Pricing) states and proves Girsanov’s Theorem an the change of measure, the fundamental theorems of asset pricing, dividend-paying stocks, forwards and futures. Chapter 6 (Connections with Partial Differential Equations) develops the links between stochastic calculus and partial differential equations. Chapter 7 (Exotic Options) presents three types of exotic options an a geometric Brownian motion asset (barrier options, lookback options, and Asian options), and works out a detailed analysis for one option of each type. Chapter 8 (American Derivative Securities) extends the material of Chapter 4 in Volume 1 for the continuous-time setting. The topics comprised are: stopping time, perpetual American put, finite-expiration American put, and the American call. Chapter 9 (Change of Numeraire) defines the foreign and domestic risk-neutral measures, the forward measures, and works out a multidimensional market model. Chapter 10 (Term-Structure Models) investigates several important models for interest-rates: affine-yield modeln, the Heath-Jarrow-Morton model, and forward LIBOR models. The final Chapter 11 (Introduction to Jump Processes) analyzes the jump-diffusion processes and presents Poisson and compound Poisson processes, stochastic integrals and stochastic calculus with respect to jump processes, changing of measures for (compound) Poisson processes, and pricing an European call in a jump model.

The three functional appendixes of the Volume II are dedicated to: (A) Advanced topics in probability; (B) Existence of conditional expectations; (C) Proof of the second fundamental Theorem of asset pricing.

Here it is a short characterization of the former book of the two-volume series. Volume I introduces the (discrete-time) binomial asset-pricing model as a paradigm of practice and a prologue to the much more complex concepts and results needed in the continuous-time theory of stochastic calculus exposed in the present Volume II. Volume I represents a prerequisite for Computational Finance courses based on Volume II, and prepares the reader for the more general setting in this latter volume by treating several different concepts in a simpler, discrete-time, and also less technical style, e.g. martingales, Markov processes, change of measure, risk-neutral pricing etc. A brief review on the contents of the six (6) chapters in Volume I: Chapter 1 (The Binomial No-Arbitrage Pricing Model) presents the no-arbitrage method of option pricing in a binomial model, and the fundamental concept of risk-neutral pricing in a first mathematical setting. Chapter 2 (Probability Theory on Coin Toss Space) introduces martingales, Markov processes, and the risk-neutral pricing formula for European derivative securities. Chapter 3 (State Prices) discusses the change of measure associated with risk-neutral pricing of European derivative securities, and its application to solving the problem of optimal investment in a binomial model. Chapter 4 (American Derivative Securities) investigates derivative securities whose owner can choose the exercise time. Chapter 5 (Random Walk) explains the reflection principle for random walk, while Chapter 6 (Interest-Rate-Dependent Assets) considers models with random interest rates, examines the difference between forward and futures prices, and introduces the concept of a forward measure. The Appendix of Volume I provides the Proof of Fundamental Properties of Conditional Expectations.

Volume II comprises 11 (eleven) chapters and 3 (three) appendixes. Chapter 1 (General Probability Theory) presents probability spaces, Lebesgue integrals, and change of measure. Chapter 2 (Information and Conditioning) introduces independence, conditional expectations, and their properties. These two first chapters of Volume II provide the measure-theoretic foundation for probability theory required for a treatment of continuous-time models. Chapter 3 (Brownian Motion) studies Brownian motion and its properties, including quadratic variation that is important for stochastic calculus. The core of Volume II is Chapter 4 (Stochastic Calculus), which contains the Itō integral, the Itō-Doeblin formula and its consequences, the characterization of Brownian motion in terms of its quadratic variation (Lévy’s theorem), the Black-Scholes-Merton equation for European call price, and the Brownian Bridge. Chapter 5 (Risk-Neutral Pricing) states and proves Girsanov’s Theorem an the change of measure, the fundamental theorems of asset pricing, dividend-paying stocks, forwards and futures. Chapter 6 (Connections with Partial Differential Equations) develops the links between stochastic calculus and partial differential equations. Chapter 7 (Exotic Options) presents three types of exotic options an a geometric Brownian motion asset (barrier options, lookback options, and Asian options), and works out a detailed analysis for one option of each type. Chapter 8 (American Derivative Securities) extends the material of Chapter 4 in Volume 1 for the continuous-time setting. The topics comprised are: stopping time, perpetual American put, finite-expiration American put, and the American call. Chapter 9 (Change of Numeraire) defines the foreign and domestic risk-neutral measures, the forward measures, and works out a multidimensional market model. Chapter 10 (Term-Structure Models) investigates several important models for interest-rates: affine-yield modeln, the Heath-Jarrow-Morton model, and forward LIBOR models. The final Chapter 11 (Introduction to Jump Processes) analyzes the jump-diffusion processes and presents Poisson and compound Poisson processes, stochastic integrals and stochastic calculus with respect to jump processes, changing of measures for (compound) Poisson processes, and pricing an European call in a jump model.

The three functional appendixes of the Volume II are dedicated to: (A) Advanced topics in probability; (B) Existence of conditional expectations; (C) Proof of the second fundamental Theorem of asset pricing.

Reviewer: Neculai Curteanu (Iaşi)

### MSC:

91-02 | Research exposition (monographs, survey articles) pertaining to game theory, economics, and finance |

91Gxx | Actuarial science and mathematical finance |

60H30 | Applications of stochastic analysis (to PDEs, etc.) |

91G20 | Derivative securities (option pricing, hedging, etc.) |

91G60 | Numerical methods (including Monte Carlo methods) |