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  3. Mastering Mathematical Finance

A series of short books that cover all the core topics and the most common electives offered in master's programs in mathematical/quantitative finance. The books are closely coordinated but each self-contained, so that they can be used efficiently in combination but also individually.

  • General Editors: Marek Capi?ski, University of Science and Technology, Krakow, Ekkehard Kopp, University of Hull, Tomasz Zastawniak, University of York

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8 results in Mastering Mathematical Finance

Credit Risk
  • Marek Capiński, Tomasz Zastawniak
  • Published online:
    05 January 2017
    Print publication:
    14 November 2016
  • Modelling credit risk accurately is central to the practice of mathematical finance. The majority of available texts are aimed at an advanced level, and are more suitable for PhD students and researchers. This volume of the Mastering Mathematical Finance series addresses the need for a course intended for master's students, final-year undergraduates, and practitioners. The book focuses on the two mainstream modelling approaches to credit risk, namely structural models and reduced-form models, and onpricing selected credit risk derivatives. Balancing rigorous theory with examples, it takes readers through a natural development of mathematical ideas and financial intuition.

Stochastic Interest Rates
  • Daragh McInerney, Tomasz Zastawniak
  • Published online:
    05 June 2015
    Print publication:
    10 August 2015
  • This volume in the Mastering Mathematical Finance series strikes just the right balance between mathematical rigour and practical application. Existing books on the challenging subject of stochastic interest rate models are often too advanced for Master's students or fail to include practical examples. Stochastic Interest Rates covers practical topics such as calibration, numerical implementation and model limitations in detail. The authors provide numerous exercises and carefully chosen examples to help students acquire the necessary skills to deal with interest rate modelling in a real-world setting. In addition, the book's webpage at www.cambridge.org/9781107002579 provides solutions to all of the exercises as well as the computer code (and associated spreadsheets) for all numerical work, which allows students to verify the results.

Portfolio Theory and Risk Management
  • Maciej J. Capiński, Ekkehard Kopp
  • Published online:
    05 May 2015
    Print publication:
    07 August 2014
  • With its emphasis on examples, exercises and calculations, this book suits advanced undergraduates as well as postgraduates and practitioners. It provides a clear treatment of the scope and limitations of mean-variance portfolio theory and introduces popular modern risk measures. Proofs are given in detail, assuming only modest mathematical background, but with attention to clarity and rigour. The discussion of VaR and its more robust generalizations, such as AVaR, brings recent developments in risk measures within range of some undergraduate courses and includes a novel discussion of reducing VaR and AVaR by means of hedging techniques. A moderate pace, careful motivation and more than 70 exercises give students confidence in handling risk assessments in modern finance. Solutions and additional materials for instructors are available at www.cambridge.org/9781107003675.

Probability for Finance
  • Ekkehard Kopp, Jan Malczak, Tomasz Zastawniak
  • Published online:
    05 June 2014
    Print publication:
    21 November 2013
  • Students and instructors alike will benefit from this rigorous, unfussy text, which keeps a clear focus on the basic probabilistic concepts required for an understanding of financial market models, including independence and conditioning. Assuming only some calculus and linear algebra, the text develops key results of measure and integration, which are applied to probability spaces and random variables, culminating in central limit theory. Consequently it provides essential prerequisites to graduate-level study of modern finance and, more generally, to the study of stochastic processes. Results are proved carefully and the key concepts are motivated by concrete examples drawn from financial market models. Students can test their understanding through the large number of exercises and worked examples that are integral to the text.

Stochastic Calculus for Finance
  • Marek Capiński, Ekkehard Kopp, Janusz Traple
  • Published online:
    05 November 2012
    Print publication:
    23 August 2012
  • This book focuses specifically on the key results in stochastic processes that have become essential for finance practitioners to understand. The authors study the Wiener process and Itô integrals in some detail, with a focus on results needed for the Black–Scholes option pricing model. After developing the required martingale properties of this process, the construction of the integral and the Itô formula (proved in detail) become the centrepiece, both for theory and applications, and to provide concrete examples of stochastic differential equations used in finance. Finally, proofs of the existence, uniqueness and the Markov property of solutions of (general) stochastic equations complete the book. Using careful exposition and detailed proofs, this book is a far more accessible introduction to Itô calculus than most texts. Students, practitioners and researchers will benefit from its rigorous, but unfussy, approach to technical issues. Solutions to the exercises are available online.

The Black–Scholes Model
  • Marek Capiński, Ekkehard Kopp
  • Published online:
    05 November 2012
    Print publication:
    13 September 2012
  • The Black–Scholes option pricing model is the first and by far the best-known continuous-time mathematical model used in mathematical finance. Here, it provides a sufficiently complex, yet tractable, testbed for exploring the basic methodology of option pricing. The discussion of extended markets, the careful attention paid to the requirements for admissible trading strategies, the development of pricing formulae for many widely traded instruments and the additional complications offered by multi-stock models will appeal to a wide class of instructors. Students, practitioners and researchers alike will benefit from the book's rigorous, but unfussy, approach to technical issues. It highlights potential pitfalls, gives clear motivation for results and techniques and includes carefully chosen examples and exercises, all of which make it suitable for self-study.

Numerical Methods in Finance with C++
  • Maciej J. Capiński, Tomasz Zastawniak
  • Published online:
    05 August 2012
    Print publication:
    02 August 2012
  • Driven by concrete computational problems in quantitative finance, this book provides aspiring quant developers with the numerical techniques and programming skills they need. The authors start from scratch, so the reader does not need any previous experience of C++. Beginning with straightforward option pricing on binomial trees, the book gradually progresses towards more advanced topics, including nonlinear solvers, Monte Carlo techniques for path-dependent derivative securities, finite difference methods for partial differential equations, and American option pricing by solving a linear complementarity problem. Further material, including solutions to all exercises and C++ code, is available online. The book is ideal preparation for work as an entry-level quant programmer and it gives readers the confidence to progress to more advanced skill sets involving C++ design patterns as applied in finance.

Discrete Models of Financial Markets
  • Marek Capiński, Ekkehard Kopp
  • Published online:
    05 June 2012
    Print publication:
    23 February 2012
  • This book explains in simple settings the fundamental ideas of financial market modelling and derivative pricing, using the no-arbitrage principle. Relatively elementary mathematics leads to powerful notions and techniques - such as viability, completeness, self-financing and replicating strategies, arbitrage and equivalent martingale measures - which are directly applicable in practice. The general methods are applied in detail to pricing and hedging European and American options within the Cox–Ross–Rubinstein (CRR) binomial tree model. A simple approach to discrete interest rate models is included, which, though elementary, has some novel features. All proofs are written in a user-friendly manner, with each step carefully explained and following a natural flow of thought. In this way the student learns how to tackle new problems.