Martingale Methods in Financial Modelling (Stochastic Modelling and Applied Probability)

Book Description

In the 2nd edition some sections of Part I are omitted for better readability, and a brand new chapter is devoted to volatility risk. As a consequence, hedging of plain-vanilla options and valuation of exotic options are no longer limited to the Black-Scholes framework with constant volatility.

The theme of stochastic volatility also reappears systematically in the second part of the book, which has been revised fundamentally, presenting much more detailed analyses of the various interest-rate models available: the authors' perspective throughout is that the choice of a model should be based on the reality of how a particular sector of the financial market functions, never neglecting to examine liquid primary and derivative assets and identifying the sources of trading risk associated. This long-awaited new edition of an outstandingly successful, well-established book, concentrating on the most pertinent and widely accepted modelling approaches, provides the reader with a text focused on practical rather than theoretical aspects of financial modelling.

Customer Reviews:

Excellent introductory book to financial math.......2006-11-03

This book takes you through the math of finance step-by-step, passing through very simple examples first and then slowly adding complexity to the models studied. It is written very clearly and the prerequisites to reading this book are only some basic notions of probabilities (sigma-fields, probability measures).

Sometimes, the problem with math books is that they are "dry" and contain only a succession of theorems and proofs. In this one, the authors make a point of explaining in detail how different theorems and models relate to each other, and make extensive comparisons between them so that you get a better feel for how they work in practice.

The book is primarily a math book and can be light on market specifics. Do not buy this book as a practical "howto" in derivatives trading.

At the Forefront of Modern Mathematical Finance.......2005-05-23

This advanced text provides an excellent account of the current state-of-the art of options pricing/hedging models and interest rate term structure models. The book is accessible to both advanced practitioners of mathematical finance as well as to pure researchers in the field.

The book is in written in a mathematical style and contains rigorous proofs of many results. However, the main focus of the text is to describe the frontier of knowledge in the subject. Each section contains copious references to the literature and is so current that several references are to working papers. Many sections detail open problems and other areas suitable for scholarly research.

In their second edition, the authors provide an extremely useful critique of each modeling paradigm that they investigate. They also provide evidence for their position in the form of literature references which instruct the reader as to the shortcomings/limitations of a particular model. This information should prove quite valuable to model practitioners and implementers.

The authors assume an advanced background from the field of stochastic analysis, although they do provide an appendix which summarizes key results needed from the field. For the stochastic calculus prerequisites, I recommend Rogers & Williams "Diffusions, Markov Processes and Martingales" volumes I and II. Suitable prerequisites are also covered by Karatzas and Shreve in "Brownian Motion and Stochastic Calculus" 2nd edition. A good foundation in arbitrage pricing theory is also needed. I recommend the nice treatment by Bjork in "Arbitrage Theory in Continuous Time" 2nd edition.

The book is divided into two parts. The first part deals with options pricing in equity markets. Chapter 1 sets premlinaries required for the arbitrage theoretic framework, while Chapter 2 has a very nice treatment of discrete time models and finite financial markets.

In Chapter 3, the authors develop the Black-Scholes model along with the Bachelier model using arbitrage techniques. The models are compared and used as benchmark continuous time models and form the basis for all subsequent analysis.

Chapter 4 provides a nice survey of techniques used to price/hedge options in foreign equity and currency markets. The authors assume familarity of the basic workings of foriegn markets.

Chapter 5 is a terrific chapter on valuing American-style options. The American call option is thoroughly studied and approximation techniques for the American put option are introduced. The explicit derivations of the formulas are referenced to the literature.

Chapter 6 provides an introduction to exotic options, although the authors vary their use of the term 'exotic' to meaning 'not a standard European-style or American-style' in this chapter to meaning 'no readily available liquid market' in Chapter 7. The descriptions are quite accessible and the basic properties of the options are described along with pricing formulas (assuming the Black-Scholes framework).

Chapter 7 provides as complete an accounting as I have ever seen of the generalizations of the Black-Scholes model and motivates this from the point of view of volatility surfaces. Many of the well-known models are studied in detail, such as CEV, local volatility, and mixture models. The strengths and weaknesses of each model are analyzed. The stochastic volatility models of Wiggins (via Orenstien-Uhlenbeck processes), Hull-White, and Heston are studied, as is the SABR model. The chapter wraps up with a study of the SIV models, describes how the stochastic volatility models can be obtained via limits of GARCH models and surveys Jump-diffusion processes and Levy processes.

The second part of the book is concerned with term structure models and interest rate derivatives. The authors are quite well-know for their many contributions to this study and their treatment is authoritative.

Martingales & Finance.......2003-04-12

I have used this book for two courses in my MSc degree in Financial Maths...well this book is hard to understand at first glance, but, once you are introduced with a good course on stochastic analysis and applied probability, this is an illuminating book...I particularly enjoyed the part on foreing equity derivatives and exotic derivatives.....Harmed with patience this is definitely the book by which you can effectively gain a sound a knowledge on modern mathematical finance theory....reading in conjunction with Bingham-Kiesel book, could help understanding the foundation of the subject.

yes, but ..........2000-03-17

I've been using this book on and off over the last year. At first I was very impressed with the level of detail in the mathematics, especially as it was the only book at the time focussing on risk-neutral methods and covering BGM. But I've become increasing disillusioned with it of late. It's difficult to explain, but although the whole book is written in traditional theorem-proof style, there are no real proofs! (I have a PhD in math and have done research for 10 years so I should know a little about proofs.) The only "proofs" provided are basically symbol shifting, but the heart of the math is strangely absent. This is especially strange given the Springer series in which it appears.

In short, if you want a catalogue of methods this book does the job, but if you want a deeper understanding try Lars Nielsens book.

excellent book for post-John-Hull readers.......1999-08-17

This book covers essentially everything needed for a serious financial math study. It captures the spirit of modern financial math. For people with math, physics or engineering background, when you feel comfortable woth John Hull's books, then this book is right one, and a must one.

Arbitrage Theory in Continuous Time (Oxford Finance)

Average customer rating:

Nicely Prepared Intermediate-Level Treatment
intuitive introduction to option pricing
Hell, I should have rated it 5 stars!
Good introductory book
An FE Bible

Arbitrage Theory in Continuous Time (Oxford Finance)
Tomas Bjork
Manufacturer: Oxford University Press, USA
ProductGroup: Book
Binding: Hardcover

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Stochastic Calculus for Finance II: Continuous-Time Models (Springer Finance)
Monte Carlo Methods in Financial Engineering (Stochastic Modelling and Applied Probability)
Stochastic Calculus for Finance I: The Binomial Asset Pricing Model (Springer Finance)
Heard on the Street: Quantitative Questions from Wall Street Job Interviews
Options, Futures and Other Derivatives (6th Edition)

ASIN: 0199271267

Book Description

The second edition of this popular introduction to the classical underpinnings of the mathematics behind finance continues to combine sound mathematical principles with economic applications. Concentrating on the probabilistic theory of continuous arbitrage pricing of financial derivatives, including stochastic optimal control theory and Merton's fund separation theory, the book is designed for graduate students and combines necessary mathematical background with a solid economic focus. It includes a solved example for every new technique presented, contains numerous exercises, and suggests further reading in each chapter. In this substantially extended new edition Bjork has added separate and complete chapters on measure theory, probability theory, Girsanov transformations, LIBOR and swap market models, and martingale representations, providing two full treatments of arbitrage pricing: the classical delta-hedging and the modern martingales. More advanced areas of study are clearly marked to help students and teachers use the book as it suits their needs.

Customer Reviews:

Nicely Prepared Intermediate-Level Treatment.......2005-05-06

The author has put together an excellent text that will take readers of an elementary text like Hull's "Options, Futures and Other Derivatives" to the next level. In the author's treatment, the power of stochastic calculus is brought to bear on the options pricing problem from the point of view of modern martingale theory, if not the complete mathematical rigor needed to establish all the results.

The text contains 26 chapters and 3 appendices. There is simply too much here to give a blow-by-blow account. So I'll try to hit the highlights.

The author gives intuitive definitions of some of the more heavy concepts from measure theory/Lebesgue integration, measure-theoretic probability theory and basic stochastic analysis. For the rigor, one need only look to the appendices, but the treatment is intuitive enough that can still follow along with only the occasionally glance to the back of the book.

Readers of Hull's text will find the first couple of chapters quite familiar, but starting in Chapter 4, stochastic integrals are (somewhat) formally introduced, along with the multi-dimensional version of Ito's change of variable rule. This is not overkill as the development of multi-factor term structure models later in the book benefits from this early development.
We note that these formulas are stated without proof, although they are motivated intuitively.

In the next chapter, stochastic differential equations are introduced and the Feynman-Kac representation is established as a nice application of Ito's rule. The chapter winds up with an intuitive treatment of Kolmogorov's forward & backward equations.

For the remainder of the first half of the text, readers of Hull will feel themselves in quite familiar territory, as the author develops the solution for the options pricing problem, studies the Greek letters and establishes parity using the now classical approach.

The second half of the text delves into martingale methods for mathematical finance. As a consequence, the sophistication level jumps considerably. The reader is well-advised to get the basic analytical toolkit in hand before delving too far into the second half of the book. I recommend Rudin's "Real and Complex Analysis" 3rd edition.

Heavy machinery is pulled in from functional analysis to establish the first and second fundamental theorems of mathematical finance. Without some basic understanding of Hilbert and Banach space theory, the reader will understand very little of this treatment.

The next highlight is the Girsanov Theorem. The author actual provides a proof in the scalar case, and presents (without proof) the Novikov condition to test when the Girsanov transformation is indeed a martingale (so the theorem holds). As a nice application, the Black-Scholes theory is revisted and re-established via these martingale results.

Another highlight is the study of the Hamilton-Jacobi-Bellman model for stochastic control, along with a small catalogue of cases under which the HJB equations can be solved. As a nice application, Merton's mutual fund theorem is established.

The last several chapters of the book deal with martingale methods for term structure models. There is a nice survey and study of the 1-factor short rate models before loading up and doing the k-factor model framework of Heath-Jarrow-Morton.
The martingale setting makes for a very rigorous treatment.

The book ends with a really nice treatment of the Libor Market and Swap Market Models. Pure finance students may feel that the mathematics at the end unnecessarily overwhelms the intuition, but students of mathematical finance will appreciate the analytical treatment and may even feel inspired to implement their own LMM.

There are a ton of terrific exercises at the end of each chapter. The exercises really solidify the understanding of the presentation and they make great technical interview questions as well.

intuitive introduction to option pricing.......2004-11-10

I agree with several reviewers above that the book is written in a style very helpful for students to understand the material.

It doesn't contain a lot of small details of financial markets like Hull's book, but the approach is very systematic. The derivations of formula for Barrier options is a nice example, Hull only lists a set of formula. The focus is on the theory, not on the practice. (No numerical method in the book). Bjork's book is very valuable for a student with very good math skills but want to learn the reasoning style for option pricing. It is a quick and enjoyable read.

A huge plus side of the book is to describe strategy before writing down all the proofs. This helps greatly. It can be contrasted with Duffie's book "Dynamic Asset Pricing Theory", which is written like a dry math book (well, I have to admit that Duffie's book is not an intro book)

Only thing I can think of that can be improved is typo in the book, too many wrong formula, especially in the second half of the book, luckily enough, they are obviously wrong so that one can still understand the topics. I also find that using SEK and mentioning street name of Britain are amusing for a student in U.S.

Hell, I should have rated it 5 stars!.......2002-05-26

If you're going to be introduced to Derivatives pricing and Quantitative finance in continuous time, you need some basics in probability theory, an elementary introduction to stochastic calculus and you need "bjork". It tells you the equation and how to understand it.

It's the best source for a complete understanding of the basics of arbitrage free pricing in continuous time; whether it's in complete or incomplete markets.

The best feature of this book is how the author invariably provides an "intuitive interpretation or explanation" to convey critical concepts. {Things like market price of risk in the context of interest rate modelling, change of measure etc...}

Why I rated the book 4 instead of 5?
I will not forgive "Tomas bjork" not to have covered the Libor Market Model; it's "THE" model and therefore should be covered in great details by any book of this calibre. A new edition of this book with the libor market model is needed.
Having said that, the coverage he gives to the popular short rate models is worth every read!

Guy,
Msc Financial Engineering at ISMA Center, Reading - UK.

Good introductory book.......2002-05-25

It is a good book to read as an introduction to the field. The author is successful in conveying the intuition behind the models instead of striving for complete mathematical rigor. I recommend this book if you want to quickly get acquainted with derivatives pricing but are a bit afraid of the higher math seen in other books.

An FE Bible.......2001-11-08

The central text for IOE 552(financial Engineering I) at the University of Michigan. Halfway through the course and I really understand the application of Ito's Lemma and the Feynman-Kac stochastic representation theorem. This book has just the right mixture of narative story telling, and mathematical rigor. The derivations are accessible to those with a semester of advanced calculus and a semester of probability. Over and over, Bjork shows that the secret of success in Financial Engineering is "RAIL" which stands for the "Relentless Application of Ito's Lemma".

An Arbitrage Guide to Financial Markets (The Wiley Finance Series)

Average customer rating:

An excellent book by a well known professional
Excellent undertaking!

An Arbitrage Guide to Financial Markets (The Wiley Finance Series)
Robert Dubil
Manufacturer: Wiley
ProductGroup: Book
Binding: Hardcover

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ASIN: 0470853328

Book Description

An Arbitrage Guide to Financial Markets is the first book to explicitly show the linkages of markets for equities, currencies, fixed income and commodities. Using a unique structural approach, it dissects all markets the same way: into spot, forward and contingent dimensions, bringing out the simplicity and the commonalities of all markets. The book shuns stochastic calculus in favor of cash flow details of arbitrage trades. All math is simple, but there is lots of it. The book reflects the relative value mentality of an institutional trader seeking profit from misalignments of various market segments.

The book is aimed at entrants into investment banking and dealing businesses, existing personnel in non-trading jobs, and people outside of the financial services industry trying to gain a view into what drives dealers in today’s highly integrated marketplace. A committed reader is guaranteed to leave with a deep understanding of all current issues.

"This is an excellent introduction to the financial markets by an author with a strong academic approach and practical insights from trading experience. At a time when the proliferation of financial instruments and the increased use of sophisticated mathematics in their analysis, makes an introduction to financial markets intimidating to most, this book is very useful. It provides an insight into the core concepts across markets and uses mathematics at an accessible level. It equips readers to understand the fundamentals of markets, valuation and trading. I would highly recommend it to anyone looking to understand the essentials of successfully trading, structuring or using the entire range of financial instruments available today."
—Varun Gosain, Principal, Constellation Capital Management, New York

"Robert Dubil, drawing from his extensive prior trading experience, has made a significant contribution by writing an easy to understand book about the complex world of today’s financial markets, using basic mathematical concepts. The book is filled with insights and real life examples about how traders approach the market and is required reading for anyone with an interest in understanding markets or a career in trading."
—George Handjinicolaou, Partner, Etolian Capital, New York

"This book provides an excellent guide to the current state of the financial markets. It combines academic rigour with the author’s practical experience of the financial sector, giving both students and practitioners an insight into the arbitrage pricing mechanism."
—Zenji Nakamura, Managing Director, Europe Fixed Income Division, Nomura International plc, London

Download Description

"An Arbitrage Guide to Financial Markets is the first book to explicitly show the linkages of markets for equities, currencies, fixed income and commodities. Using a unique structural approach, it dissects all markets the same way: into spot, forward and contingent dimensions, bringing out the simplicity and the commonalities of all markets. The book shuns stochastic calculus in favor of cash flow details of arbitrage trades. All math is simple, but there is lots of it. The book reflects the relative value mentality of an institutional trader seeking profit from misalignments of various market segments.

""This is an excellent introduction to the financial markets by an author with a strong academic approach and practical insights from trading experience. At a time when the proliferation of financial instruments and the increased use of sophisticated mathematics in their analysis, makes an introduction to financial markets intimidating to most, this book is very useful. It provides an insight into the core concepts across markets and uses mathematics at an accessible level. It equips readers to understand the fundamentals of markets, valuation and trading. I would highly recommend it to anyone looking to understand the essentials of successfully trading, structuring or using the entire range of financial instruments available today.""
—Varun Gosain, Principal, Constellation Capital Management, New York

""Robert Dubil, drawing from his extensive prior trading experience, has made a significant contribution by writing an easy to understand book about the complex world of today’s financial markets, using basic mathematical concepts. The book is filled with insights and real life examples about how traders approach the market and is required reading for anyone with an interest in understanding markets or a career in trading.""
—George Handjinicolaou, Partner, Etolian Capital, New York

""This book provides an excellent guide to the current state of the financial markets. It combines academic rigour with the author’s practical experience of the financial sector, giving both students and practitioners an insight into the arbitrage pricing mechanism.""
—Zenji Nakamura, Managing Director, Europe Fixed Income Division, Nomura International plc, London "

Customer Reviews:

An excellent book by a well known professional.......2005-04-15

From someone who not only read the book but also worked for the author in the risk management field, I highly recommend this book, specifically for the clarity of style and good explanations.

Excellent undertaking!.......2004-12-27

Probably the single best introduction to financial markets' mechanics. The complex workings of the financial world are decomposed into simple building blocks, that have their foundation upon the principle of risk-sharing. Maybe of all books about hedge fund strategies, this is one of the clearest and most useful expositions of the principles upon which absolute return strategies are based - even though it has no magic 'hedge fund' words in its title to capture the reader's attention. Amid a flood of useless marketing stuff about hedge funds (including some of the works of Dr. Nicholas) and extremely boring expositions of minute details about the institutional and regulatory details of various markets this is a very good achievement indeed. Although truly sophisticated readers would probably want to look elsewhere, this book gives you something essential, which painfully misses from many fields of modern knowledge - a broad picture, without which we risk not to see the wood for the trees.

The Mathematics of Arbitrage (Springer Finance)

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The Mathematics of Arbitrage (Springer Finance)
Freddy Delbaen , and Walter Schachermayer
Manufacturer: Springer
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Binding: Hardcover

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Similar Items:

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Mathematical Concepts of Quantum Mechanics (Universitext)
The Mathematical Theory of Finite Element Methods

ASIN: 3540219927

Book Description

This long-awaited book aims at a rigorous mathematical treatment of the theory of pricing and hedging of derivative securities by the principle of 'no arbitrage'. The first part presents a relatively elementary introduction, restricting itself to the case of finite probability spaces. The second part consists of an updated edition of seven original research papers by the authors, which analyse the topic in the general framework of semi-martingale theory.

Arbitrage-free premium calculation for extreme losses using the shot noise process and the Esscher transform [An article from: Insurance Mathematics and Economics]

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Arbitrage-free premium calculation for extreme losses using the shot noise process and the Esscher transform [An article from: Insurance Mathematics and Economics]
J.W. Jang , and Y. Krvavych
Manufacturer: Elsevier
ProductGroup: Book
Binding: Digital

Elsevier | By Publisher | e-Docs | Formats | Books
ASIN: B000RQYIRI

Book Description

This digital document is a journal article from Insurance Mathematics and Economics, published by Elsevier in 2004. The article is delivered in HTML format and is available in your Amazon.com Media Library immediately after purchase. You can view it with any web browser.

Description:
We consider the classical compound Poisson model of insurance risk, with the additional economic assumption of a positive interest rate. Insurance premiums, that are the present values of the aggregate claims, are priced by enforcing a no-arbitrage condition between the insurance and reinsurance markets. We note a duality result relating the aggregate accumulated claims and the shot noise process and so we apply the piecewise deterministic Markov processes theory. It is assumed that the claim sizes are Loggamma, Frechet and truncated Gumbel to deal with heavy-tail losses in practice. We also use an exponential distribution for the case of non-heavy-tail losses. In order to obtain an arbitrage-free premium, we use an equivalent martingale probability measure obtained via the Esscher transform. In case of Loggamma and Frechet distribution for claim sizes, which allow us to derive the explicit forms for the insurance premium calculations, we find that the arbitrage-free premiums can only be obtained by levying the loading in terms of claim arrival rate, not in terms of claim size measure. It is due to the non-existence of the Laplace transforms of Loggamma and Frechet distribution for claim sizes after changing measure. We find that if claim size follows truncated Gumbel distribution, the security loading can be levied either in terms of claim arrival rate or in terms of claim size measure (or both). However, it is not possible for us to obtain the explicit form for the insurance premium calculation. Using the analytical/explicit forms for four different claim size distributions, we also provide a several numerical examples.

Optimal stopping and American options with discrete dividends and exogenous risk [An article from: Insurance Mathematics and Economics]

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Optimal stopping and American options with discrete dividends and exogenous risk [An article from: Insurance Mathematics and Economics]
A. Battauz , and M. Pratelli
Manufacturer: Elsevier
ProductGroup: Book
Binding: Digital
ASIN: B000RQYIFK

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Portfolio optimizations in incomplete financial markets

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Portfolio optimizations in incomplete financial markets
Walter Schachermayer
Manufacturer: Edizioni della Normale
ProductGroup: Book
Binding: Paperback

General | Mathematics | Science | Subjects | Books

Book Description

These Lecture Notes are based on a course given in June 2001 at the Cattedra Galileiana of Scuola Normale Superiore di Pisa. The course consisted of a short introduction into the basic concepts of Mathematical Finance, focusing on the notion of “no arbitrage”, and subsequently applying these concepts to portfolio optimization. To avoid technical difficulties I mainly dealt with the situation where the underlying probability space is finite and only sketched the difficulties arising in the general case. We then pass to the scheme of utility optimisation for general semi-martingale models. Some topics of this course are not standard: for example, in the treatment of the general existence theorem for the optimal portfolio, we give a direct proof which is not relying on duality theory. Similarly, the treatment of the asymptotic elasticity of utility functions and a related counter-example are original to these notes.

Static-arbitrage optimal subreplicating strategies for basket options [An article from: Insurance Mathematics and Economics]

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Static-arbitrage optimal subreplicating strategies for basket options [An article from: Insurance Mathematics and Economics]
D. Hobson , P. Laurence , and T.H. Wang
Manufacturer: Elsevier
ProductGroup: Book
Binding: Digital
ASIN: B000RR56JQ

Book Description

This digital document is a journal article from Insurance Mathematics and Economics, published by Elsevier in . The article is delivered in HTML format and is available in your Amazon.com Media Library immediately after purchase. You can view it with any web browser.

Description:
In this paper we investigate the possible values of basket options. Instead of postulating a model and pricing the basket option using that model, we consider the set of all models which are consistent with the observed prices of vanilla options of all strikes. In the case of basket options on two components we find, within this class, the model for which the price of the basket option is smallest. This price, as discovered by Rapuch and Roncalli, is associated to the lower Frechet copula. We complement their result in this paper by describing an optimal subreplicating strategy. This strategy is associated with an explicit portfolio which consists of being long and short a series of calls with strikes chosen as the zeros of an auxiliary function.

Universal strategies for diffusion markets and possibility of asymptotic arbitrage [An article from: Insurance Mathematics and Economics]

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Universal strategies for diffusion markets and possibility of asymptotic arbitrage [An article from: Insurance Mathematics and Economics]
N.G. Dokuchaev , and A.V. Savkin
Manufacturer: Elsevier
ProductGroup: Book
Binding: Digital
ASIN: B000RQYINC

Book Description

What kind of new asset will push up the CML? [An article from: Insurance Mathematics and Economics]

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What kind of new asset will push up the CML? [An article from: Insurance Mathematics and Economics]
B. Zhang
Manufacturer: Elsevier
ProductGroup: Book
Binding: Digital
ASIN: B000RQYIOQ

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