Book Description
The 2nd edition of this successful book has several new features. The calibration discussion of the basic LIBOR market model has been enriched considerably, with an analysis of the impact of the swaptions interpolation technique and of the exogenous instantaneous correlation on the calibration outputs. A discussion of historical estimation of the instantaneous correlation matrix and of rank reduction has been added, and a LIBOR-model consistent swaption-volatility interpolation technique has been introduced.
The old sections devoted to the smile issue in the LIBOR market model have been enlarged into several new chapters. New sections on local-volatility dynamics, and on stochastic volatility models have been added, with a thorough treatment of the recently developed uncertain-volatility approach. Examples of calibrations to real market data are now considered.
The fast-growing interest for hybrid products has led to new chapters. A special focus here is devoted to the pricing of inflation-linked derivatives.
The three final new chapters of this second edition are devoted to credit. Since Credit Derivatives are increasingly fundamental, and since in the reduced-form modeling framework much of the technique involved is analogous to interest-rate modeling, Credit Derivatives -- mostly Credit Default Swaps (CDS), CDS Options and Constant Maturity CDS - are discussed, building on the basic short rate-models and market models introduced earlier for the default-free market. Counterparty risk in interest rate payoff valuation is also considered, motivated by the recent Basel II framework developments.
Customer Reviews:
Best book on interest rate models.......2002-12-14
This is the best book available on interest rate models. Very detailed. Much more focused and readable than Rebonato's book. More pragmatic and explicit than Musiela and Rutkowski. Not as theoretical as Hunt and Kennedy. James and Webber also looks very good, but I'm not that familiar with it. All other books have only bits and pieces on interest rates.
The best book I have read on the subject.......2002-05-06
With all the due respect to the other authors I would say that if one is interested in a good theoretical book whihc is also good on the implementation side then the book of Brigo and Mercurion is definetly the best book I have ever read on the subject.
Anyone interested in implementing the LMM/BGM/MSS model in practice is well advised to read it.
I would just say that this is certainly a must have in the field.
New stuff and nice overview: hard to beat!.......2002-01-17
In the late nineties I went through Brigo's innovative work on stochastic nonlinear filtering with differential geometry techniques. I was favorably impressed by results and style, particularly in his dissertation and in his 'geometry in present day science' very readable overview. Interesting results are found and nicely told with accurate - but not pointlessly complicated - advanced mathematics for the problems at hand, I reasoned.
I've followed a similar path from control to finance, and having worked with interest rate models, I couldn't help but order this Brigo-Mercurio book. I had high expectations 'cause these two guys are working in a bank on the real thing.
Sure enough I'm not disappointed.
1-factor models are handled with great care, a ton of formulas and recipes are given. I've never seen this kind of analysis of pricing with Gaussian 1-f models. The new upgrade of the CIR model is interesting and accurate. "CIR++" is now my favorite 1-f model. I like the treatment of lognormal 1-f models and the explanation of Monte Carlo and trees -- the flow-chart for Bermudan swaptions is crystal clear! Plots of market implied structures and volatility calibration are useful additions.
The chapter on 2-f extensions has one of the best discussions on volatility, and two tons of useful formulas/recipes. Two dimensional trees!
The HJM chapter size is OK. I agree - the useful models embedded in HJM are short rate models and market models.
Market models - these three chapters alone are worth the book. You'll find yourself nodding as you read the guided tour. They make it look easy all the time. The exposition is focused, clear, intuitive, detailed. There's also new stuff, just check the calibration discussion! Smile modeling begins with a brilliant tour and ends with Brigo-Mercurio's new approach - the mixing dynamics - deserving a whole chapter if expanded.
The detailed explanation on products is a much welcome original addition. Cross currency derivatives!
Quotes - as in Brigo's old work - are a pleasant diversion while reading. The 500 and more pages are a treat given the competitive price.
Still there's room for improvements - more "CIR2++"! Something on 3-f models. Historical estimation of the correlation matrix and low-rank optimized approximations. Expand smile modeling! More hedging. Something on structured products. Cross currency libor model. chapter 9 - other interest rate models - sounds out of place and can be suppressed for other things.
This book rings true and has useful teachings for students, academics and practitioners. Although it requires some background in stochastic calculus, it's hard to beat on the pricing front. Kudos to Brigo and Mercurio! It only harms there aren't enough books like this.
Nicely written overview of interest rate models.......2001-12-15
This recent book, written by two Italian "quants" Mercurio & Brigo, gives a nice and accessible overview of interest rate models which is a compromise between the practitioner viewpoint, expressed for ex. in Rebonato's book "Interet Rate option models"
and the theoretical viewpoint such as the one in Musiela & Rutkowski.
The authors, themselves PhDs in quantitative finance/ applied maths, wrote this book while working as quants in an Italian bank and this first hand contact with the market gave them a
practical view on the subject which markes this book very interesting.
The book contains a "rational" catalogue of models used in practice ( as opposed to models which are impossible to implement!).
In contrast with academic books on interest rate modeling which deal with HJM formulation, there is a lot of emphasis here on LIBOR and Swap market models
(BGM -Jamshidian models) which reflects the current market practice. This is a positive point since there are not many books with details on implementing and using these "market models".
Part II: Interest rate models in practice is particularly useful because it deals with implementation and calibration which, as any practitioner knows, are important and usually delicate issues.
However calibration issues are dealt with somewhat lightly, especially recent developments on modeling cap/swaption smiles
are not included here.
This book can also be used for a graduate level/PhD course on interest rate models.
There are a lot of numerical examples in the book and mathematics is kept to the necessary level while keeping the
approach both rigorous and understandable.
Overall, it is one of the best books written on the subject.
I highly recommend it to PhD students, quants and researchers interested in this field.
Well written and useful book.......2001-11-04
In my humble opinion, this is the best book on Interest Rate modeling out there. The writing style is clear and focused and the appendices are fantastic. The book is rigorous but someone with some background in Stochastic Calculus will find it easy to follow. If you need refresher, dont worry the authors have you covered, see the appendix on Stochastic Calculus. Not an introductory book. Very exciting book.
Book Description
There are many mathematics textbooks on real analysis, but they focus on topics not readily helpful for studying economic theory or they are inaccessible to most graduate students of economics. Real Analysis with Economic Applications aims to fill this gap by providing an ideal textbook and reference on real analysis tailored specifically to the concerns of such students.
The emphasis throughout is on topics directly relevant to economic theory. In addition to addressing the usual topics of real analysis, this book discusses the elements of order theory, convex analysis, optimization, correspondences, linear and nonlinear functional analysis, fixed-point theory, dynamic programming, and calculus of variations. Efe Ok complements the mathematical development with applications that provide concise introductions to various topics from economic theory, including individual decision theory and games, welfare economics, information theory, general equilibrium and finance, and intertemporal economics. Moreover, apart from direct applications to economic theory, his book includes numerous fixed point theorems and applications to functional equations and optimization theory.
The book is rigorous, but accessible to those who are relatively new to the ways of real analysis. The formal exposition is accompanied by discussions that describe the basic ideas in relatively heuristic terms, and by more than 1,000 exercises of varying difficulty.
This book will be an indispensable resource in courses on mathematics for economists and as a reference for graduate students working on economic theory.
Customer Reviews:
Great book for mathematical economics.......2007-05-12
This is a very interesting book that explains real analysis focusing on economics issues and, I must say, it does its job beautifully and with no lack of rigour. When it comes to the mathematical aspects of microeconomics, the book turns out to be even better. A great book that will help very much Mas-Colell's Microeconomic Theory readers.
Book Description
Modelling with the Itô integral or stochastic differential equations has become increasingly important in various applied fields, including physics, biology, chemistry and finance. However, stochastic calculus is based on a deep mathematical theory.
This book is suitable for the reader without a deep mathematical background. It gives an elementary introduction to that area of probability theory, without burdening the reader with a great deal of measure theory. Applications are taken from stochastic finance. In particular, the Black-Scholes option pricing formula is derived. The book can serve as a text for a course on stochastic calculus for non-mathematicians or as elementary reading material for anyone who wants to learn about Itô calculus and/or stochastic finance.
Customer Reviews:
Great Introduction.......2007-09-01
This book provides an excellent mild introduction of stochastic calculus and stochastic differential equations to someone like me who do not have a first mathematics degree (haven't done measure theories). Although the final chapter on application to finance is not as good as other financial maths books such as Joshi's Concepts and Baxter&Rennie's Financial Calculus. Overall, this book sets some firm grounds for further studies on stochastic calculus & financial maths. In addition, the price is low for this book with a hardcover.
Not for beginners or intermediates.......2007-03-04
This book may be fine if you have at least an undergraduate degree in math. I have an engineering degree with a minor in math, have read many books on quantitative finance, read math books and work math problems for furn, have several years' work experience in analyzing and hedging with derivatives, am taking a course in quantitative finance, and have worked many problems in stochastic calculus. I was actually MORE confused AFTER reading this book (I'm not exaggerating). This book should definitely not have "elementary" in the title.
But whatever your level, there are other books that cover the topic much more clearly and comprehensively. Start with Nefti's Intro to the Mathematics of Financial Derivatives; it's the best. Then read Joshi's Concepts and Practices of Mathematical Finance. Then you may be able to understand Steele's Stochastic Calculus and Financial Applications. Steele doesn't pretend to be a book for beginners, but it is actually more comprehensible to beginners than Mikosch's book.
But if you are more comfortable with reams of mathematical notation and do not require much explanation of that notation, Mikosch may be just what you want.
A well organised and high standard little book.......2007-01-17
It is amazing that the author can expose the difficult topics step-by-step clearly in such a little book of less than 200 pages. Mikosch starts the book by introducing basic probability theories and stochastic processes which are prerequisite for the development of stochastic integrals, techniques for solving stochastic differential equations and applications in modern finance. However, reader should be aware that although the book is recommended for those without "deep" mathematical background, it is not suitable for average advanced undergraduates and practitioners without serious statistics and advanced claculus trainings. In my opinion, this book is suitable for those whos have finished the book like Financial Calculus by Baxter and Rennie and are looking for more formal details of the subject matter.
Best introduction.......2007-01-08
if you're looking for an introduction to stochastic calculus that uses english and pictures instead of mathematical symbols, this is the place to start. only problem is that it doesn't cover very much, just the very basics. much easier than oksendel. i wish it covered as much.
Best start.......2006-11-30
This is an excellent book to start with before reading any book on continuous time finance.
The author actually spends only the last dozen of pages related to finance, but you get all the preparation you need before learning about derivatives pricing.
Average customer rating:
- Go elsewhere for Econ theory
- Much better than the other reviewers indicate
- Pretty good despite the criticisms
- Worst Microeconomics Book
- A good book for undergraduation
|
Microeconomics With Calculus (2nd Edition)
Brian R. Binger , and
Elizabeth Hoffman
Manufacturer: Addison Wesley
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Microeconomic Analysis, Third Edition
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ASIN: 0321012259 |
Customer Reviews:
Go elsewhere for Econ theory.......2007-09-18
This book is outstanding at taking a difficult subject and making it even more inaccessible. Textbooks like this are why kids don't want to study Econ today. Dry writing, poor examples, and errors make this book a chore to learn from. If it were possible to give it zero stars, I would.
A better resource for your rusty Calc will be Haeussler, Paul, and Wood's Introductory Mathematical Analysis - which includes a section on the Lagrangian method. This is a really great (and fun, yes, fun) calc book.
A better microeconomic theory book is the Wetzstein Microeconomic Theory text.
Finally, for lighter introduction check out any of the recent econ books such as "The Undercover Economist", "Undressing the naked science", etc.
There are much better ways to make micro interesting to learn than from this book...
Much better than the other reviewers indicate.......2003-11-21
Microeconomics with Calculus is a great textbook. this is the ideal book for all those economics students who grow weary of wordy explanations of economic models when a touch of calculus say so much more. a great book for people unassociated with economics, but who have backgrounds in any math-related disciplines.
Pretty good despite the criticisms.......2001-05-15
This book isn't perfect, but it is much better than the other reviewers indicate. The best thing about it is that it is more or less self-contained. All of the necessary math can be found in the first chapters, and there are lots of worked-out examples. It's pretty well written too. Problems include: too many typos and an ugly typeface. The graphs are nicely done, though. Its most obvious competitors are Intermediate Microeconomics by Varian and Microeconomic Theory: Basic Principles and Extensions by Nicholson. Varian hides all the math in brief appendices and a supplementary workbook, and Nicholson's book is good but much too expensive.
Hopefully a third edition will address some of the problems with this book. In the meantime, it is a reasonable choice for a microeconomics text.
Worst Microeconomics Book.......2000-12-05
This is simply the worst economics book I have ever read. There are typos everywhere, the author does a horrible job of tying the economic theory to the math, and the presentation of economic material gives very few applied examples. This book makes a good sleeping aid.
A good book for undergraduation.......2000-06-20
This book is a very nice book to be used in undergraduation level. For those who got a good basis in calculus, this book fits perfectly. After understanding all this book , the stundent will be ready for an advanced text. .
Average customer rating:
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Economics With Calculus
Michael C. Lovell
Manufacturer: World Scientific Publishing Company
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ASIN: 9812388575 |
Book Description
This textbook provides a calculus-based introduction to economics. Students blessed with a working knowledge of the calculus will find that this text facilitates their study of the basic analytical framework of economics. The textbook examines a wide range of micro and macro topics, including prices and markets, equity versus efficiency, Rawls versus Bentham, accounting and the theory of the firm, optimal lot size and just in time, monopoly and competition, exchange rates and the balance of payments, inflation and unemployment, fiscal and monetary policy, IS-LM analysis, aggregate demand and supply, speculation and rational expectations, growth and development, exhaustible resources and over-fishing. While the content is similar to that of conventional introductory economics textbook, the assumption that the reader knows and enjoys the calculus distinguishes this book from the traditional text.
Book Description
One of the problems in economics to which economists have devoted a considerable amount of attention in recent years has been to ensure consistency in the models they employ. Assuming markets to be generally in some state of equilibrium, it must be asked under what circumstances such an equilibrium is possible. The fundamental mathematical tools used to address this concern are fixed point theorems. These outline the conditions under which sets of assumptions have a solution. This book gives the reader access to the mathematical techniques involved and goes on to apply fixed point theorems to proving the existence of equilibria for economics and for cooperative and non-cooperative games. Special emphasis is given to economics and games in cases where the preferences of agents may not be transitive. In addition, the author presents new proofs of old results in order to further clarify the results. He also proposes new results, notably in the last chapter, that refer to the core of a game without transitivity. This book will be useful as a text or reference work for mathematical economists and graduate and advanced undergraduate students.
Customer Reviews:
An excellent survey of more than fixed point theorems........2000-06-02
The author compiles theorems, definitions, and properties on a number of topics. There's some convex analysis, maximization of binary relations, fixed point and selection theorems, and various sufficient conditions for existence of equilibria in games and economies.
A nice feature is that the author explains the equivalence or interconnectedness of theorems from different classes -- comparing the KMM lemma to Brouwer's fixed pt. thm., for instance.
This book is extremely useful for its many variations on common principles. Researchers can benefit from consulting it when they have a problem which does not satisfy the usual criteria -- for instance, if you ever have to ask, "I have a correspondence which satisfies all the conditions for Kakutani's theorem except that it is l.h.c. rather than u.h.c.; am I still able to guarantee a fixed point?"
Grad students in micro/game theory can benefit from the survey of theorems, and familiarizing themselves with the many ways they can get to desired results.
The book is compact and thorough with little exposition. Definitions are very clear, and the author is very good at noting when definitions vary, or the same term has multiple definitions in the literature. Though it presents itself as a mathematical text, its audience is clearly economists -- the conditions given and situations described are clearly ones that economists will recognize, and the applications are economic.
A very nice book for people who already know the applications of such theorems and who need minimal explanation.
the best buy.......2000-03-26
The author, one of the most prominet professors in mathematical economics surveyed fixed point theorems. The theorems are often used to prove the existence of solutions in game theory and economics. If you major in mathematical economics or game theory, you can't avoid to buy this book.
Book Description
Updated and expanded to include the optional use of graphing calculators, this combination textbook and workbook is a good teach-yourself refresher course for men and women who took a calculus course in school, have since forgotten most of what they learned, and now need some practical calculus for business purposes or advanced education. The book is also very useful as a supplementary text for students who are taking calculus and finding it a struggle. Each progressive work unit offers clear instruction and worked-out examples. Special emphasis has been placed on business and economic applications. Topics covered include functions and their graphs, derivatives, optimization problems, exponential and logarithmic functions, integration, and partial derivatives.
Customer Reviews:
Great, easy to understand review.......2007-08-17
For a calculus textbook, this is really easy to follow. There is minimal theoretical explanation and lots of examples to help you learn the material. It's been 12 years since I took calculus, and this book almost makes it feel easy! Highly recommended reading for someone going back to school who needs to brush up.
Lightweight, your mileage might vary...........2007-05-13
I bought this and its companion book, Forgotten Algebra, in the hopes of shoring up math skills for engineering and physics classes. This is not a useful book for such. It gets about as far as the chain rule and some simple multivariable stuff, with an emphasis on biz apps. My mistake, the book is likely fine for most readers just wanting a quick refresh of the very bssics.
A+ Book Review!.......2007-03-20
If you're studying calculus, you should read this book first before delving in your textbook.
Best Calculus Book Ever.......2006-12-15
I got this book to relearn calculus 12 years after I finnished my undergrad work in Math at Ohio State. This book is a great resource for anyone returning to academics.
The best purchase in your life?.......2006-11-23
Are you studying calculus? This book is a must if you are. Just like milk and cereal or buns and hamburger patties. This book and the study of calculus go together. I took calculus several years ago. I purchased this book as a refresher. I have never seen something written so well on a particular subject. All the bases are covered in this book! Even if you don't remember a bit of your algebra. I actually learned more from this book than I did from my instructor I had in college. It will be the best $2.00 you've ever spent in you life if you pick up a copy of this book.
Average customer rating:
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Nonsmooth Approach to Optimization Problems with Equilibrium (Nonconvex Optimization and Its Applications)
J. Outrata ,
M. Kocvara , and
J. Zowe
Manufacturer: Springer
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ASIN: 0792351703 |
Book Description
This book presents an in-depth study and a solution technique for an important class of optimization problems. This class is characterized by special constraints: parameter-dependent convex programs, variational inequalities or complementarity problems. All these so-called equilibrium constraints are mostly treated in a convenient form of generalized equations. The book begins with a chapter on auxiliary results followed by a description of the main numerical tools: a bundle method of nonsmooth optimization and a nonsmooth variant of Newton's method. Following this, stability and sensitivity theory for generalized equations is presented, based on the concept of strong regularity. This enables one to apply the generalized differential calculus for Lipschitz maps to derive optimality conditions and to arrive at a solution method. A large part of the book focuses on applications coming from continuum mechanics and mathematical economy. A series of nonacademic problems is introduced and analyzed in detail. Each problem is accompanied with examples that show the efficiency of the solution method.
This book is addressed to applied mathematicians and engineers working in continuum mechanics, operations research and economic modelling. Students interested in optimization will also find the book useful.
Average customer rating:
- Material every quantitative financial analyst should know.
- Good Treatment of Continuous Time Martingales
- Elegant Math Book on Finance - you need the math to read
- It is indeed meant for learning
- Not meant for learning
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Continuous Stochastic Calculus with Applications to Finance
Michael Meyer
Manufacturer: Chapman & Hall/CRC
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ASIN: 1584882344 |
Book Description
The prolonged boom in the US and European stock markets has led to increased interest in the mathematics of security markets, most notably in the theory of stochastic integration. This text gives a rigorous development of the theory of stochastic integration as it applies to the valuation of derivative securities. It includes all the tools necessary for readers to understand how the stochastic integral is constructed with respect to a general continuous martingale. The author develops the stochastic calculus from first principles, but at a relaxed pace that includes proofs that are detailed, but streamlined to applications to finance. The treatment requires minimal prerequisites-a basic knowledge of measure theoretic probability and Hilbert space theory-and devotes an entire chapter to application in finances, including the Black Scholes market, pricing contingent claims, the general market model, pricing of random payoffs, and interest rate derivatives. Continuous Stochastic Calculus with Application to Finance is your first opportunity to explore stochastic integration at a reasonable and practical mathematical level. It offers a treatment well balanced between aesthetic appeal, degree of generality, depth, and ease of reading.
Customer Reviews:
Material every quantitative financial analyst should know........2006-01-25
Time spent to read the book in detail: Four weeks
The book, 295 pages, is ordered as follows:
Chapter 1 (First 50 pages):
These cover discreet time martingale theory.
Expectation/Conditional expectation: The coverage here is unusual and I found it irritating. The author defines conditional expectation of variables in e(P) - the space of extended random variables for which the expectation is defined - i.e. either E(X+) or E(X-) is defined - rather than the more traditional space L^1(R) - the space of integrable random variables. The source of irritation is that the former is not a vector space. Thus given a variable X in e(P) and another variable Y, in general X+Y will not be defined, for example if EX+ = infinity, EY= - infinity. As a result, one is constantly having to worry about whether one can add variables or not, a real pain. Perhaps an example might help:
Suppose I have two variables X1 AND X2. If I am in the space L^1 then I know both are finite almost everywhere (a.e) and so I can create a third variable Y through addition by setting say Y = X1+X2. In the treatment here however, I have to be careful since it is not a priori clear that X1+X2 is defined a.e. What I need is - one of the proofs in the book - that E(X1)+E(X2) be defined (i.e. it is not the case that one is + infinity the other -infinity). If both E(X1)and E(X2) are finite this reduces to the L^1 case. However, because the Author chooses to work in e(P), we still have, in order to show even this basic result, quite a bit of boring work to do. Specifically: if E(X1) = +infinity then we must have, recall the definition of e(P), that E(X1^+)= +infinity AND E(X1-)
< -infinity and also, because E(X1)+E(X2) is defined E(X2)> -infinity and so , since X2 is in e(P), that E(X2^-)
< -infinity. Now since,
(X1+X2)^-
<= (X1)^- +(X2)^-, we have
E(X1+X2)- less than infinity which shows that a)X1+X2 is defined a.e. and b) it is in e(P).A little more work shows that, E(X1)+E(X2) =E(X1)+E(X2).
When one introduces conditioning the above irritation continues. We have that if X is in e(P) that the conditional expectation E(X|L) exist and is in , not as is standard in the literatureL^1, but rather, in e(P). Consequently we can no longer carry out simple operations, normally done without thinking, such as E(X1|L)+ E(X2|L)= E(X1+X2|L), but rather have to pause to check if as in the example above that E(X1|L)+ E(X2|L) is defined etc, etc.
Submartingale , Supermartingales ,Martingales: The definitions here again are a little unusual. The variables for both Sub and Super martingales are taken to be, yet again, in e(P). This in turn forces the definition:
A submartingale is an adapted process X = (Xn,Fn) such that:
1) E(Xn^+)
<¥ ( The Standard in the literature is to have E(Xn)
<¥
2) E( Xn+1|Fn)>=Xn
Likewise for a supermartingale we get:
A supermartingale is an adapted process X = (Xn,Fn) such that:
1) E(Xn^-)
<¥ ( The Standard in the literature is to have E(Xn)
<¥
2) E( Xn+1|Fn)
<=Xn
These definitions, along with the fact that a martingale is both a supermartingale and submartingale, lead then to the standard - as appears in the literature - definition of a martingale.
Stopping Times, Upcrossing Lemmas, Modes of Convergence: The treatment here is quite nice - modulo the e(P)- inconvenience. The proofs are all given in detail. And the level is at that of Chung's "A Course in Probability Theory", Chapter 9.
Optional Sampling Theorem, Maximal Inequalities: A very rigorous treatment of the Optional Sampling Theorem (OST) is given. The need for closure is emphasized in order for OST to be applied in its full generality. In the absence of closure - the author emphasizes why - it is shown how the OST still applies if the optional times are taken to be bounded. The author then uses these results to show how stopped smartingales - super, sub and marts - are smartingales. Finally, Doobs, submartingle and L^p inequalities are derived.
Chapter 1 (Next 50 pages)
These cover continuous time martingale theory under the assumption that the probability space is complete and the filtration augmented and right continuous.
The treatment here - most of the hard work has already been done in the discreet case - uses the standard bootstrapping technique based on sequences of optional times taking only countable values, along with the assumption of right continuity of paths to generalize the discreet time results - through passing to limits - to analogous ones for a continuous time, i.e. where the index set is a subset of [0, ¥], setting. The Upcrossing lemmas, Convergence results, OST and Doobs inequalities are all derived
Next follows a superb treatment of local martingales.
At this point, and for what follows, the treatment switches to smartingales, with continuous paths.
It is now shown that for any bounded - continuous - martingale M, there exists a unique continuous bounded variation (increasing) process starting at 0 -denoted by [M], such that the process M^2-[M] is a closed martingale. Moreover, it is shown that this process is the limit in L^2 of the Quadratic variation of M. This result is then generalized to the case where M is a local martingale where it is shown that M^2-[M] is also a local martingale and where [M] is now only the limit in probability of the quadratic variation. Next the covariation process for two local martingales [M N] is defined and it is shown that MN -[MN] is again a local martingale.
Finally, integration with respect to integrators of bounded variation is defined for a suitable class of integrands and the "Kunita Watanabe", inequality derived.
All of the above is then extended to the case of Semi Martingales.
Chapter 2 (29 Pages) Brownian Motion
Definition of. Existence is shown. The Weak Markov properties derived. I found the notation in this chapter to be rather cumbersome. One would be better served by skipping this chapter, replacing it instead, by chapter 2 in Karatzas and Shreve's "Brownian Motion and Stochastic Calculus" (KS).
Chapter 3 (80 Pages) Stochastic Integration
This chapter, my favourite in the book, is a detailed discussion of integration with respect to continuous semi-martingales. The approach is modern. The chapter starts with a detailed definition of stochastic integration with respect to a continuous local martingale M. The level of rigour, is at the level of sections 3.1 and 3.2 of KS. However, the approach is different and in my opinion more elegant. Leveraging on the material in chapter 1 the stochastic integral for a square integrable - with respect to the induced product measure-progressively measurable, r.v X is defined to be the unique square integrable local martingale, starting at 0, I, such that for any other continuous local martingale N we have:
[I, N] = X DOT [M,N].
This is then extended to the case where X is only locally pathwise integrable with respect to [M], which is then extended to the case where M is a continuous semi martingale.
It is then shown how in the case where X is simple predictable the above definition yields that suggested by one's intuition, that the space of simple predictable variables is dense in the space of square integrable - with respect to the induced product measure - predictable processes, and that I in this case is an L^2 - this is the usual approach - limit of , with respect to P, of simple integrals.
Following this, is a derivation of Ito's lemma - this says that semimartingales are preserved under smooth transformation. It is then shown that given a P semimartingale X, a probability measure Q equivalent to P, X is a Q semi martingale and its Compensator under Q given by Uq = Up + [logM,X], where M is the Radoyn Nikodym derivative of Q with respect to P. It is then an easy step to conclude that the local martingale component of X under Q is related to that under P by:
LMq = LMp - [logM,X].
Thus the Girsanov theorem is proved. In the case where M is of the form,
M = exp( L - [L]/2), where L is a continuous local martingale, conditions on L , those of Novikov and other weaker one, necessary to make M a martingale are given and proven.
Finally, the Chapter concludes with a detailed section on the Martingale representation theorem. Most of this section is very similar to that in section 3.4 of KS. However, while the treatment there leaves a lot of work for the reader, many of the key results are buried in the exercises, the results here are all spelled out and detailed proofs furnished.
Chapter 4 (84 Pages) Applications to Finance:
I only read, the first 40 pages - the section dealing with the Black and Scholes Economy and that with The General Market Model. The treatment of the Black and Scholes economy - 17 pages -is standard and concise. The General Market model is at the level of chapters 4 and 5 in Nielson Pricing and Hedging of Derivatives Securities"(N). Because however, the Author has spent the time to develop the machinery in detail, unlike in the case Nielsen's 106 pages of "hand waving", the pace is a lot faster and the treatment more general. Moreover, results like, no free lunch with limited risk implying the existence of local martingale measure, based the work of Schaechermayer, something not alluded to in Nielson, are covered here The final 40 or so- which I have not read-pages are devoted to applications of the general theory to pricing specific derivatives.
Good Treatment of Continuous Time Martingales .......2005-06-10
Chapter 1: This is a summary of what every probabilist should know about Continuous Time Martingales. Essentially it does, although in a rather terse fashion, and with no examples, for Continuous Time Martingales, what David Williams book, "Probability and Martingales", does for the discreet time case. By restricting himself to the continuous case, as opposed to the more general cadlag processes, the author is able to provide a simple proof of the Doob Meyer Decomposition. The coverage in this chapter is more extensive than that of Chapter 1 in Karatzas and Shreeve and perhaps closer to ChapterII in Rogers and Williams.
Chapter 2: Essentially a brief introduction to Brownian Motion. I would advize the reader to skip this Chapter and replace it with chapter 2 of Karatzas and Shreves "Stochastic Calculus and Brownian Motion". The coverage there is more rigorous.
Chapter 3:This chapter covers Stochatic Integration with respect to a Continous Time Local Martingales. The coverage here mirrors that of chapter three in Karatazs and Shreve though the notation is perhaps closer in spirit to Chapter 4 of Rogers and Williams, Diffusions, Markhov Processes and Martingales. The construction of the Stochastic Integral is then followed by the usual suspects: Ito's Lemma which says that the SemiMartingale property is preserved under smooth transformations. The Martingale Representation Theorem this says that in the case where the integral is with respect to Brownian Motion, then the integral viewed as a mapping from the space of measurable adapted processes that are square integrable with respect to the product measure onto the space of continuous square integrable martingales is surjective. And last but not least Girsanovs theorem which allows one, modulu the satisfaction of the Novikov Condition, to alter the "drift term" in semi martingales through changing to an equivalent measure.
Chapter 4: I would advice the reader to replace this with chapters 4 and 5 in Nielsen's "Pricing and Hedging of Derivative Securities" for the general theory and chapter 6 for the Black and Scholes Economy. The coverage there is the best I have seen.
Elegant Math Book on Finance - you need the math to read.......2004-01-22
This is a math book first and foremost. It uses advanced mathematical techniques to discuss aspects of randomness that can be used to understand finance. Please don't mistake it for a course to teach concepts in basic finance.
It is a very elegant and sophisticated book for those who are very well versed in the necessary mathematics in stochastic calculus and in particular Martingale theory to show them how these tools can be applied to problems in finance.
If you have the math background and are interested in this topic you will get a lot from this book. If you don't have the math, don't bother. This book will be opaque.
It is indeed meant for learning.......2004-01-02
I completely disagree with Student.
This book is indeed meant for learning. Just do not take it as your first entry into Stochastic Calculus. Take it as a second reading. It is complete, thorough and well, very well written.
It will teach you. A lot. All theorems are cross-referenced, so you will not have any "it is obvious that" etc. Theorems are proved, over and over again, until they hammer themselves in your head.
It is a fine achievement, if you want something quick and dirty read something else.
Not meant for learning.......2001-02-03
Some books are meant to teach, and to elucidate new material; this book is not one of them. It seems the purpose of this book was rather to record for prosperity all theorems related to Stochastic Calculus. Instead of developing any intuition on the subject, the author seems to think the purpose of writing is to use the most elegant proofs with the most modern of mathematical jargon. In short, the book consists of stated lemmas and theorems with terse, undeveloped proofs. This book will not teach you anything.
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