Stochastic Calculus and Financial Applications

Book Description

The Wharton School course on which the book is based is designed for energetic students who have had some experience with probability and statistics, but who have not had advanced courses in stochastic processes. Even though the course assumes only a modest background, it moves quickly and - in the end - students can expect to have the tools that are deep enough and rich enough to be relied upon throughout their professional careers. The course begins with simple random walk and the analysis of gambling games. This material is used to motivate the theory of martingales, and, after reaching a decent level of confidence with discrete processes, the course takes up the more demanding development of continuous time stochastic process, especially Brownian motion. The construction of Brownian motion is given in detail, and enough material on the subtle properties of Brownian paths is developed so that the student should sense of when intuition can be trusted and when it cannot. The course then takes up the It¿ integral and aims to provide a development that is honest and complete without being pedantic. With the It¿ integral in hand, the course focuses more on models. Stochastic processes of importance in Finance and Economics are developed in concert with the tools of stochastic calculus that are needed in order to solve problems of practical importance. The financial notion of replication is developed, and the Black-Scholes PDE is derived by three different methods. The course then introduces enough of the theory of the diffusion equation to be able to solve the Black-Scholes PDE and prove the uniqueness of the solution.

Customer Reviews:

A Beautiful MATH Book.......2006-06-21

Before I write this review, it's only fair to disclose that before even hearing of it I already had a very solid background in (graduate-level) analysis, which as another reader astutely pointed out is often considered "calculus" in the math community (I think the classic Calculus by Shlomo Steinberg, which can be found free online, has been used at Harvard for decades, while Tom Apostol's "Calculus," a misnomer to say the least, is the standard text at Stanford and Cal Tech - both are really books on advanced calculus and elementary real analysis). Part of the reason I am writing this is to clarify the distinction - many people aspiring towards quantitative roles on Wall Street don't know exactly what the mathematical prerequisites are for a particular subject or presentation, and hopefully I can help clarify this for other readers who, like myself, sought books like this one to learn the basics of mathematical finance.

On that note, Steele's book is a MATH book. By contrast, the wonderful book by Baxter & Rennie emphasizes core ideas with emphasis on the relationship between the three primary tools of the discipline (Martingale Representation, Ito-Doeblin Calculus, and the Feynman-Kac formula) while Shreve's classic emphasizes actual development of key models and techniques. Even Oksendal, which is aimed at a slightly more sophisticated mathematical audience, emphasizes applications at the expense of elegance.

In contrast, Steele's book is a math book aimed at Wharton (read: finance and economics doctoral students, likely in their second year) students with varied interests. Students taking this course probably have already taken a rigorous course in asset pricing theory from the academic viewpoint and need to fill in the blanks with the continuous-time techniques to extend these techniques and to understand stochastic calculus at the level necessary for research in economics/finance.

With that in mind, the book is versatile enough to be appreciated by different audiences. Steele certainly takes care give a clear, well-motivated presentation which explains to the reader WHY he is giving a concept, proof, or problem, and breaks the book up into small, digestible chapters. The problems are neither overly difficult nor disconnected from the text, although doing them is not an essential part of understanding the overall view. Furthermore, Steele clearly takes delight in the beauty of stochastic calculus, as demonstrated by Chapter 5 - Richness of Paths, which discusses the "interesting" properties of Brownian motion. For anyone who sat through a difficult analysis class thinking the whole purpose of the course was to annoy and taunt the student with irrelevant counterexamples (remember constructing a continuous yet non-differentiable function using limits?), this chapter will be especially fun.

In the first part of the book, Steele covers the basics of the random walk and martingales, introducing important theorems such as the upcrossing (downcrossing) lemma, submartingales and the Doob Decomposition theorem, the basic martingale inequalities, stopping times, and conditional probability (for those who are familiar with Williams' Probability with Martingales, the treatment is similiar). He then covers Brownian motion from both the standard perspective (a Brownian motion is a process such that...) and more intuitively as a limit of random walks (i.e. the "wavelet" construction/proof), using this subject as an opportunity to extend the martingale concepts to continuous-time.

In what could roughly be called the "second" part of the book, Steele develops the Ito integral as a martingale and as a process. Steele provides a lot of detail to the subject, perhaps in mind with the view that readers using stochastic calculus with more general underlying processes will have to understand the difference between a martingale and "just" a local martingale. He then quickly but sufficiently covers the standard topics of Ito calculus - Ito's lemma, quadratic variation, and the basic SDE, although in the Picard-type existence/uniqueness proof of SDEs he shows why the careful description of the Ito integral is not simply technical.

The next part of the book covers the "standard" topics in financial mathematics that would appeal to quant finance students . The chapter on arbitrage covers the basic Black-Scholes-Merton equation and its generalization to arbitrage pricing, although Steele (appropriately) addresses Black and Scholes CAPM derivation of their options pricing formula, which gives the finance/economics reader a historical perspective. The chapter on diffusions is excellent and gives all of the necessary elements for handling "nice" parabolic second-order equations. He even sneaks in Green's functions, series expansions, and the Maximum Principle without making uninterested readers have to learn them to follow the presentation.

In the last few chapters, he covers Martingale Representation, Girsanov's Theorem and their applications to more advanced topics in pricing, such as forward measures. The problems in this part of the book are nice because they help the reader understand the intuition behind a particular mathematical principle but not necessarily its application to a well-recognized model. The final chapter on the Feynman-Kac formula gives a very intuitive proof of its topic which helps the reader understand what is meant by "killing" a process and hopefully how that translates into finance; other books often just do a coefficient-matching proof, which really doesn't capture what's really going on.

I emphasize again that while the book is designed to serve a different purpose than texts such as Shreve or Baxter & Rennie, it can help readers of different backgrounds understand the basic elements needed for more advanced stochastic analysis and gain an appreciation for both the beauty of the subject and the underlying intuition liking the math to the finance. The prerequisite, though, is at least a (rigorous undergrad) course in real analysis, probably some familiarity with measure theory, probability, and L(p) spaces (or at least L(1,2,inf) spaces), and at least basic familiarity with the elements of stochastic calculus (Ito's lemma and computations with "box calculus", for example). For readers seeking a more comprehensive treatment of quantitative finance, this book is reasonably good mathematical preparation to understand Musiela/Rutkowski, and for doctoral students, understanding most of the topics in this book with a brief introduction to dynamic programming in the continuous-time setting is sufficient background to read Merton's book (consumption-investment problems) as well as understand the basics of derivative pricing.

Good book.......2006-04-23

This is a good start.

One thing about mathematical prerequisitives and a pet peeve.
In general, when mathematicians state that a minimal prerequistive is calculus, they are not refering to the calculus that a science major such as a physicist would study... as David Hilbert once said, Physics is too hard for physicists...this is engineering calculus... this is geared toward usage and application( they are consumers of math). What a mathematician is refering to when they mention calculus is actually analysis...the study of limits, etc. You should be comfortable with topological concepts such as compact sets, open and closed sets, limits, epsilon-delta notation, etc, etc.
So you should keep this in mind. So if you have had a course in classical analysis and a course in probability which makes use of this background, you should have no problem with this book.
But if you are a typical engineer, physicist ( whether PHd or not ) and have never been exposed to the concept of a compact set, or group theory, etc, etc, then you need to do some homework. You will of course have the brain power...you just need the lingo and the concepts...you have probably learned the material but just don't know the math speak for it.
The type of "calculus" book I am thinking of are books such as 'Elementary Classical Analysis' by Jerrold Marsden ( my freshman "calculus" book, or 'Real Analysis' by Royden ( my sophomore "calculus" book ). By all means don't give up...but if you are planning a serious career in quantitative finance, you should master the concepts in these books. They will go a long way to help you master modern economics & finance. This book will too.

I Hate It When Books Lie About Mathematical Requriements.......2003-05-03

The book says that its only prerequisites are calculus and probability. This is not true. To be able to understand everything that's going on, you'll need to have a very good grasp of subjects like measure-theoretic probability, Hilbert spaces, and functional analysis. I quit reading the book in the early chapters, when Steele starts talking about things like "spans" and "denseness" for function spaces. I don't know where you went to school, but at my school, I didn't learn these subjects in my intro calculus and probability classes. To summarize, don't buy this book if you don't know measure theory.

If you want to learn quant finance at an elementary level, Baxter and Rennie is much, much better. Moreover, if you're comfortable with measure theory,and you want to learn the math that's necessary for option pricing, you'd be better off buying Oksendal's excellent book, which is at least as rigorous as Steele's book but much more clear.

Riskfree profit !!.......2003-03-09

The book is at the interface of three areas, math, statistics, and finance. While connections between the first two have a long history, it was the connection to finance that caught my attention. Coming from math myself, I needed first to take a closer look at the book to orient myself. The mathematical subjects, smooth sailing, include stochastic differential equations (SDE) as they relate to PDEs; and the ideas from probability and statistics include Brownian motion, martingales, stochastic processes, and the Feynman-Kac connection. Browsing the chapters I found them to be a lovely presentation of ideas with which I am familiar. For me, it was chapter 10 that turned out to have stuff that I wasn't familiar with. That is the finance part, and it is based on a model for Option Pricing developed in 1973 by Fischer Black and Myron Scholes. An arbitrage opportunity [simplified] amounts to the simultaneous purchase and sale of related securities which is guaranteed to produce a *riskless* profit. It was after reading more in this chapter I understood why the book is used in a course at the Wharton School at the University of Pennsylvania. I am impressed with the level of math in this course. Part of the motivation in the applications to finance is that arbitrage enforces the price of most derivative securities. And I learned from ch 10 that the SDE of the Black-Scholes model governs the processes which represent the two variables S, the price of a stock, and B the price of a bond, both S and B representing stochastic variables depending of time t, i.e., both stochastic processes. In the model, S is a geometric Brownian motion, and B is a deterministic process with exponential growth. The two are determined as solutions to the SDE of Black-Scholes.

Review from a grad student not at Wharton.......2003-01-29

Reading Steele's book without attending has classes at Wharton leaves the reader looking for explanations to equations. Ideas are not clearly explained and problems are not worked out in detail with a descriptive process of how to solve the problem. The brief explanations in this book intended for a reader with knowledge of calculus and probability but not having a background in Stochastic calculus do not provide a sufficient basis for the reader to learn the material.

Continuous Stochastic Calculus with Applications to Finance

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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

Continuous Stochastic Calculus with Applications to Finance
Michael Meyer
Manufacturer: Chapman & Hall/CRC
ProductGroup: Book
Binding: Hardcover

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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.

Conservation Laws and Symmetry: Applications to Economics and Finance (Research Monographs in Japan-U.S. Business and Economics)

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Conservation Laws and Symmetry: Applications to Economics and Finance (Research Monographs in Japan-U.S. Business and Economics)

Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover

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ASIN: 0792390725

Random Evolutions and their Applications: New Trends (Mathematics and Its Applications)

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Random Evolutions and their Applications: New Trends (Mathematics and Its Applications)
A. Swishchuk
Manufacturer: Springer
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ASIN: 0792362640

Book Description

This book is devoted to new trends in random evolution and their applications to the stochastic evolutionary system. It contains new developments such as an analogue of Dynkin's formula, boundary value problems, stability and control of random evolutions, stochastic evolutionary equations, and driven martingale measures. In addition, it treats statistics of random evolutions processes, statistics of financial stochastic models, and stochastic stability and control of financial markets.
Audience: This volume will be of interest to research and applied mathematicians working in the fields of applied probability, stochastic processes, and random evolutions, as well as experts in statistics, finance and insurance.

Stochastic Modeling and Optimization: With Applications in Queues, Finance, and Supply Chains (Springer Series in Operations Research)

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Stochastic Modeling and Optimization: With Applications in Queues, Finance, and Supply Chains (Springer Series in Operations Research)

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ASIN: 0387955828

Book Description

This book covers the broad range of research in stochastic models and optimization. Applications covered include networks, financial engineering, production planning and supply chain management. Each contribution is aimed at graduate students working in operations research, probability, and statistics.

Stochastic Processes and Applications to Mathematical Finance

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Stochastic Processes and Applications to Mathematical Finance

Manufacturer: World Scientific Publishing Company
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Based around recent lectures given at the prestigious Ritsumeikan conference, the tutorial and expository articles contained in this volume are an essential guide for practitioners and graduates alike who use stochastic calculus in finance.

Book Description

This book surveys new advances in Markov-switching models with applications to business cycle research and finance. The extensive editors' introduction surveys the existing methods and new results of the last decade. Individual chapters study features of the U.S. and European business cycles, with particular focus on the role of monetary policy, oil shocks, co-movements among key variables, and the short-run versus long-run consequences of an economic recession. The book also features extensive analysis of currency crises and the possibility of bubbles or fads in stock prices. A concluding chapter offers useful new results on testing for this kind of regime-switching behaviour. Overall, the book provides a state-of-the-art overview of methods and results for estimation and uses of Markov-switching time-series models.

Pricing American options with stochastic interest rates: International money market applications (Working paper / Mitsui Life Financial Research Center, ... Administration, University of Michigan)

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Pricing American options with stochastic interest rates: International money market applications (Working paper / Mitsui Life Financial Research Center, ... Administration, University of Michigan)
Kaushik I Amin
Manufacturer: Mitsui Life Financial Research Center, School of Business Administration, University of Michigan
ProductGroup: Book
Binding: Unknown Binding

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