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Continuous Stochastic Calculus with Applications to Finance (Hardcover)

by Michael Meyer (Author) "Preliminaries. Let (F, P) be a probability space, R = [-, +] denote the extended real line and B(R) and B (Rn) the Borel o-fields..." (more)
Key Phrases: local deflator, bounded variation process, bounded optional times, Optional Sampling Theorem, Black Scholes, Strong Law of Large Numbers (more...)
4.0 out of 5 stars See all reviews (5 customer reviews)
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Editorial Reviews

Review
As a reference or a second book on stochastic calculus, Meyer is outstanding. In a formal, highly rigorous manner, he develops stochastic calculus, all the while focusing on topics of primary interest to financial engineers. … I highly recommend Meyer. It is an excellent introduction and reference on stochastic calculus.
- Glyn A. Holton of Contingency Analysis

As a reference or a second book on stochastic calculus, Meyer is outstanding. In a formal, highly rigorous manner, he develops stochastic calculus, all the while focusing on topics of primary interest to financial engineers. … I highly recommend Meyer. It is an excellent introduction and reference on stochastic calculus.
- Glyn A. Holton of Contingency Analysis

Product 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.

Product Details

  • Hardcover: 336 pages
  • Publisher: Chapman & Hall/CRC; 1 edition (October 25, 2000)
  • Language: English
  • ISBN-10: 1584882344
  • ISBN-13: 978-1584882343
  • Product Dimensions: 9.2 x 5.9 x 1 inches
  • Shipping Weight: 1.4 pounds (View shipping rates and policies)
  • Average Customer Review: 4.0 out of 5 stars See all reviews (5 customer reviews)
  • Amazon.com Sales Rank: #2,357,583 in Books (See Bestsellers in Books)

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Inside This Book (learn more)
First Sentence:
Preliminaries. Let (F, P) be a probability space, R = [-, +] denote the extended real line and B(R) and B (Rn) the Borel o-fields on R and Rn respectively. Read the first page
Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
local deflator, bounded variation process, bounded optional times, scalar semimartingale, trading strategy investing, spot martingale measure, stochastic product rule, finite dimensional rectangles, submartingale sequence, uniformly bounded martingale, continuous local martingale, finite dimensional cylinders, strictly positive linear functional, local martingale measure, exceptional null set, continuous semimartingale, augmented filtration, covariation process, progressively measurable processes, reversed submartingale, semimartingale decomposition, nondecreasing nature, numeraire asset, reducing sequence, continuous increasing process
Key Phrases - Capitalized Phrases (CAPs): (learn more)
Optional Sampling Theorem, Black Scholes, Strong Law of Large Numbers, Upcrossing Lemma, Law of One Price, Martingale Theory, Consequences of Ito, Rd-valued Brownian, Real Analysis, Submartingale Convergence Theorem, The Novikov
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4 of 4 people found the following review helpful:
4.0 out of 5 stars Good Treatment of Continuous Time Martingales , June 9, 2005
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.

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3 of 3 people found the following review helpful:
5.0 out of 5 stars It is indeed meant for learning, January 2, 2004
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.

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4 of 5 people found the following review helpful:
5.0 out of 5 stars Material every quantitative financial analyst should know., January 24, 2006
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.
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4.0 out of 5 stars Elegant Math Book on Finance - you need the math to read
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. Read more
Published on January 21, 2004 by Craig Matteson

2.0 out of 5 stars Not meant for learning
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... Read more
Published on February 2, 2001 by Student

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