1、Financial Derivatives Pricing, Applications, and Mathematics Jamil Baz, George Chacko0.1 Financial Derivatives This book offers a succinct account of the principles of nancial derivatives pricing. The rst chapter provides readers with an intuitive exposition of basic random calculus. Concepts such a
2、s volatility and time, random walks, geometric Brownian motion, and It os lemma are discussed heuristically. The second chapter develops generic pricing techniques for assets and derivatives, determining the notion of a stochastic discount factor or pricing kernel, and then uses this concept to pric
3、e conventional and exotic derivatives. The third chapter applies the pricing concepts to the special case of interest rate markets, namely, bonds and swaps, and discuss factor models and termstructureconsistent models. The fourth chapter deals with a variety of mathematical topics that underlie deri
4、vatives pricing and portfolio allocation decisions, such as mean reverting processes and jump processes, and discusses related tools of stochastic calculus, such as Kolmogorov equations, martingales techniques, stochastic control, and partial differential equations. Jamil Baz is Head of Global Fixed
5、 Income Research at Deutsche Bank, London. Prior to this appointment, he was a Managing Director at Lehman Brothers. Dr. Baz is a Research Fellow at Oxford University, where he teaches nancial economics. He has degrees from the Ecole des Hautes Etudes Commerciales (Dilpl ome), the London School of E
6、conomics (MSc), MIT (SM), and Harvard University (AM, PhD). George Chacko is Associate Professor in the Finance Faculty of Harvard University Business School. He has degrees from MIT (SB), the University of Chicago (MBA), and Harvard University (PhD). Professor Chackos research continues to focus on
7、 optimal portfolio choice and consumption decisions in a dynamic framework. 10.2 Acknowledgements We are as ever in many peoples debt. Both authors are lucky to have worked with or been taught by eminent experts such as John Campbell, Sanjiv Das, Jerome Detemple, Ken Froot, Andrew Lo, Franco Modigli
8、ani, Vasant Naik, Michael Pascutti, Lester Seigel, Peter Tufano, Luis Viceira, and JeanLuc Vila. A list, by no means exhaustive, of colleagues who have read or inuenced this manuscript includes Richard Bateson, Eric Briys, Robert Campbell, Marcel Cassard, Didier Cossin, Franc ois Degeorge, Lev Dynki
9、n, David Folkerts Landau, Vincent Koen, Ravi Mattu, Christine MiqueuBaz, Arun Muralidhar, Prafulla Nabar, Brian Pinto, David Prieul, Vlad Putyatin, Nassim Taleb, Michele Toscani, Sadek Wahba, and Francis Yared. Special thanks are due to Tarek Nassar, Saurav Sen, Feng Li, and Dee Luther for diligent
10、help with the manuscript. The biggest debt claimant to this work is undoubtedly Robert Merton, whose inuence pervades this manuscript, including the footnotes; as such, because there is no free lunch, he must take full responsibility for all serious mistakes, details of which should be forwarded dir
11、ectly to him. 20.3 Introduction This book is about risk and derivative securities. In our opinion, no one has described the issue more eloquently than Jorge Luis Borges, an intrepid Argentinian writer. He tells a ctional story of a lottery in ancient Babylonia. The lottery is peculiar because it is
12、compulsory. All subjects are required to play and to accept the outcome. If they lose, they stand to lose their wealth, their lives, or their loved ones. If they win, they will get mountains of gold, the spouse of their choice, and other wonderful goodies. It is easy to see how this story is a metap
13、hor of our lives. We are shaped daily by doses of randomness. This is where the providential nancial engineer intervenes. The engineers thoughts are along the following lines: to confront all this randomness, one needs articial randomness of opposite sign, called derivative securities. And the engin
14、eer calls the ratio of these two random quantities a hedge ratio. Financial engineering is about combining the Tinker Toys of capital markets and nancial institu tions to create custom riskreturn proles for economic agents. An important element of the nancial engineering process is the valuation of
15、the Tinker Toys; this is the central ingredient this book pro vides. We have written this with a view to the following two objectives: to introduce readers with a modicum of mathematical background to the valuation of derivatives to give them the tools and intuition to expand upon these results when
16、 necessary By and large, textbooks on derivatives fall into two categories: the rst is targeted toward MBA students and advanced undergraduates, and the second aims at nance or mathematics PhD students. The former tend to score high on breadth of coverage but do not go in depth into any specic area
17、of derivatives. The latter tend to be highly vigorous and therefore limit the audience. While this book is closer to the second category, it strives to simplify the mathematical presentation and make it accessible to a wider audience. Concepts such as measure, functional spaces, and Lebesgue integra
18、ls are avoided altogether in the interest of all those who have a good knowledge of mathematics but yet have not ventured into advanced mathematics. The target audience includes advanced undergraduates in mathematics, economics, and nance; graduate students in quantitative nance masters programs as
19、well as PhD students in the aforemen tioned disciplines; and practitioners afliated with an interest derivatives pricing and mathematical curiosity. The book assumes elementary knowledge of nance at the level of the Brealey and Myers cor porate nance textbook. Notions such as discounting, net presen
20、t value, spot and forward rates, and basic option pricing in a binomial model should be familiar to the reader. However, very little knowl edge of economics is assumed, as we develop the required utility theory from rst principles. The level of mathematical preparation required to get through this b
21、ook successfully comprises knowledge of differential and integral calculus, probability, and statistics. In calculus, readers need to know basic differentiation and integration rules and Taylor series expansions, and should have had the standard yearlong sequence in probability and statistics. This
22、includes conventional, discrete, and continuous probability distributions and related notions, such as their moment generating functions and characteristic functions. The outline runs as follows: 1. Chapter 1 provides readers with the mathematical background to understand the valuation con cepts and
23、 the valuation concepts developed in chapters 2 and 3. It provides an intuitive exposition of basic random calculus. Concepts such as volatility and time, random walks, geometric Brownian mo tion, and It os lemma are exposed heuristically and given, where possible, an intuitive interpretation. 3This
24、 chapter also offers a few appetizers that we call paradoxes of nance: these paradoxes explain why forward exchange rates are biased predictors of future rates; why stock investing looks like a free lunch; and why success in portfolio management might have more to do with luck than with skill. 2. Ch
25、apter 2 develops generic pricing techniques for assets and derivatives. The chapter starts from basic concepts of utility theory and builds on these concepts to derive the notion of a stochastic discount factor, or pricing kernel. Pricing kernels are then used as the basis for the derivation of all
26、subsequent pricing results, including the BlackScholes/Merton model. We also show how pricing kernels relate to the hedging, or dynamic replication, approach that is the origin of all modern valua tion principles. The chapter concludes with several applications to equity derivatives to demonstrate t
27、he power of the tools that are developed. 3. Chapter 3 specializes the pricing concepts of Chapter 3 to interest rate markets; namely bonds, swaps, and other interest rate derivatives. It starts with elementary concepts such as yieldtomaturity, zerocoupon rates, and forward rates; then moves on to n
28、aive measures of interest rate risk such dura tion and convexity and their underlying assumptions. An overview of interest rate derivatives precedes pricing models for interest rate instruments. These models fall into two conventional families: factor models, to which the notion of price of risk is
29、central, and termstructureconsistent models, which are partial equilibrium models of derivatives pricing. The chapter ends with an interpretation of interest rates as options. 4. Chapter 4 is an expansion of the mathematical results in Chapter 1. It deals with a variety of mathematical topics that u
30、nderlie derivatives pricing and portfolio allocation decisions. It describes in some detail random processes such as random walks, arithmetic and geometric Brownian motion, meanreverting processes and jump processes. This chapter also includes an exposition of the rules of It?calculus and contrasts
31、it with the competing Stratonovitch calculus. Related tools of stochastic calculus such as Kolmogorov equations and martingales are also discussed. The last two sections elaborate on techniques widely used to solve portfolio choice and option pricing problems: dynamic programming and partial differe
32、ntial equations. We think that one virtue of the book is that the chapters are largely independent. Chapter 1 is essential to the understanding of the continuoustime sections in Chapters 2 and 3. Chapter 4 may be read independently, though previous chapters illuminate the concepts developed in each
33、chapter much more completely. Why Chapter 4 is at the end and not the beginning of this book is an almost aesthetic undertaking: Some nance experts think of mathematics as a way to learn nance. Our point of view is different. We feel that the joy of learning is in the process and not in the outcome.
34、 We also feel that nance can be a great way to learn mathematics. 4Contents 0.1 Financial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0.2 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 0.3 Introduction . . . . . . .
35、. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1 Preliminary Mathematics 1 1.1 Random Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Another Take on V olatility and Time . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 A First G
36、lance at It os lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Continuous Time: Brownian Motion; More on It os lemma . . . . . . . . . . . . . . 5 1.5 TwoDemensional Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.6 Bivariate It os lemma . . . . . . .
37、. . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.7 Three Paradoxes of Finance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.7.1 Paradox 1: Siegels Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.7.2 Paradox 2: The Stock, FreeLunch Paradox . . . . . .
38、. . . . . . . . . . . . 9 1.7.3 Paradox 3: The Skill Versus Luck Paradox . . . . . . . . . . . . . . . . . . . 10 2 Principles of Financial Valuation 12 2.1 Uncertainty, Utility Theory, and Risk . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Risk and the Equilibrium Pricing of Securities
39、 . . . . . . . . . . . . . . . . . . . . . 16 2.3 The Binomial OptionPricing Model . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4 Limiting OptionPricing Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.5 ContinuousTime Models . . . . . . . . . . . . . . . . . . . . .
40、. . . . . . . . . . . 27 2.5.1 The BlackScholes/Merton Model Pricing Kernel Approach . . . . . . . . . 27 2.5.2 The BlackScholes/Merton Model Probabilistic Approach . . . . . . . . . . 33 2.5.3 The BlackScholes/Merton Model Hedging Approach . . . . . . . . . . . . 35 2.6 Exotic Options . . . . . . .
41、 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.6.1 Digital Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.6.2 Power Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.6.3 Asian Options . . . . . . . . . . . . . . .
42、. . . . . . . . . . . . . . . . . . . 39 2.6.4 Barrier Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3 Interest Rate Models 46 3.1 Interest Rate Derivatives: Not So Simple . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2 Bonds and Yields . . . . . . . . . . . . .
43、 . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2.1 Prices and Yields to Maturity . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2.2 Discount Factors, ZeroCoupon Rates, and Coupon Bias . . . . . . . . . . . 49 3.2.3 Forward Rates . . . . . . . . . . . . . . . . . . . . . . . . . . .
44、. . . . . . . 50 3.3 Naive Models of Interest Rate Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.3.1 Duration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.3.2 Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 53.3
45、.3 The Free Lunch in the Duration Model . . . . . . . . . . . . . . . . . . . . . 62 3.4 An Overview of Interest Rate Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 65 3.4.1 Bonds with Embedded Options . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.4.2 Forward Rate Agreements .
46、. . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.4.3 Eurostrip Futures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.4.4 The Convexity Adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.4.5 Swaps . . . . . . . . . . . . . . . . . . . . . . . . . .
47、. . . . . . . . . . . . 71 3.4.6 Caps and Floors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.4.7 Swaptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.5 Yield Curve Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48、 . 73 3.5.1 The CMS Swap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.5.2 The Quanto Swap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.6 Factor Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.6.1 A General Sing
49、leFactor Model . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.6.2 The Merton Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.6.3 The Vasicek Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.6.4 The CoxIngersollRoss Model . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.6.5 RiskNeutral Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.7 TermStructureConsistent Models . . . . . . . . . . . . . . . . . . . . . . . . . . .
