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E-grāmata: Factor Model Approach to Derivative Pricing

(Department of Finance, Mihaylo College of Business and Economics, California State University, Fullerton, USA)
  • Formāts: 294 pages
  • Izdošanas datums: 19-Dec-2016
  • Izdevniecība: Chapman & Hall/CRC
  • Valoda: eng
  • ISBN-13: 9781498763356
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Factor Model Approach to Derivative PricingPalielināt
  • Formāts: 294 pages
  • Izdošanas datums: 19-Dec-2016
  • Izdevniecība: Chapman & Hall/CRC
  • Valoda: eng
  • ISBN-13: 9781498763356
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Written in a highly accessible style, A Factor Model Approach to Derivative Pricing lays a clear and structured foundation for the pricing of derivative securities based upon simple factor model related absence of arbitrage ideas. This unique and unifying approach provides for a broad treatment of topics and models, including equity, interest-rate, and credit derivatives, as well as hedging and tree-based computational methods, but without reliance on the heavy prerequisites that often accompany such topics.



Whether being used as text for an intermediate level course in derivatives, or by researchers and practitioners who are seeking a better understanding of the fundamental ideas that underlie derivative pricing, readers will appreciate the books ability to unify many disparate topics and models under a single conceptual theme.

Recenzijas

"The approach that the author has taken should provide a new insight for derivative pricing, which is quite useful to learn the topic for students entering graduate studies or starting senior undergraduate projects. In particular, the book is recommended for students with engineering background as well as those in business schools." Yuji Yamada, Professor and Chair of Master's Program in Systems Management, Faculty of Business Sciences, University of Tsukuba, Japan

"This book will be of tremendous help to those who either are new to the field and unsure where to start, or have started but are frustrated by not being able to apply what they learn. This is one of a kind book that provides a "step by step" recipe for developing the broadest possible set of derivating pricing models with the minimum reliance on sophisticated mathematical analysis. The style of the book is truly educational also - not only does it provide intuition behind every use of mathematics, but it also ensures the intuitions are coherent. The factor model approach not only simplifies the steps of derivations, but I believe provides also a great opportunity for students with or without finance background to appreciate the beauty of this fundamental model in finance. Well thought-out exercises are provided in each chapter, which makes the book a great choice for advanced undergraduate or graduate courses." Jonathan Yu-Meng Li, Assistant Professor of Risk Management, University of Ottawa, Canada

List of Figures xv
Preface xix
Notation xxi
Chapter 1 Building Blocks and Stochastic Differential Equation Models 1(18)
1.1 Brownian Motion And Poisson Processes
1(7)
1.1.1 Gaussian Random Variables
2(1)
1.1.2 Brownian Motion
2(2)
1.1.3 Poisson Random Variables
4(1)
1.1.4 Poisson Processes
5(1)
1.1.5 Increments of Brownian Motion and Poisson Processes
6(2)
1.1.5.1 Brownian Motion Increment
6(1)
1.1.5.2 Poisson Process Increment
7(1)
1.2 Stochastic Differential Equations
8(6)
1.2.1 The Differential
8(2)
1.2.1.1 The Problem with Jumps
8(1)
1.2.1.2 Altering the Differential for Jumps
9(1)
1.2.2 Compound Poisson Process
10(1)
1.2.3 Ito Stochastic Differential Equations
11(1)
1.2.3.1 Instantaneous Statistics
11(1)
1.2.4 Poisson Driven Differential Equations
12(2)
1.3 Summary
14(5)
Chapter 2 Ito's Lemma 19(14)
2.1 Chain Rule Of Ordinary Calculus
19(2)
2.1.1 Multivariable Taylor Series Expansions
20(1)
2.2 Ito's Lemma For Brownian Motion
21(3)
2.2.1 Replacing dz2 by dt
22(2)
2.2.1.1 Computing the Mean
22(1)
2.2.1.2 Computing the Variance
23(1)
2.2.2 Discussion of Ito's Lemma
24(1)
2.3 Ito's Lemma For Poisson Processes
24(2)
2.3.1 Interpretation of Ito's Lemma for Poisson
26(1)
2.4 More Versions Of Ito's Lemma
26(2)
2.4.1 Ito's Lemma for Compound Poisson Processes
26(1)
2.4.2 Ito's Lemma for Brownian and Compound Poisson
26(1)
2.4.3 Ito's Lemma for Vector Processes
27(1)
2.5 Ito's Lemma, The Product Rule, And A Rectangle
28(2)
2.6 Summary
30(3)
Chapter 3 Stochastic Differential Equations 33(12)
3.1 Geometric Brownian Motion
33(2)
3.1.1 Stock Price Interpretation
34(1)
3.2 Geometric Poisson Motion
35(1)
3.2.1 Conditional Lognormal Version
35(1)
3.3 Jump Diffusion Model
36(1)
3.3.1 Jump Diffusion as a Stock Price Model
37(1)
3.4 A More General Stochastic Differential Equation
37(2)
3.4.1 Ornstein—Uhlenbeck Process and Mean Reversion
38(1)
3.5 Cox-Ingersoll-Ross Process
39(3)
3.6 Summary
42(3)
Chapter 4 The Factor Model Approach to Arbitrage Pricing 45(18)
4.1 Returns And Factor Models
45(2)
4.1.1 Returns from Price Changes
45(1)
4.1.2 Stochastic Differential Equations and Factor Models
46(1)
4.2 Factor Approach To Arbitrage Using Returns
47(6)
4.2.1 Arbitrage
49(1)
4.2.2 Null and Range Space Relationship
50(1)
4.2.3 A Useful Absence of Arbitrage Condition
51(1)
4.2.4 Interpretations
52(1)
4.2.4.1 Market Price of Risk
52(1)
4.2.4.2 Market Price of Time
52(1)
4.2.5 A Problem with Returns
53(1)
4.3 Factor Approach Using Prices And Value Changes
53(3)
4.3.1 Prices, Value Changes, and Arbitrage
53(2)
4.3.2 Profit/Loss and Arbitrage
55(1)
4.4 Two Standard Examples
56(3)
4.4.1 A Stock under Geometric Brownian Motion
56(1)
4.4.1.1 Using the Return APT
56(1)
4.4.1.2 Using the Price APT
57(1)
4.4.2 Futures Contracts
57(2)
4.5 Summary
59(4)
Chapter 5 Constructing a Factor Model Pricing Framework 63(20)
5.1 Classification Of Quantities
63(2)
5.1.1 Factors
64(1)
5.1.2 Underlying Variables
64(1)
5.1.3 Tradables
64(1)
5.1.4 Importance of Tradables
64(1)
5.2 Derivatives
65(2)
5.2.1 Examples of Derivatives
66(1)
5.3 Factor Models For Underlying Variables And Tradables
67(2)
5.3.1 Direct Factor Models
67(1)
5.3.2 Factor Models via Ito's Lemma
68(1)
5.4 Tradable Tables
69(1)
5.4.1 Structure of a Tradable Table
69(1)
5.5 Applying The Price Apt
70(4)
5.5.1 Relative Pricing and Marketed Tradables
71(1)
5.5.2 Pricing the Derivative
72(2)
5.6 Underdetermined And Overdetermined Systems
74(2)
5.6.1 Underdetermined and Incompleteness
74(1)
5.6.2 Overdetermined and Calibration
75(1)
5.7 Three Step Procedure
76(1)
5.8 Solving The Price Apt Equation For A Specific Derivative
76(2)
5.9 Summary
78(5)
Chapter 6 Equity Derivatives 83(36)
6.1 Black—Scholes Model
83(7)
6.1.1 Solution for European Call and Put Options
86(3)
6.1.2 Implied Volatility
89(1)
6.2 Continuous Dividends
90(4)
6.2.1 European Calls and Puts under Continuous Dividends
92(2)
6.2.2 What Pays a Continuous Dividend?
94(1)
6.3 Cash Dividends
94(3)
6.3.1 European Calls and Puts under Cash Dividends
96(1)
6.4 Poisson Processes
97(2)
6.4.1 Closed Form Solution for European Call Option
99(1)
6.5 Options On Futures
99(3)
6.5.1 Solution for European Call and Put Options
101(1)
6.6 Jump Diffusion Model
102(6)
6.6.1 Bankruptcy!
105(1)
6.6.2 Lognormal Jumps
105(3)
6.7 Exchange One Asset For Another
108(6)
6.7.1 Closed Form Solution
109(1)
6.7.2 Using a Different Currency (Change of Numeraire)
110(2)
6.7.3 Recovering the Closed Form Solution
112(2)
6.8 Stochastic Volatility
114(1)
6.9 Summary
115(4)
Chapter 7 Interest Rate and Credit Derivatives 119(58)
7.1 Interest Rate Modeling And Derivatives
119(1)
7.2 Modeling The Term Structure Of Interest Rates
120(7)
7.2.1 Spot Rates
120(1)
7.2.2 Zero-Coupon Bonds
121(1)
7.2.3 Forward Rates
122(1)
7.2.4 Discount Factors
123(3)
7.2.5 Instantaneous Rates
126(1)
7.2.6 Two Choices for Underlying Variables
126(1)
7.3 Models Based On The Short Rate
127(9)
7.3.1 The Short Rate of Interest and a Money Market Account
127(1)
7.3.2 Single Factor Short Rate Models
128(6)
7.3.2.1 Vasicek Model
130(3)
7.3.2.2 Cox-Ingersoll-Ross Model
133(1)
7.3.2.3 Additional Single Factor Short Rate Models
134(1)
7.3.3 Multifactor Short Rate Models
134(2)
7.4 Models Based On Forward Rates
136(8)
7.4.1 An Important Detour on Forwards
136(8)
7.4.1.1 Market Price of Risk for Forwards
137(3)
7.4.1.2 Pricing Options on Forwards
140(1)
7.4.1.3 Simplified Pricing in Units of the Bond
141(3)
7.5 Libor Market Model
144(16)
7.5.1 Discount Factors are Forward Prices
144(1)
7.5.2 LMM with Discount Factors as Underlying Variables
145(6)
7.5.2.1 Calibration of the Market Prices of Risk
147(1)
7.5.2.2 Using the Forward Analogy
147(1)
7.5.2.3 Formal Three Step Approach
148(3)
7.5.3 Forward Rates as Underlying Variables
151(4)
7.5.4 Pricing a Caplet in the Right Units
155(7)
7.5.4.1 LMM Caplet Formula
158(1)
7.5.4.2 From Caplets to Floorlets and Beyond
159(1)
7.6 Heath-Jarrow-Morton Model
160(2)
7.7 Credit Derivatives
162(15)
7.7.1 Defaultable Bonds
162(4)
7.7.1.1 Different Types of Recovery
164(2)
7.7.2 Defaultable Bonds with Random Intensity of Default
166(2)
7.8 Summary
168(1)
7.9 Appendix: Heath-Jarrow-Morton Derivation
168(9)
Chapter 8 Hedging 177(24)
8.1 Hedging From A Factor Perspective
177(12)
8.1.1 Description Using a Tradable Table
178(1)
8.1.2 Relationship between Hedging and Arbitrage
179(2)
8.1.2.1 Creating a No-Cost Hedge
179(1)
8.1.2.2 Simple Explanation
180(1)
8.1.3 Hedging Examples
181(5)
8.1.3.1 Hedging in Black-Scholes
181(1)
8.1.3.2 Pricing from the Black-Scholes Hedge
181(1)
8.1.3.3 Hedging in Bonds
182(3)
8.1.3.4 Pricing from the Bond Hedge
185(1)
8.1.4 Hedging under Incompleteness
186(2)
8.1.5 A Question of Consistency
188(1)
8.2 Hedging From An Underlying Variable Perspective
189(3)
8.2.1 Black-Scholes Hedging
189(1)
8.2.2 Hedging Bonds
190(1)
8.2.3 Derivatives Imply Small Changes
190(2)
8.3 Higher order Approximations
192(6)
8.3.1 The Greeks
192(2)
8.3.2 Delta-Gamma Hedge
194(2)
8.3.3 Determining What the Error Looks Like
196(2)
8.4 Summary
198(3)
Chapter 9 Computation of Solutions 201(40)
9.1 Discretizing The Modeling Paradigm
201(5)
9.1.1 Discretizing Factor Models
202(1)
9.1.2 Discretizing Geometric Brownian Motion
203(1)
9.1.2.1 Direct Discretization of the Factor Model
203(1)
9.1.2.2 Discretizing the Exact Solution
203(1)
9.1.3 More General Modeling Induced by Binary Factor
204(1)
9.1.4 From the Binary Model to a Factor Model
204(2)
9.1.4.1 Note on Time Scaling of the Factor
205(1)
9.2 Three Step Procedure Under Binary Modeling
206(4)
9.2.1 Single Step Calibration Equation
207(1)
9.2.2 Single Step Pricing Equation
207(2)
9.2.2.1 Interpreting the Pricing Equation
208(1)
9.2.3 Risk Neutral Parameterization
209(1)
9.3 Final Step: Pricing On Multistep Binomial Trees
210(3)
9.3.1 Collapsing Trees to Recombining Lattices
212(1)
9.4 Pricing Options On A Stock Lattice
213(5)
9.4.1 Modeling the Stock
214(1)
9.4.2 Calibration Using Marketed Tradables
215(1)
9.4.3 Pricing a European Call Using the Calibrated Lattice
216(1)
9.4.4 Pricing American Options
217(1)
9.5 Pricing On A Cir Short Rate Tree
218(8)
9.5.1 Modeling and Classification of Discretized Variables
219(1)
9.5.2 Representation of Marketed Tradables
220(2)
9.5.3 Calibration Using Marketed Tradables
222(1)
9.5.4 Market Price of Risk Calibration Algorithm
223(1)
9.5.5 Pricing Derivatives on CIR Calibrated Trees
224(2)
9.6 Single Factor Libor Market Model
226(10)
9.6.1 LMM Forward Rates and Calibration
227(1)
9.6.2 Modeling and Classification of Discretized Variables
227(1)
9.6.3 Specification of Bond Prices and Forward Rates
228(1)
9.6.4 Calibration Using Marketed Tradables
229(2)
9.6.5 Calibration Equation for the Forward Rate Parameters
231(2)
9.6.5.1 Solving for an
232(1)
9.6.6 Calibration Algorithm for LMM
233(1)
9.6.7 Pricing Derivatives with LMM Calibrated Trees
234(2)
9.7 Additional Computational Methods
236(1)
9.8 Summary
237(4)
Chapter 10 The Road to Risk Neutrality 241(22)
10.1 Do The Factors Matter?
241(5)
10.1.1 Brownian Factors
242(2)
10.1.1.1 Adding a Drift Works
243(1)
10.1.1.2 Changing the Variance Does Not Work
243(1)
10.1.2 Poisson Factors
244(2)
10.1.2.1 Changing the Intensity Works
245(1)
10.1.2.2 Adding a Drift Does Not Work
245(1)
10.2 Risk Neutral Representations
246(4)
10.2.1 Risk Neutrality under Brownian Factors
246(1)
10.2.2 Market Price of Risk Interpretation
247(1)
10.2.3 Risk Neutrality under Poisson Factors
248(2)
10.2.4 Market Price of Risk under Poisson Factors
250(1)
10.3 Pricing As An Expectation
250(1)
10.4 Applications Of Risk Neutral Pricing
251(10)
10.4.1 How to Apply Risk Neutral Pricing
251(2)
10.4.2 Black-Scholes
253(2)
10.4.3 Poisson Model
255(1)
10.4.4 Vasicek Single Factor Short Rate Model
256(2)
10.4.5 Heath-Jarrow--Morton Calibration
258(3)
10.5 Summary
261(2)
Bibliography 263(4)
Index 267
James A. Primbs holds undergraduate degrees in Mathematics and Electrical Engineering from UC Davis, an MS degree in Electrical Engineering from Stanford, and a PhD in Control and Dynamical System from Caltech. From 2001-2012 he served as an Assistant and then a Consulting Associate Professor in the Management Science and Engineering department at Stanford University. From 2012 to 2014 he was an Associate Professor in the Systems Engineering department at UT Dallas. He is currently an Associate Professor of Finance in the Mihaylo College of Business and Economics at California State University, Fullerton. He has won teaching awards at both the undergraduate and graduate level, given short courses to and consulted for the financial industry, and organized numerous conference tutorials and workshops, especially in the application of systems and control methods to finance. He is active in INFORMS where he has held various officer positions in the Section on Finance. His research interests involve the use of systems, optimization, and control theory in finance.
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