Financial Markets in Continuous Time

By: Rose-Anne Dana, Monique Jeanblanc, A. Kennedy (Translator)

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Paperback | 12 July 2007

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In modern financial practice, asset prices are modelled by means of stochastic processes, and continuous-time stochastic calculus thus plays a central role in financial modelling. This approach has its roots in the foundational work of the Nobel laureates Black, Scholes and Merton. Asset prices are further assumed to be rationalizable, that is, determined by equality of demand and supply on some market. This approach has its roots in the foundational work on General Equilibrium of the Nobel laureates Arrow and Debreu and in the work of McKenzie. This book has four parts. The first brings together a number of results from discrete-time models. The second develops stochastic continuous-time models for the valuation of financial assets (the Black-Scholes formula and its extensions), for optimal portfolio and consumption choice, and for obtaining the yield curve and pricing interest rate products. The third part recalls some concepts and results of general equilibrium theory, and applies this in financial markets. The last part is more advanced and tackles market incompleteness and the valuation of exotic options in a complete market.

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From the reviews:

This is a high-level but well-written summary of the modern essentials of mathematical finance, including excellent chapters on the yield curve, pricing interest rate products, exotic options and incomplete markets.

D. L. McLeish, Short Book Reviews, December 2003

"Financial Markets in Continuous Time is a well-written textbook for graduate students in mathematical finance. ... Graduate students in finance, mathematics, financial engineering, and risk management would benefit from the book in grasping the key financial concepts, mathematical tools, and theories of this discipline. ... This book ... covers the most important advances in mathematical finance that form the foundation for much of the continuing growth of the discipline." (Thomas S. Y. Ho, SIAM Review, Vol. 46 (3), 2004)

"Dana and Jeanblanc have successfully converted a finance problem into a completely mathematical form. ... This book is suited to advanced mathematics students who want to develop mathematics within the framework of finance. ... This book looks like mathematics dressed with some financial terms. However, this dressing is so nice that each chapter sounds really interesting. ... I would like to recommend this book to postgraduate students or researchers in mathematics or theoretical physics when they want to re-direct their research in finance." (Myungshik Kim, Bulletin of the Irish Mathematical Society, Vol. 51, 2003)

"The objective of this book is to develop the continuous time theory of the valuation of asset prices and the theory of equilibrium of financial markets. ... This is a high-level but well-written summary of the modern essentials of mathematical finance, including excellent chapters on the yield curve, pricing interest rate products, exotic options and incomplete markets." (D.L. McLeish, Short Book Reviews, Vol. 23 (3), 2003)

"This book gives an introduction to the theory of financial markets andits applications for the pricing and hedging of financial instruments. For an introductory text, the range of topics covered is amazingly broad. ... Many ideas and concepts are first introduced and studied in very simple discrete models, and it is particularly remarkable how many interesting developments are already explained in the first chapter ... . The material is chosen well and covers a considerable part of the subject area ... . " (Martin Schweizer, Zentralblatt MATH, Vol.1014, 2003)

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Non-Fiction Economics Accounting & Finance Finance Investment & Securities Mathematics Probability & Statistics

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Published: 26th November 2002

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The Discrete Case	p. 1
A Model with Two Dates and Two States of the World	p. 1
The Model	p. 1
Hedging Portfolio, Value of the Option	p. 2
The Risk-Neutral Measure, Put-Call Parity	p. 4
No Arbitrage Opportunities	p. 5
The Risk Attached to an Option	p. 6
Incomplete Markets	p. 8
A One-Period Model with (d + 1) Assets and k States of the World	p. 12
No Arbitrage Opportunities	p. 13
Complete Markets	p. 18
Valuation by Arbitrage in the Case of a Complete Market	p. 19
Incomplete Markets: the Arbitrage Interval	p. 20
Optimal Consumption and Portfolio Choice in a One-Agent Model	p. 22
The Maximization Problem	p. 23
An Equilibrium Model with a Representative Agent	p. 28
The Von Neumann-Morgenstern Model, Risk Aversion	p. 30
Optimal Choice in the VNM Model	p. 32
Equilibrium Models with Complete Financial Markets	p. 36
Dynamic Models in Discrete Time	p. 43
A Model with a Finite Horizon	p. 44
Arbitrage with a Finite Horizon	p. 45
Arbitrage Opportunities	p. 45
Arbitrage and Martingales	p. 46
Trees	p. 49
Complete Markets with a Finite Horizon	p. 53
Characterization	p. 54
Valuation	p. 55
The Complete Market Case	p. 56
An Example	p. 57
The Binomial Model	p. 57
Option Valuation	p. 59
Approaching the Black-Scholes Model	p. 60
Maximization of the Final Wealth	p. 64
Optimal Choice of Consumption and Portfolio	p. 68
Infinite Horizon	p. 73
The Black-Scholes Formula	p. 81
Stochastic Calculus	p. 81
Brownian Motion and the Stochastic Integral	p. 82
Ito Processes. Girsanov's Theorem	p. 84
Ito's Lemma	p. 85
Multidimensional Processes	p. 87
Multidimensional Ito's Lemma	p. 88
Examples	p. 89
Arbitrage and Valuation	p. 90
Financing Strategies	p. 90
Arbitrage and the Martingale Measure	p. 92
Valuation	p. 94
The Black-Scholes Formula: the One-Dimensional Case	p. 95
The Model	p. 95
The Black-Scholes Formula	p. 96
The Risk-Neutral Measure	p. 99
Explicit Calculations	p. 101
Comments on the Black-Scholes Formula	p. 103
Extension of the Black-Scholes Formula	p. 107
Financing Strategies	p. 107
The State Variable	p. 108
The Black-Scholes Formula	p. 109
Special Case	p. 111
The Risk-Neutral Measure	p. 111
Example	p. 113
Applications of the Black-Scholes Formula	p. 113
Portfolios Optimizing Wealth and Consumption	p. 127
The Model	p. 127
Optimization	p. 130
Solution in the Case of Constant Coefficients	p. 130
Dynamic Programming	p. 130
The Hamilton-Jacobi-Bellman Equation	p. 131
A Special Case	p. 136
Admissible Strategies	p. 137
Existence of an Optimal Pair	p. 141
Construction of an Optimal Pair	p. 142
The Value Function	p. 144
A Special Case	p. 145
Solution in the Case of Deterministic Coefficients	p. 147
The Value Function and Partial Differential Equations	p. 148
Optimal Wealth	p. 149
Obtaining the Optimal Portfolio	p. 150
Market Completeness and NAO	p. 151
The Yield Curve	p. 159
Discrete-Time Model	p. 159
Continuous-Time Model	p. 154
Definitions	p. 164
Change of Numeraire	p. 166
Valuation of an Option on a Coupon Bond	p. 171
The Heath-Jarrow-Morton Model	p. 172
The Model	p. 172
The Linear Gaussian Case	p. 174
When the Spot Rate is Given	p. 179
The Vasicek Model	p. 181
The Ornstein-Uhlenbeck Process	p. 181
Determining P(t,T) when q is Constant	p. 183
The Cox-Ingersoll-Ross Model	p. 185
The Cox-Ingersoll-Ross Process	p. 185
Valuation of a Zero Coupon Bond	p. 187
Equilibrium of Financial Markets in Discrete Time	p. 191
Equilibrium in a Static Exchange Economy	p. 192
The Demand Approach	p. 194
The Negishi Method	p. 196
Pareto Optima	p. 196
Two Characterizations of Pareto Optima	p. 197
Existence of an Equilibrium	p. 200
The Theory of Contingent Markets	p. 201
The Arrow-Radner Equilibrium Exchange Economy with Financial Markets with Two Dates	p. 203
The Complete Markets Case	p. 205
The CAPM	p. 208
Equilibrium of Financial Markets in Continuous Time. The Complete Markets Case	p. 217
The Model	p. 217
The Financial Market	p. 218
The Economy	p. 219
Admissible Pairs	p. 220
Definition and Existence of a Radner Equilibrium	p. 222
Existence of a Contingent Arrow-Debreu Equilibrium	p. 224
Aggregate Utility	p. 224
Definition and Characterization of Pareto Optima	p. 225
Existence and Characterization of a Contingent Arrow-Debreu Equilibrium	p. 228
Existence of a Radner Equilibrium	p. 228
Applications	p. 230
Arbitrage Price of Real Secondary Assets. Lucas' Formula	p. 230
CCAPM (Consumption-based Capital Asset Pricing Model)	p. 232
Incomplete Markets	p. 237
Incomplete Markets	p. 237
The Case of Constant Coefficients	p. 237
No-Arbitrage Markets	p. 239
The Price Range	p. 240
Superhedging	p. 242
The Minimal Probability Measure	p. 243
Valuation Using Utility Functions	p. 244
Transaction Costs	p. 245
Stochastic Volatility	p. 245
The Robustness of the Black-Scholes Formula	p. 246
Wealth Optimization	p. 247
Exotic Options	p. 249
The Hitting Time and Supremum for Brownian Motion	p. 250
Distribution of the Pair (B[subscript t], M[subscript t])	p. 250
Distribution of Sup and of the Hitting Time	p. 251
Distribution of Inf	p. 252
Laplace Tranforms	p. 252
Hitting Time for a Double Barrier	p. 253
Drifted Brownian Motion	p. 254
The Laplace Transform of a Hitting Time	p. 254
Distribution of the Pair (Maximum, Minimum)	p. 255
Evaluation of E(e [superscript -r T[subscript y]] 11 [subscript T[subscript y[less than a]]])	p. 255
Barrier Options	p. 256
Down-and-Out Options	p. 256
Down-and-In Options	p. 257
Up-and-Out and Up-and-In Options	p. 257
Intermediate Calculations	p. 257
The Value of the Compensation	p. 260
Valuation of a DIC Option	p. 261
Up-and-In Options	p. 263
P. Carr 's Symmetry	p. 264
Double Barriers	p. 267
Lookback Options	p. 269
Other Options	p. 271
Options Linked to the Hitting Time of a Barrier	p. 271
Options Linked to Occupation Times	p. 272
Other Products	p. 274
Asian Options or Average Rate Options	p. 274
Products Depending on an Interim Date	p. 275
Still More Products	p. 275
Brownian Motion	p. 279
Historical Background	p. 279
Intuition	p. 279
Random Walk	p. 280
The Stochastic Integral	p. 283
Ito's Formula	p. 285
Numerical Methods	p. 287
Finite Difference	p. 288
Method	p. 289
The Implicit Scheme Case	p. 290
Solving the System	p. 291
Other Schemes	p. 292
Extrapolation Methods	p. 292
The Heat Equation	p. 293
Approximations	p. 293
Simulation	p. 294
Simulation of the Uniform Distribution on [0, 1]	p. 295
Simulation of Discrete Variables	p. 295
Simulation of a Random Variable	p. 295
Simulation of an Expectation	p. 296
Simulation of a Brownian Motion	p. 297
Simulation of Solutions to Stochastic Differential Equations	p. 298
Calculating E(f(X[subscript t]))	p. 299
References	p. 301
Index	p. 323
Table of Contents provided by Ingram. All Rights Reserved.

Financial Markets in Continuous Time

At a Glance

Paperback

Industry Reviews

Shipping

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Defective items

You Can Find This Book In

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