ISBN-10:
0198775180
ISBN-13:
9780198775188
Pub. Date:
01/28/1998
Publisher:
Oxford University Press, USA
Arbitrage Theory in Continuous Time

Arbitrage Theory in Continuous Time

by Tomas Bjork

Hardcover

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Product Details

ISBN-13: 9780198775188
Publisher: Oxford University Press, USA
Publication date: 01/28/1998
Edition description: Older Edition
Pages: 328
Product dimensions: 9.40(w) x 6.20(h) x 1.10(d)
Lexile: 1520L (what's this?)

About the Author



Tomas Björk is Professor of Mathematical Finance at the Stockholm School of Economics. His background is in probability theory and he was formerly at the Mathematics Department of the Royal Institute of Technology in Stockholm. He is co-editor of Mathematical Finance and Associate Editor of Finance and Stochastics. He has published numerous journal articles on mathematical finance in general, and in particular on interest rate theory.

Table of Contents

1Introduction1
1.1Problem Formulation1
2The Binomial Model5
2.1The One Period Model5
2.1.1Model Description5
2.1.2Portfolios and Arbitrage6
2.1.3Contingent Claims9
2.1.4Risk Neutral Valuation11
2.2The Multiperiod Model15
2.2.1Portfolios and Arbitrage15
2.2.2Contingent Claims17
2.3Exercises25
2.4Notes25
3A More General One Period Model26
3.1The Model26
3.2Absence of Arbitrage27
3.3Martingale Pricing30
3.4Completeness31
3.5Stochastic Discount Factors34
3.6Exercises35
4Stochastic Integrals36
4.1Introduction36
4.2Information38
4.3Stochastic Integrals40
4.4Martingales42
4.5Stochastic Calculus and the Ito Formula45
4.6Examples50
4.7The Multidimensional Ito Formula53
4.8Correlated Wiener Processes55
4.9Exercises59
4.10Notes61
5Differential Equations62
5.1Stochastic Differential Equations62
5.2Geometric Brownian Motion63
5.3The Linear SDE66
5.4The Infinitesimal Operator67
5.5Partial Differential Equations68
5.6The Kolmogorov Equations72
5.7Exercises75
5.8Notes79
6Portfolio Dynamics80
6.1Introduction80
6.2Self-financing Portfolios83
6.3Dividends85
6.4Exercise87
7Arbitrage Pricing88
7.1Introduction88
7.2Contingent Claims and Arbitrage89
7.3The Black-Scholes Equation94
7.4Risk Neutral Valuation98
7.5The Black-Scholes Formula100
7.6Options on Futures102
7.6.1Forward Contracts102
7.6.2Futures Contracts and the Black Formula103
7.7Volatility104
7.7.1Historic Volatility105
7.7.2Implied Volatility106
7.8American options106
7.9Exercises108
7.10Notes110
8Completeness and Hedging111
8.1Introduction111
8.2Completeness in the Black-Scholes Model112
8.3Completeness-Absence of Arbitrage117
8.4Exercise118
8.5Notes120
9Parity Relations and Delta Hedging121
9.1Parity Relations121
9.2The Greeks123
9.3Delta and Gamma Hedging126
9.4Exercises130
10The Martingale Approach to Arbitrage Theory133
10.1The Case with Zero Interest Rate133
10.2Absence of Arbitrage136
10.2.1A Rough Sketch of the Proof137
10.2.2Precise Results140
10.3The General Case142
10.4Completeness145
10.5Martingale Pricing147
10.6Stochastic Discount Factors149
10.7Summary for the Working Economist150
10.8Notes153
11The Mathematics of the Martingale Approach154
11.1Stochastic Integral Representations154
11.2The Girsanov Theorem: Heuristics158
11.3The Girsanov Theorem160
11.4The Converse of the Girsanov Theorem164
11.5Girsanov Transformations and Stochastic Differentials164
11.6Maximum Likelihood Estimation165
11.7Exercises167
11.8Notes168
12Black-Scholes from a Martingale Point of View169
12.1Absence of Arbitrage169
12.2Pricing171
12.3Completeness172
13Multidimensional Models: Classical Approach175
13.1Introduction175
13.2Pricing177
13.3Risk Neutral Valuation183
13.4Reducing the State Space184
13.5Hedging188
13.6Exercises191
14Multidimensional Models: Martingale Approach192
14.1Absence of Arbitrage193
14.2Completeness195
14.3Hedging196
14.4Pricing198
14.5Markovian Models and PDEs199
14.6Market Prices of Risk200
14.7Stochastic Discount Factors201
14.8The Hansen-Jagannathan Bounds201
14.9Exercises204
14.10Notes204
15Incomplete Markets205
15.1Introduction205
15.2A Scalar Nonpriced Underlying Asset205
15.3The Multidimensional Case214
15.4A Stochastic Short Rate218
15.5The Martingale Approach219
15.6Summing Up220
15.7Exercises223
15.8Notes224
16Dividends225
16.1Discrete Dividends225
16.1.1Price Dynamics and Dividend Structure225
16.1.2Pricing Contingent Claims226
16.2Continuous Dividends231
16.2.1Continuous Dividend Yield232
16.2.2The General Case235
16.3Exercises237
17Currency Derivatives239
17.1Pure Currency Contracts239
17.2Domestic and Foreign Equity Markets242
17.3Domestic and Foreign Market Prices of Risk248
17.4Exercises252
17.5Notes253
18Barrier Options254
18.1Mathematical Background254
18.2Out Contracts256
18.2.1Down-and-Out Contracts256
18.2.2Up-and-Out Contracts260
18.2.3Examples261
18.3In Contracts265
18.4Ladders267
18.5Lookbacks268
18.6Exercises270
18.7Notes270
19Stochastic Optimal Control271
19.1An Example271
19.2The Formal Problem272
19.3The Hamilton-Jacobi-Bellman Equation275
19.4Handling the HJB Equation283
19.5The Linear Regulator284
19.6Optimal Consumption and Investment286
19.6.1A Generalization286
19.6.2Optimal Consumption288
19.7The Mutual Fund Theorems291
19.7.1The Case with No Risk Free Asset291
19.7.2The Case with a Risk Free Asset295
19.8Exercises297
19.9Notes301
20Bonds and Interest Rates302
20.1Zero Coupon Bonds302
20.2Interest Rates303
20.2.1Definitions303
20.2.2Relations between df(t,T), dp(t,T), and dr(t)305
20.2.3An Alternative View of the Money Account308
20.3Coupon Bonds, Swaps, and Yields309
20.3.1Fixed Coupon Bonds310
20.3.2Floating Rate Bonds310
20.3.3Interest Rate Swaps312
20.3.4Yield and Duration313
20.4Exercises314
20.5Notes315
21Short Rate Models316
21.1Generalities316
21.2The Term Structure Equation319
21.3Exercises324
21.4Notes325
22Martingale Models for the Short Rate326
22.1Q-dynamics326
22.2Inversion of the Yield Curve327
22.3Affine Term Structures329
22.3.1Definition and Existence329
22.3.2A Probabilistic Discussion331
22.4Some Standard Models333
22.4.1The Vasicek Model333
22.4.2The Ho-Lee Model334
22.4.3The CIR Model335
22.4.4The Hull-White Model335
22.5Exercises338
22.6Notes339
23Forward Rate Models340
23.1The Heath-Jarrow-Morton Framework340
23.2Martingale Modeling342
23.3The Musiela Parameterization344
23.4Exercises345
23.5Notes347
24Change of Numeraire348
24.1Introduction348
24.2Generalities349
24.3Changing the Numeraire353
24.4Forward Measures355
24.4.1Using the T-bond as Numeraire355
24.4.2An Expectation Hypothesis357
24.5A General Option Pricing Formula358
24.6The Hull-White Model361
24.7The General Gaussian Model363
24.8Caps and Floors365
24.9Exercises366
24.10Notes366
25LIBOR and Swap Market Models368
25.1Caps: Definition and Market Practice369
25.2The LIBOR Market Model371
25.3Pricing Caps in the LIBOR Model372
25.4Terminal Measure Dynamics and Existence373
25.5Calibration and Simulation376
25.6The Discrete Savings Account378
25.7Swaps379
25.8Swaptions: Definition and Market Practice381
25.9The Swap Market Models382
25.10Pricing Swaptions in the Swap Market Model383
25.11Drift Conditions for the Regular Swap Market Model384
25.12Concluding Comment387
25.13Exercises388
25.14Notes388
26Forwards and Futures389
26.1Forward Contracts389
26.2Futures Contracts391
26.3Exercises394
26.4Notes394
AMeasure and Integration395
A.1Sets and Mappings395
A.2Measures and Sigma Algebras397
A.3Integration399
A.4Sigma-Algebras and Partitions404
A.5Sets of Measure Zero405
A.6The L[superscript p] Spaces406
A.7Hilbert Spaces407
A.8Sigma-Algebras and Generators410
A.9Product measures414
A.10The Lebesgue Integral415
A.11The Radon-Nikodym Theorem416
A.12Exercises419
A.13Notes421
BProbability Theory422
B.1Random Variables and Processes422
B.2Partitions and Information425
B.3Sigma-algebras and Information427
B.4Independence430
B.5Conditional Expectations432
B.6Equivalent Probability Measures438
B.7Exercises441
B.8Notes442
CMartingales and Stopping Times443
C.1Martingales443
C.2Discrete Stochastic Integrals446
C.3Likelihood Processes447
C.4Stopping Times448
C.5Exercises451
References453
Index461

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