Martingale methods in financial modelling:
Gespeichert in:
| Beteiligte Personen: | , |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2009
|
| Ausgabe: | 2. ed., corr. 3. print. |
| Schriftenreihe: | Stochastic modelling and applied probability
36 |
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018970515&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Beschreibung: | Literaturverz. S. [623] - 672 |
| Umfang: | XIX, 715 S. |
| ISBN: | 3540209662 9783540209669 |
Internformat
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| 245 | 1 | 0 | |a Martingale methods in financial modelling |c Marek Musiela ; Marek Rutkowski |
| 250 | |a 2. ed., corr. 3. print. | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2009 | |
| 300 | |a XIX, 715 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a Stochastic modelling and applied probability |v 36 | |
| 500 | |a Literaturverz. S. [623] - 672 | ||
| 650 | 7 | |a Derivativos (modelos matemáticos) |2 larpcal | |
| 650 | 7 | |a Finanças (modelos matemáticos) |2 larpcal | |
| 650 | 7 | |a Matemática aplicada |2 larpcal | |
| 650 | 7 | |a Opções financeiras (modelos matemáticos) |2 larpcal | |
| 650 | 7 | |a Taxa de juros (modelos matemáticos) |2 larpcal | |
| 650 | 4 | |a Mathematisches Modell | |
| 650 | 4 | |a Options (Finance) |x Mathematical models | |
| 650 | 4 | |a Derivative securities |x Mathematical models | |
| 650 | 4 | |a Interest rates |x Mathematical models | |
| 650 | 4 | |a Fixed-income securities |x Mathematical models | |
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| adam_text | Titel: Martingale methods in financial modelling
Autor: Musiela, Marek
Jahr: 2009
Contents
Preface to the Second Edition v
Note on the Second Printing vii
Preface to the First Edition ix
Part I Spot and Futures Markets
1 An Introduction to Financial Derivatives.......................... 3
1.1 Options................................................... 3
1.2 Futures Contracts and Options................................ 5
1.3 Forward Contracts.......................................... 6
1.4 Call and Put Spot Options................................... 8
1.4.1 One-period Spot Market.............................. 9
1.4.2 Replicating Portfolios ................................ 10
1.4.3 Martingale Measure for a Spot Market .................. 12
1.4.4 Absence of Arbitrage................................. 13
1.4.5 Optimality of Replication............................. 15
1.4.6 Change of a Numeraire............................... 17
1.4.7 Put Option.......................................... 18
1.5 Forward Contracts.......................................... 19
1.5.1 Forward Price....................................... 20
1.6 Futures Call and Put Options................................. 21
1.6.1 Futures Contracts and Futures Prices.................... 21
1.6.2 One-period Futures Market............................ 22
1.6.3 Martingale Measure for a Futures Market................ 23
1.6.4 Absence of Arbitrage................................. 24
1.6.5 One-period Spot/Futures Market ....................... 26
1.7 Options of American Style................................... 26
1.8 Universal No-arbitrage Inequalities............................ 31
Contents
Discrete-time Security Markets.................................. 35
2.1 The Cox-Ross-Rubinstein Model............................. 36
2.1.1 Binomial Lattice for the Stock Price .................... 36
2.1.2 Recursive Pricing Procedure........................... 38
2.1.3 CRR Option Pricing Formula.......................... 43
2.2 Martingale Properties of the CRR Model....................... 46
2.2.1 Martingale Measures................................. 47
2.2.2 Risk-neutral Valuation Formula........................ 50
2.2.3 Change of a Numeraire............................... 51
2.3 The Black-Scholes Option Pricing Formula..................... 53
2.4 Valuation of American Options............................... 58
2.4.1 American Call Options ............................... 58
2.4.2 American Put Options................................ 60
2.4.3 American Claims.................................... 61
2.5 Options on a Dividend-paying Stock .......................... 63
2.6 Security Markets in Discrete Time............................ 65
2.6.1 Finite Spot Markets.................................. 66
2.6.2 Self-financing Trading Strategies....................... 66
2.6.3 Replication and Arbitrage Opportunities................. 68
2.6.4 Arbitrage Price...................................... 69
2.6.5 Risk-neutral Valuation Formula........................ 70
2.6.6 Existence of a Martingale Measure ..................... 73
2.6.7 Completeness of a Finite Market....................... 75
2.6.8 Separating Hyperplane Theorem....................... 77
2.6.9 Change of a Numeraire............................... 78
2.6.10 Discrete-time Models with Infinite State Space........... 79
2.7 Finite Futures Markets...................................... 80
2.7.1 Self-financing Futures Strategies....................... 81
2.7.2 Martingale Measures for a Futures Market............... 82
2.7.3 Risk-neutral Valuation Formula........................ 84
2.7.4 Futures Prices Versus Forward Prices ................... 85
2.8 American Contingent Claims................................. 87
2.8.1 Optimal Stopping Problems ........................... 90
2.8.2 Valuation and Hedging of American Claims.............. 97
2.8.3 American Call and Put................................ 101
2.9 Game Contingent Claims.................................... 101
2.9.1 Dynkin Games...................................... 102
2.9.2 Valuation and Hedging of Game Contingent Claims....... 108
Benchmark Models in Continuous Tinie.......................... 113
3.1 The Black-Scholes Model ................................... 114
3.1.1 Risk-free Bond...................................... 114
3.1.2 Stock Price......................................... 114
3.1.3 Self-financing Trading Strategies....................... 118
3.1.4 Martingale Measure for the Black-Scholes Model......... 120
Contents xiii
3.1.5 Black-Scholes Option Pricing Formula.................. 125
3.1.6 Case of Time-dependent Coefficients.................... 131
3.1.7 Merton s Model ..................................... 132
3.1.8 Put-Call Parity for Spot Options........................ 134
3.1.9 Black-Scholes PDE.................................. 134
3.1.10 A Riskless Portfolio Method........................... 137
3.1.11 Black-Scholes Sensitivities............................ 140
3.1.12 Market Imperfections................................. 144
3.1.13 Numerical Methods.................................. 145
3.2 A Dividend-paying Stock.................................... 147
3.2.1 Case ofaConstant Dividend Yield...................... 148
3.2.2 Case of Known Dividends............................. 151
3.3 Bachelier Model ........................................... 154
3.3.1 Bachelier Option Pricing Formula...................... 155
3.3.2 Bachelier s PDE..................................... 157
3.3.3 Bachelier Sensitivities................................ 158
3.4 Black Model............................................... 159
3.4.1 Self-financing Futures Strategies ....................... 160
3.4.2 Martingale Measure for the Futures Market.............. 160
3.4.3 Black s Futures Option Formula........................ 161
3.4.4 Options on Forward Contracts ......................... 165
3.4.5 Forward and Futures Prices............................ 167
3.5 Robustness of the Black-Scholes Approach..................... 168
3.5.1 Uncertain Volatility.................................. 168
3.5.2 European Call and Put Options......................... 169
3.5.3 Convex Path-independent European Claims.............. 172
3.5.4 General Path-independent European Claims.............. 177
Foreign Market Derivatives..................................... 181
4.1 Cross-currency Market Model................................ 181
4.1.1 Domestic Martingale Measure......................... 182
4.1.2 Foreign Martingale Measure........................... 184
4.1.3 Foreign Stock Price Dynamics......................... 185
4.2 Currency Forward Contracts and Options....................... 186
4.2.1 Forward Exchange Rate............................... 186
4.2.2 Currency Option Valuation Formula .................... 187
4.3 Foreign Equity Forward Contracts ............................ 191
4.3.1 Forward Price of a Foreign Stock....................... 191
4.3.2 Quanto Forward Contracts............................. 192
4.4 Foreign Market Futures Contracts............................. 194
4.5 Foreign Equity Options...................................... 197
4.5.1 Options Struck in a Foreign Currency................... 198
4.5.2 Options Struck in Domestic Currency................... 199
4.5.3 Quanto Options...................................... 200
4.5.4 Equity-linked Foreign Exchange Options................ 202
Contents
5 American Options......................................205
5.1 Valuation of American Claims................................ 206
5.2 American Call and Put Options............................... 213
5.3 Early Exercise Representation of an American Put............... 216
5.4 Analytical Approach........................................ 219
5.5 Approximations of the American Put Price..................... 222
5.6 Option on a Dividend-paying Stock........................... 224
5.7 Game Contingent Claims.................................... 226
6 Exotic Options.................................... 229
6.1 Packages.................................................. 230
6.2 Forward-start Options....................................... 231
6.3 Chooser Options........................................... 232
6.4 Compound Options......................................... 233
6.5 Digital Options ............................................ 234
6.6 Barrier Options............................................ 235
6.7 Lookback Options.......................................... 238
6.8 Asian Options............................................. 242
6.9 Basket Options............................................. 245
6.10 Quantile Options........................................... 249
6.11 Other Exotic Options ....................................... 251
7 Volatility Risk........................................ 253
7.1 Implied Volatilities of Traded Options......................... 254
7.1.1 Historical Volatility.................................. 255
7.1.2 Implied Volatility.................................... 255
7.1.3 Implied Volatility Versus Historical Volatility............. 256
7.1.4 Approximate Formulas ............................... 257
7.1.5 Implied Volatility Surface............................. 259
7.1.6 Asymptotic Behavior of the Implied Volatility............ 261
7.1.7 Marked-to-Market Models ............................ 264
7.1.8 VegaHedging....................................... 265
7.1.9 Correlated Brownian Motions.......................... 267
7.1.10 Forward-start Options................................ 269
7.2 Extensions of the Black-Scholes Model........................ 273
7.2.1 CEV Model......................................... 273
7.2.2 Shifted Lognormal Models............................ 277
7.3 Local Volatility Models ..................................... 278
7.3.1 Implied Risk-Neutral Probability Law................... 278
7.3.2 Local Volatility...................................... 281
7.3.3 Mixture Models..................................... 287
7.3.4 Advantages and Drawbacks of LV Models............... 290
7.4 Stochastic Volatility Models.................................. 291
7.4.1 PDE Approach...................................... 292
7.4.2 Examples of SV Models.............................. 293
Contents xv
7.4.3 HuIl and White Model................................ 294
7.4.4 Heston s Model...................................... 299
7.4.5 SABR Model ....................................... 301
7.5 Dynamical Models of Volatility Surfaces....................... 302
7.5.1 Dynamics of the Local Volatility Surface ................ 303
7.5.2 Dynamics of the Implied Volatility Surface............... 303
7.6 Alternative Approaches ..................................... 307
7.6.1 Modelling of Asset Returns............................ 308
7.6.2 Modelling of Volatility and Realized Variance............ 313
8 Continuous-time Security Markets............................... 315
8.1 Standard Market Models .................................... 316
8.1.1 Standard Spot Market ................................ 316
8.1.2 Futures Market...................................... 325
8.1.3 Choice of a Numeraire................................ 327
8.1.4 Existence of a Martingale Measure ..................... 330
8.1.5 Fundamental Theorem of Asset Pricing.................. 332
8.2 Multidimensional Black-Scholes Model........................ 333
8.2.1 Market Completeness................................. 335
8.2.2 Variance-minimizing Hedging......................... 337
8.2.3 Risk-minimizing Hedging............................. 338
8.2.4 Market Imperfections................................. 345
Part II Fixed-income Markets
9 Interest Rates and Related Contracts............................. 351
9.1 Zero-coupon Bonds......................................... 351
9.1.1 Term Structure of Interest Rates........................ 352
9.1.2 Forward Interest Rates................................ 353
9.1.3 Short-term Interest Rate............................... 354
9.2 Coupon-bearing Bonds...................................... 354
9.2.1 Yield-to-Maturity.................................... 355
9.2.2 Market Conventions.................................. 357
9.3 Interest Rate Futures........................................ 358
9.3.1 Treasury Bond Futures................................ 358
9.3.2 Bond Options....................................... 359
9.3.3 Treasury Bill Futures................................. 360
9.3.4 Eurodollar Futures................................... 362
9.4 Interest Rate Swaps......................................... 363
9.4.1 Forward Rate Agreements............................. 364
9.5 Stochastic Models of Bond Prices............................. 366
9.5.1 Arbitrage-free Family of Bond Prices................... 366
9.5.2 Expectations Hypotheses.............................. 367
9.5.3 Case of Itö Processes................................. 368
xvi Contents
9.5.4 Market Price for Interest Rate Risk..................... 371
9.6 Forward Measure Approach.................................. 372
9.6.1 Forward Price....................................... 373
9.6.2 Forward Martingale Measure.......................... 375
9.6.3 Forward Processes................................... 378
9.6.4 Choice of a Numeraire................................ 379
10 Short-Term Rate Models........................................ 383
10.1 Single-factor Models........................................ 384
10.1.1 Time-homogeneous Models........................... 384
10.1.2 Time-inhomogeneous Models.......................... 394
10.1.3 Model Choice....................................... 399
10.1.4 American Bond Options .............................. 401
10.1.5 Options on Coupon-bearing Bonds ..................... 402
10.2 Multi-factor Models........................................ 402
10.2.1 State Variables ...................................... 403
10.2.2 Affine Models....................................... 404
10.2.3 Yield Models........................................ 404
10.3 Extended CIR Model ....................................... 406
10.3.1 Squared Bessel Process............................... 407
10.3.2 Model Construction.................................. 407
10.3.3 Change of a Probability Measure....................... 408
10.3.4 Zero-coupon Bond................................... 409
10.3.5 Case of Constant Coefficients.......................... 410
10.3.6 Case of Piecewise Constant Coefficients................. 411
10.3.7 Dynamics of Zero-coupon Bond........................ 412
10.3.8 Transition Densities.................................. 414
10.3.9 Bond Option........................................ 415
11 Models of Instantaneous Forward Rates.......................... 417
11.1 Heath-Jarrow-Morton Methodology........................... 418
11.1.1 Ho and Lee Model................................... 419
11.1.2 Heath-Jarrow-Morton Model .......................... 419
11.1.3 Absence of Arbitrage................................. 421
11.1.4 Short-term Interest Rate............................... 427
11.2 Gaussian HJM Model....................................... 428
11.2.1 Markovian Case..................................... 430
11.3 European Spot Options...................................... 434
11.3.1 Bond Options....................................... 435
11.3.2 Stock Options....................................... 438
11.3.3 Option on a Coupon-bearing Bond...................... 441
11.3.4 Pricing of General Contingent Claims................... 444
11.3.5 Replication of Options................................ 446
11.4 Volatilities and Correlations.................................. 449
11.4.1 Volatilities.......................................... 449
Contents xvii
11.4.2 Correlations......................................... 451
11.5 Futures Price.............................................. 452
11.5.1 Futures Options ..................................... 453
11.6 PDE Approach to Interest Rate Derivatives..................... 457
11.6.1 PDEs for Spot Derivatives............................. 457
11.6.2 PDEs for Futures Derivatives.......................... 461
11.7 Recent Developments....................................... 465
12 Market LD30R Models......................................... 469
12.1 Forward and Futures LIBORs................................ 471
12.1.1 One-period Swap Settled in Arrears..................... 471
12.1.2 One-period Swap Settled in Advance.................... 473
12.1.3 Eurodollar Futures................................... 474
12.1.4 LIBOR in the Gaussian HJM Model.................... 475
12.2 Interest Rate Caps and Floors ................................ 477
12.3 Valuation in the Gaussian HJM Model......................... 479
12.3.1 Plain-vanilla Caps and Floors.......................... 479
12.3.2 Exotic Caps......................................... 481
12.3.3 Captions............................................ 483
12.4 LIBOR Market Models...................................... 484
12.4.1 Black s Formula for Caps............................. 484
12.4.2 Miltersen. Sandmann and Sondermann Approach......... 486
12.4.3 Brace. Gatarek and Musiela Approach................... 486
12.4.4 Musiela and Rutkowski Approach...................... 489
12.4.5 SDEs for LIBORs under the Forward Measure............ 492
12.4.6 Jamshidian s Approach............................... 495
12.4.7 Alternative Derivation of Jamshidian s SDE.............. 498
12.5 Properties of the Lognormal LIBOR Model..................... 500
12.5.1 Transition Density of the LIBOR....................... 501
12.5.2 Transition Density of the Forward Bond Price............ 503
12.6 Valuation in the Lognormal LIBOR Model..................... 506
12.6.1 Pricing of Caps and Floors ............................ 506
12.6.2 Hedging of Caps and Floors........................... 508
12.6.3 Valuation of European Claims ......................... 510
12.6.4 Bond Options....................................... 513
12.7 Extensions of the LLM Model................................ 515
13 Alternative Market Models...................................... 517
13.1 Swaps and Swaptions....................................... 518
13.1.1 Forward Swap Rates ................................. 518
13.1.2 Swaptions.......................................... 522
13.1.3 Exotic Swap Derivatives.............................. 524
13.2 Valuation in the Gaussian HJM Model......................... 527
13.2.1 Swaptions.......................................... 527
13.2.2 CMS Spread Options................................. 527
xviii Contents
13.2.3 Yield Curve Swaps................................... 529
13.3 Co-terminal Forward Swap Rates............................. 530
13.3.1 Jamshidian s Model.................................. 535
13.3.2 Valuation of Co-terminal Swaptions..................... 538
13.3.3 Hedging of Swaptions................................ 539
13.3.4 Bermudan Swaptions................................. 540
13.4 Co-initial Forward Swap Rates............................... 541
13.4.1 Valuation of Co-initial Swaptions....................... 544
13.4.2 Valuation of Exotic Options........................... 545
13.5 Co-sliding Forward Swap Rates .............................. 546
13.5.1 Modelling of Co-sliding Swap Rates.................... 547
13.5.2 Valuation of Co-sliding Swaptions...................... 551
13.6 Swap Rate Model Versus LIBOR Model....................... 552
13.6.1 Swaptions in the LLM Model.......................... 553
13.6.2 Caplets in the Co-terminal Swap Market Model........... 557
13.7 Markov-functional Models................................... 558
13.7.1 Terminal Swap Rate Model............................ 559
13.7.2 Calibration of Markov-functional Models................ 562
13.8 Flesaker and Hughston Approach............................. 565
13.8.1 Rational Lognormal Model............................ 568
13.8.2 Valuation of Caps and Swaptions....................... 569
14 Cross-currency Derivatives...................................... 573
14.1 Arbitrage-free Cross-currency Markets......................... 574
14.1.1 Forward Price of a Foreign Asset....................... 576
14.1.2 Valuation of Foreign Contingent Claims................. 580
14.1.3 Cross-currency Rates................................. 581
14.2 Gaussian Model............................................ 581
14.2.1 Currency Options.................................... 582
14.2.2 Foreign Equity Options............................... 583
14.2.3 Cross-currency Swaps................................ 588
14.2.4 Cross-currency Swaptions............................. 599
14.2.5 BasketCaps......................................... 602
14.3 Model of Forward LD30R Rates.............................. 603
14.3.1 Quanto Cap......................................... 604
14.3.2 Cross-currency Swap................................. 606
14.4 Concluding Remarks........................................ 607
Part III APPENDIX
A An Overview of Ito Stochastic Calculus........................... 611
A.l Conditional Expectation..................................... 611
A.2 Filtrations and Adapted Processes............................. 615
A.3 Martingales ............................................... 616
Contents xix
A.4 Standard Brownian Motion .................................. 617
A.5 Stopping Times and Martingales.............................. 621
A.6 Ito Stochastic Integral....................................... 622
A.7 Continuous Local Martingales................................ 625
A.8 Continuous Semimartingales................................. 628
A.9 Ito s Lemma............................................... 630
A. 10 Levy s Characterization Theorem............................. 633
A.l 1 Martingale Representation Property........................... 634
A.12 Stochastic Differential Equations ............................. 636
A.13 Stochastic Exponential...................................... 639
A.14 Radon-Nikodym Density.................................... 640
A.15 Girsanov s Theorem........................................ 641
A. 16 Martingale Measures........................................ 645
A.17 Feynman-Kac Formula...................................... 646
A.18 First Passage Times......................................... 649
References......................................................... 657
Index............................................................. 707
|
| any_adam_object | 1 |
| author | Musiela, Marek 1950- Rutkowski, Marek |
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| dewey-ones | 332 - Financial economics |
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| dewey-search | 332/.01/5118 22 |
| dewey-sort | 3332 11 45118 222 |
| dewey-tens | 330 - Economics |
| discipline | Mathematik Wirtschaftswissenschaften |
| edition | 2. ed., corr. 3. print. |
| format | Book |
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| id | DE-604.BV036079416 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-20T14:06:22Z |
| institution | BVB |
| isbn | 3540209662 9783540209669 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-018970515 |
| oclc_num | 457561572 |
| open_access_boolean | |
| owner | DE-91G DE-BY-TUM DE-188 DE-473 DE-BY-UBG DE-634 |
| owner_facet | DE-91G DE-BY-TUM DE-188 DE-473 DE-BY-UBG DE-634 |
| physical | XIX, 715 S. |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Springer |
| record_format | marc |
| series | Stochastic modelling and applied probability |
| series2 | Stochastic modelling and applied probability |
| spellingShingle | Musiela, Marek 1950- Rutkowski, Marek Martingale methods in financial modelling Stochastic modelling and applied probability Derivativos (modelos matemáticos) larpcal Finanças (modelos matemáticos) larpcal Matemática aplicada larpcal Opções financeiras (modelos matemáticos) larpcal Taxa de juros (modelos matemáticos) larpcal Mathematisches Modell Options (Finance) Mathematical models Derivative securities Mathematical models Interest rates Mathematical models Fixed-income securities Mathematical models Finance Mathematical models Modellierung (DE-588)4170297-9 gnd Finanzmathematik (DE-588)4017195-4 gnd Kapitalmarkttheorie (DE-588)4137411-3 gnd Martingal (DE-588)4126466-6 gnd Optionspreistheorie (DE-588)4135346-8 gnd Martingaltheorie (DE-588)4168982-3 gnd |
| subject_GND | (DE-588)4170297-9 (DE-588)4017195-4 (DE-588)4137411-3 (DE-588)4126466-6 (DE-588)4135346-8 (DE-588)4168982-3 |
| title | Martingale methods in financial modelling |
| title_auth | Martingale methods in financial modelling |
| title_exact_search | Martingale methods in financial modelling |
| title_full | Martingale methods in financial modelling Marek Musiela ; Marek Rutkowski |
| title_fullStr | Martingale methods in financial modelling Marek Musiela ; Marek Rutkowski |
| title_full_unstemmed | Martingale methods in financial modelling Marek Musiela ; Marek Rutkowski |
| title_short | Martingale methods in financial modelling |
| title_sort | martingale methods in financial modelling |
| topic | Derivativos (modelos matemáticos) larpcal Finanças (modelos matemáticos) larpcal Matemática aplicada larpcal Opções financeiras (modelos matemáticos) larpcal Taxa de juros (modelos matemáticos) larpcal Mathematisches Modell Options (Finance) Mathematical models Derivative securities Mathematical models Interest rates Mathematical models Fixed-income securities Mathematical models Finance Mathematical models Modellierung (DE-588)4170297-9 gnd Finanzmathematik (DE-588)4017195-4 gnd Kapitalmarkttheorie (DE-588)4137411-3 gnd Martingal (DE-588)4126466-6 gnd Optionspreistheorie (DE-588)4135346-8 gnd Martingaltheorie (DE-588)4168982-3 gnd |
| topic_facet | Derivativos (modelos matemáticos) Finanças (modelos matemáticos) Matemática aplicada Opções financeiras (modelos matemáticos) Taxa de juros (modelos matemáticos) Mathematisches Modell Options (Finance) Mathematical models Derivative securities Mathematical models Interest rates Mathematical models Fixed-income securities Mathematical models Finance Mathematical models Modellierung Finanzmathematik Kapitalmarkttheorie Martingal Optionspreistheorie Martingaltheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018970515&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000895226 |
| work_keys_str_mv | AT musielamarek martingalemethodsinfinancialmodelling AT rutkowskimarek martingalemethodsinfinancialmodelling |
Inhaltsverzeichnis
Paper/Kapitel scannen lassen
Paper/Kapitel scannen lassen
Handapparate (nicht verfügbar)
| Signatur: |
0048 WIR 170f 2006 A 2217(2,2009) Lageplan |
|---|---|
| Exemplar 1 | Dauerhaft ausgeliehen Ausgeliehen – Rückgabe bis: 31.12.9999 |
Teilbibliothek Mathematik & Informatik
| Signatur: |
0102 WIR 170f 2006 A 2217(2,2009) Lageplan |
|---|---|
| Exemplar 1 | Ausleihbar Am Standort |