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Author Index
Abramowitz, M., 14, 16, 18, 639, 657,659, 667
Ait-Sahalia, Y., 145, 183, 444, 667Albanese, C., 549, 667Andersen, L., 444, 667Andreasen, J., 444, 667Ansel, J. P., 520, 667Arrow, K. J., 408, 667Artzner, P., 69, 667
Bachelier, L., 109, 142, 667Bajeux-Besnainou, I., 321, 667Bakshi, G., 304, 667Balakrishnan, N., 16, 545, 660, 674Barndorff-Nielsen, O., 18, 19, 83, 84,
90, 126, 178, 259, 485, 569, 667,668
Barraquand, J., 577, 668Barrett, R., 607, 668Basle, 394, 668Becherer, 321, 668Beckers, S., 444, 456, 457, 668Berry, M., 607, 668Bjork, T., 260, 569, 668Black, F., 134, 142, 144, 281, 288, 337,
440, 441, 485, 668Blaesild, P., 83, 667Blattberg, R. C., 19, 84, 668Bochner, S., 81, 668Bollerslev, T., 238, 239, 441, 668, 671Borodin, A. N., 146, 147, 157, 158, 308,
668Bouchaud, J. P., 508, 668, 676Boyle, P. P., 551, 577, 668, 674
Breeden, D. T., 466, 468, 668Breiman, L., 372, 386, 403, 408, 668Breymann, W., 52, 494, 504, 505, 669Brigo, D., 381, 444, 509, 669Browne, S., 386, 669Bru, M.-F., 544, 669Bruti-Liberati, N., 522, 554, 569, 586,
669Buchen, P. W., 38, 49, 543, 669Buhlmann, H., 319, 321, 669
Cao, C., 304, 667Carr, P., 84, 509, 669Chan, T. F., 607, 668Chen, O. X., 549, 667Chen, Z., 304, 667Cherny, A. S., 271, 669Chesney, M., 304, 473, 475, 669, 674Chou, R., 441, 668Christensen, M. M., 260, 405, 520, 521,
535, 669Clark, P. K., 81, 669Clewlow, L., 442, 670Cochrane, J. H., 74, 339, 670Constatinides, G. M., 339, 670Cont, R., 183, 441, 443, 444, 481, 508,
668, 670Corsi, F., 178, 485, 670Courtadon, G., 145, 670Cover, T., 386, 670Cox, J. C., 103, 112, 143, 144, 444, 446,
456, 457, 591, 592, 596, 670Craddock, M., 148, 149, 670Curran, M., 576, 670
E. Platen, D. Heath, A Benchmark Approach to Quantitative Finance,Springer Finance,© Springer-Verlag Berlin Heidelberg 2006, Corrected printing 2010
684 Author Index
Dacorogna, M., 74, 178, 485, 670Davis, M. H. A., 428, 670da Fonseca, J., 443, 444, 670Delbaen, F., 69, 143, 312, 337, 377, 389,
445, 455, 456, 667, 670Demmel, J., 607, 668Derman, E., 144, 442, 461, 466, 468,
668, 670Dewynne, J., 600, 681De Vries, C. G., 74, 670Dias, A., 52, 494, 669Dickson, D. C. M., 131, 671Dimson, E., 344, 490, 671Dongarra, J., 607, 668Doob, J. L., 106, 172, 173, 671Dothan, L. U., 144, 671Duffie, D., 144, 316, 339, 569, 671Dumas, B., 444, 671Dupire, B., 461, 466, 468, 671
Eber, J. M., 69, 667Eberlein, E., 20, 84, 90, 126, 259, 671Eijkhout, V., 607, 668Einstein, A., 109, 671Elliott, R. J., 172, 173, 175, 199, 591,
596, 671, 681El Karoui, N., 350, 352, 508, 668, 672Embrechts, P., 52, 68, 69, 95, 97, 494,
669, 671, 676Engle, R. F., 238, 239, 478, 671
Fama, E. F., 82, 85, 671Fang, K. T., 68, 671Feller, W., 81, 672Fergusson, K., XI, 85, 89, 473, 494, 672Fernholz, E. R., 389, 401, 672Fischer, P., 567, 672Fishman, G. S., 570, 573, 672Fleming, J., 444, 671Fleming, W. H., 143, 672Follmer, H., 69, 141, 143, 208, 337, 435,
672Frey, R., 68, 441, 477, 672, 676Friedman, A., 362, 672
Gaines, J. G., 564, 672Geman, H., 84, 90, 126, 669, 672Geman, S., 350, 352, 672Gerber, H. U., 129–131, 319, 672
Ghysels, E., 441, 672Gibson, M. S., 70, 673Glasserman, P., 551, 569, 578, 673Going-Jaeschke, A., 310, 673Gonedes, N., 19, 84, 668Gorovoi, V., 144, 673Grandell, J., 129, 130, 673
Hakansson, N. H., 372, 386, 403, 408,420, 423, 673
Hansen, L. P., 74, 339, 673Harris, C. J., 569, 676Harrison, J. M., 319, 337, 377, 436, 673Harvey, A., 441, 672Heath, D., XI, 69, 362, 383, 384, 435,
444, 446, 448, 468, 470, 473, 474,478, 489, 494, 503, 505, 509, 591,667, 673
Heston, S. L., 473, 475, 674Heyde, C. C., 78, 199, 674Ho, T. S. Y., 144, 241, 674Hofmann, N., 435, 473, 674Howison, S., 600, 681Hui, C. H., 444, 676Hull, J., 145, 178, 473, 475, 674Hulley, H., 501, 502, 536, 540, 545–547,
674Hurst, S. R., 18, 37, 82, 85, 88, 473,
474, 478, 494, 673, 674
Ikeda, N., 259, 271, 364, 674Ingersoll, J. E., 112, 144, 543, 670, 674
Jackel, P., 578, 674Jacod, J., 174, 674Jagannathan, R., 339, 673Jamshidian, F., 384, 674Jarrow, R., 383, 384, 673Jeanblanc, M., 304, 674Johnson, H., 473, 674Johnson, N. L., 16, 545, 660, 674Joy, C., 577, 674
Kabanov, Y., 260, 569, 668Kahaner, D., 552, 674Kan, R., 144, 316, 671Kani, I., 442, 461, 466, 468, 670Karasinski, P., 144, 668Karatzas, I., 106, 172, 175, 189, 195,
229, 271, 304, 306, 322, 338, 344,
Author Index 685
348, 387, 389, 422, 425, 433, 445,452, 672, 675
Kardaras, C., 389, 672
Keller, U., 20, 84, 90, 126, 259, 671Kelly, J. R., 169, 320, 372, 408, 423,
675
Kelly, L., 440, 493, 504, 505, 669, 675Kessler, M., 473, 493, 675
Khanna, A., 407, 408, 675Kloeden, P. E., 189, 195, 238, 239, 248,
273, 551, 552, 558, 563–565, 567,573, 578, 583, 584, 586, 675
Kluppelberg, C., 95, 97, 671
Konstandatos, O., 38, 49, 543, 669Kopp, P. E., 596, 671
Korn, R., 422, 675Kotz, S., 16, 68, 545, 660, 671, 674
Kou, S. G., 128, 257, 675Kreps, D. M., 319, 337, 377, 436, 673
Kroner, K., 441, 668Krylov, N. V., 247, 675
Kubilius, K., 569, 586, 675Kuchler, U., 20, 84, 675
Kulldorff, M., 407, 408, 675Kunita, H., 363, 675
Lamberton, D., 610, 675
Lapeyre, B., 610, 675Larsen, K., 520, 669
Latane, H., 372, 386, 403, 408, 675Le, T., 402, 444, 675
Ledoit, O., 444, 675Lee, S. B., 144, 241, 674
Lewis, A. L., 444, 445, 448, 458, 478,675
Linetsky, V., 144, 673
Lintner, J., 332, 415, 675Litzenberger, R., 466, 468, 668
Liu, S., 199, 674Lo, A., 444, 667
Lo, C. F., 444, 676Loewenstein, M., 376, 377, 455, 676
Long, J. B., 169, 320–322, 325, 372,676
Longstaff, F. A., 145, 676Luenberger, D. G., 412, 676
Lukacs, E., 37, 676Lyons, T. J., 564, 672
Madan, D., 21, 84, 90, 126, 259, 568,669, 672, 676
Maghsoodi, Y., 569, 676Mandelbrot, B., 82, 85, 676Margrabe, W., 547, 676Markowitz, H., 85, 88, 372, 386, 403,
406, 407, 410, 412, 494, 533, 676Marsh, P., 344, 490, 671Marsh, T. A., 144, 676Maruyama, G., 560, 676Matacz, A., 508, 676McNeil, A., 68, 69, 671, 676Melino, A., 473, 475, 676Mendel, J., 73, 677Mercurio, F., 381, 444, 509, 669Merener, N., 569, 673Merton, R. C., 134, 144, 226, 238, 241,
254, 257, 288, 319, 332, 403, 407,408, 415, 419, 424, 535, 536, 539,547, 677
Mikosch, T., 95, 97, 671Mikulevicius, R., 569, 586, 677Miller, S., 144, 452, 456, 457, 499, 501,
502, 536, 540, 542, 545–547, 674,677
Milne, F., 259, 676Milstein, G. N., 563, 567, 578, 589, 677Miltersen, K. R., 496, 677Miyahara, Y., 259, 677Moler, C., 552, 674Morgenstern, O., 419, 681Morton, A., 383, 384, 673Morton, K., 600, 605, 679Mossin, J., 332, 415, 677Muller, U., 74, 178, 485, 670Mulvey, J. M., 386, 681Musiela, M., 381, 384, 542, 677
Nash, S., 552, 674Nelsen, R., 51, 677Nelson, D. B., 239, 477, 478, 677Neumann, K., 20, 84, 675Neyman, J., 80, 677Ng, K. W., 68, 671Niederreiter, H., 577, 578, 677Nielsen, L. T., 408, 677Nikitopoulos-Sklibosios, C., 522, 569,
669Novikov, A. A., 259, 348, 677
686 Author Index
Ord, J. K., 85, 680Osborne, M. F. M., 82, 485, 677
Pan, J., 569, 671Paskov, S., 577, 677Pearson, E. S., 80, 677Pearson, N., 144, 677Picard, P., 131, 677Pictet, O. V., 74, 670Pivitt, B., 553, 678Platen, E., XI, 18, 19, 68, 70, 82, 85,
86, 88, 89, 92, 143–145, 148, 149,169, 189, 195, 238, 239, 248, 253,260, 273, 321, 322, 334, 335, 373,376, 382, 388–391, 394, 396, 402,405, 407, 413, 415, 420, 422, 423,435, 442, 444, 446, 448, 452, 456,457, 461, 468, 470, 473, 474, 478,488, 489, 492–494, 496, 497, 499,501–505, 509, 518, 520–523, 526,528, 535, 536, 540, 542, 545–547,551, 552, 554, 558, 563–565, 567,569, 573, 578, 580, 581, 583–586,591, 597, 599, 604, 669, 670,672–675, 677, 678, 681
Pliska, S. R., 319, 337, 377, 436, 596,673, 679
Portait, R., 321, 667Potters, M., 508, 668Pozo, R., 607, 668Praetz, P. D., 19, 84, 679Prakasa Rao, B. L. S., 473, 493, 679Pratt, J. W., 408, 679Protter, P., 125, 127, 174, 176, 184,
189, 191, 192, 195, 197, 199, 201,202, 211, 225, 229, 262, 309, 679
Rachev, S. T., 82, 674Randall, C., 600, 680Rao, C. R., 80, 679Rapisarda, F., 509, 669Raviart, P. A., 611, 679Rebolledo, R., 143, 569, 678Renault, E., 441, 672Revuz, D., 147, 184, 263, 304, 306, 308,
328, 348, 486, 679Richtmeyer, R., 600, 605, 679Ripley, B. D., 577, 679Rochet, J. C., 350, 352, 672
Rogers, L. C. G., 186, 273, 679Rogers, L. C. G., 339, 679Romine, C., 607, 668Rosenfeld, E. R., 144, 676Ross, S. A., 103, 112, 144, 319, 337,
444, 446, 456, 457, 591, 592, 596,670, 679
Ross, S. M., 576, 679Rubinstein, M., 103, 386, 442, 591, 592,
596, 670, 679Runggaldier, W., 260, 569, 668Rutkowski, M., 381, 384, 542, 677
Sagna, N., 508, 668Salminen, P., 146, 147, 157, 158, 308,
668Samuelson, P. A., 82, 134, 142, 403,
485, 539, 679Sandmann, K., 145, 680Santa-Clara, P., 444, 675Schachermayer, W., 337, 377, 389, 455,
670Schiedt, A., 69, 672Scholes, M., 134, 142, 281, 288, 337,
441, 485, 668Schonbucher, P. J., 444, 549, 680Schroder, M., 143, 444, 446, 456, 457,
680Schurz, H., 552, 567, 578, 675, 677Schwartz, E., 496, 677Schweizer, M., 141, 143, 337, 362, 435,
436, 442, 473, 672–674, 678, 680Scott, L. O., 473, 475, 669, 680Seneta, E., 21, 84, 90, 126, 259, 568,
676Shanno, D., 473, 674Sharpe, W. F., 332, 403, 406–409, 412,
415, 680Shaw, W., 17, 456, 457, 557, 600, 605,
606, 680Shephard, N., 126, 178, 259, 485, 668Sheu, S.-J., 143, 672Shiga, T., 305, 680Shirakawa, H., 143, 312, 445, 456, 670Shiryaev, A. N., 8, 29, 33, 50, 93, 94,
174, 674, 680Shiu, E. S. W., 131, 672Shreve, S. E., 106, 172, 175, 189, 195,
229, 271, 304, 306, 322, 338, 344,
Author Index 687
348, 387, 422, 425, 433, 445, 452,675
Singleton, K., 569, 671Sklar, A., 51, 680Smith, G. D., 600, 680Sondermann, D., 145, 337, 672, 680Sørensen, M., 20, 84, 493, 675Stahl, G., 68, 335, 391, 394, 496, 526,
678Staunton, M., 344, 490, 671Stegun, I. A., 14, 16, 18, 639, 657, 659,
667Stein, E. M., 473, 475, 680Stein, J. C., 473, 475, 680Straumann, D., 68, 69, 671Streller, A., 20, 84, 675Stricker, C., 520, 667Stroock, D. W., 160, 364, 680Stuart, A., 85, 680Stutzer, M. J., 386, 680Sun, T. S., 144, 677
Talay, D., 583, 680Tan, K. S., 577, 674Tanaka, H., 263, 680Tankov, P., 183, 441, 481, 670Tavella, D., 600, 680Taylor, H. M., 82, 676Taylor, S. J., 442, 680Thomas, J. M., 611, 679Thorp, E. O., 372, 386, 403, 408, 680Tobin, J., 403, 408, 412, 680, 681Toy, W., 144, 668Traub, J., 577, 677Tubaro, L., 583, 680Turnbull, S., 473, 475, 676
Usmen, N., 85, 88, 494, 676
van der Hoek, J., 199, 591, 596, 671,681
van der Vorst, H., 607, 668Varadhan, S. R. S., 160, 364, 680Vasicek, O. A., 107, 144, 241, 243, 681Vassalou, M., 408, 677von Neumann, J., 419, 681
Wagner, W., 558, 565, 678, 681Watanabe, S., 259, 263, 271, 305, 364,
674, 680, 681Wei, L., 131, 681Whaley, R., 444, 671White, A., 145, 473, 475, 674Wiggins, J. B., 473, 475, 681Willard, G. A., 376, 377, 455, 676Williams, D., 186, 273, 679Williams, J. B., 372, 681Wilmott, P., 600, 681Wolf, M., 444, 675Wu, L., 509, 669Wu, R., 131, 681
Xu, X., 442, 670, 680
Yamada, T., 263, 681Yor, M., 84, 90, 126, 147, 184, 263, 304,
306, 308, 310, 328, 348, 486, 669,672–674, 679
Yuen, P. H., 444, 676
Zhao, Y., 422, 681Ziemba, W. T., 372, 386, 408, 420, 422,
423, 673, 681Zumbach, G., 178, 485, 670
Index
3/2 model, 4783/2 volatility model, 493N -point random variable, 553d-dimensional linear SDE, 247k-sigma rule, 15pth moment, 36pth variation, 183
accumulation index, 400actuarial
diagnostics, 130pricing, 329
approach, 319formula, 331, 380
adapted, 164additivity property, 31, 44, 305affine model, 144, 158affine process, 316aggregate diffusion coefficient, 405almost surely, 6, 56, 92American option, 277, 280, 599annuity, 384antithetic variate Monte Carlo
estimator, 573appreciation rate, 216, 238, 369, 409arbitrage, 284, 376, 520
pricing theory, 320, 337, 377ARCH diffusion model, 477, 480asset-or-nothing binary, 543asymptotically normal, 571asymptotically unbiased, 71at-the-money, 278autocovariance matrix, 44
autoregressive conditional heteroscedas-tic, 441
Bachelier model, 142balanced implicit methods, 567Bayes formula, 7Bayes rule, 346Bayes’ Theorem, 7benchmark approach, 320, 375benchmarked P&L, 451benchmarked security, 324Bernoulli trials, 62Berry-Esseen inequality, 61Bessel function
modified of the first kind, 16, 147modified of the third kind, 18
Bessel process, 271, 487best fit, 89beta, 335binomial
model, 591option pricing formula, 596tree, 104, 112volatility, 594
Black model, 144Black-Derman-Toy model, 144Black-Karasinski model, 144Black-Scholes
formula, 288model, 36, 134, 142, 216, 219, 246,
281multi-asset, 249
PDE, 286
E. Platen, D. Heath, A Benchmark Approach to Quantitative Finance,Springer Finance,© Springer-Verlag Berlin Heidelberg 2006, Corrected printing 2010
690 Index
bond-or-nothing binary, 545Borel-Cantelli Lemma, 93boundary condition, 608Box-Muller method, 554Brownian motion, 100, 108butterfly spread, 279
Cameron-Martin Girsanov Theorem,340
candidate Radon-Nikodym derivative,499
capital asset pricing model, 332intertemporal, 319, 332
capital market line, 409Cauchy distribution, 17Cauchy-Schwartz inequality, 32Central Limit Theorem, 61central Student t distributed, 17CEV model, 143chain rule, 205change of numeraire, 350, 352
pricing formula, 356Chapman-Kolmogorov equation, 136characteristic function, 36, 37, 49Chebyshev inequality, 32chi-square
distribution, 15random variable, 556
Cholesky decomposition, 41, 45CIR model, 144claim amount distribution, 128coherent risk measure, 69collection of events, 4commutativity, 249
condition, 564compensated jump measure, 568complement of a set, 3complementary binomial distribution,
597complete market, 436conditional
density, 33expectation, 32, 33probability, 6
confidence interval, 63, 64Gaussian, 64Student t, 65
confidence level, 64consistency property, 101
continuation region, 360continuous financial market, 369continuous hedging, 282continuous random variable, 11control variate
estimator, 574method, 587
convergesin distribution, 61, 92in mean order p, 92in probability, 57, 92with probability one, 92
copula, 50bivariate t, 52function, 51Gaussian, 51Student t, 52
correlation, 40countably additive probabilities, 5counting process, 514Courtadon model, 145covariance, 40
matrix, 41, 44covariation, 180, 195, 221
approximate, 180Cramer-Lundberg model, 123, 129, 257Cramer-Rao lower bound, 73Crank-Nicolson method, 606credit rating, 118criterion
generalized least-squares, 77cumulants, 35curve fitting, 76
default intensity, 117defaultable zero coupon bond, 548deflator, 339degree of implicitness, 566, 605delta, 291density
function, 11generalized inverse Gaussian, 83hyperbolic, 21Student t, 84, 90symmetric generalized hyperbolic, 83variance gamma, 21
derivative securities, 277determinant, 41diffusion
Index 691
coefficient, 141, 192process, 141, 156, 160
vector, 160Dirac delta function, 161direct solvers, 606Dirichlet condition, 610discounted
BS-PDE, 285drift, 404GOP drift, 485underlying market, 283
discrete real valued random variable, 10distribution
binomial, 62elliptic, 68function, 8hyperbolic, 84marginal, 39, 46non-central chi-square, 307nondegenerate, 95normal-inverse Gaussian mixture, 84normal-variance gamma mixture, 84symmetric generalized hyperbolic, 18
Diversification Theorem, 396diversified portfolio, 393drift
coefficient, 141, 192implicit Euler scheme, 566implicit Milstein scheme, 566implicit simplified Euler scheme, 583implicit strong order 1.0 Runge-Kutta
schemes, 566Dupire formula, 468
efficient growth rate, 411efficient portfolio, 406, 410elementary outcomes, 2empty set, 4equal value index, 399equi-value weighted index, 399equidistant time discretization, 560equivalent measure, 338ergodicity, 119, 156estimate, 70
least-squares, 35estimation
error, 71least-squares, 75
estimator, 70
best linear unbiased, 78biased, 71consistent, 71efficient, 72maximum likelihood, 79nonlinear least-squares, 78
Euclidean norm, 50Euler scheme, 560, 563Euler-Maruyama scheme, 238European call option, 277, 455European option, 280European put option, 278event, 3
driven trading uncertainty, 514factor, 69
EWI, 399existence, 261expectation, conditional, 32, 33expected
excess return, 221, 332rate of return, 216, 409utility maximization, 421
expiration date, 277explicit strong order 1.0 scheme, 565explicit weak order 2.0 scheme, 581exponential
density, 554distribution, 13Levy model, 126, 259
exponentials of semimartingale, 260extended Vasicek model, 144extrapolation, 582Extreme Value Theorem, 95
fair, 326, 378price, 326put-call parity, 458zero coupon bond, 452
family ofdrift implicit
Euler schemes, 566Milstein schemes, 566simplified Euler schemes, 583
implicit weak Euler schemes, 584weak order 1.0 predictor-corrector
methods, 584Fatou’s Lemma, 94Feynman-Kac formula, 356
general, 361
692 Index
Fibonacci generatorlagged, 552
filtered probability space, 164filtration, 164finite difference method, 600, 604first exit time, 360first hitting time, 172first moment, 22Fisher information, 73Fisher-Tippett Theorem, 95Fokker-Planck equation, 145, 154, 476Follmer-Schweizer decomposition, 435forward contract, 385, 542forward price, 385, 542forward rate, 381, 497
equation, 382, 522fractional Kelly strategy, 408free snack, 455frictionless market, 282fully implicit finite difference method,
605fundamental matrix, 248fundamental solution, 241
gains from trade, 187gamma, 291
distributed, 15function, 15
Gauss-Seidel method, 607Gaussian
confidence interval, 64density
bivariate, 41, 42function, 13multivariate, 46
distribution, 13multivariate, 48
jointly, 41random variable, 13
shifted, 38shift, 38
multivariate, 47, 49general market risk, 335, 390generalized least-squares criterion, 77generalized volatility process, 515geometric Brownian motion, 134, 137,
142, 215, 561Girsanov Theorem, 347GOP, 372, 387, 518
greeks, 290Gronwall inequality, 264growth optimal portfolio, 320, 322, 372growth rate, 134, 221, 372
Heath-Jarrow-Morton equation, 383hedgable part, 434hedge portfolio, 281hedge ratio, 285hedge simulation, 298hedging strategy, 298, 432Heston model, 475Ho-Lee model, 144Hull-White model, 145hypothetical risk neutral measure, 447
i.i.d., 55ICAPM, 334
formula, 335pricing rule, 336relationship, 416
imaginary unit, 36implied transition density, 466implied volatility, 441, 442
surface, 442in-the-money, 278incomplete gamma function, 16incomplete market, 436independent, 7, 8indicator function, 9indistinguishability, 263inequality, 50
Berry-Esseen, 61Cauchy-Schwartz, 32Chebyshev, 32Doob, 173Gronwall, 264Jensen’s, 31Lyapunov, 31Markov, 32maximal martingale, 173
information set, 163inner product, 50insurance premium, 123insurance risk, 128integrable, 23
p-integrable, 23integral
Ito, 187, 191, 199
Index 693
Riemann-Stieltjes, 192stochastic, 187
integrand, 187integration-by-parts, 213intensity
matrix, 115measure, 125parameter, 10
interest rate term structure, 381intertemporal capital asset pricing
model, 319, 332, 415intrinsic value, 278inverse Fourier transform, 37inverse gamma density, 476, 493inverse matrix, 41inverse transform method, 554invertible matrix, 41iterative solvers, 606Ito differential, 192, 212Ito formula, 206, 208, 212, 224, 230
for semimartingales, 224Ito integral, 187, 191, 199, 247
properties, 194Ito process with jumps, 258
Jensen’s inequality, 31joint distribution, 39, 45jump adapted
scheme, 569simplified Euler scheme, 586time discretization, 569, 586
jump diffusion, 258market, 513, 519
jump martingale, 514jump size, 181
Kelly strategy, 408knock-out-barrier option, 360Kolmogorov
backward equation, 115, 146, 161forward equation, 115, 145, 160
kurtosis, 27excess, 29sample, 58
Lagrange multiplier, 412Laplace transform, 35Law of
iterated conditional expectations, 33
Large Numbers, 56least-squares estimate, 35Lebesgue almost everywhere, 11Lebesgue’s Dominated Convergence
Theorem, 94leptokurtic, 28, 82level of confidence, 64leverage effect, 440Levy measure, 127Levy process, 126, 202, 568Levy’s Theorem, 227likelihood
function, 79, 89ratio, 88
test, 80limited liability, 284, 376linear congruential pseudo-random
number generators, 552linear growth bound, 264linear SDE
with multiplicative noise, 243with additive noise, 241
Lipschitz condition, 264LLN
strong, 56weak, 57
local martingale, 183strict, 309
local volatility, 462model, 461
locally optimal, 405, 533locally riskless, 368log-likelihood function, 79log-return, 1
Student t, 67log-utility, 420, 423lognormal, 36
asset price model, 14model, 82, 134, 144
logstable model, 82long in a security, 279long term growth rate, 386, 522Longstaff model, 145Lyapunov inequality, 31
marginal distribution, 39, 46mark set, 125, 568marked point process, 124market
694 Index
activity, 505complete, 436incomplete, 436jump diffusion, 513, 519portfolio, 334, 416price
of event risk, 515of risk, 370, 515of risk contribution, 497
Markov chaincontinuous time, 113discrete time, 110, 111
Markov inequality, 32Markov process, 110, 272
continuous, 133, 135, 141Markov property, 110, 111, 135
continuous time, 114Markowitz efficient frontier, 409Marsaglia method, 556martingale, 166
local, 183Representation Theorem, 433square integrable, 166strict local, 184
matrix, 41autocovariance, 44covariance, 41, 44fundamental, 248intensity, 115invertible, 41positive definite, 73random, 44regular, 41sparse, 606transition probability, 114
maturity date, 278maximal element, 389maximum step size, 560mean, 10, 22
reversion level, 243vector, 41
mean-square error, 72mean-variance optimality, 406measurable function, 8measure transformation method, 588Merton model, 255, 536, 539Merton’s jump diffusion model, 254method of moments, 74Milstein scheme, 563
minimalequivalent martingale measure, 435market model, 143, 253, 483, 488, 539
MMM, 143multi-asset stylized, 529multi-currency, 494with random scaling, 502
model3/2, 478affine, 144, 158ARCH diffusion, 477, 480Bachelier, 142Black, 144Black-Derman-Toy, 144Black-Karasinski, 144Black-Scholes, 134, 142, 281CIR, 144constant elasticity of variance, 143,
444Courtadon, 145Cramer-Lundberg, 123, 129, 257exponential Levy, 126extended Vasicek, 144Heston, 475Ho-Lee, 144Hull-White, 145lognormal, 82, 134, 144logstable, 82Longstaff, 145Merton, 255minimal market, 143, 253, 483, 488,
539modified CEV, 445Ornstein-Uhlenbeck, geometric, 143Pearson-Sun, 144Platen, 145Sandmann-Sondermann, 145Scott, 475variance gamma, 84, 90Vasicek, 243Wiggins, 475
modified Bessel functionof the first kind, 16, 147of the third kind, 18
modified CEV model, 445modified trapezoidal method, 608modulus, 552moment, 22, 30
pth, 36
Index 695
central, 30equations, 244first empirical, 58generating function, 35normalized, 31normalized central, 30second central, 24, 30
Monotone Convergence Theorem, 94Monte Carlo
estimator, raw, 570method, classical, 570simulation, 551
Morgan Stanley capital weighted worldstock accumulation index, 334
MSCI, 344, 417multi-asset stylized MMM, 529multi-currency MMM, 494mutual fund, 535mutually exclusive, 5
natural filtration, 164negative skew, 442net growth rate, 253, 489net present value, 331NFLVR, 377no arbitrage, 319no free lunch with vanishing risk, 377non-central chi-square distribution, 16,
307, 500, 545nonnegative portfolio, 375nonreplicable payoff, 427normal variance mixture model, 68, 81normal-inverse Gaussian mixture
distribution, 19normalized GOP, 490normalized index, 254normally distributed, 14Novikov condition, 348null event, 6null hypothesis, 66numeraire
change, 320invariance, 286pairs, 353portfolio, 325
numerical stability, 566
observation vector, 70operational risk, 124
optimization, 412option price, 286Optional Sampling Theorem, 172order of strong convergence, 563Ornstein-Uhlenbeck process, 139, 242,
244geometric, 139standard, 138
out-of-the-money, 278outcome, 2overbetting, 412
parameter vector, 70partial integro differential equation,
360, 544partition, 7payoff, 378
function, 277rate, 358
Pearson-Sun model, 144perfect hedge, 284Platen model, 145Poisson
jump measure, 226, 568measure, 125, 127, 200probability, 10random variable, 10
Poisson process, 120compensated, 170compound, 123standard, 121transformed, 122
Portfolio Selection Theorem, 407positive definite, 73power utility, 420, 423predictor-corrector method, 608pricing function, 280pricing kernel, 339primary security account, 368, 514probability, 2, 4
binomial, 62of ruin, 130space, 6
filtered, 164process
affine, 316Bessel, 271counting, 514diffusion, 141, 156, 160
696 Index
Levy, 126, 202, 568Markov, 110, 272Ornstein-Uhlenbeck, 139Poisson, 120predictable, 172square root, 146, 250, 311, 312squared Bessel, 271, 304, 311Wiener, 109, 137
profit and loss, 283pseudo-random numbers, 552pure jump process, 222put-call parity, 295
quadratic variation, 174approximate, 174
quantile, 66quasi Monte Carlo, 577
Radon-Nikodym derivative, 338, 345random
bit generator, 554matrix, 44measure, 125number generator, natural, 553variable, 8
independent, 43vector, 44
random walkGaussian, 104symmetric, 103
real worldmartingale decomposition, 434martingale representation, 327pricing, 320
formula, 326, 430realization, 105reference level, 243, 253regular matrix, 41relative arbitrage, 389relative frequency, 2replicated payoff, 281replicating portfolio, 431return, 2rho, 293Richardson extrapolation, 583Riemann-Stieltjes integral, 192, 247right-continuous, 10risk aversion coefficient, 408, 422risk neutral, 320
pricing, 337, 380formula, 341
probability measure, 337risk premium, 221, 332, 406
samplekurtosis, 58mean, 55path, 105skewness, 58space, 2variance, 58
Sandmann-Sondermann model, 145savings account, 219, 384savings bond, 453scale measure, 157scaling process, 503scaling property, 305scenario simulation, 551, 560Scott model, 475SDE, 209
with jumps, 257self-financing, 353
portfolio, 282, 370strategy, 282
semimartingale, 197special, 198
sensitivity, 290sequence of
approximate GOPs, 395, 528CFMs, 391
regular, 394diversified portfolios, 393, 526DPs, 526JDMs, 523
regular, 527set of real numbers, 8Sharpe ratio, 406short in a security, 279short rate model, 143short-selling, 279sigma-algebra, 5, 163, 164
predictable, 171simplified weak
Euler scheme, 579order 2.0 Taylor scheme, 580
simulationMonte Carlo, 551scenario, 551, 560
Index 697
skewed, 26skewness, 25
sample, 58smile, 442sparse matrix, 606spatial discretization, 601specific generalized volatility, 525specific market risk, 335, 391specific volatility, 390speed of adjustment, 243, 253square integrable, 23square root process, 146, 250, 311, 312
generalized, 148squared Bessel process, 271, 304, 311,
446, 486time transformed, 311
SR process, 146standard deviation, 24standard Gaussian random variable, 13state price density, 339state space, 100stationary density, 138, 154stationary probability, 116statistical error, 579stochastic
differential equation, 209, 237, 239discount factor, 339integral, 187
multiple, 559integrals
multiple, 564process, 100, 105
continuous time, 100indistinguishable, 106measurable, 106moments, 107stationary, 107stationary independent increments,
108, 126volatility, 254, 439
model, 472stopping time, 170
inaccessible, 172strategy, 370stratified sampling, 576strict local martingale, 309strict supermartingale, 329strictly positive portfolio, 371strike price, 277
strongapproximation, 563consistency, 570order 1.5 Taylor scheme, 564solution, 263uniqueness, 264
Student t, 17density, 19distribution, p-dimensional, 49log-return, 494random variable, 557
stylized MMM, 478sub-sigma-algebra, 33, 164submartingale, 168subordination, 81successive overrelaxation method, 607supermartingale, 168, 184, 186, 375, 520
strict, 168sure event, 3survival probability, 130systematic
error, 579outperformance, 523risk, 335risk parameter, 335, 415
systematically outperformance, 388
Tanaka’s SDE, 263terminal condition, 285theta, 292
method, 605time of ruin, 129time set, 100time transformed squared Bessel
process, 311total market price of risk, 406, 534total probability, 8total specific volatility, 394tracking rate, 395, 527
expected, 528tradable martingale representation, 432trading uncertainty, 368trajectory, 105transition density, 137
lognormal, 140transition probability, 111
matrix, 114transpose, 41tree method, 591
698 Index
Two Fund Separation Theorem, 408two-point random variable, 10
unbiased, 570underlying value, 404, 486unfair portfolio, 328unhedgable part, 434uniformly distributed, 12unique weak solution, 263uniqueness, 261, 263
in law, 263of strong solution, 263
utilityfunction, 420indifference price, 428indifference pricing, 428
formula, 429
Value at Risk, 65value line index, 399VaR analysis, 67variance, 24, 30
reduction, 573sample, 58
Vasicek model, 107, 144, 243vector, 41
stochastic differential equation, 247vega, 293
volatility, 178, 216, 238, 368implied, 441, 442local, 462parametrization, 483stochastic, 439
von Neumann condition, 610
Wagner-Platen expansion, 558, 580weak
approximation, 579convergence, 578error, 579order, 578
2.0 extrapolation, 5832.0 predictor-corrector method, 5852.0 Taylor scheme, 5804.0 extrapolation, 583
Wiener process, 109, 137multi-dimensional, 210transformed, 109, 142
Wiggins model, 475with probability one, 6world stock index, 85, 401WSI, 401
Yamada condition, 270
zero coupon bond, 330, 381
Recommended