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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