Index

Lp integrability, 23 asset price, 191 ,27 assign, 397 χ 2,28 asymmetric double exponential, δ, 249 210 δ-hedging, 249 asymptotically unbiased estimator, δ-method, 70 58 γ greek, 259 asynchronous covariance estimator, G-stable convergence, 103 363 ρ greek, 259 attributes, 405 σ-algebra, 13 θ Greek, 258 backtest ϕ-, 102 , 419 :, 396 basket option, 8, 278 ?, 394 BAWAmericanApproxOption, 299 absorbing state, 97 Bayes’ rule, 16 adapted, 81 benchmark, 341 addBBands, 422 Bermudan option, 286 additive, 194 Bernoulli AEAsian, 270 sample, 79 American option, 7, 285 Bessel function, 32 AmericanPutExp, 291 besselK, 180 AmericanPutImp, 296 best linear unbiased estimator, 59 apply, 409 Beta, 30 arbitrage free, 223 bias, 58 arbitrage opportunity, 229 bilateral gamma, 179 args, 429 binary, 252 as, 405 binomial, 25 as.data.frameCOPYRIGHTED, 405 Borel, MATERIAL 51 as.Date, 426 Brown, 104 as.integer, 405 Brownian motion, 9, 104 as.ts, 425 multidimensional, 145, 278 as.xts, 426 translated, 232 as.zoo, 425 two-sided, 351

Option Pricing and Estimation of Financial Models with R, First Edition. Stefano M. Iacus. © 2011 John Wiley & Sons, Ltd. Published 2011 by John Wiley & Sons, Ltd. 448 INDEX BSAmericanApproxOption, expectation, 55 300 probability, 15 Burkholder-Davis-Gundy confint,70 inequality, 89, 127 consistent estimator, 60 contingent claim, 221 c, 396 continuous cad-l` ag` process, 108 mapping theorem, 50 call option, 5 , 19 Cantelli, 51 time cat, 393 , 99 Cauchy, 30 convergence Cauchy functional equation, 135 almost sure, 49 Cauchy-Schwarz-Bunyakovsky in r-th mean, 49 inequality, 47 in distribution, 48 cce, 364 in probability, 48 CGMY mean square, 49 process, 209 weak, 48 change point, 350 convolution, 35 Chapman-Kolomogorov equation, COS, 329 95, 98 counting process, 108 character, 397 covariance, 22 characteristic asynchronous estimator, 363 exponent, 38, 40 function, 84 function, 23 CPoint, 359 de Finetti, 132 cpoint, 275, 350, 352 triplet, 38 CRAN, 395 charSeries, 421 CreditMetrics, 419 chartSeries, 274 cumulative distribution function, 19 Chebyshev inequality, 46 Chebyshev-Cantelli inequality, 46 data.frame, 401 Chebyshev-Markov inequality, 46 dbeta,31 clusterSetupRNG, 412 dbinom,25 coefficient dcauchy,30 diffusion, 128 dcCIR, 201 drift, 128 dchisq,29 compensated Poisson random de Finetti characteristic function, measure, 143 132 compensator, 87, 113 De Morgan’s laws, 15 complementary, 13 delta Greek, 249 complete additivity, 14 deltat, 423 completeness, 223 density complex conjugate, 23 function, 19 complex, 398 method, 252 , 110 transition, 107 conditional dependence, 79 INDEX 449 dexp,27 Euler’s formula, 73 dgamma,28 European option, 7, 222 dgh,33 events, 14 diffusion elementary, 14 coefficient, 128 exercise price, 3 equation, 108 existence of solutions, 129 process, 128 exotic option, 7 digital option, 252 expected value, 21 Dirac delta, 38, 117, 331 expiry date, 3 discrete exponential Fourier transform, 43 law, 26 random variable, 19 martingale, 86 dist, 378 extension space, 103 distribution function, 19 dlnorm,31 fair value, 4 dmvnorm,35 family of experiments, 57 dnig,32 fAsianOptions, 272, 420 dnorm,28 fast Fourier transform, 43 Doob fBasics, 205, 420 maximal L2 inequality, 88 fExoticOptions, 272, 420 maximal Lp inequality, 88 filtration, 81 maximal inequality, 88 natural, 81 Doob-Meyer decomposition, 87 fImport, 273 dopar, 243, 413 finite dimensional doSNOW, 244, 414 distribution, 80 doubling strategies, 229 first passage time, 89 dpois,25 first-order variation, 83 drift, 8, 192 Fisher information, 62 coefficient, 128 fOptions, 240, 260, 299, 420 dsCIR, 201 for, 410 dstable,41 foreach, 243, 413 dt,29 format, 429 dtw, 378 forward, 2 dunif,26 fPortfolio, 419 fraction early exercise premium, 297 optimal, 343 efficiency, 63 fractions, 342 end, 423 FRED, 442 Epps effect, 363 fredImport, 444 equivalent measure, 17, 261 fredSeries, 444 Esscher transform, 315 frequency, 423 estimating function, 66 fTrading, 419 estimator function, 408 consistence, 60 future, 2 EUCdist, 378 fWEBDATA, 443 450 INDEX Galton, 31 hedge ratio, 251 gamma, 179, 259 hedging, 6, 223 GBSCharacteristics, 260 help, 394 GBSOption, 240, 321 help.search, 394 GBSVolatility, 275 hessian, 165 generalized Hessian matrix, 75 hyperbolic holder, 3, 223 process, 207 homogeneous tempered , 209 Poisson process, 110 generalized hyperbolic hybrid diffusion system, 147 distribution, 32 hypFit, 205 generating function moment, 24 i.i.d., 24, 25, 57 geometric IBrokers, 419 Brownian motion, 8, 191, 210 implied volatility, 275 translated, 232 independent, 16, 20 telegraph process, 211 index, 425 get.zoo.data, 178 index of stability, 40 getDoParWorkers, 244, 414 inequality getSymbols, 421, 442 Burkholder-Davis-Gundy, 89, GH, 32 127 process, 207 Cauchy-Schwarz-Bunyakovsky, ghFit, 205 47 Girsanov, 130, 141 Chebyshev, 46 global Lipschitz condition, 129 Chebyshev-Cantelli, 46 Google, 442 Chebyshev-Markov, 46 grad, 165 Doob Greek maximal, 88 delta, 249 maximal L2,88 gamma, 259 maximal Lp,88 kappa, 259 Holder,¨ 47 rho, 259 Hajek-R´ enyi,´ 88 theta, 258 Jensen, 47, 57 vega, 259 Kolmogorov, 47, 89 growth infinite activity, 138 optimal portfolio, 341 infinitely divisible, 37 rate, 342 infinitesimal generator, 100 information, 19 Holder¨ inequality, 47 install.packages, 395 Hajek-R´ enyi´ instantaneous state, 101 inequality, 88 integer, 397 Hayashi-Yoshida estimator, 363 integrable random variables, 21 head, 404 intensity function, 110 heat equation, 108, 233 invariant distribution, 97 INDEX 451 inverse image, 18 load, 394 inversion theorem, 43 , 131 irts, 427 localizing sequence, 131 ISOdate, 429 location, 40 Itoˆ log-likelihood, 61 formula, 124, 126, 146 log-normal, 31, 193 integral, 9, 119 log-returns, 193 process, 123 logLik,69 Ito-Lˆ evy´ Lorentz, 30 decomposition, 137 lower,70 process, 143 ls, 396 LSM, 310 Jensen’s inequality, 47, 57 , 134 makeCluster, 411 kappa, 259 makeSOCKcluster, 411 Kolmogorov marked , 109 inequality, 47, 89 market price of risk, 343 Kous process, 210 Markov chain, 92 L-BFGS-B,70 continuous time, 99 Levy,´ 39 operator, 375 exponent, 38 process, 91 jump diffusion, 134 property measure, 133, 136 strong, 96, 99 process, 133 switching, 147, 183 Levy-Khintchine´ formula, 38 marks, 109 Lamperti transform, 130 martingale, 84 lapply, 409 exponential, 86 large numbers local, 131 strong law, 51 measure, 261 weak law, 51 MASS,35 lasso, 371 matrix, 396 law of total probability, 16 MatrixExp, 101 least squares method, 66 maximum likelihood estimator, 64 levels, 405 MCAsian, 268 Lewis method, 170 MCdelta, 256 likelihood, 61 MCdelta2, 257 Lindeberg’s condition, 53 MCPrice, 242, 243 linear mean growth condition, 129 reverting, 200 list, 399 square error, 58 listFinCenter, 432 measurability, 18 lm, 398 measurable function, 18 452 INDEX measure normal inverse equivalent, 17 Gaussian Levy,´ 136 process, 208 martingale, 261 NULL, 397 of probability, 14 numDeriv, 165 random, 136 numeraire, 341 risk neutral, 225, 261 numeric, 396 Meixner nws, 412 distribution, 33 process, 208 OANDA, 442 memoryless, 27 oandaImport, 444 merge, 437 oandaSeries, 444 method, 70, 406 opefimor,1 method of moments, 65 optim, 168 midori, 248 option, 2 mixed moment, 22 American, 7, 285 mixing property, 101 basket, 8, 278 MLE, 64 Bermudan, 286 mle, 68, 169, 202, 406 binary, 252 modified Bessel function, 117, 331 call, 5 MOdist, 377 digital, 252 moment, 21 European, 7, 222 generating function, 24, 86 exotic, 7 mixed, 22 path-dependent, 7 MSE, 58 put, 5 msm, 101, 185 vanilla, 3 mts, 423 Ornstein-Uhlenbeck, 152 multicore, 244 multicore, 413 par, 394 multidimensional path, 80 Brownian motion, 145 path-dependent option, 7 geometric Brownian motion, 278 payoff, 5 mvrnorm,35 function, 221 pbeta,31 Newton-Raphson, 167 pbinom,25 NIG, 32, 179 pcauchy,30 process, 208 pcCIR, 201 nigFit, 205 pchisq,29 nlm, 167 PerformanceAnalytics, 419 non-arbitrage, 6 pexp,27 nonhomogeneous pgamma,28 Poisson process, 110 pgh,33 normal plnorm,31 gamma, 179 pmvnorm,35 inverse Gaussian, 32, 179 pnig,32 INDEX 453 pnorm,28 tempered stable, 207 point process, 108 Variance Gamma, 179, 208, 323 Poisson, 25 Wiener, 104 process, 109 psCIR, 201 random measure, 136 pstable,41 compensated, 143 pt,29 random variable, 25 punif,26 polyroot, 166 put option, 5 portfolio put-call party, 239 hedging, 229 self-financing, 228 qbeta,31 strategy, 228 qbinom,25 portfolio, 419 qcauchy,30 POSIXct, 429 qcCIR, 201 POSIXlt, 429 qchisq,29 ppois,25 qexp,27 predictable, 83 qgamma,28 probability, 14 qgh,33 measure, 14 qlnorm,31 space, 14 qmle, 198 extension, 103 qmvnorm,35 transition, 92, 98 qnig,32 Process qnorm,28 Kou, 210 qpois,25 process qsCIR, 201 Brownian, 104 qstable,41 CGMY, 209 qt,29 continuous time, 80 quadratic counting, 108 variation, 83 diffusion, 128 quantile, 20 discrete time, 79 quantmod, 274 gamma, 179 qunif,26 generalized quotes, 440 hyperbolic, 207 tempered stable, 209 R-Forge, 395 Levy,´ 133 Radon-Nikodym,´ 20, 130, 141 Markov, 91 random switching, 147, 183 experiment, 14 Meixner, 208 measure, 136 normal inverse Gaussian, 208 variable, 18 point, 108 Bernoulli, 24 marked, 109 Beta, 30 Poisson, 109 Binomial, 25 predictable, 83 Cauchy, 30 telegraph, 114 Chi-square, 28 454 INDEX random (Continued) rnorm,28 continuous, 19 RollGeskeWhaleyOption, 300 discrete, 19 roundoff error, 163 exponential, 26 rpois,25 gamma, 27 rpvm, 412 Gaussian, 28 RQuantLib, 420 generalized hyperbolic, 32 rsCIR, 201 inverse Gaussian, 32 RSiteSearch, 394 Levy,´ 39 rstable,41 log-normal, 31 rt,29 Meixner, 33 runif,26 Poisson, 25 Student’s t,29 uniform, 26 sample walk, 85, 92 space, 14 rate sapply, 409 Poisson process, 110 save, 394 rates, 101 save.image, 394 rbeta,31 scale, 40 rbind, 434 score function, 62 rbinom,25 sde, 175, 275, 377, 395 RBloomberg, 444 sde.sim, 175, 201 rc.hy, 364 , 131 rcauchy,30 setModel, 178 rcCIR, 201 sfInit, 413 rchisq,29 sfStop, 246, 413, 415 realized, 364 showMethods, 407 registerDoSEQ, 246, 415 sigma-algebra, 13 registerDoSNOW, 414 sign function, 40 return, 408 simMarkov, 184 returns, 274 simMSdiff, 185, 335 rev, 439 simulate, 178 rexp,27 skewness, 40 rgamma, 28, 180 Slutsky’s theorem, 50 rgh,33 snow, 411 rho, 259 snowfall, 244, 412 Richardson’s extrapolation, 164 solution risk existence and uniqueness, 129 free strategy, 6 strong, 128 neutral measure, 225, 261 weak, 128 rlnorm,31 solve, 294 Rmetrics, 240, 420 sort, 439 Rmpi, 412 square integrability, 23 rmvnorm, 35, 180 stable, 179 rnig,32 convergence, 103 INDEX 455 in distribution, 148 total variation, 83 law, 38 trajectory of a process, 79 stableFit, 205 transform standard Esscher, 315 error, 160 Lamperti, 130 normal, 28 transient, 97 start, 423 transition state space, 80 density, 107 stats4, 68, 406 matrix, 92 stochastic probability, 92, 98 differential equation, 9, 128 translated exponential, 326 Brownian motion, 232 integral, 9, 119, 121 geometric Brownian motion, 232 process, 79 trivial σ-algebra, 13 covariance function, 84 truncation error, 163 increments, 84 TS , 83 process, 207 total variation, 83 ts, 423 stochastically continuous, 133 tseries, 427 stopCluster, 412 TTR, 419 , 89 ttrTests, 419 str, 398 TurnbullWakemanAsian- strike price, 3, 222 ApproxOption, 272 strong two-sided Brownian motion, 351 , 51 , 96, 99 unbiased estimator, 58 solution, 128 underlying asset, 3 strptime, 430 uniform, 26 subsampling, 366 uniqueness of solutions, 129 summary,70 uniroot, 166 Sys.getlocale, 431 Sys.setlocale, 431 vanilla, 3 VaR, 419 tail, 404 variance, 21 telegraph variance–covariance matrix, 22 equation, 115 variance , 114 process, 208 geometric, 211 vcov,69 tempered stable, 179 vector, 397 process, 207 vega, 259 theta, 258 velocity process, 114 thinning, 170 VG, 179, 208, 323 time, 423 volatility, 4, 8, 192 timeDate, 420, 428 matrix, 278 timeSeries, 274, 428 smile, 276 456 INDEX volatility window, 424 implied, 275 writer, 3, 223 Vsmc, 338 xts, 426 wave equation, 115 weak Yahoo, 442 convergence, 48 yahooImport, 443 law of large numbers, 51 yahooSeries, 273, 443 solution, 128 yuima, 177, 270 while, 410 yuima-data, 178 Wiener, 104 process, 9 zoo, 178, 424 wild, 168 zooreg, 425