Index

absorbing boundary, see diffusion, forward rate dynamics, 514 absorbing barrier regularity issues, 512–513 accrual factor, see year fraction short rate state dynamics, 512 ADI, see PDE, ADI scheme one-factor, 429–442 adjusters method, see out-of-model PDE, 442 adjustment, adjusters method regularity issues, 430 affine short rate model, 429–442, short rate domain, 430 510–518 short rate dynamics, 429 bond reconstitution formula, 431, short rate state dynamics, 435 513–515 rate , 438 calibration, 439–441 affine approximation, 438 multi-pass bootstrap, 440 time averaging, 438 calibration to , 435–437 time-dependent, 431 characteristic function, 432 volatility skew range, 431 European , 437 volatility smile, 430 Fourier integration, 437 almost surely, 4 Gram-Charlier expansion, 437 American capped , 936 extended transform, 431 American swaption, 893–898 constant parameters, 432, 434 accrued current coupon, 893 piecewise constant parameters, 434 approximating with Bermudan Feller condition, 319, 430 swaption, see Bermudan swap- importance sampling, 1065 tion, approximating American moment-generating function, 432, swaption 437 discontinuity of value in Monte Carlo, 442 time, 893 multi-factor, 510–518 PDE, 895–897 bond dynamics, 514 extra state variable, 896–897 bond reconstitution formula, proxy Libor rate method, 895–896 513–515 American/Bermudan , 30–42 existence and uniqueness, 513 Bellman principle, 32, 33, 69 exponential affine, 511 Black-Scholes model, 837 Feller condition, 513 capped, 936 forward rate correlation, 514 conditional on no exercise, 31 xxii Index

continuation region, 33 averaging, see calibration, time discontinuity at expiry, 39 averaging duality, 36 averaging cash flow, 201, 720–721 early exercise boundary, 37 convexity adjustment, 720 early exercise premium, 36, 39, 42 averaging swap, see averaging cash exercise never optimal, 36 flow exercise policy, 30 see exercise region, 33 Bachelier model, Normal model see exercise value, 30 backbone, volatility smile, high contact condition, 38 backbone see hold value, 32 backward Kolmogorov equation, integral representation, 39, 41 Kolmogorov backward equation lower bound, see Monte Carlo, lower balance-guarantee swap, 898 see bound for American option band swap, flexi-swap marginal exercise value decomposi- “bang-bang”, 900 tion, 41 , 44 Monte Carlo, 158–165 Broadie adjustment for sampling frequency, see Monte Carlo, confidence interval for value, 164 sampling extremes, adjusting random tree, 164 barrier for sampling frequency stochastic mesh, 165 continuous barrier, 64 PDE jump condition, 34 discrete barrier, 66 perfect foresight bias, 160 importance sampling, 1074–1077 short-maturity asymptotics, 39 Markovian projection, see Marko- smooth pasting condition, 37, 38 vian projection, barrier option supermartingale, 31 Monte Carlo, see Monte Carlo, see upper bound, Monte Carlo, barrier option upper bound for American option on capped straddle, 937 see annuity mapping function, termi- one-touch, 939 nal swap rate model, annuity pathwise differentiation method, mapping function 1041–1044 see annuity measure, measure, annuity recursion, 1043 arbitrage opportunity, 8 payoff smoothing, see payoff arbitrage pricing, 11 smoothing, barrier option arithmetic put-call symmetry, 940 PDE jump condition, 66 Arrow-Debreu security, 21, 76, 78, 79, rebate, 64 456, 460, 1048 semi-static replication, 939 backward Kolmogorov equation, 456 step-down, 64 forward Kolmogorov equation, 456 step-up, 64 art of derivatives trading, 980 tube Monte Carlo, 1025 , 70 up-and-out, 44, 64, 66, 124, 126, , 920 1134 Monte Carlo, see Monte Carlo, basis point, 169 Asian option basis risk, see yield curve, basis risk PDE, see PDE, Asian option , 205, 1146 ATM backbone, see volatility smile, Black model, 922 ATM backbone displaced log-normal approximation, autocorrelation, see inter-temporal 1147 correlation model, 1145 Index xxiii

Monte Carlo, see Monte Carlo, Markov-functional model, 879 Asian option on basket quadratic Gaussian model, 886 slope of volatility smile, 1148 quasi-Gaussian model, 879, 886, model, 1149 889 BDT model, see Black-Derman-Toy representative swaption for model accreting Bermudan, 884 Bermudan cancelable swap, see representative swaption for Bermudan swaption; cancelable amortizing Bermudan, 883, 884 note super-replication, 888–892 Bermudan option, see Ameri- upper bound, 889, 890, 907 can/Bermudan option non-vanilla, see Bermudan swaption, Bermudan swaption, 207, 873–918 non-standard accreting, see Bermudan swaption, PDE jump condition, see Amer- non-standard ican/Bermudan option, PDE American, see American swaption jump condition amortizing, see Bermudan swaption, strike, 873 non-standard survival measure, 1047 approximating American swaption, vanilla, 878 894 zero-coupon, 892–893 bullet, see Bermudan swaption, Bermudan swaption calibration vanilla adjusters method, 955 carry, 906, 913 local projection method, 552, impact on exercise decision, 913 874–878 control variate, 1090 Gaussian short rate model, 875 exercise fee, 897 non-standard Bermudan, see exercise value, XXXVIII, 208, 873 Bermudan swaption, non- flexi-swap, see flexi-swap standard gamma-theta mismatch, 912 quadratic Gaussian model, 875 hold value, XXXVIII, 208 quasi-Gaussian model, 875 lockout, 207, 873 smile calibration, 876–878 mid-coupon, 895, 897–898 at-the-money, 876 no-call, see Bermudan swaption, exercise boundary, 877 lockout strike, 876 non-standard, 878–898 Bermudan swaption calibration by payoff matching, pathwise differentiation method, 882, 883 1044–1050 calibration by PVBP matching, forward induction, 1049–1050 882–884 performance, 1050 calibration by tenor matching, 881 survival density, 1048 calibration to basket, 885–887 survival measure, 1047 calibration to representative portfolio replication for hedging, 911 swaption, 882 Principal Components Analysis, 911 calibration to row of European robust hedging, 910–913 , 886 static hedging, 911 Gaussian short rate model, 886 Bermudan swaption valuation, 820–871 global calibration, 879, 881 control variate, 1086 Libor market model, 885 non-linear, 1089 local projection method, 879, 881 sampled at exercise time, 1087 lower bound, 891, 907 fast pricing, 914 xxiv Index

impact of forward volatilities, 874 Boltzman-Gibbs distribution, see impact of inter-temporal correlation, out-of-model adjustment, 552, 875 path re-weighting method, impact of mean reversion, 552, 874 Boltzman-Gibbs distribution impact of the number of factors, 875 Bond Market Association, see BMA Monte Carlo, 903–910 index exercise strategy, 904 box smoothing method, see payoff explanatory variables, 903 smoothing, box smoothing parametric lower bound, 904–910 break-even rate, see forward swap rate regression lower bound, 903 Broadie adjustment for sampling Bermudanality, 877 frequency of barriers, see Monte Bessel function of the first kind, 282 Carlo, sampling extremes, Bessel process, 281, 282 adjusting barrier for sampling best-of option, see MAX-option frequency best-of-calls option, 780 Brownian bridge, 125, 645, 646 BGM model, see Libor market model conditional moments, 129 Black model, XXXVIII, 22, 24, 202, Libor market model, see Libor 279, 283 market model valuation, Monte Asian option, see Asian option, Carlo, Brownian bridge Black model path construction, see Brownian basket option, see basket option, motion, path construction by Black model Brownian bridge , 24 sampling extremes, see Monte CMS spread, 774 Carlo, sampling extremes, with delta, 350, 696 Brownian bridge effects of volatility mis-specification, Brownian motion, 4 987 geometric, 16 Fourier integration, 329 Haar function decomposition, gamma-vega, 981 see Brownian motion, path log-likelihood ratio, 1060 construction by Brownian bridge moment-generating function, 329 Ito integral, see Ito integral PDE, 25 Karhunen-Loeve decomposition, stochastic interest rates, 28, 30 see Brownian motion, path strike-specific volatility, 696 construction by Principal time-dependent parameters, 27, Components 983–985 path construction, 106 vega, 696 path construction by Brownian use in calibration, 702 bridge, 128, 129 with dividends, 28 path construction by Principal Black shadow rate model, 450 Components, 130 Black-Derman-Toy model, 443–445 Stratonovich integral, see mean-fleeting, 445 Stratonovich integral short rate dynamics, 444 BSM model, see Black model Black-Karasinski model, 445 0 Black-Scholes model, see Black model C , XXXVIII 1 Black-Scholes-Merton model, see Black C , XXXVIII model C2, XXXVIII BMA index, 192, 265 Cn, XXXVIII BMA rate, 192 calibration, 299 Index xxv

calibration norm, 628–631 time-dependent, 304 fit, 632 effective parameter, 305 regularity, 632 volatility skew, 284 cold start, 631 characteristic function, 20 forward induction, 443, 456, 953 Cheyette model, see quasi-Gaussian Levenberg-Marquardt, 631 model local projection method, see local chi-square distribution, 100 projection method Monte Carlo, 100, 102 Markovian projection method, see non-central, see non-central Markovian projection chi-square distribution most likely path, 990 PDF, 100 stochastic optimization method, 953 chooser cap, see flexi-cap time averaging, 301, 307, 363, chooser swap, see flexi-swap 370–381, 548, 581, 666 CIR model, see Cox-Ingersol-Ross algorithm, 376–381 model non-zero correlation, 376 CLE, 213, 216, 626, 815–871, 873 skew, 373–374 accreting at coupon rate, 216, 868 volatility, 371–373 carry, 857, 906, 913 volatility of variance, 374–376 impact on exercise decision, 847, callable Libor exotic, see CLE 857 callable zero, see Bermudan swaption, definition, 820 zero-coupon exercise value, XXXVIII, 215, 820 cancelable note, 214, 827, 828 hold value, XXXVIII, 215, 820, 821 ATM, 858 lockout, 213 carry, 856, 913 marginal exercise value decomposi- cancelable swap, see cancelable note tion, 822 cap, 186, 202 multi-tranche, 217 caplet volatility from cap volatility, no-call, see CLE, lockout 704 optimal exercise, 822 interpolation, 705 single-rate, 862 precision norm, 705 smooth function of Monte Carlo relaxation, 706 path, 1029 smoothness norm, 706 snowball, 216, 870 splitting scheme, 706 CLE calibration, 815–820 digital, 203, 209 local projection method, 862–868 valuation formula, 202 calibration targets, 863 Capital Asset Pricing Model, 357 core swap rate analog, 865 capped floater, 209 local models, 864–865 Cauchy distribution, 98, 101 quadratic Gaussian model, 865 Monte Carlo, 98 quasi-Gaussian model, 864 certificate of deposit, 194 two-factor Gaussian model, 864 CEV model, 280–286 two-strike calibration, 865 attainability of zero, 280 vega, 867 displaced, 285 low-dimensional models, 862–868 European call option value, 282, 283 model choice, 819 explosion, 280 single-rate, 862–863 regularization, 284 to forward volatility, 819 relation to Bessel process, 281 CLE greeks, 1036–1040 strict supermartingale, 280 as sum of coupon greeks, 1037 xxvi Index

discontinuity in Monte Carlo, 1041 reuse exercise boundary, see freezing exercise boundary, 833, CLE greeks, freezing exercise 1039, 1040 boundary freezing exercise time, 1038–1040 ridge regression, see CLE regression, likelihood ratio method, see Tikhonov regularization likelihood ratio method robust implementation, 858–862 pathwise differentiation method, singular value decomposition, 104 1035–1040, 1058–1060 stabilization, 859 computational complexity, 1052 state variables, 848–849 forward induction, 1049–1050 Libor market model, 849 survival density, 1048 SVD decomposition, 860, 861 survival measure, 1047 connection to Tikhonov regulariza- perturbation method, 1040, 1059 tion, 861 computational complexity, 1053 Tikhonov regularization, 162, 255, portfolio replication for hedging, 911 859–861 recursion, 1036 connection to SVD, 861 source of noise, 1040 truncated SVD decomposition, 162, tube Monte Carlo, 1029 860, 861 CLE regression, 823–862 two-step, 857 automatic selection of regression upper bound, 837–848 variables, 855 alternative methods, 847 boundary optimization, 831 computational cost, 841 improvements to algorithm, cancelable note, 827–828 845–847 choice of regression variables, nested simulation algorithm, 848–854 837–847 decision only, 828–830 non-analytic exercise values, discrepancy principle, 859 843–845 excluding suboptimal points, 856 simulation within a simulation, see exercise value, 825–827 CLE regression, upper bound, explanatory variables, 850–854 nested simulation algorithm classification, 851 CLE valuation, 215, 820–871 CMS spread, 851 as cancelable note, 827 core swap rate, 851 boundary optimization, 831 stochastic volatility, 854 confidence interval for value, 842 with convexity, 852–854 control variate, see Bermudan general-to-specific approach, 856 swaption valuation, control generalized cross-validation, 859 variate L-curve method, 859 discontinuous function of Monte Libor market model, 849, 850 Carlo path, 1041 state variables, 849 duality, 836, 1093 lower bound, 831–833 multiplicative, 1093 perfect foresight bias, 832 duality gap, 839, 842, 908, 909 pseudo-inverse method, 860 in stochastic volatility models, 910 quadratic Gaussian model, 849 exercise policy consistency condi- quasi-Gaussian model, 849 tions, 833 regression operator, 824 fast pricing, 916 regression variables, 823 Hamilton-Jacobi-Bellman equation, rescaling, 861 821 Index xxvii

impact of forward volatility, 818 out-of-model adjustment, 963, 964 impact of inter-temporal correlation, quasi-Gaussian model, 728–729 863 replication method, 722–724 impact of volatility smile dynamics, stochastic volatility model, 738 819 swap-yield TSR model, 726 Libor market model, 824 vega hedging, see terminal swap lower bound, 834, 841, 845, 848 rate model, linear TSR model, by regression, see CLE regression, vega hedging lower bound hedging portfolio, 723 iterative improvement, 833 quanto, see quanto CMS iterative improvement by nested CMS cap, 207, 695 simulation, 835 impact of CMS convexity on quality test, 1060 volatility smile, 739 see LS method, CLE regression link to European swaptions, 739 Monte Carlo, 823–862, 903 CMS digital spread option, 789 optimal exercise policy, 833, 835, dimensionality reduction, 789 1039 CMS floor, 207 PDE, 868–871 CMS rate, 206 accreting at coupon rate, 868 distribution in forward measure, path-dependent, 868–871 734–737 similarity reduction, 869 CMS spread option, 210, 211, 619, 688, snowball, 870 763, 774 perfect foresight bias, 832 by integration, 775 policy fixing, 846 copula method, 774–782 recursion, 821 dimensionality reduction, 787 regression method, see CLE floating digital, 790 regression Gaussian copula, 775 tube Monte Carlo, 1029 correlation impact, 776 upper bound, 836–848 vega to swaptions, 776 cancelable note, 844 nested simulation algorithm, 839, implied copula, 779 908 implied correlation, 776 non-analytic exercise values, Libor market model, 617–619, 634, 843–845 690, 806 weighted coupon decomposition, 916 closed-form approximation, 808 CMS, 206 Libor market model calibration, 634 annuity to forward measure change, local volatility model, 1145 734–737 Margrabe formula, 810 convexity adjustment, 721–744 Markovian projection, 1145, 1149 annuity mapping function, see multi-stochastic volatility, see terminal swap rate model, multi-stochastic volatility model annuity mapping function non-standard gearing, 775, 789 correcting arbitrage, 732–733 dimensionality reduction, 789 density integration method, 736 Normal spread volatility, 774 impact of mean reversion, 733–734 one-dimensional integration, 787 impact of volatility smile, 733 out-of-model adjustment, 964, 966 impact on , 774 power Gaussian copula, 779 Libor market model, 729–731 quadratic Gaussian model, 808 linear TSR model, 726–728 closed-form approximations, 808 xxviii Index

risk management with one-factor proxy model, see control variate, model, 971 model-based stochastic volatility convexity adjustment correlation impact, 805 averaging swap, see Libor-with-delay, stochastic volatility de-correlation, convexity adjustment 962 CMS, see CMS, convexity adjust- stochastic volatility model, 1149 ment correlation impact, 803 futures, see ED future, convexity vega in Libor market model, 1116 adjustment CMS swap, 206, 695 Libor-in-arrears, see Libor-in- valuation formula, 207 arrears, convexity adjustment CMS-linked cash flow, 721–744 Libor-with-delay, see Libor-with- direct integration method, 734 delay, convexity adjustment replication method, 723 moment explosion, 759–762 coherent risk measure, see risk second moment, 759 measure, coherent copula, 768 collateral, 192, 266 Archimedean, 770 complementary Gamma function, 281 Monte Carlo, 798 complete market, 11 Clayton, 770 compounded rate, 200 conditional CDF, 790 conditional expected value, 19 Frechet bounds, 769 iterated conditional expectations, see Gaussian, 766 iterated conditional expectations CMS spread option, see CMS projection approximation, see spread option, Gaussian copula Markovian projection, con- integration, 787 ditional expected value by joint CDF, 767 projection joint PDF, 767, 775 constant elasticity of variance model, mixture, 772 see CEV model Monte Carlo, 797 constant maturity swap, see CMS Gumbel, 770, 771 swap implied, 779 contingent claim, see independence, 768 security mixture, 772 continuity correction, see payoff Monte Carlo, 798 smoothing, continuity correction perfect anti-dependence, 769 control variate, 146–149, 330, 652, 653, perfect dependence, 768 1077–1094 power Gaussian, 773, 778 adjusters method, 955 parameter impact, 779 construction from MC upper bound, product, 773 1093 Monte Carlo, 798 dynamic, 148, 653, 1090–1093 reflection, 771 regression-based, 1091 Monte Carlo, 798 efficiency, 147 Sklar’s theorem, 769 impact on risk stability, 1093 copula density, 770 instrument-based, 1086–1090 copula method, 766 model-based, 675, 1077–1086 CMS spread option, see CMS spread non-linear controls, 147–149 option, copula method path re-weighting method, 961 dimensionality reduction, 787–796 proxy Markov LM model, 1078 by conditioning, 791–795 Index xxix

by measure change, 795–796 bucketed interest rate deltas, 251, forward swaption straddle, 949 1045 integration, 784–796 forward rate, 253 inverse CDF caching, 785 Jacobian method, see risk sensitivi- singularities, 786 ties, Jacobian method limitations, 799–800 par-point, 251, 252, 256, 257, 993 mapping function, 793 parallel, 257 Monte Carlo, 797–799 with backbone, 1120–1122 observation lag, 782 delta hedge, 18 quanto options, 747 density process, 9 , 934 derivative security, 11 core correlations, see inter-temporal attainable, 11 correlation pricing, 11 core volatilities, 863, 874 diffusion, 4, 15 correlation extractor, see Libor market absorbing barrier, 281, 289 model, correlation extractor displaced, 285 correlation risk sensitivity, 1119 Feller boundary classification, 280 correlation smile, 776 Feller condition, 319 Cox-Ingersol-Ross model, 430 Fubini’s theorem, 407 multi-factor, 518 integration by parts, 120 see two-factor, 516 Ito integral, Ito integral Crank-Nicolson scheme, see PDE, Ito process, 4 Crank-Nicolson scheme local time, 26, 294 credit risk, 260, 975 Ornstein-Uhlenbeck process, 411 polynomial growth condition, 19 credit value adjustment, 266, 914 predictable process, 7 cross-currency , see floating- scale measure, 280 floating cross-currency basis SDE, 15 swap generator, 19 cross-currency basis swap spread, 262, linear, 16 265 locally deterministic, 172, 539 CRX basis swap, see floating-floating strong Markov, 15 cross-currency basis swap strong solution, 15 CRX spread, see cross-currency basis weak solution, 15 swap spread speed measure, 280 cumulant-generating function, 154 diffusion invariance principle, 14 curve cap, 211, 764 discount bond, XXXVIII, 23, 167 range accrual, see range accrual, valuation formula, 172 curve cap discount curve, see yield curve CVA, see credit value adjustment displaced CEV model, see CEV model, displaced date rolling convention, 224 displaced log-normal model, 285 day count convention, 223–226 basket option, 1147 30/360, 225 canonical form, 286 Actual/360, 224 explicit solution to SDE, 312 Actual/365.25, 224 Fourier integration, 328 day count fraction, see year fraction implied correlation, 809 deflator, 9 moment matching, 920 delta, 18, 132, 355, 980 moment-generating function, 329 xxx Index

time-dependent, 304 Eurodollar , see ED effective skew, 305 future explicit solution to SDE, 307 European call option, 24 range for process, 306 at-the-money, 24 Dupire local volatility, 1131 Fourier integration, 324 proof by Tanaka extension, 294, in-the-money, 24 1131 out-of-the-money, 24 duration, 246 probability density from, see volatil- DVF model, see local volatility model ity smile, probability density Dybvig parameterization, see short from rate model, Dybvig parameteriza- European digital call option, 60 tion European option Fourier integration, 326 early exercise, 30 European , 24 ED future, 168–170, 196–197, 695, at-the-money, 24 748–759 in-the-money, 24 convexity adjustment, 187, 197, out-of-the-money, 24 748–759 European swaption, 203, 695–703 from market inputs, 751 cash-settled, 205, 742–744 Gaussian HJM model, 186 payoff, 743 impact of volatility smile, 750, 756 put-call parity, 743 Libor market model, 751, 756 replication method, 742, 743 replication method, 751, 755 core swaptions, 422, 817 delivery arbitrage, 170 coterminal swaptions, see European futures rate, 169 swaption, core swaptions definition, 196 diagonal swaptions, see European instantaneous, 170, 172, 173 swaption, core swaptions martingale in risk-neutral measure, forward swaption straddle, see 172, 749 forward swaption straddle, 943 martingale in spot Libor measure, midcurve, 223 749 non-standard, see Bermudan simple, 169 swaption, non-standard to forward rate, 754, 758 Black formula, 887 mark to market, 169 physically-settled, 205 yield curve construction, 231, 992 SV model calibration, 701–702 ED futures contract, see ED future swap-settled, 205, 743 effective volatility swaption grid, 205, 701 local volatility model, see local swaption strip, 421 volatility model, effective tenor, 204 volatility valuation formula, 204 stochastic volatility model, see volatility cube, 696 stochastic volatility model, European-style option, 95 effective volatility replication method, 337 envelope theorem, 1038 valuation by volatility mixing, 339 Eonia, 193, 200 exchange market, 193 equivalent martingale measure, see Chicago Mercantile Exchange, 196 measure, equivalent martingale London International Financial Esscher transform, see exponential Futures and Options Exchange, twisting 196 Index xxxi

March`e`aTerme International de purely local bounds, 899 France, 196 “flip-flop”, 210 exotic swap, 205, 208, 209, 820, 951 floating digital, 790, 792 CMS spread, 764 dimensionality reduction, 790 CMS-based, 210 floating digital spread option, 790 digital CMS spread, 764 dimensionality reduction, 790 global cap, 219 floating-floating cross-currency basis global floor, 219 swap, 262, 264, 265 knock-out, 218 floating-floating single-currency basis Libor-based, 209 swap, 201, 268 multi-rate, 210, 764 floor, see cap path-dependent, 212 Fokker-Plank equation, see Kol- principal amount, 208 mogorov forward equation range accrual, see range accrual Fong-Vasicek model, 452–453, 515 snowball, 212 spread-based, 210 bond reconstitution formula, 452 structured coupon, 208–211 forward CMS straddle, 941, 944, 945 see expectations hypothesis, 173 swaption, forward swaption expected hedging P&L, 988 straddle see exponential distribution, 98 volatility, forward volatility Monte Carlo, 98 , 195 exponential integral, 334 forward Kolmogorov equation, see exponential twisting, 154 Kolmogorov forward equation extra state variable method, see PDE, forward Libor model, see Libor market path-dependent options model forward Libor rate, XXXVIII, 168, “The Fed Experiment”, 450 191, 192, 196 Federal funds future, 201 accrual end date, 224 Federal funds rate, 192, 200, 201, 266 accrual period, 224 effective, 192 accrual start date, 224 target, 192 martingale in forward measure, 174 Federal funds/Libor basis swap, 201, tenor, 168 266 variance by replication method, 756 see Feller condition, diffusion, Feller year fraction, see year fraction condition forward par rate, see forward swap Feynman-Kac solution, 21 rate FFT, see stochastic volatility model, , 24, 168 Fourier integration forward rate, 167 filtration, 3, 4 usual condition, 3 continuously compounded, flexi-cap, 71 XXXVIII, 168 flexi-swap, 898–903 instantaneous, XXXVIII, 169 decomposition into Bermudan simple, 168 swaptions, 899 tenor, 168 local projection method, 899 volatility hump, 416, 492 marginal exercise value decomposi- , see forward tion, 901 contract narrow band limit, 902 forward starting option, 222 PDE, 899, 901 forward swap rate, XXXVIII, 171, 199 xxxii Index

distribution in forward measure, fundamental theorem of arbitrage, 10 see CMS rate, distribution in fundamental theorem of derivatives forward measure trading, 987 expiry, 171 futures contract, see ED future fixing date, 171 futures rate, see ED future, futures linking forward and annuity measure, rate 735 fuzzy logic, see payoff smoothing, market-implied variance, 555 fuzzy logic martingale in swap measure, 178 FX rate, 179, 745, 746 non-standard, 879 dynamics in domestic risk-neutral decomposition, 880 measure, 180 tenor, 171 forward, 178 weighted average of Libor rates, 171, martingale in domestic forward 256 measure, 180 forward swaption straddle, 223, 945–950 Gˆateaux derivative, 253 copula method, 949 gamma, 980 relation to CMS spread option, 948 pathwise differentiation method, see triangulation, see forward volatility, pathwise differentiation method, triangulation gamma vanilla model, 946 payoff smoothing, 1019 vega exposure, 948 relationship to vega, 981 volatility, see forward volatility gamma distribution, 100 forward volatility, 222 Monte Carlo, 100, 102 connection to inter-temporal PDF, 100 correlations, see inter-temporal Gamma function, XXXVII correlation, connection to incomplete, see incomplete Gamma forward volatilities function hedging, 912 quick approximation, 1153 impact of rate correlation, 918 Gauss-Hermite quadrature, see impact of volatility smile, 945 quadrature, Gauss-Hermite Libor rate, see volatility, forward Gaussian copula, see copula, Gaussian volatility of Libor rate Gaussian distribution, XXXVII triangulation, 948 conditional distribution, 646 forward volatility derivative, 220, 222 cumulant-generating function, 154 forward swaption straddle, see imaginary mean, 796 forward swaption straddle inverse CDF, 99, 165 implied Normal volatility contract, linear transform, 103 223 measure change, 795 midcurve swaption, see European multi-dimensional PDF, 103 swaption, midcurve quadratic form, 522 volatility swap, see volatility swap moment-generating function, 522, forward yield, see forward rate 533 Fourier transform, 325 moments, 534 inverse, 325 Gaussian HJM model, 184–187 FRA, see forward contract caplet, 186 Frobenius norm, see matrix, Frobenius ED future convexity adjustment, see norm ED future, convexity adjustment, fundamental matrix, 484 Gaussian HJM model Index xxxiii

time-stationary, 416 correlation stationarity, 488 zero-coupon , 185 European swaption, 500–505 Gaussian multi-factor short rate model, European swaption by Jamshidian see Gaussian short rate model, decomposition, 503 multi-factor factors and loadings, see Gaussian Gaussian one-factor short rate model, short rate model, multi-factor, see Gaussian short rate model statistical approach Gaussian short rate model, 406, forward rate correlation, 488–489 413–429, 478–510 forward rate volatility, 482 as special case of affine model, 430 Gaussian swap rate approximation, Bermudan swaption, see Bermudan 504–505 swaption calibration, local loadings, 499 projection method, Gaussian mean reversion matrix diagonaliza- short rate model tion, 487–488 bond dynamics, 415 Monte Carlo, 508–509 bond reconstitution formula, 414 PDE, 510 efficient calculation, 415 rotations, 484 calibration, 421 separability, 478–485 bootstrap, 422 short rate dynamics, 479 calibration to yield curve, 414 short rate state distribution, 485, European swaption, 418, 421 509 Jamshidian decomposition, 418 short rate state dynamics, 479–485 fast pricing of Bermudan swaptions, short rate state dynamics, 914 integrated, 485, 509 forward rate dynamics, 413 single Brownian motion, 496 forward rate volatility, 413 statistical approach, 495–500 dynamics, 417 swap rate volatility, 505 humped volatility structure, 416 PDE, 423–425 in spot measure, 428 boundary conditions from PDE, in terminal measure, 428 424 mean reversion, see mean reversion short rate distribution, 426 mean reversion calibration, see short rate dynamics, 413 mean reversion calibration short rate state dynamics, 414, 425 Monte Carlo, 425–429 integrated, 425 approximate, 427 swap rate dynamics in annuity Euler scheme, 427 measure, 420 exact, 425 swap rate volatility, 420 other measures, 428 time-stationary, 416 multi-factor, 478–510 two-factor, 489–495 benchmark rate parameterization, bond reconstitution formula, 490, 506–508 500 benchmark rates, 506 CLE, see CLE calibration, local benchmark tenors, 506 projection method, two-factor bond reconstitution formula, 478, Gaussian model 481, 483 correlated Brownian motions, 490 bond volatility, 479 correlation stationarity, 491 calibration, 506 doubly mean-reverting form, 493 classic development, 485–488 European swaption by Jamshidian correlated Brownian motions, 489 decomposition, 500–504 xxxiv Index

forward rate correlation, 490–491 shadow delta, see volatility smile, forward rate dynamics, 490 shadow delta hedging forward rate volatility, 490–491, sub-replicate, 717 493, 494 super-replicate, 717, 979 short rate state conditional zero-beta, 357 distribution, 502 Hermite matrix, 270 short rate state correlation, 490 Heston model, see stochastic volatility short rate state dynamics, 490 model single Brownian motion, 495 HJM model, 181–189 volatility hump, 492–493 bond dynamics, 181 Gaussian two-factor short rate model, forward bond dynamics, 182 see Gaussian short rate model, forward rate dynamics, 182 two-factor Gaussian, see Gaussian HJM model generalized trigger product, 1074 Gaussian Markov, 187–189 importance sampling, 1074–1077 short rate dynamics, 188 pathwise differentiation method, log-normal, 189 1041–1044 Markovian, 405 payoff smoothing, 1074–1077 separable, 413 trigger variable, 1074 short rate dynamics, 183 see tube Monte Carlo, barrier stochastic basis, see HJM model, option, tube Monte Carlo two-curve Girsanov’s theorem, 12, 13 two-curve, 678–681 Gaussian distribution, 795 forward rate spread dynamics, 679 Gram-Charlier expansion, 368, 437 Gaussian basis spread, 681 see greeks, risk sensitivities index bond dynamics, 680 Green’s function, 20 index forward rate dynamics, 680 grid shifting, see payoff smoothing, index short rate dynamics, 680 grid shifting quanto correction, 681 GSR model, see Gaussian short rate Ho-Lee model, 406–410 model bond dynamics, 409 Gy¨ongy theorem, see Markovian bond reconstitution formula, 408 projection, Gy¨ongy theorem calibration to yield curve, 407 drawbacks, 410 H2 , 5 forward rate dynamics, 409 Hagan and Woodward parameteriza- short rate dynamics, 408 see tion, short rate model, Hagan hybrid differentiation method, 1061 and Woodward parameterization hat smoothing method, see payoff smoothing, hat smoothing implied volatility, see volatility, Heath-Jarrow-Morton model, see HJM implied model importance sampling, 146, 149–158, hedge, 251 1063–1077 best hedging strategy, 355 application to payoff smoothing, beta, 357 1067 minimum variance, 355–357 barrier option, see barrier option, model-independent, 716 importance sampling semi-static, see replication method, density formulation, 149 semi-static efficiency, 151 Index xxxv

generalized trigger product, see mean reversion calibration to, see generalized trigger product, mean reversion calibration, to importance sampling inter-temporal correlations least-squares, 154 interbank money market, 192 likelihood ratio, 150, 153, 155 International Swaps and Derivatives rare events, 154 Association, 192, 266 approximately optimal mean shift intrinsic value, 27 in multi-variate case, 158 inverse floater, 209 asymptotic optimality, 158 iterated conditional expectations, 176 Ito integral, 4, 5 efficiency, 156 Ito isometry, 5 minimal variance, 155 Ito’s lemma, 6 multi-variate, 156 Ito-Taylor expansion, 118 SDE, 151–154 see short rate model, short rate Jacobian, see risk sensitivities, model, importance sampling Jacobian method survival measure, 1067 Jamshidian decomposition simulation under, 1072, 1074, 1076 American/Bermudan option, see TARN, see TARN, importance American/Bermudan option, sampling Jamshidian decomposition incomplete Gamma function, XXXVII, European swaption, see Gaussian 281 short rate model, European swap- index, 206 tion, Jamshidian decomposition index option, see basket option infinitesimal operator of SDE, see Kolmogorov backward equation, 19, 20 diffusion, SDE, generator Kolmogorov forward equation, 20, 386, infinitesimal perturbation analysis, 136 457, 1048 information theory, 957 correct boundary conditions, 386 instantaneous futures rate, see ED fu- discrete consistency with backward ture, futures rate, instantaneous equation, 458 Kullback-Leibler relative entropy, 957 integration by parts for diffusion kurtosis, 375 process, see diffusion, integration by parts 1 L , XXXVIII, 4 inter-temporal correlation, 422, 552, 2 L , XXXVIII, 4 818, 863, 874 ladder, 985 connection to forward volatilities, ladder swap, see ratchet swap 818 Lagrange basis functions, see PDE, hedging, 875, 912 Lagrange basis; payoff smoothing, impact of mean reversion, 552 Lagrange basis impact of volatility smile, 945 Lagrange multiplier, 249, 958 impact on Bermudan swaption, least squares method, see CLE see Bermudan swaption valua- regression tion, impact of inter-temporal LIA, see Libor-in-arrears correlation Libor curve, see yield curve impact on CLEs, see CLE valua- Libor market model, 449, 589–692, tion, impact of inter-temporal 729, 866, 910 correlation annuity mapping function, 730, 731 impact on TARNs, 929 asset-based adjustment, 963 xxxvi Index

back stub, 655–660 link to HJM, 595 arbitrage-free, 657–659 local volatility, 596–598 from Gaussian model, 659–660 CEV, 597 simple, 656–657 displaced log-normal, 597 choosing number of factors, 612 existence and uniqueness, 597, 598 CLE, 819 LCEV, 597 CMS convexity adjustment, 964 log-normal, 597 correlation extractor, 863 Markov, 674–675, 1078–1086 deflated bond dynamics, 649 as control variate, 1084 delta with backbone, 1120–1122 Brownian bridge, 1079 drift approximation, 644 calibration, 1082 Brownian bridge, 1079 one-factor, 1079 drift freezing, 1052 one-factor reconstitution formula, exercise boundary, 910 1080 exercise strategy, 907 separable volatility, 1080 expected value of Libor rate in two-factor, 1081 annuity measure, 669 two-factor reconstitution formula, front stub, 660–666 1081 exogenous volatility, 661–664 Markovian projection, 666, 668, from Gaussian model, 665–666 1139 simple interpolation, 664–665 model risk, 627 zero volatility, 660–661 multi-stochastic volatility, 688–692, in hybrid measure, 640 962 index function, see tenor structure, caplet, 690 index function CMS spread option, 690 Libor rate correlation, 601–612, 757 European swaption, 690 correlation PCA, 609 moment-generating function, 690 covariance PCA, 624 Musiela parameterization, 602 historical estimation, 604 pathwise derivative majorization, 611 forward Libor rate, 1051 parametric form, 606, 607 forward swap rate, 1055 PCA, 602–604 numeraire, 1054 poor man’s correlation PCA, 612 structured coupon, 1055 regularization, 608 stub bond, 1054 Libor rate dynamics, 591–601 pathwise differentiation method, annuity measure, 731 1051–1058 in forward measures, 592–593 computational complexity, 1052 in hybrid measure, 595 PCA, see Principal Components in spot measure, 594 Analysis in terminal measure, 594, 639 portfolio replication, 912 Libor rate inter-temporal correlation, stochastic basis, see Libor market 757 model, two-curve Libor rate volatility stochastic variance dynamics, 688 from volatility norm, 623–625 stochastic volatility, 599–601 functional form, 620 moment-generating function, 687 grid-based, 620–621 non-zero correlation, 686 interpolation, 622–623 stub volatility, 662, 666 Libor rate volatility link to HJM swap rate correlation, 618–619 forward rate volatility, 596 swap rate dynamics, 615, 667 Index xxxvii

approximate, 616 antithetic variates, 652 time-stationary, 621 Brownian bridge, 645 tool to extract forward volatility, choice of numeraire, 640 819 control variate, 652 two-curve, 682–686 discretization bias, 637 deterministic spread, 685 Euler scheme, 636 European swaption, 684 front stub, 662 Libor rate dynamics, 683 high-order schemes, 648 Monte Carlo, 684 importance sampling, 653 swap rate dynamics, 684 lagging predictor-corrector, 642 vega, see vega, Libor market model large time steps, 639, 644–647 Libor market model calibration, log-Euler scheme, 636 620–635 martingale discretization, 648–651 algorithm, 631, 634, 674 Milstein scheme, 648 bootstrap, 633 predictor-corrector, 641, 642, 645, for vega, 1111 651 cascade, see Libor market model survival measure, 1072, 1075 calibration, bootstrap two-curve, 684 choice of instruments, 625 variance reduction, 651–653 effective skew, 670 multi-rate vanilla derivative, 806 effective volatility, 669 PDE, see Libor market model, global, 626 Markov grid-based, see Libor market model TARN, see TARN, Libor market calibration, global model local, 626 volatility swap, see volatility swap, objective function, 628 Libor market model PCA, 624 Libor rate, see forward Libor rate row-by-row, 631, 632 Libor-in-arrears, 200, 714–717 to spread options, 633, 806 convexity adjustment, 715 volatility skew, 635 replication method, 716 volatility smile, 672 sub-replicating portfolio, 717 Libor market model valuation super-replicating portfolio, 717 Bermudan swaption, see Bermudan Libor-with-delay, 717–721 swaption valuation, Monte Carlo convexity adjustment, 718 caplet, 613 Libor market model, 718, 720 CLE, see CLE valuation, Libor quasi-Gaussian model, 718, 719 market model replication method, 718, 720 CMS convexity adjustment, see swap-yield TSR model, 718 CMS, convexity adjustment, likelihood ratio method, 139–142, Libor market model 1060–1061 CMS spread option, see CMS spread discontinuous payoff, 138 option, Libor market model exploding variance, 1061 curve interpolation, 655–666 for Euler scheme, 141–142 European swaption, 614, 616, 666 for Milstein scheme, 142 Libor-with-delay, see Libor-with- log-likelihood ratio, 140 delay, Libor market model score function, 140 Monte Carlo, 635 vega, 1124 analysis of computational effort, linear regression, 146 637 Lipschitz function, 137 xxxviii Index

LM model, see Libor market model quadratic volatility, see quadratic local projection method, 558, 862, 863, volatility model 953, 1097 range-bound, 287 Bermudan swaption, see Bermudan small-noise expansion, see volatility, swaption calibration, local small-noise expansion projection method smile dynamics, 279, 350, 352 CLE, see CLE calibration, local time-dependent, 299–312 projection method separable, 300 non-standard Bermudan swap- log-normal distribution, XXXVII, 16 tion, see Bermudan swaption, moment matching, see moment non-standard, local projection matching method moments, 16 TARN, see TARN, local projection Monte Carlo, 101 method Longstaff-Schwartz method, see CLE volatility swap, see volatility swap, regression local projection method Longstaff-Schwartz model, 516–517 local stochastic volatility model, 316, bond reconstitution formula, 516 1137–1145 , 124 see calibration, see Markovian projec- Monte Carlo, Monte Carlo, tion, LSV calibration lookback option see Markovian projection, see Marko- LS method, CLE regression see vian projection, LSV calibration LSV model, local stochastic local time, see diffusion, local time volatility model see local volatility model, 277–312 LVF model, local volatility model approximation with displaced Malliavin calculus, 142, 1042, 1060 log-normal model, 286 Margrabe formula for spread option, asymptotic expansion, 295–299 810 basket option, see Markovian mark-to-model, 816 projection, basket option in LV Markov process, 15 model see see Feynman-Kac theorem, CEV, CEV model Feynman-Kac solution see displaced log-normal, displaced strong, 15 log-normal model transition density, 20 effective convexity, 307–312 Markov-functional model, 470–476 effective skew, 301–312 calibration to yield curve, 473 effective volatility, 301 criticism, 476 expansion around displaced Libor parameterization, 471 log-normal model, 296 log-normal, 472 expansion around Gaussian model, no-arbitrage condition, 471 298 non-standard Bermudan swaption, forward equation for call options, 879 293 numeraire, 470 PDE, 292–295 numeraire mapping, 470 simultaneous for multiple Libor parameterization, 471 parameters, 293 non-parametric, 474 space discretization, 292 swap parameterization, 474 transform to constant diffusion PDE, 475 coefficient, 88, 292 state process, 470 Index xxxix

swap parameterization, 473 quadratic volatility, see quadratic transition density, 470 volatility model, strict super- Markovian projection, 803, 1129–1156 martingale average option, 1133 SV model, see SV model with barrier option, 1134 general variance process, strict basket option in LV model, supermartingale 1145–1148 matrix basket option in SV model, exponential, 484 1149–1152 Frobenius norm, 105, 608, 609, 624, CMS spread option, 1145 625, 849 conditional expected value by infinity norm, 53 Gaussian approximation, positive semi-definite, 103 1134–1135 Cholesky decomposition, 103 conditional expected value by rank-deficient, 106 projection, 725, 1136–1137 spectral norm, 53 displaced Heston model, 1149, 1151 stiffness, 1111 non-perturbative approximation, tri-diagonal, 47 1151 MAX-option, 906 displaced log-normal model, 1136, mean reversion, 316, 411, 550, 571 1146 effects, 550–552 Gy¨ongy theorem, 1130 inter-temporal correlation, 552 LSV calibration, 1139–1145 swaption volatility ratio, 551 mean reversion calibration, 550–558, mapping function, 1142 571 proxy model, 1143–1145 to inter-temporal correlations, quadratic volatility model, 1137, 555–557 1148 to row of European swaptions, 553, quasi-Gaussian model, see quasi- 886 Gaussian model, Markovian to volatility ratios, 552–555 projection mean-reverting square-root process, spread option, 1151 see square-root process stochastic volatility model, 1138 measure, XXXVII martingale, 5 absolutely continuous, 1067 Doob-Meyer decomposition, 35 annuity, 178, 204 exponential, 12 change of numeraire, see numeraire, Doleans exponential, XXXVII, 12 change of numeraire local, 5 domestic, 744 bounded, 288 equivalent, 9, 1067 martingale representation theorem, equivalent martingale, 8, 9, 14, 171 6 foreign, 744 Novikov condition, 12 hybrid, 176 optional sampling theorem, 35 local martingale, 10 Snell envelope, 31, 821 risk-neutral, XXXVII, 23, 172 square-integrable, 5 domestic and foreign, 179, 180 stopping time, see stopping time spot, XXXVII, 175 submartingale, 5 survival density, 1047 supermartingale, 5, 360 survival for Bermudan swaption, CEV, see CEV model, strict see Bermudan swaption, survival supermartingale measure xl Index

survival in importance sampling, see region of stability, 111 importance sampling, survival weak convergence order, 111 measure Euler-Maruyama scheme, see Monte T-forward, XXXVII, 29, 174 Carlo, Euler scheme domestic and foreign, 180 Heun scheme, 116 terminal, 176 higher-order schemes, 116 min-max volatility swap, 222, 938 implicit Euler scheme, 113 capped, 940 region of stability, 114 semi-static replication, 939 implicit Milstein scheme, 390 moment explosion, 323, 343, 344, 361, log-Euler scheme, 112, 113 759, 760 lookback option, 125 impact on convexity adjustment, see low-discrepancy sequence, see convexity adjustment, moment Monte Carlo, random number explosion generation, quasi-random SABR model, see SABR model, lower bound for American option, moment explosion 34, 35, 164 stochastic volatility model, see parametric, 159, 161 stochastic volatility model, regression-based, 161 moment explosion mean-square error, 123 SV model with general variance Milstein scheme, 119, 121 process, see SV model with multi-dimensional, 121 general variance process, moment modified trapezoidal scheme, see explosion Monte Carlo, Heun scheme moment matching, 887, 919–923 optimal root-mean-square error, 123 Asian option, 920 perfect foresight bias, see Ameri- basket option, 922 can/Bermudan option, perfect moment-generating function, 13 foresight bias Monte Carlo, 95–165 predictor-corrector, 115, 116 A-stable scheme, 110 convergence order, 116 Asian option, 107 random number generation, 97 Asian option on basket, 107 acceptance-rejection method, average rate option, see Monte 99–101 Carlo, Asian option Box-Muller method for Gaussian barrier option, 124–128 distribution, 99 adjusting barrier for sampling composition method, 101–102 frequency, 128 conditional Gaussian, 1066 double-barrier knock-out, 124 correlated Gaussian, 103 bias, 122 correlated Gaussian by Cholesky bias/standard error trade-off, 123 decomposition, 103 Brownian motion, see Brownian correlated Gaussian by eigenvalue motion decomposition, 104 calibration by stochastic optimiza- inverse transform method, 98 tion method, 953 linear congruential generator, 97 central limit theorem, 96 Marsaglia polar method for convergence rate, 97 Gaussian distribution, 99 discretization bias, 426 Mersenne twister, 98 efficiency, 144 period, 98 Euler scheme, 110, 111 pseudo-random, 97, 130 linear SDE, 112 quasi-random, 129 Index xli

Sobol, 129 non-central chi-square distribution, region of stability, 110 284 Richardson extrapolation, 122, 468 asymptotics, 392 sample mean, 96 CDF, 102, 319 sampling extremes, 124–128 in CEV model, 283 adjusting barrier for sampling in delta-gamma VaR/cVaR, 998 frequency, 128, 937, 970 in LS model, 517 with Brownian bridge, 125 two-dimensional, 517 SDE discretization, 108 Normal model, XXXVIII, 283 CMS spread, 774 second-order scheme, 119, 121 vega to swaptions, 775 seed, 97 numeraire, 10, 171 standard error, 97, 122 change of numeraire, 12 for digital option, 133 Girsanov’s theorem, see Girsanov’s for greeks, 132, 135 theorem strong convergence order, 111 discrete money market account, strong law of large numbers, 96 XXXVIII, 175 strongly consistent, 109 money market account, XXXVIII, third-order scheme, 468 22, 28, 172 upper bound for American option, 34–36, 163, 164 OIS, see overnight index swap variance reduction, see variance one-dimensional integral for spread reduction option, 787 weak convergence, 109 operator calculus, 998–999 weak convergence order, 110 OTC market, see over-the-counter weakly consistent, 109 market most likely path, see volatility, implied, out-of-model adjustment, 951–971 most likely path approximation adjusters method, 954–956 multi-rate vanilla derivative, 763–813 algorithm, 955 copula method, see copula method as control variate, 955 volatility adjustment, 956 Libor market model, 807 asset-based adjustment, 963–964 observation lag, 782 CMS spread option, 964 stochastic volatility, see multi- coupon calibration, 952–954 stochastic volatility model delta-adjustment method, 956 term structure models, 806 extended calibration, 953 multi-stochastic volatility model, fee adjustment method, 967–969 800–806, 1149 additive, 968 correlation impact, 803 blended, 968 measure change by CMS caplet impact on derivatives, 968 calibration, 802 multiplicative, 968 measure change by drift adjustment, issues, 961, 964 801 mapping function adjustment, 965 Monte Carlo market adjustment, 965 Quadratic-Exponential scheme, path re-weighting method, 956–961 803 as control variate, 961 multi-rate vanilla derivative, Boltzman-Gibbs distribution, 959 800–806 Boltzman-Gibbs weights, 959 multi-tranche, see CLE, multi-tranche dual, 961 xlii Index

inappropriate use, 958 CLE, see CLE greeks, pathwise partition function, 958 differentiation method risk sensitivities, 961 computational complexity, 1052, PDE for coupon values, 953 1053 proxy model method, 961 discontinuous payoff, 1042, 1061 spread adjustment method, 966 European option, 1054 strike adjustment method, 969–971 gamma, 1050, 1056 impact on derivatives, 970 generalized trigger product, see over-the-counter market, 193 generalized trigger product, overhedge, 1023 pathwise differentiation method overlay curve, see yield curve, overlay Libor market model, see Li- curve bor market model, pathwise overnight index swap, 193, 200, 266 differentiation method money market account, 1046 Monte Carlo models, 1051–1060 P&L, 696, 991–995 pathwise delta approximation, 1059 P&L analysis, 986 see PDE models, 1044–1050 P&L attribution, P&L explain sensitivity path generation, 138–139 P&L explain, 993–995 TARN, see TARN, pathwise see bump-and-do-not-reset explain, differentiation method P&L explain, waterfall explain vega, 1050, 1056 bump-and-reset explain, 994–995 payoff smoothing, 1001–1034 waterfall explain, 993–994 adaptive integration, 1006 see P&L explanation, P&L explain adding singularity to grid, 78, 1007 P&L of hedged book, 987–990 barrier option, 1074–1077 see P&L predict, P&L prediction benefits, 1012 analysis Bermudan swaption, see CLE greeks, P&L prediction analysis, 258, 991–993 tube Monte Carlo first-order, 991 box smoothing, 1015–1018 second-order, 991 multiple dimensions, 1020 unpredicted P&L, 991 on discrete grid, 1015 par rate, see forward swap rate by importance sampling, 1065–1077 parameter averaging, see calibration, CLE, see CLE greeks, tube Monte time averaging Carlo partial differential equation, see PDE continuity correction, 59, 1012 partition function, 958 fuzzy logic, 1028 pathwise delta approximation, see gamma, 1019 pathwise differentiation method, grid shifting, 1007 pathwise delta approximation hat smoothing, 1019 pathwise differentiation method, integration, 1012 135–139, 1035–1060 Lagrange basis, 59, 1019 adjoint method, 1056 locality, 1019 computational complexity, 1053, Monte Carlo, 1022–1030 1057 moving average, 1012, 1013 barrier option, see barrier option, choice of window, 1014 pathwise differentiation method multiple dimensions, 1019–1022 Bermudan swaption, see Bermudan box smoothing, 1020 swaption greeks, pathwise dominant dimension, 1022 differentiation method one dimension, 1014 Index xliii

partial analytical integration, 76–78, extra state variable method, see 1010 PDE, path-dependent options partial coupons, 1028 for implied volatility, see volatility, PDE, 1012 implied, PDE for piecewise smooth function on a grid, forward equation, see Kolmogorov 1016 forward equation singularity removal, 1009 fully implicit scheme, 50 TARN, see TARN, payoff smoothing; greeks off grid, 1005 TARN, tube Monte Carlo L-stable scheme, 55 tube Monte Carlo, see tube Monte Lagrange basis, 58, 59 Carlo Lax equivalence theorem, 56 PCA, see Principal Components local volatility model, see local Analysis volatility model, PDE PDE, 18, 43–93 mesh refinement, 73, 79 A-stable scheme, 55 equidistant blocks, 74 ADI scheme, 43, 82–85 non-equidistant, 75 boundary conditions, 85 multi-dimensional, 92 Asian option, 70 multi-exercise, 71 backward induction, 51 multi-level time-stepping, 58 Black-Scholes, see Black model, non-equidistant discretization, 56 PDE Nyquist frequency, 59 boundary conditions odd-even effect, 59 for barrier options, 64 operator splitting, 82 orthogonalization, 86 from PDE itself, 385, 424 drawbacks, 88 linear at boundary, 48 partial analytical integration, log-linear at boundary, 48 see payoff smoothing, partial Cauchy problem, 18, 44 analytical integration centering, 561 path-dependent options, 69, 71, 868, conditional stability, 55 870, 896, 899, 932, 934 consistent scheme, 56 Peaceman-Rachford scheme, 84 convection-dominated, 61–64 boundary conditions, 85 convergent scheme, 56 predictor-corrector scheme, 89–92 coupon-paying, 67 quantization error, 59 see Craig-Sneyd scheme, PDE, Rannacher stepping, 58–61, 67, 457 predictor-corrector scheme semi-Lagrangian methods, 64 Crank-Nicolson scheme, 50 Shannon Sampling Theorem, 59 American options, 69 similarity reduction, 71 not strongly A-stable, 55 sinh transform, 384 oscillations, 55, 58 smoothing, 58–61 Dirichlet problem, 44, 64 continuity correction, 59 space discretization, 46 grid dimensioning, 1002 dividends, 67, 68 grid shifting, 60, 1002 domain truncation, 44 space discretization, 45 stability of greeks, 1002 stable scheme, 53 Douglas-Rachford scheme, 85, 91 strongly A-stable scheme, 55 boundary conditions, 85 time discretization, 49 early exercise, 69 theta scheme, 50 exponentially fitted schemes, 63 two-dimensional, 80 xliv Index

two-dimensional with mixed bond reconstitution formula, 520 derivatives, 86, 89 calibration, 531–532 upwinding, 62 multi-pass bootstrap, 531 variable transform, 44 CLE, see CLE calibration, local von Neumann method, 53–56 projection method, quadratic amplification factor, 54 Gaussian model stability criterion, 54 CMS spread option, see CMS spread well-posed, 56 option, quadratic Gaussian Poisson distribution, 102 model portfolio replication, see Bermudan curve factor, 523 swaption greeks, portfolio European swaption, 526–531 replication for hedging approximations, 528 power Gaussian copula, see copula, exact, 527 power Gaussian Fourier integration, 529 predictor-corrector, 89, 115, 382, 641 rank-2 approximation, 530 Monte Carlo, see Monte Carlo, Fourier integration, 530 predictor-corrector mean-reverting state variables, 519 PDE, see PDE, predictor-corrector moment-generating function, 529 scheme Monte Carlo, 533 present value of a basis point, see one-factor, 441 swap, annuity parameterization, 523–526 principal component, 105 PDE, 533 Principal Components Analysis, 105, quadratic approximation to swap 106, 498, 602–604 rate, 529 principal factor, 105 short rate, 519 product integral, 484 short rate in SV form, 525 Profit-And-Loss, see P&L short rate state distribution pseudo-Gaussian model, see quasi- in annuity measure, 526 Gaussian model in forward measure, 521 pseudo-random number generator, see short rate state dynamics, 441, 519 Monte Carlo, random number in forward measure, 521 generation, pseudo-random in annuity measure, 526 put-call parity, 24 smile generation, 523–524 PVBP, see swap, annuity spanned stochastic volatility, 523, 532 QG model, see quadratic Gaussian TARN, see TARN, local projection model method, quadratic Gaussian qG model, see quasi-Gaussian model model quadratic covariation, XXXVII, 7 volatility factor, 523 quadratic Gaussian model, 441, volatility smile, 531 518–533 volatility swap, see volatility swap, as affine model, 519 quadratic Gaussian model benchmark rate parameterization, quadratic variation, XXXVII, 7 525 quadratic volatility model, 287–291 Bermudan swaption, see Bermudan European call option value, 290 swaption calibration, local European put option value, 290, 291 projection method, quadratic Markovian projection, 1137 Gaussian model measure change, 289 bond dynamics, 521 small-noise expansion, 308 Index xlv

smile dynamics, 350 mean reversion calibration, see strict supermartingale, 288 mean reversion calibration time-dependent, 308 Monte Carlo, 563 Quadratic-Exponential scheme, see Euler scheme, 563 square-root process, Monte Carlo, multi-factor, 572–583 Quadratic-Exponential scheme benchmark rate correlations, 582 multi-dimensional, see multi- benchmark rate parameterization, stochastic volatility model, 574 Monte Carlo, Quadratic- bond reconstitution formula, 574 Exponential scheme calibration to spread options, 582 quadrature, 531, 786 correlation smile, 582 Gauss-Hermite, 531, 787 loadings, 582 Gauss-Legendre, 786 local volatility, 574 Gauss-Lobatto, 786 Monte Carlo, 583 quanto CMS, 744–748 PDE, 582 annuity mapping function, 748 short rate state distribution in convexity adjustment, 747–748 annuity measure, 577 copula method, 747 short rate state dynamics, 573 quanto adjustment, 745 stochastic volatility, 574–583 replication method, 746 swap rate dynamics, 576–581 quasi-Gaussian model, 537–587 swap rate dynamics by Markovian see Bermudan swaption, Bermudan projection, 577 swaption calibration, local pro- one-factor local volatility, 539 jection method, quasi-Gaussian short rate state dynamics, 539 model PDE, 560–563 bond reconstitution formula, 538 convection-dominated, 561 calibration, 581 domain truncation, 562 CEV local volatility, 545 space discretization, 561 CLE, see CLE calibration, local pro- short rate state distribution, 559 jection method, quasi-Gaussian short rate state dynamics, 538 model CMS convexity adjustment, see in annuity measure, 542, 543 CMS, convexity adjustment, in forward measure, 583 quasi-Gaussian model single-state approximation, 563–567 density approximation, 583 small-time asymptotics, 559 direct integration, 558, 583 stochastic volatility, 567–572 Libor-with-delay, see Libor-with- bond reconstitution formula, 568 delay, quasi-Gaussian model calibration, 570–571 linear local volatility, 545–548 Monte Carlo, 572 calibration, 548 non-zero correlation, 572 European swaption, 547 PDE, 572 for swaption strip, 547 swap rate dynamics, 568–570 swap rate dynamics, 546 unspanned, 568 swap rate inter-temporal correla- swap rate dynamics, 540–545, 549 tion, 555 approximate, 541–545 swap rate variance ratio, 553 approximate linear, 542 Markovian projection, 541, 564, 577, approximate quadratic, 545 1139 swap rate variance, 544 mean reversion, see mean reversion swap rate volatility, 540 xlvi Index

TARN, see TARN, local projection running maximum, 124 method, quasi-Gaussian model running minimum, 124 volatility swap, see volatility swap, quasi-Gaussian model SABR model, 343–345, 357, 951, 1121 ad-hoc improvements, 703 Radon-Nikodym derivative, 9, 1067 density tail, 760 range accrual, 211 moment explosion, 344 CMS, 211 volatility smile expansion, 345 CMS spread, 211, 764 SALI tree, see tree, SALI curve cap, 212, 764 sausage Monte Carlo, see tube Monte dual, 212, 764 Carlo floating, 764 SDE, see diffusion, SDE product-of-ranges, 212 SDE discretization, see Monte Carlo, ratchet swap, 212 SDE discretization relative entropy, 957 Sharpe ratio, 22 replication method, 337, 722 shifted log-normal model, see displaced CMS, see CMS, convexity adjust- log-normal model ment, replication method short rate, 169 European option, see European-style option, replication method short rate model, 172 see Libor-in-arrears, see Libor-in- affine, affine short rate model see arrears, replication method affine one-factor, affine short Libor-with-delay, see Libor-with- rate model, one-factor see delay, replication method Black-Derman-Toy, Black- semi-static, 939 Derman-Toy model reserve, 986 calibration to yield curve, 455 rho, 980 forward induction, 456 Riccati, 364 forward-from-backward induction, Riemann zeta function, 128 458 see risk limit, 986 Cox-Ingersol-Ross, Cox-Ingersol- risk measure, 996 Ross model coherent, 996 Dybvig parameterization, 461–463, risk sensitivities, 1093 466 common definitions, 980 HJM representation, 462 delta, see delta econometric, 449 grid dimensioning for stability, 1002 empirical estimation, 449 grid shifting for stability, 1002 forward volatility impact on Jacobian method, 254–258, 985, 986, Bermudan swaption, 876 1105, 1106, 1111, 1118, 1119, Gaussian approximation, 1064 1121 Gaussian model for basis spread, off PDE grid, 1005 681 perturbation approach, 1050 Gaussian short rate, see Gaussian vega, see vega short rate model root search, 99 Hagan and Woodward parameteriza- Newton-Raphson method, 99, 116, tion, 463–466 235 Ho-Lee, see Ho-Lee model secant method, 235 importance sampling, 1063–1065 Runge-Kutta method, 116, 365, 432, log-normal, 443–449 434, 514 issues, 445 Index xlvii

Sandmann-Sondermann transform, smile vega, see vega, smile vega 446 snowball, see CLE, snowball Monte Carlo, 467–469 snowbear, 213 Euler scheme, 467 snowrange, 213 Milstein scheme, 467 snowstorm, 213 payoff construction issues, 468 Sonia, 193, 200 SDE discretization, 467 spline, 230, 270–275 variance reduction, 468 Catmull-Rom, 238, 240, 271, 272 multi-factor, 477 cubic C2, 273–274 path independence, 444 cubic smoothing, 248 PDE, 454–455 exponential tension spline, 243 domain truncation, 454 Hermite cubic, 238, 270–273 power-type, 449 interpolating, 248 quadratic Gaussian, see quadratic Kochanek-Bartels, 272 Gaussian model least-squares regression, 248 quasi-Gaussian, see quasi-Gaussian natural, 241 model natural cubic, 273 time-stationary, 416 shape preserving, 275 volatility calibration, 459–461 smoothing, 234 multi-pass bootstrap, 461 TCB, see spline, Kochanek-Bartels shout option, 935 tension, 240, 243, 244, 246, 247, 250, on capped coupon, 935 272, 274–275 optimal stopping time, 936 convergence to piecewise linear, similarity reduction, 71, 869 275 CLE, see CLE valuation, PDE, tension factor, 243 similarity reduction spot Libor measure, see measure, spot PDE, see PDE, similarity reduction spot rate, see short rate single-rate vanilla derivative, 695–762 square-root process, 315 approximately single-rate, 707 E(√z), 1153, 1155 cap, see cap basic properties, 318–320 CMS cap, see CMS cap boundary behavior, 319 CMS floor, see CMS floor conditional CDF, 319 CMS swap, see CMS swap conditional moments, 319 ED future, see ED future Feller condition, 319 European swaption, see European moment-generating function, 322, swaption 342, 364, 372 futures contract, see ED future time-dependent parameters, 364 Libor-in-arrears, see Libor-in-arrears moments, 375 Libor-with-delay, see Libor-with- Monte Carlo, 388–394 delay Euler scheme, 389 range accrual, see range accrual exact simulation, 388 singular value, 860 full truncation scheme, 389 singular value decomposition, see CLE higher-order schemes, 389 regression, SVD decomposition log-normal approximation, 390 truncated, see CLE regression, moment-matching schemes, 390 truncated SVD decomposition Quadratic-Exponential scheme, singularity removal, see payoff 392, 394 smoothing, singularity removal truncated Gaussian scheme, 391 skew vega, see vega, skew vega multi-dimensional, 1152 xlviii Index

PDF, 1153, 1156 level parameter, 317 stationary distribution, 320, 383 link between forward and annuity static replication, 210, 717 measures, 739 CMS, see CMS, convexity adjust- LSV, see local stochastic volatility ment, replication method model European option, see European-style martingale property, 320 option, replication method mean reversion speed, 316, 317, 348 Libor-in-arrears, see Libor-in- half-life, 318 arrears, replication method measure change, 322 Libor-with-delay, see Libor-with- moment explosion, 323 delay, replication method moment-generating function, 321, stochastic optimization method, 953 324, 327 stochastic volatility model, 315–402, branch cut, 330 569, 570, 1140 singularities, 329 as interpolation rule, 701 time-dependent parameters, 364 ATM volatility, 348 Monte Carlo, 387–397 basket option, see Markovian Broadie-Kaya scheme, 394 projection, basket option in SV Broadie-Kaya simplified scheme, model 396 calibration, 701–702 exact scheme, 394 calibration norm, 702 martingale correction, 397 normalization, 702 Taylor-type schemes, 396 caplet calibration, 705 variance process, see square-root CEV type, see SABR model process, Monte Carlo CMS convexity adjustment, 738 multi-dimensional, see multi- correlation, 347 stochastic volatility model dampening constant, 325 PDE, 381–387 delta, 697 boundary conditions for stochastic effective skew, 373 variance, 385 effective volatility, 371, 372 boundary conditions from PDE effective volatility of variance, 375 itself, 385 European option, 327 discretizing spot, 387 control variate, 328 discretizing stochastic variance, volatility mixing, 339 383 explicit solution, 320 for forward Kolmogorov equation, for CMS rate, 738–742 386 dynamics in forward measure, 739 predictor-corrector, 382 Fourier integration, 324–339 quadratic discretization, 384 arbitrary European payoffs, 336, range for spot, 386 338 range for stochastic variance, 382 convolution, 325 sinh transform, see PDE, sinh direct integration, 330 transform discrete, 330 sinh-quadratic discretization, 384 FFT, 330 variable transform, 383, 384 for variance, 339–343 process for variance, see square-root integration bounds, 330 process strip of convergence, 329 skew, 317, 346 with control variate, 328, 330 smile dynamics, 347–349, 351, 353, hedging, 353–358 354 Index xlix

SV volatility, 317 fixed-floating, 198, 199, 230, 231 time-dependent, 363–402 valuation formula, 199 asymptotic expansion, 366–370 fixing dates, 198 averaging, see calibration, time legs, 197 averaging Libor-in-arrears, see Libor-in-arrears Fourier integration, 363, 366 Libor-with-delay, see Libor-with- volatility of variance, 316, 317, 346 delay volatility of volatility, 318 par rate, see forward swap rate stopping time, 6 payer, 203 straddle, 223 payment dates, 198 strategy, 7 receiver, 203 doubling, 10 swap rate, see forward swap rate gains process, 8 swap market model, 617, 675–677 permissible, 9 swap measure, see measure, annuity replicating, 11 swap rate, see forward swap rate self-financing, 8, 17 swaption grid, see European swaption, Stratonovich integral, 5 swaption grid , 24 structured note, see exotic swap Tanaka extension of Ito’s lemma, 7, 26, structured swap, see exotic swap 294, 1131 Student’s t-distribution, 101 targeted redemption note, see TARN Monte Carlo, 101 TARN, 217, 218, 925–933 survival measure cap at trigger, 219 Bermudan swaption, see Bermudan global model, 927 swaption, survival measure impact of inter-temporal correlation, importance sampling, see im- see inter-temporal correlation, portance sampling, survival impact on TARNs measure importance sampling, 1068–1077 SV model, see stochastic volatility one-step survival conditioning, model 1069 SV model with general variance removing first digital, 1068 process, 359–361 leverage, 927 martingale properties, 360 Libor market model, 927 moment explosion, 361 lifetime cap, see TARN, cap at properties, 359 trigger stationary distribution, 360 lifetime floor, see TARN, make strict supermartingale, 360 whole SVD, see CLE regression, SVD local projection method, 928–931 decomposition Gaussian short rate model, 929 SVI model, see volatility smile, SVI Markov-functional model, 931 swap, 197 quadratic Gaussian model, 931 accreting, 200 quasi-Gaussian model, 931 amortizing, 200 make whole, 219 annuity, XXXVIII, 199 Markov-functional model, 473 annuity factor, 170 multi-factor quasi-Gaussian model, averaging, see averaging cash flow 927 cash-settled, 744 partial analytical integration, 1011 CMS, see CMS swap pathwise differentiation method, effective date, 225 1044 l Index

payoff smoothing, 1011, 1029, vega hedging, 712 1068–1077 loading from Gaussian model, 712 PDE, 931–933 no-arbitrage condition, 708 cap at trigger, 933 PDF of swap rate in forward make whole, 933 measure, 737 Monte Carlo pre-simulation, 933 from CMS caplets, 737 upper bound for extra state reasonableness, 708 variable, 932 swap rate distribution in forward tube Monte Carlo, 1029 measure, 736 valuation formula, 218 swap-yield TSR model, 713–714 volatility smile, 927, 929–931 CMS convexity adjustment, see tenor structure, XXXVIII, 170 CMS, convexity adjustment, index function, 591 swap-yield TSR model tension spline, see spline, tension theta, 980, 992 term parameters, 378 rolling yield curve, 992 term structure model, 202, 277 Tikhonov regularization, see CLE re- terminal swap rate model, 707–714 gression, Tikhonov regularization annuity mapping function, 708, 713, time decay, 52 722, 724–725, 728, 730, 732 time value, 27 as conditional expected value, “tip-top”, see “flip-flop” 724–725 tower rule, see iterated conditional calibration to market, 728 expectations forward swap rate condition, 733 tree, 423 forward value condition, 732 binomial, 444, 456 in measure change, 735 SALI, 78 linear approximation, 728 trinomial, 51, 456 LM model, see Libor market truncated Gaussian scheme, see model, annuity mapping function square-root process, Monte Carlo, mean reversion, see CMS, convex- truncated Gaussian scheme ity adjustment, impact of mean TSR model, see terminal swap rate reversion model multi-rate, 765 tube Monte Carlo, 1022–1030 swap rate squared condition, 733 barrier option, see barrier option, CMS convexity adjustment, see tube Monte Carlo CMS, convexity adjustment, Bermudan swaption, see CLE greeks, linear TSR model tube Monte Carlo consistency condition, 708 CLE, see CLE greeks, tube Monte exponential TSR model, 712–713 Carlo Libor-with-delay, see Libor-with- digital option, 1024 delay, swap-yield TSR model discrete knock-in barrier, 1028 linear TSR model, 709 generalized trigger product, see CMS convexity adjustment, see barrier option, tube Monte Carlo CMS, convexity adjustment, partial coupons, 1028 linear TSR model TARN, see TARN, tube Monte forward CMS straddle, 941 Carlo mean reversion parameterization, 710 underhedge, 1023 swap rate distribution in forward uniform distribution, XXXVII, 768 measure, 736, 737 universal law of volatility, 1137 Index li upwinding, see PDE, upwinding bucketed shocks, 1099 CMS spread option, 1116, 1120 value-at-risk, 499, 975, 996–998 constant Libor correlations, 1120 conditional, 996 constant Libor correlations, 1115, delta VaR, 998 1120 delta-gamma VaR/cVaR, 998 constant term swap correlations, Gaussian, 997 1116, 1118–1120 historical, 996 cumulative shocks, 1099 vanilla derivative, 695–813 direct method, 1098–1102, 1110 multi-rate, see multi-rate vanilla Bermudan swaption, 1103 derivative European swaption, 1102 single-rate, see single-rate vanilla second-order effects, 1111 derivative European swaption, 1113, 1114 vanilla model, 202, 277, 315, 1121, flat shock, 1099 1129 forward swaption straddle, 948 for multi-rate derivative, see “good”, 1102–1105 multi-rate vanilla derivative hybrid method, 1111–1113 for single-rate derivative, see algorithm, 1112 single-rate vanilla derivative Bermudan swaption, 1114 local volatility model, see local CMS spread option, 1116 volatility model European swaption, 1113, 1114 stochastic volatility model, see in LM model stochastic volatility model coverage, 884 vanna, 980 indirect method, 1105–1111, 1121 VaR, see value-at-risk Bermudan swaption, 1109 variance reduction, 143–158 European swaption, 1108 antithetic variates, 144 least-squares problem, 1106 efficiency, 145 locality, 1107 non-Gaussian, 145 smoothing, 1107 common random number scheme, Jacobian method, see vega, indi- 132, 134 rect method; risk sensitivities, conditional Monte Carlo, 127 Jacobian method control variate, see control variate Libor market model, 1095–1125 from hedging strategy, see control bootstrap calibration, 1111, 1112 variate, dynamic multi-factor, 1115 importance sampling, see impor- projection, 1123 tance sampling local projection method, 867 moment matching, 146 local vs. global, 1097 systematic sampling, 145 locality, 1104 Vasicek model, 411–413 benchmark set locality, 1104 bond reconstitution formula, 412 exotic locality, 1104 bond volatility, 413 full set locality, 1104 forward rate volatility, 413 market vega, 984, 1096, 1110 short rate distribution, 411 model vega, 984, 1096, 1124–1125 short rate dynamics, 411 pathwise differentiation method, see yield curve shapes, 412 pathwise differentiation method, vega, 355, 980, 1095–1125 vega additivity, 1103 projection, 1122–1124 Bermudan swaption, 1114 relationship to gamma, 981 lii Index

row shocks, 1099 Gaussian backbone, 698 running cumulative shocks, 1099 impact on forward volatilities, see scaling, 1103 forward volatility, impact of skew vega, 1113–1115 volatility smile smile vega, 1113–1115 impact on inter-temporal cor- volatility, 27 relations, see inter-temporal average convexity, 307 correlation, impact of volatility Bachelier, see volatility, Normal smile basis point, see volatility, Normal probability density from, 278 Black, XXXVIII, 204 SABR, see SABR model bp, see volatility, Normal shadow delta hedging, 697 CEV, 280, 623 skew vega, 1114 Dupire’s, see Dupire local volatility skew-dominated, 352 factor volatility, 499 slope, 279 forward volatility of Libor rate, 817 smile vega, 1114 Gaussian, see volatility, Normal SVI, 703, 951, 1121 implied, 278 upward sloping, 281 as average of realized, 989 vega, 1114 effects of mis-specification, 987 volatility structure, 815 most likely path approximation, volatility swap, 220, 221, 933–945 990 capped, 937 PDE for, 296 CMS spread, 221 local, see Dupire local volatility copula method, see copula method, Normal, 204, 283, 623 volatility swap Normal for CMS spread option, 774 fixed-expiry, 221, 940 separable, 300 fixed-tenor, 221, 940 small-noise expansion, 307 impact of forward volatility, 944 spanned stochastic volatility, 452 impact of volatility smile dynamics, spot volatility, 817 941 spread, 774 Libor market model, 933, 934 strike-dependent, 775 local projection method, 934 stochastic, see stochastic volatility min-max, see min-max volatility model swap unspanned stochastic volatility, 443 PDE, 934 “volatility squeeze”, 422 quadratic Gaussian model, 941 volatility cube, see European swaption, quasi-Gaussian model, 941 volatility cube with barrier, 222 volatility derivative, see forward with shout, 221, 935 volatility derivative volga, 980 volatility skew, 279 Volterra integral equation, 436 volatility smile, 279, 315 vomma, 980 ATM backbone, 699, 700 backbone, 696 Wiener process, see Brownian motion adjustable, 697–700 curvature, 1138 year fraction, 224 dynamics, 279, 348, 696–700, 818 yield curve, 191, 230, 231, 233 sticky delta, 350, 697 base index curve, 268 sticky strike, 352, 697 basis risk, 270 forward skew, 944 benchmark set, 230 Index liii

forecasting curve, see yield curve, curve overlays, 258 index curve FX forwards, 259 index curve, 261, 267, 677 Hermite spline, 238–240 index-discounting basis, 197, 261 iterative solution, 239 instantaneous forward curve, 233 Jacobian rebuild, 256 joint evolution of discount and multi-index curve group, 230, 265 forward curves, 677 non-parametric fitting, 245–250 multi-index curve group, 267–270 norm specification, 245 overlay curve, 259 optimization algorithm, 245 perturbation locality, 230, 251–253, separate discount and forward 258 curves, 260 Principal Components Analysis, see spline, see spline Principal Components Analysis spline fitting, 234–244 ringing, 235, 242, 243, 252 tension spline, 243–244 smooth, 258 yield curve risk, 250–258 spread curve, 269, 884 cumulative shifts, 256, 257 tenor basis, 230, 267 forward rate approach, 252 TOY effect, 258 Jacobian method, see risk sensitivi- yield curve construction, 229–275 ties, Jacobian method benchmark set, 231 par-point approach, 251 bootstrapping, 234 rolling for theta, 992 flat forward, 236 waterfall approach, see yield curve linear yield, 235 risk, cumulative shifts constrained optimization, 248 yield curve spread option, see CMS cross-currency, 259 spread option cross-currency arbitrage, 260 cubic spline C2, 240–243 zero-coupon bond, see discount bond problems, 242 zero-coupon bond option, 185