Index
1D elements, finite element methods 86-8, 98, 107-9, 111-13
2D elements, finite element methods 88-90, 93-104, 109, 112-14
3D elements, finite element methods 98, 100, 109-10, 114-15
3D IR/FX models, Bermudan callable steepener cross-currency swaps 189-91
9/11 196
acceptance-rejection methods, random number generators 154-6, 160
accompanying software 4
accrued interest 73
Acklam algorithm 157
advanced equity models 3, 193-207
advanced Monte Carlo techniques 3, 5, 161-77
see also Monte Carlo . . .
advancing (moving) unstructured-mesh generation algorithm 84-5
algebraic multigrid methods (AMGs) 126-7
Amdahl’s law 287-8
American options 3, 7-8, 12, 15, 179-91, 202-7
see also Bermudan . . .
definition 7-8, 179
Monte Carlo simulations 179-91
prices 3, 179-91, 202-3
amplification matrices 37-8
analytical solutions 14-15, 52-3, 58-9, 168, 206, 262-4
see also Black-Scholes PDE
annual compounding 39-40
antithetic variates, Monte Carlo simulations 161-3
APIs (application programming interfaces) 285-95
Apple 7
approximation tools 24-9, 43-5, 58-70, 81-115, 133-4, 145-6, 156-60, 180-91, 209-16, 257
see also backward difference...; central difference...; forward difference...
arbitrage 2, 9, 11, 13-14, 21, 45-6, 194-6
Archimedean copulas 222, 223-9, 231-4
see also Clayton...; Frank...; Gumbel...
arithmetic Asian options 164-6
artificial boundary conditions 2, 77-9, 210-16
Asian (arithmetic average rate) options 71-2, 164-6
asset allocations 253
asymptotic analysis 145-6
see also approximation tools
at-the-money (ATM) options 44-5, 245
auditing 298
autocorrelation 217
Bachelier, Louis 19
back-substitution methods 118-20
backward difference quotient 25-7, 29, 66-70, 263-4
Bank of International Settlement (BIS) 39
barrier options 12, 15-16, 30, 77-8, 86, 101-3, 167-8, 202, 210-16, 245-51, 290-5
see also double . . . ; knockout . . .
Dirichlet boundary conditions 77-8, 101-3, 210-16
parallel architectures 290-5
basis functions 185-91
basket default swaps 234-7
Bates model 200-1
behavioral psychology 239
Bermudan options 3, 7-8, 12, 74-5, 179-80, 202-7
see also American . . .
bonds 7, 180-91
callability possibilities 74, 181-91
definition 7-8, 179
Monte Carlo simulations 179-91
prices 3, 179-91, 202-3
beta distributions 233-4
bibliography 301-6
BiCGStab 107, 191
bid-ask spreads 240
binary options see digital options
binomial trees 1-2, 5, 8-16, 30, 47, 51, 56
Black-Scholes model 10-11, 13-14
Cox-Ross-Rubinstein tree 11, 12-14
critique 1-2, 12-16
exotic options 14-16
forward trees 11, 12-14, 15-16
the Greeks 15-16
grid-adaptivity factors 15-16
multiperiod binomial model 9-10
no-arbitrage conditions 2, 11, 13-14
non-recombining trees 14, 47
one-period model 8-9, 10
oscillations 12-14, 30
Rendleman-Bartter tree 11, 12-14
biologically-inspired optimization techniques 260-1
bivariate standard normal distributions 155-7
Black Monday 196
Black-Karasinski interest-rate model 46, 73, 187-91, 217
see also Hull-White...
Black-Scholes model 2-3, 5, 10-14, 17-38, 39, 55, 71-2, 77-8, 165, 167-8, 180-1, 185-6, 194-207, 209-10, 239-40, 245, 250, 290-5
see also Brownian motion; finite difference methods
binomial trees 10-11, 13-14
critique 194-6
definitions 10-11, 17-23
Black-Scholes PDE 2, 3, 5, 17-38, 55, 71-2, 77-8, 165, 180-1, 185-6, 194-207, 209-10, 239-40, 245, 250, 290-5
definition 17-23
solutions 22-3
Black-Scholes SDE 10-11, 13-14, 17, 19-20, 55, 165
Black76 approximations 19, 43-5, 139-40, 245, 297
caplets 44-5, 245, 297
definition 44-5
swaptions 44-5, 139-40, 245
Bloomberg screens 7
bonds 2, 7, 39-53, 63-70, 72-9, 86-90, 105-7, 133-9, 140-6, 162-9, 180-91, 234-7, 240-51, 269, 274-6, 297-300
see also coupons; interest-rate instruments
Bermudan options 7, 180-91
definitions 39, 63-4
Boost library 156-9
bootstrapping of zero rates 42-3, 298-300
see also discount factors; forward . . .
boundary conditions 2, 5, 21, 23, 30-6, 49-53, 58-61, 65-70, 71, 77-9, 83-4, 92, 101-3, 106-7, 113, 127-31, 138-9, 193-207, 210-16
see also Dirichlet...; Neumann...
definitions 31-4, 58, 65, 77, 78, 101
finite element methods 92, 101-3, 106-7, 113, 139, 193-207
Box-Muller random number algorithm 155-6
British Bankers Association (BBA) 40
Brownian bridge method 3, 175-7, 290-5
Brownian covariance/distance see distance correlation
Brownian motion 17-20, 45-6, 55-6, 143-4, 171, 175-7, 196-207
see also Black-Scholes model; geometric...; Wiener processes
definition 17-19, 199-200
SDEs 19
bushy trees 14
business day conventions 40, 137, 139, 182
C1060 291-4
calibration 1-5, 21, 138-9, 193-4, 205, 207, 239-51, 255-68, 291-5, 297-300
see also optimization techniques; underlying assets
definition 4, 239
hybrid calibration algorithms 259, 261-4, 291-5
ill-posed/well-posed problems 243-5
instability problems 4
inverse problems 4, 5, 193-4, 239-51, 297
local volatility models 245-51
model risk/uncertainty 249-51
parallel architectures 291-5
penalty terms 244-5
regularization methods 243-7, 297
yield curve problems 240-5
call options 3, 7-16, 17-38, 50-2, 71-2, 74, 77-8, 164-6, 167-8, 179-91, 195-207, 209-16, 239-51, 255-68, 290-5
callable (redeemable) fixed rate bonds 63-4, 181-91, 297-300
cap-floor-parity 43
capacitance matrices, finite element methods 97-8
capital asset pricing model (CAPM) 218
caplets 43, 44-5, 143-4, 245, 297
caps 43-5, 73, 86, 139-40, 245, 297-300
definition 43
prices 43-4, 73, 297-300
cash cows 7
CDOs see collateralized debt obligations
central difference quotient 25-7, 29, 58-61, 263-4
central limit theorem 9-11, 17-19, 133-4, 229
CFL see Courant-Friedrichs-Levy condition
characteristic function methods 3, 5, 57-8, 169-75, 193-207, 209
Chi-Squared distributions 159, 233
Cholesky decomposition/factorization 117-18, 121-2, 130, 138-9, 158, 175-7, 232-4
chooser options 71-2, 75
CIR see Cox-Ingersoll-Ross interest-rate model
Clayton copulas 226-9, 233-4
clean values, definition 73
client-server architectures 298-300
CMSs see constant maturity swaps
coarse-grained parallelization 288, 299
coding requirements 1, 12, 50, 68, 119-22, 170, 180, 182-4, 289-95
coefficient matrices 117-18
collateralized debt obligations (CDOs) 234-7, 269
collocation weighting function 81
colour codes, reading guide 5
compound options 71-2
compounding 8, 39-40, 195-6, 241-7
see also annual...; continuous ...; quarterly...; semi-annual...
computational finance 1-5, 217, 239
see also calibration; models; risk analyses; risk management; valuations
definition 1
overview of the book 1-5
computing scalability 298-9
conditional Monte Carlo 166-8
conditional VaR (CVaR) see Expected Shortfall
confidence levels, VaR 269-76
congruential random number generators 148-52
conjugate gradient iterative method 125-6, 129, 130-1
consistent formulations 98
constant maturity swaps (CMSs) 42-5, 73, 76-7, 137-9, 188-91
constant memory 290
continuous compounding 40, 195-6, 241-7
continuous time models for equity prices 17-19
contribution VaR 274-6
control divergence problems, parallel architectures 289
control variates, Monte Carlo simulations 161, 163-6
Controllable Cellular Automata 151
convection-diffusion-reaction problems 2, 22-3, 30, 47, 53, 55-70, 90-104, 105-7, 119-31
convergence analysis 9-10, 11, 12-16, 36-8, 122-31, 134, 169-75
convexity 249, 254-5, 265
copulas 3-4, 5, 217, 221-37, 274
see also Archimedean . . . ; correlations; elliptical...; Frank...; Gaussian...; Gumbel...; t...
critique 217, 221-2
default probabilities 234-7
definition 217, 221-9
important copula functions 222-9
Monte Carlo inversion method 232-4
parameter estimates 229-34
sampling 229, 232-4
types 221-9
correlation ratios, definition 221
correlations 3-4, 5, 138-9, 141-6, 149-50, 162-9, 170-5, 196-207, 217-37, 248-9, 271-83
see also copulas; Kendall’s ...; Pearson’s...; Spearman’s...
critique 3-4, 217-37
de-correlation factors 144-6
definitions 217-21
LMM 141-6
overview 3-4, 5, 217
pitfalls 3-4, 5, 217-37
types 217-21
counter-party risk 269
coupons 7-8, 39-53, 63-70, 72-9, 137-9, 181-91, 240-51, 297-300
see also bonds
Courant-Friedrichs-Levy condition (CFL) 38, 58
Courant-Isaacson-Rees method 57
covariance 140-1, 157-60, 161-9, 218-21, 266-8, 270-1
Cox-Ingersoll-Ross interest-rate model (CIR) 46, 159, 196-7, 248
Cox-Ross-Rubinstein tree 11, 12-14
CPUs, parallel architectures 4, 285-95, 299
Crank-Nicolson semi-implicit finite difference method 29, 30, 33-6, 66-70
credit default swaps (CDSs) 235-7, 269
credit derivatives 4, 234-7, 269
credit events 39, 234-7, 240
credit ratings 39
credit risk 234-7, 269
credit spread risk 269, 274-6
cross-currency swaps 109, 189-91
CUDA framework 287-8, 289-95
dangling nodes, finite element methods 83-5
data management systems 297-300
databases 297-300
day-count conventions 40-1, 44, 139, 182, 241, 297-300
day-count fractions (DCFs) 43-5
DE see Differential Evolution
de-correlation factors 144-6
default probabilities 4, 234-7, 240, 269
see also copulas
Delauny triangulation unstructured-mesh generation technique 84-5
delta 15-16, 21-2, 29, 55-6, 81, 83, 143, 272-6, 278-83, 299
see also gamma
delta hedging 16, 21-2, 55-6
dense matrices 117
dependencies 217-37
see also copulas; correlations
derivatives 1-5, 7-16, 39-53, 71-90, 101-3, 133-60, 169
see also interest-rate . . . ; options; swaps
overview of the book 1-5
statistics 39
Differential Evolution (DE) 260-1, 263-4
diffusion-reaction problems 2, 22-3, 30, 47, 53, 55-70, 90-104
digital options 52, 71-2
digital range-accrual notes 52-3
Dirac-delta functions 81, 83
direct search methods 259-60
Direct Search Simulated Annealing (DSSA) 259-60, 263-4
direct solving methods, linear equations 3, 117-22
Dirichlet boundary conditions 30-6, 58, 77-9, 101-3, 127-8, 210-16
definition 31-4, 58, 77, 101
double barrier options 77-8, 101-3, 210-16
finite element methods 101-3
dirty values, definition 73
discount bonds
see also forward rates
LMM 140-6
discount factors 10-11, 40, 42-5, 137-9, 140-6, 165-6, 180-91, 290-5
discounting 9, 10-11, 40, 42-53, 73, 74-5, 137-9, 140-6, 165-6, 180-91, 290-5
discrete dividends 22, 71, 74-9
discretization tools 2-3, 22-38, 51-3, 56-70, 81-115, 117-31, 135-9, 162-3, 165-6, 168-9, 171-5, 180-91, 193-207, 209-16
see also upwinding techniques
dispersion 169-75
see also low-discrepancy sequence theory; variance
distance correlation 221
distributed memory systems 285-8
diversification 218
dividends 7-8, 14, 22-3, 71, 74-9, 196-207, 248-51
see also equities
double barrier options, Dirichlet boundary conditions 77-8, 101-3, 210-16
down-and-in European call options 167-8
down-branching trinomial-tree regimes 48-53, 56
drift 19-23, 48-53, 56-70, 105-7, 186-91, 198-207, 221-2, 241-5
see also expected returns
Dupire local volatility model 194-6, 246-8
see also volatility smiles
dynamic copulas 222
Egger-Engl local volatility algorithm 246-8
eigensystem calculations, PCA 276-83
electricity spot markets, mean-reversion rates 49
element matrices, finite difference methods 91-104
elliptical copulas 222-6, 233-4
see also Gaussian...; t...
elliptical distributions 217-21, 222-3
empirical copulas 229-34
equidistantly spaced grids, finite difference methods 24-38, 211-16
equities 3, 4, 7-16, 17-38, 39-53, 71-9, 165-6, 193-207, 239-51, 269, 297-300
see also dividends
advanced equity models 3, 193-207
definition 7
errors
finite difference methods 25-7, 36-8, 60-1, 214-16
Fourier-cosine series expansions 206-7
linear equations 122-31
Euler equation 31-6, 58-61, 135-9, 162-3, 165-6, 168, 191
see also explicit finite difference method
Euribor 41-3, 76, 186-91
European options 3, 7-16, 17-38, 44-5, 47, 50-2, 78, 165-6, 167-8, 179, 193-207, 210-16, 255-68
concepts 7-16, 44-5, 50-2, 78, 165-6, 167-8, 179, 193-207
definition 7-8
payoffs 7-8, 10-11,21,71-2
execution models, parallel architectures 285-9
exotic options 3, 14-16, 52-3, 71-2, 75-7, 164-6, 179-91, 202-7, 210-16, 249-51, 290-5
see also Asian...; barrier...; snowball floaters; target redemption notes
binomial trees 14-16
prices 3, 14-16, 52-3, 71-2, 75-7, 164-6, 179-91, 202-7, 210-16, 249-51, 290-5
expected returns 19-23
see also drift
Expected Shortfall (ES)
concepts 4, 218, 269-76, 280-3, 288, 299
definition 270, 272, 281
expected values 133, 134-5, 166-8
see also theta
bonds 39-53, 63-70, 297-300
options 7-16, 20-38, 50-2, 71-5, 194-6, 239-51, 255-68
explicit finite difference method 12, 17, 28-9, 30-8, 57-70, 212-16
see also binomial trees
exponential distributions 153-4, 155, 160, 199, 210-16
extreme value copulas 222, 229
see also Gumbel . . .
extreme value theory (EVT) 4, 229, 278-83
FACTORY software 297-300
Fast Fourier Transformation (FFT) 3, 193, 201-7, 297
fat tails 194-6, 217-37, 282-3
see also kurtosis; skewness
Faure sequences 171-5
feasible region, optimization techniques 253-68
fees for market data 297-8
Feller condition 46, 197, 249, 261-4
FEMs see finite element methods
financial crisis from 2007 217, 221
financial instruments 1, 2, 71-9, 179-91, 297-300
see also individual instruments
computational finance definition 1,181
large software systems 297-300
term sheets 2, 71-9, 297
fine-grained parallelization 288
finite difference methods 2, 5, 23-38, 55-70, 72, 77-9, 110, 117-31, 191, 193-207, 209-16, 262-4, 297
see also Black-Scholes PDE; partial differential equations; upwinding techniques
approximation tools 24-9, 209-16
backward difference quotient 25-7, 29, 66-70, 263-4
central difference quotient 25-7, 29, 58-61, 263-4
concepts 2, 5, 23-38, 55-70, 77, 110, 191, 193-207, 209-16, 262-4, 297
Crank-Nicolson semi-implicit method 29, 30, 33-6, 66-70
definition 23-7
element matrices 91-104
error analyses 25-7, 36-8, 60-1, 214-16
explicit method 12, 17, 28-9, 30-8, 57-70, 212-16
five-point stencils 26-7, 31-3
forward difference quotient 25-7, 29, 211-16, 263-4
heat equations 23, 30-8, 56
implicit method 12, 29, 30, 32-6, 77, 105, 117-31, 191, 212-16
overview 2, 5, 23
semi-implicit method 29, 30, 33-6, 66-70, 117-31
spatial discretization 2, 24-38, 56-70, 211-14
stability considerations 2, 30, 34-8, 49, 55-70, 77
Taylor series expansions 24-7, 28-9, 59-61, 136-9, 256-7, 272, 277
time discretization 2, 24, 27-38, 56-70, 117-31, 162-3, 191, 193-207, 209-16
finite element methods 2, 5, 16, 24, 72, 77, 78-9, 81-115, 117-31, 139, 193-207, 209-16, 245, 297
1D elements 86-8, 98, 107-9, 111-13
2D elements 88-90, 93-104, 109, 112-14
3D elements 98, 100, 109-10, 114-15
boundary conditions 92, 101-3, 106-7, 113, 139, 193-207
capacitance matrices 97-8
collocation weighting function 81
concepts 2, 5, 24, 72, 78-9, 81-115, 139, 193-207, 209-16, 245, 297
convection-diffusion-reaction problems 98, 103-4, 105-7, 119-31
dangling nodes 83-5
definition 81-3
Galerkin weighting function 82-3, 92, 97-8, 103-4, 105-7, 209
global matrices 98-100
grid generation processes 83-5, 211-12
integration variables 94-7, 112, 210
least squares weighting function 82
linear elements 86-90, 93-104, 109-10, 113-15, 117
linear shape functions 86-90, 94-104
local coordinates 96-104, 107-15
matrix assembly processes 2, 82-5, 90-104
natural coordinates 110, 111-15
overview 2, 5
quadratic elements 88-90, 93-8, 106-9, 117-31
rectangular elements 90, 98, 103, 109, 113-15
shape functions 85-90, 94-115
stabilization considerations 103-4
streamline-diffusion stabilization method 104-7
streamline-upwind-Petrov-Galerkin stabilization method 104-7
structured meshes 83-5
subdomain weighting function 82
time discretization 97-8, 103-4, 105-7, 117-31, 193-207, 209-16
triangular elements 88-90, 93-104, 109, 112-14
unstructured meshes 84-5
weighted residual methods 81-3, 97-8
first order upwind schemes 57-61, 63
see also upwinding techniques
five-point stencils, finite difference methods 26-7, 31-3
fixed income instruments see interest-rate instruments
floating-rate notes (FRNs) 40-2, 73-4, 75-6, 133, 274-6
floorlets, definition 43
floors 43-5,73,76, 86, 189
Flynn’s taxonomy 285
foreign exchange (FX) rate instruments 3, 14, 109, 137-9, 179-91, 269, 271-83, 297-300
foreign exchange risk 269
fork-join parallel execution model 286-8
forward difference quotient 25-7, 29, 211-16, 263-4
forward rates 9-10, 42-5, 139-46, 185-91, 195-6, 241-5, 297-8
see also discount bonds; interest rates; Libor market model
definition 42-3
forward start swap rates 43
forward trees 11, 12-14, 15-16
see also Cox-Ross-Rubinstein tree
Fourier transforms 3, 193, 201-7, 297
Fourier-cosine series expansions
concepts 3, 193-4, 201, 202, 203-7
definition 203-6
Frank copulas 226-9
FTSE-100 index 194-6, 249-50, 295
Galerkin weighting function 82-3, 92, 97-8, 103-4, 105-7, 209
gamma 21-2, 83, 159-60, 233-4
see also delta
gamma distributions 159-60, 233-4
Gauss-Legendre quadrature 95-6, 112
Gauss-Newton method 258
Gauss-Seidel iterative method 122, 124-5, 127-8
see also Jacobi...
Gaussian copulas 221-6, 231-7, 274
Gaussian elimination
see also LU decomposition/factorization; Thomas algorithm
linear equations 117-22
General Electric (GE) 194-6
generalized Pareto distributions 278-83
geometric Asian options 164-6
geometric Brownian motion (GBM) 19-21
see also Black-Scholes model; Brownian motion
global matrices, finite element methods 98-100
global memory 289
global minimizer of the optimization problem 253-4
Gnedenko-Pickands-Balkema-deHaan theorem (GPBdH) 278-83
Godunov’s theorem 62
Gould’s study 118
GPUs, parallel architectures 4, 285-95
gradient-based optimization method 4, 205, 249, 255, 256-8, 262-4
graphics cards 289, 291-5
the Greeks 15-16, 21-2, 83-4, 206, 239-40
see also delta; gamma; rho; theta; vega
binomial trees 15-16
types 21-2, 239-40
Green’s theorem 91-2
grid generation processes, finite element methods 83-5, 211-12
grid-adaptivity factors 15-16, 24
gsl algorithm 157-9
GTX260 291-4
Gumbel copulas 226-9, 233-4
halley it2 algorithm 158-9
Halton sequences 171-5
hardware environments 4, 5, 146-8, 155-60, 206, 285-95
see also hardware environments; software
hazard rates, default probabilities 234-7
heat equations 23, 30-8, 56, 92, 239
hedge funds 300
hedging 16, 20-3, 41-2, 45-6, 55, 142, 222, 300
Hermite polynomials 108
Hessian matrices 256-8
Heston calibration model 4, 245, 248-51, 259, 260, 261-4
concepts 4, 245, 248-51, 260, 261-4
hybrid algorithm 259, 261-4, 291-5
Heston stochastic volatility model 46, 159, 196-7, 205-7, 245, 248-51, 290-5
heuristically motivated optimization method 4, 190-1, 255-6, 258-61
higher order upwinding schemes 61-3
see also upwinding techniques
histograms, uniformly distributed random numbers 149-50
historical VaR 269, 272-4, 278
Ho-Lee interest-rate model 45-6
host memory 289
Hull-White interest-rate model 45-8, 49-53, 56, 63-70, 73, 78-9, 105-7, 117, 137-40, 146, 177, 186-91, 217, 241-5, 297
see also Black-Karasinski . . . ; trinomial trees
calibration 240-5, 297
critique 46
definition 45-6, 65, 105-6
Monte Carlo simulations 137-9, 146, 177, 186-91
upwinding techniques 63-70
hybrid algorithm, Heston calibration model 259, 261-4, 291-5
ill-posed/well-posed problems, calibration 243-5
implicit finite difference method 12, 29, 30, 32-6, 66-70, 77, 105, 117-31, 191, 212-16
implied volatilities 21, 143-6, 194-6, 239-40, 246-51
see also volatility smiles
definition 239-40
noisy data 239-40, 247-8
in advance coupons 73
in arrears coupons 73
in-the-money (ITM) options 44-5, 240, 246
individual VaR 274-6
induction methods 42-3, 298
infinite activity models 196, 199-200, 209, 249-51
see also Normal Inverse Gaussian . . . ; Variance Gamma . . .
inflation rates 185, 271-83
information flows,
upwinding techniques 56-70
initial conditions 30, 34-6, 210-16
instability problems of calibration 4
integer programming 255
integral equations of the first kind 241
integration principles, Monte Carlo simulations 133-4
integration variables, finite element methods 94-7, 112, 210
Intel E5520 CPUs 291
interbank funding 40-1 interest rate risk 269, 274-6
interest rates 2, 3, 8-9, 12-16, 22, 39-53, 64-70, 71, 133-60, 179-91, 239-51, 269-83, 297-300
see also Euribor; forward...; Libor...; rho terminology 39-43
interest-rate instruments 2, 3, 39-53, 55-70, 72-3, 75-9, 135-46, 162-9, 179-91, 274-6
see also bonds; caps; floors; models; swaps
overview 2, 5, 39
snowball floaters 75-7, 185-6
statistics 39
TARNs 76-7
terminal conditions 72-3
terminology 39-43
interest-rate models 2, 5, 39-53, 56, 63-70, 71, 73, 78-9, 105-7, 117, 135-40, 146, 162-9, 177, 186-91, 217, 239-51, 300
see also models; short-rate . . .
critique 46
types 45-53, 105, 137-46, 186-8, 217, 300
interest-rate swaps (IRSs) 41-5, 73, 137-9, 146, 181-91, 297-300
agreement terms 41-2
Black76 approximations 43-5, 139-40, 245, 297
cash-flow schematics 41-2
CMSs 42-5, 73, 76-7, 137-9, 146, 188-91
definition 41-3
forward rates 42-3, 297-8
replication methods 41-2
valuations 42, 43-5, 73, 137-9, 146, 181-91, 297-300
interface conditions 2, 5, 22, 71, 75-9, 210-16
overview 2, 5
snowball floaters 75-7
TARNs 76-7
interior point optimization methods 4, 254-5, 268
International Swaps and Derivatives Association (ISDA) 41-2
Internet 298
interpolation techniques 26-7, 68-70, 72, 81-115, 155-60, 185-91, 214-16
see also polynomial-type . . .
intranets 298-300
introduction 1-5
inverse problems 4, 5, 193-4, 239-51, 297
overview 4, 5
parameter calibration 4, 5, 193-4, 239-51
inverse transform method, random number generators 152-4, 156-7
investment banks 240
iterative solvers methods, linear equations 3, 101, 117, 122-31
Itô’s lemma 17, 19, 45-6, 55-6, 136-9, 195-6, 245
Jacobi iterative method 122, 124, 127-30
see also Gauss-Seidel...
Jacobian matrices 94-5, 122, 124, 127, 258
joint cumulative distribution function 222-37
joint distributions 222-37
jump-diffusion models 160, 171, 196-207, 209-16
see also Brownian motion; Kou . . . ; Merton . . . ; Poisson process
definitions 196, 198-9
jumps 160, 171, 196-201, 209-16, 245-51
‘junior quants’ 1
Kendall’s rank correlation coefficient 217, 220-1, 226-9, 231-4
Khronos group 289
knockout barrier options 15-16, 77-8, 101-3
Kou jump-diffusion model 199, 212-14
Kronecker delta 143-4
Krylov subspace methods 122, 125-6
kurtosis 194-7, 200
see also fat tails; skewness
Lagrange interpolation 26-7
Laplace operators 26-8, 129, 233-4
large software systems 4, 5, 297-300
see also software
law of large numbers 133-4
Lax-Wendroff upwinding scheme 61-3
leapfrog-like methods 61
Least Squares Monte Carlo algorithm (LSMC) 3, 5, 179-91
examples 186-91
overview 3, 5
least squares weighting function 82
Lehman Brothers 196
Levenberg-Marquardt convergent algorithm (LMA) 256, 257-8, 261-4, 293-5
Levy models 165-6, 196, 209-16
Libor 3, 40-3, 45, 139-46, 177
Libor market model (LMM) 3, 40-3, 45, 139-46, 177
see also forward rates; stochastic interest rates
correlations 141-6
definition 139-46
setup processes 141-2, 145
steepener valuations 146, 177, 188-91
volatility function 142-3
likelihood function 231-4, 278-83
linear correlation see Pearson’s linear correlation coefficient
linear elements, finite element methods 86-90, 93-104, 109-10, 113-15, 117
linear equations 2-3, 5, 28-9, 107, 117-31, 158, 209-16, 243-5
see also partial (integro) differential equations
Cholesky decomposition/factorization 117-18, 121-2, 130, 138-9, 158, 175-7, 232-4
conjugate gradient iterative method 125-6, 129, 130-1
direct solving methods 3, 117-22
errors 122-31
Gauss-Seidel iterative method 122, 124-5, 127-8
Gaussian elimination 117-22 iterative solvers methods 3, 101, 117, 122-31
Jacobi iterative method 122, 124, 127-30
Krylov subspace methods 122, 125-6
LU decomposition/factorization 120-2, 130
matrix decomposition 123-5
multigrid methods 117-18, 126-8, 130
preconditioning procedures 129-31
solving 2-3, 5, 29, 107, 117-31
SOR 122, 125
stabilized bi-conjugate gradient iterative method 126, 130-1
Thomas algorithm 119-20
Linear Programming (LP) 253-5, 268
linear shape functions, finite element methods 86-90, 94-104
Lipschitz continuous conditions 63
LMA see Levenberg-Marquardt convergent algorithm
LMM see Libor market model
load balancing and computing scalability 298-9
local coordinates, finite element methods 96-104, 107-15
local minimizer of the optimization problem 253-4
local volatility models 194-6, 245-51
localization errors 210-16
Log-Likelihood function 231-4
log-normal distributions 11, 17-20, 44-5, 46, 74-5, 193-6, 201
log-returns 17-20, 193-7
long positions 43-5
Longstaff and Schwartz LSMC approaches 179-81, 185-91
low-discrepancy sequence theory 3, 161, 169-76, 290-5
see also pseudo random numbers; Quasi Monte Carlo method
definition 169-74
lower bound algorithms, LSMC 186-91
LSMC see Least Squares Monte Carlo algorithm
LU decomposition/factorization, linear equations 120-2, 130
lumped formulations 98
marginal distributions 217-37, 273-4
market data
calibration 239-51, 255-6, 272-3, 297-300
fees 297-8
software 298-300
market risk 269-83
see also risk...
Markowitz, Harry 265
Marsaglia ‘Diehard’ tests, random number generators 150, 151
Marsaglia-Bray random number algorithm 156
MathConsult 1
see also UnRisk software package
Mathematica 12, 50, 148-9, 151, 299
mathematical finance, overview of the book 1-5
matrix assembly processes, finite element methods 2, 82-5, 90-104
matrix decomposition 123-5
Maximum-Likelihood method 231, 278-83
mean excess function 282-3
mean reversion 2, 5, 45-53, 56-70, 78-9, 105-7, 189-91, 197, 217, 241-5
see also upwinding techniques
concepts 2, 5, 45-53, 78-9, 105-7, 197, 217, 241-5
overview 2, 5
means 11-12, 18-23, 45-53, 133-60, 218-21, 240, 282-3
measures
physical measures 9, 10-11
risk measurements 218, 221-2, 269-83, 297-300
risk-neutral measures 9, 10-11, 135-9, 165-6, 193-207, 209-16
memory
parallel architectures 285-95
types 289-90
Mersenne Twister random number generator 151-2, 171-5, 291-5
Merton jump-diffusion model (MJD) 160, 198-9
Merton, Robert 19
meta-heuristic methods 256, 258-61
Metropolis algorithm 259
Milstein scheme 136
MIMD (multiple instruction, multiple data) 285-9
‘min-mod’ limiter 63
MJD see Merton jump-diffusion model
models
see also calibration
Bates model 200-1
Black-Karasinski interest-rate model 46, 73, 187-91, 217
Black-Scholes model 2, 3, 5, 10-11, 13-14, 17-38, 39, 55, 71-2, 77-8, 165, 167-8, 180-1, 185-6, 194-207, 209-10, 239-40, 245, 250, 290-5
Black76 approximations 19, 43-5, 139-40, 245, 297
CIR 46, 159, 196-7, 248
continuous time models for equity prices 17-19
critique 45-6
Heston calibration model 4, 245, 248-51, 259, 260, 261-4
Heston stochastic volatility model 46, 159, 196-7, 205-7, 245, 248-51, 290-5
Ho-Lee interest-rate model 45-6
Hull-White interest-rate model 45-8, 49-53, 56, 63-70, 73, 78-9, 105-7, 117, 137-40, 146, 177, 186-91, 217, 241-5, 297
infinite activity models 196, 199-200, 209, 249-51
jump-diffusion models 160, 171, 196-207, 209-16
Kou jump-diffusion model 199, 212-14
large software systems 4, 5, 297-300
Levy models 165-6, 196, 209-16
LMM 3, 40-3, 45, 139-46, 177
local volatility models 194-6, 245-51
Merton jump-diffusion model 160, 198-9
one-factor short rate interest-rate models 2, 39, 45-53, 63-70, 71, 73, 86, 105-7, 117, 135-9, 185, 241-5
overview of the book 1-5
two-factor interest-rate models 52, 78-9, 105-7, 117, 137-40, 146, 177, 245
upwinding techniques 2, 5, 55-70, 211-16
Vasicek interest-rate model 2, 45-6, 48-51, 78-9, 136, 162-3, 186-91
money markets 298
moneyness 44-5, 240, 245-6
see also at-the...; in-the...; out-of-the...
monotonicity concepts 62, 152, 162, 218-37
Monte Carlo methods 2, 3, 5, 72, 133-60, 161-77, 179-91, 209, 232-4, 245, 269, 273-4, 278, 288, 290-5, 297
see also advanced...; Least Squares ...; Quasi...
copula sampling 232-4
overview 2, 3, 5, 133-4
parallel architectures 288, 290-5
Monte Carlo simulations 2, 3, 5, 72, 133-60, 161-77, 179-91, 209, 232-4, 245, 269, 273-4, 278, 288, 290-5, 297
advanced techniques 161-77
American options 179-91
antithetic variates 161-3
Bermudan options 179-91
Brownian bridge method 3, 175-7, 290-5
conditional Monte Carlo 166-8
control variates 161, 163-6
definition 133-4
derivatives’ pricing 133, 134-9
Euler scheme 135-9, 162-3, 165-6, 191
integration principles 133-4
Longstaff and Schwartz LSMC approaches 179-81, 185-91
Milstein scheme 136
overview 3, 5, 133-4
random number generators 3, 146-60, 169-75
steepener valuations 137-9, 146, 177, 188-91
two-factor Hull-White interest-rate model 137-40, 146, 177, 186-91
uncertainty sources 133
variance-reduction efficiency techniques 152, 161-9
Monte Carlo VaR 269, 273-4, 278, 299
MPI (message passing interface) 285-8
multi-library approaches, software 299-300
multi-moment correlation measures 221
multicore CPUs, parallel architectures 4, 285-95, 299
multigrid methods 117-18, 126-8, 130
multiperiod binomial model 9-10
multistep optimization procedures 232-4
multivariate distribution functions 222-37, 274
multivariate normals, random number generators 157-9
n-th to default baskets 234-7, 269
National Institute of Standards and Technology (NIST) 150
natural coordinates, finite element methods 110, 111-15
Neumann boundary conditions 30-6, 65-70, 78-9, 101, 103, 106-7, 139, 213-14
artificial boundary conditions 78-9
definition 31, 32, 65, 78, 101, 103
finite element methods 101, 103, 106-7, 139
Newton optimization 256-8
Newton-Cotes formulae 214-16
Newton-Raphson technique 158-9
NIG see Normal Inverse Gaussian model
nine-point prolongation 128-9
NLP see Nonlinear Programming
no-arbitrage conditions 2, 11, 13-14, 21, 45-6, 105-7, 140-6, 194-6
noisy data, implied volatilities 239-40, 247-8
non-puttable fixed rate bonds, upwinding techniques 69-70
non-recombining trees, binomial trees 14, 47
non-scalar risk factors 271, 274, 276-83
Nonlinear Programming (NLP) 255-8
normal distributions 3, 17-20, 45-6, 155-9, 162-3, 193-6, 198-9, 217-37, 271, 278, 290-5
normal equations 117
Normal Inverse Gaussian model (NIG) 196, 199-200, 249-51
nr it5 algorithm 158-9
numerical integration techniques 94-7, 112, 193-207
numerical methods
see also finite...; Fourier...; Monte Carlo...; trees
overview of the book 1-5
P(I)DEs 3, 5, 193, 209-16
uses 3, 14-15, 22-4, 47-53, 56, 65-70, 81, 133, 193, 209-16, 245, 297
numerical quadrature methods 3, 72, 95-6, 112, 154-5, 161, 169-75, 193, 201-7, 211-16
NVIDIA 289, 291
objective function, optimization techniques 253-68
one-factor short rate interest-rate models 2, 39, 45-53, 63-70, 71, 73, 86, 105-7, 117, 135-9, 185, 241-5
see also models
one-period binomial tree model 8-9, 10
open architecture and multi-library approaches 299-300
OpenCL framework 287-9
OpenMP framework 286-8, 291-5
optimal exercise rule, definition 179-81
optimal portfolio selection, definition 265-8
optimization techniques 4, 5, 205, 232-4, 246-7, 249, 253-68, 291-5, 297
see also calibration
definition 4, 253-5
hybrid algorithm for Heston calibration model 259, 261-4, 291-5
overview 4, 5, 253-5
parallel architectures 291-5
portfolios 265-8
terminology 253-5
options 2, 3, 7-38, 43-5, 50-2, 63-70, 71-2, 74-5, 138-46, 164-6, 167-8, 171, 179-91, 193-207, 209-16, 239-51, 255-68, 290-5, 297-300
see also American . . . ; barrier . . . ; Bermudan...; call...; chooser...; compound . . . ; digital . . . ; European . . . ; exotic...; put...; swaptions
Black-Scholes model 2, 3, 5, 10-11, 13-14, 17-38, 71-2, 77-8, 165, 167-8, 180-1, 185-6, 194-207, 209-10, 250, 290-5
definitions 7-8, 240
payoffs 7-9, 10-11, 21, 52-3, 71-2, 77-8, 140-6, 165-6, 290-5
prices 2, 3, 7-16, 17-38, 50-2, 71-2, 74-5, 138-9, 164-6, 167-8, 179-91, 193-207, 209-16, 239-51, 255-68, 290-5, 297-300
replication methods 8-9
terminal conditions 71-2, 139
types 7-8, 12, 71-2
oscillations 12-14, 30, 35-6, 56-70, 103-4, 197
see also stabilization considerations
OTC see over-the-counter derivatives
out-of-the-money options 44-5, 240, 246
outliers 219-37
over-the-counter derivatives (OTC) 39, 41, 297
see also swaps
statistics 39
overview of the book 1-5
parallel architectures 4, 5, 160, 206, 285-95, 299
definition 285-8
different levels 288
hybrid calibration algorithms 291-5
overview 4, 5, 285
QMC valuations 288, 290-5
valuations 288-95, 299
parameters 1-5, 21, 138-9, 193-4, 205, 207, 229-34, 239-51, 269-71, 277
see also calibration
copulas 229-34
parametric (variance-covariance) VaR 269-71, 277
Pareto distributions 278-83
partial differential equations (PDEs) 2, 5, 17-38, 45,55-6,71-9,81-115, 117-31, 135-9, 185-91, 193-207, 209-16, 239-40, 246-51
see also Black-Scholes PDE; finite...
short-rate models 55-6, 186-91
partial (integro) differential equations (P(I)DEs) 2-3, 5, 117-31, 193, 209-16
concepts 2-3, 5, 117, 193, 209-16
definition 117
numerical solution methods 3, 5, 193, 209-16
path-dependency 2, 71-9, 167-8
payer swaps 41-3
payer swaptions 44-5
payoffs, options 7-9, 10-11, 21, 52-3, 71-2, 77-8, 140-6, 165-6, 290-5
PCA see principal component analysis
PDEs see partial differential equations
Pearson’s linear correlation coefficient 217, 218-21
penalty terms, calibration 244-5
periodicity considerations for random number generators 147-8
physical measures 9, 10-11
P(I)DEs see partial (integro) differential equations
pivots 119-22
Poisson process 126-7, 129, 160, 171, 196, 198-201, 209-16
see also jump . . .
polynomial time algorithms 255
polynomial-type interpolation functions 81-115, 185-91, 214-16
portability considerations for random number generators 148
portfolios of financial instruments 1, 8-9, 20-38, 217, 265-8, 269, 276, 297-300
computational finance definition 1
optimization techniques 265-8
power sum40 algorithm 158-9
power sum100 algorithm 158-9
preconditioning procedures, linear equations 129-31
predictor-corrector methods 29
present value (PV) 40, 69-70, 73
prices
see also valuations
American options 3, 179-91, 202-3
Asian (arithmetic average rate) options 72, 164-6
Bermudan options 3, 179-91, 202-3
binomial trees 1-2, 5, 8-16
Black-Scholes model 2, 3, 5, 10-11, 13-14, 17-38, 71-2, 77-8, 165, 167-8, 180-1, 185-6, 194-207, 209-10, 250, 290-5
calibration 1-5, 21, 138-9, 193-4, 205, 207, 239-51, 291-5
caplets 44, 143-4, 245
caps 43-4, 73, 297-300
characteristic function methods 3, 5, 193-207
exotic options 3, 14-16, 52-3, 71-2, 75-7, 164-6, 179-91, 202-7, 210-16, 249-51, 290-5
Monte Carlo simulations 133, 134-9, 146-60, 161-77, 179-91, 245, 288, 290-5
options 2, 3, 7-16, 17-38, 50-2, 71-2, 74-5, 138-9, 164-6, 167-8, 179-91, 193-207, 209-16, 239-51, 255-68, 290-5, 297-300
parallel architectures 288-95
prime numbers 148-52
principal component analysis (PCA) 177, 271, 272, 274, 276-83
concepts 276-83
definition 272
probabilities 3, 8-9, 17-23, 193-4, 230-7, 259-68, 270-83
probability density functions 3, 193-4, 278-83
problem size direct methods 3
pseudo random numbers 3, 146-7, 151-2, 160, 161, 169-75
see also low-discrepancy sequence theory; Monte Carlo simulations
pseudo-Maximum-Likelihood method 231
put options 2, 3, 7-16, 17-38, 50, 63-70, 71-2, 75, 179-91, 205-7, 245-51, 255-68, 290-5
puttable (retractable) fixed rate bonds, upwinding techniques 2, 63-70
QMC see Quasi Monte Carlo method
Quad/Octree unstructured-mesh generation technique 84-5
quadratic elements, finite element methods 88-90, 93-8, 106-9, 117-31
Quadratic Programming (QP) 254-5, 257-8, 265-8, 293-4
quantile-quantile plots 194-6
quantitative methods 1-5, 21, 81, 86, 159-60, 217, 239
quarterly compounding 39-40
Quasi Monte Carlo method (QMC) 3, 72, 154-5, 161, 169-77, 193, 209, 288-95
concepts 3, 154-5, 161, 169-77, 193, 209, 288, 290-5
definition 161, 169-75
parallel architectures 288, 290-5
Quasi-Newton methods 257-8
random number generators 3, 146-60, 169-75, 290-5
acceptance-rejection methods 154-6, 160
Box-Muller random number algorithm 155-6
commonly used distributions 155-60, 162-3
congruential random number generators 148-52
definition 146-7
inverse transform method 152-4, 156-7
Mersenne Twister random number generator 151-2, 171-5, 291-5
properties 147-50
recent developments 151-2, 171-5
tests 150-1, 169-75
transformation of variables 152-5
random permutation of digits technique 174
random shift technique 174
random vectors 150-1, 174-5
random walks 176-7, 245
randomizing QMC 174-5
randomness considerations for random number generators 148, 174-5
rank correlation coefficients see Kendall’s . . . ; Spearman’s . . .
reading guide 5
real-world examples, overview of the book 1
Rebonato’s formulation of the LMM 140-6
receiver swaps 41-3
receiver swaptions 44-5
rectangle (mid-point) rule 215-16
rectangular elements, finite element methods 90, 98, 103, 109, 113-15
recursion 180-91
regression analysis 179-91
regularization methods, calibration 243-7, 297
Rendleman-Bartter tree 11, 12-14
replication methods
IRSs 41-2
options 8-9
reproducibility considerations for random number generators 147-8
return/risk ratios 218
returns 11-12, 17-38, 193-6, 218, 266-8
Reuters 7
reverse floaters 73-4, 186-91
rho 22
see also interest rates
Riemann integral 18, 214-16
risk analyses, overview of the book 1-5
risk management 1-5, 217, 218, 221-2, 239, 249-51, 253, 269-83, 297-300
see also Expected Shortfall; Value at Risk
concepts 1, 4, 5, 218, 221-2, 269-83, 297-300
definition 269, 298-9
large software systems 4, 5, 297-300
overview of the book 1-5
risk measurements 218, 221-2, 269-83, 297-300
risk types 269
risk-free rates 9, 10, 21, 55-6, 71, 167-8, 202-7, 212-16, 240-51
risk-neutral measures 9, 10-11, 135-9, 165-6, 193-207, 209-16
Runge-Kutta methods 29
sampling, copulas 229, 232-4
scalar risk factors 271, 274, 276-83
scenario index 272-3, 277-83
scenario sensitivity 269-70, 272-83, 298-300
scrambled net technique 174
SDEs see stochastic differential equations
security aspects of software 298-9
self-fulfilling prophecies 239
semi-annual compounding 39-40
semi-implicit finite difference method 29, 30, 33-6, 66-70, 117-31
sensitivity analysis 269-70
Sequential Quadratic Programming (SQP) 257-8
series expansions 3, 24-7, 28-9, 59-61, 136-9, 193-4, 201-7
settlement day conventions 40
SGI Altix 4700 CPUs 291
shape functions, finite element methods 85-90, 94-115
shared memory systems 285-8, 290
short positions 20-3
short-rate models 2, 39, 45-53, 55-70, 71, 73, 86, 139, 185, 186-91, 241-5
see also models
concepts 45-53, 55-70, 71, 73, 139, 185, 186-91
definition 45-6
PDEs 55-6, 186-91
upwinding techniques 56-70
SIMD (single instruction, multiple data) 285-9
Simpson rule 215-6
Simulated Annealing (SA) 259-60, 263-4
skewness 194-7, 199-200
see also fat tails
Sklar’s theorem 222, 231
snowball floaters 75-7, 185-6
Sobol sequences 169, 173-5, 177, 262, 290-5
see also Quasi Monte Carlo method
software 1, 4, 5, 12, 50, 68, 119-22, 146-8, 155-60, 170, 180, 182-4, 285-95, 297-300
see also hardware environments; parallel architectures
accompanying software 4
coding requirements 1,12, 50, 68, 119-22, 170, 180, 182-4, 289-95
concepts 1, 4, 297-300
databases 297-300
large software systems 4, 5, 297-300
load balancing and computing scalability 298-9
market data 298-300
open architecture and multi-library approaches 299-300
security aspects 298-9
UnRisk software package 1, 297-300
user administration 298-300
solvers methods 3, 101, 117-31
SOR see successive over-relaxation iterative method
space grids 24-38, 57-8, 65-70, 107, 127-31, 211-14
spatial discretization, finite difference methods 2, 24-38,56-70,211-14
Spearman’s rank correlation coefficient 217, 218-20
speed considerations for random number generators 148, 186
SQP see Sequential Quadratic Programming
stabilization considerations 2, 30, 34-8, 48-53, 55-70, 77, 103-4, 126, 130-1
see also finite . . . ; oscillations
stabilized bi-conjugate gradient iterative method 126, 130-1
stable and robust schemes 1, 297-300
standard deviations 19-23, 133-60, 194-9, 218-37, 240, 271-83
see also variances; volatilities
steepeners 137-9, 146, 177, 188-91
steepest descent method 256-7, 258
stochastic calculus 17-21, 45-6, 55-6, 195-6
stochastic differential equations (SDEs) 3, 10-11, 13-14, 17-20, 133, 135-9, 165-6, 168, 193-207, 209-16
see also Black-Scholes SDE
concepts 17-20, 135-9, 195-6, 209-16
discretization 135-9, 165-6
stochastic interest rates 3, 45-6, 77-9, 139-46, 180-91
see also Libor market model
stochastic processes 3, 18-23, 39-53, 55-6, 77-9, 105-7, 133, 180-91, 221-2, 235-7, 245-51
see also Brownian motion
stock exchanges 7-8
stopping criterion, Fourier-cosine series expansions 206-7
stratified sampling 168-9
streamline-diffusion stabilization method (SD) 104-7
streamline-upwind-Petrov-Galerkin stabilization method (SUPG) 104-7
stress tests 239, 298-300
strike prices 7-16, 20-38, 50-2, 71-2, 138-9, 164-6, 194-6, 206, 210-16, 239-51, 255-68, 290-5, 298-300
structured meshes, finite element methods 83-5
subdomain weighting function 82
subgrid scale stabilization method (SGS) 104
successive over-relaxation iterative method (SOR) 122, 125
‘super-bee’ limiter 63
SUPG see streamline-upwind-Petrov-Galerkin stabilization method
swaps 41-5, 73, 76-7, 109, 137-9, 146, 181-91, 234-7, 245-51, 269, 271, 297-300
see also interest-rate . . .
concepts 41-5, 109, 137-9, 181-91, 269, 271, 297-300
valuations 42, 43-5, 73, 137-9, 146, 181-91, 297-300
swaptions 43-5, 47, 138-46, 245-51, 297-300
Black76 approximations 44-5, 139-40, 245, 297
definition 44-5
t-copulas 223, 225, 231, 232-7
target redemption notes (TARNs) 76-7
Taylor series expansions 24-7, 28-9, 59-61, 136-9, 256-7, 272, 277
term sheets 2, 71-9, 297
terminal conditions 2, 5, 20, 23, 71-9, 139, 193-207, 210-16
equity options 71-2, 139
interest-rate instruments 72-3
testing functions 81-2
see also weighted residual methods
TestU01 software library 150
theta 22, 29, 32-8, 165-6
see also expiry times
Thomas algorithm, linear equations 119-20
threads, parallel architectures 286-95
Tikhonov regularization 243-7
time discretization 2, 24, 27-38, 56-70, 97-8, 103-4, 105-7, 117-31, 135-9, 162-6, 175-7, 180-91, 193-207, 209-16
time grids 24-38, 57-8, 65-70, 73, 135-9, 211-14
time series 217, 270, 272-3, 299
timestep restrictions, trinomial trees 49-53
total variance diminishing framework (TVD) 62-3
transaction costs 21, 29
transformation of variables, random number generators 152-5
trapezoidal rule 137, 211, 215-16
trees 2, 5, 8-16, 30, 39, 45, 47-53
see also binomial . . . ; trinomial . . .
triangular elements, finite element methods 88-90, 93-104, 109, 112-14
tridiagonal matrices 27, 33-6, 66-70, 117-31
trigonometric interpolation functions 81, 155-6
trinomial trees 2, 5, 45, 47-53, 56
see also Hull-White interest-rate model
critique 52-3
definition 47-9
down/up branching regimes 48-53, 56
overview 2, 5, 47
stabilization considerations 48-53, 56
timestep restrictions 49-53
trust regions 258
TVD see total variance diminishing framework
two-factor interest-rate models 52, 78-9, 105-7, 117, 137-40, 146, 177, 245
see also Hull-White . . .
uncertainty sources, Monte Carlo simulations 133
underlying assets 3-5, 7-16, 17-38, 39-53, 71-9, 86, 164-6, 181-91, 194-207, 209-16, 234-7, 239-51, 269-83, 290-5, 297-300
see also calibration
uniformly distributed random numbers 147-50, 169-75, 290-5
UnRisk FACTORY software 297-300
UnRisk software package 1, 297-300
see also MathConsult
unstructured meshes, finite element methods 84-5
up-and-out call options 15-16, 249-51, 290-5
up-branching trinomial-tree regimes 48-53, 56
upwinding techniques 2, 5, 56-70, 211-16
see also finite difference methods; mean reversion
definition 56-63
first order upwind schemes 57-61, 63
higher order upwinding schemes 61-3
Lax-Wendroff upwinding scheme 61-3
overview 2, 5, 56-7
puttable (retractable) fixed rate bond example under the Hull-White one-factor model 63-70
TVD 62-3
user administration, software 298-300
valuation pools, parallel architectures 288
valuations 1-5, 7-16, 17-38, 71-9, 133, 134-9, 162-3, 165, 167-8, 179-91, 245, 288-95, 297-300
see also prices
Black-Scholes model 2, 3, 5, 10-11, 13-14, 17-38, 71-2, 77-8, 165, 167-8, 180-1, 185-6, 194-207, 209-10, 250, 290-5
overview of the book 1-5
parallel architectures 288-95, 299
swaps 42, 43-5, 73, 137-9, 146, 181-91, 297-300
Value at Risk (VaR) 4, 217, 218, 269-76, 277-83, 288, 298-9
see also contribution . . . ; historical . . . ; individual . . . ; Monte Carlo . . . ; parametric . . .
definitions 269-76, 281
van der Corput sequences 170-5
vanilla floaters, definition 40-1
Variance Gamma model (VG) 196, 199-200, 203, 249-51
variance-reduction efficiency techniques, Monte Carlo simulations 152, 161-9
variances 11-12, 18-23, 44-53, 140-6, 152, 157-60, 161-9, 196-207, 218-21, 248-51, 265-8, 270-83
Vasicek interest-rate model 2, 45-6, 48-51, 78-9, 136, 162-3, 186-91
artificial boundary conditions 78-9
critique 46
definition 45-6
vector of unknowns, definition 117
vega 22, 239-40
see also volatilities
VG see Variance Gamma model
volatilities 10-11, 12-16, 19-23, 29, 44-5, 49-53, 65-70, 74-5, 138-46, 165-6, 168, 186-91, 194-207, 239-51, 271-83, 298-300
see also implied . . . ; standard deviations; vega
concepts 10-13, 19-23, 29, 142-3, 194-6, 199-200, 298-300
model parameters 248-51
volatility smiles 144-6, 194-7
see also Dupire local volatility model; implied volatilities
volatility of variance 197, 248-9
Volterra integral equations 241
warps, parallel architectures 289
warrants 240
Wavelet-based methods 193
weighted residual methods, finite element methods 81-3, 97-8
Wiener processes 10-11, 17-21, 45-6, 135-6, 171, 191, 199-200, 248-9
see also Brownian motion
‘wrong’ points, one-dimensional regression analysis 186
yield curves 42-5, 48-53, 73, 138-46, 165-6, 168, 207, 240-51, 271-3, 277-83, 298-300
zero-coupon bonds 42-53, 69-70, 73, 78-9, 105-7, 138-46, 162-9, 274-6, 298-300
zero-coupon curves 42-5, 73, 138-46, 274-6, 298-300
Index compiled by Terry Halliday