Zhu J. Applications of Fourier Transform to Smile Modeling: Theory and Implementation
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Index
ABR model, 295
activity rate, 183
affine jump-diffusion, 168
alpha log-stable model, 194
alpha-stable distribution, 200
Andersen and Andreasen’s weights, 289
approximation algorithm, 101
arbitrage-free, 10
arithmetic average Asian options, 244, 252
arithmetic Brownian motion, 258
Arrow-Debreu price, 32
ARV models, 47
Asian interest rate options, 254
Asian option, 244
at-the-period, 245
Attari’s formula, 80
autoregressive random variance models, 47
backward change, 276
barrier option, 226
Bermudan swaption, 285
binary option, 266
Black-Scholes PDE, 6
Black-Scholes formula, 6
Black76 formula, 229
BMS model, 312
BNS model, 193
Bochner’s procedure, 178
bootstrapping method, 291
Box-Mueller algorithm, 114
branch cut, 100
Broadie-Kaya Scheme, 122
Brownian motion, 2
butterfly volatility, 19
caching technique, 89
calendar spread, 257
call option, 5
call spread, 266
caplet, 275
caplet volatility, 281
caps, 275
Carr and Madan’s formula, 78
cascade calibration, 291
characteristic function, 25
chi-square distribution, 49
Cholesky-decomposition, 131
chooser option, 267
CMS, 291
co-initial swaptions, 285
co-sliding swaptions, 285
co-terminal swaptions, 285
complex number, 99
complex power function, 100
compound jump process, 156
computation accuracy, 94
constant elasticity of variance, 46, 296
constant maturity swap, 291
convexity adjustment, 292
correlation matrix, 130, 287
correlation option, 257
Cox-Ingosoll-Ross model, 138
crack futures spread, 257
cross-correlation, 130
curvature of smile, 84
Del
´
eans exponential, 24
Delta, 81
Delta measure, 24
Delta volatility, 19
density measure, 174
deterministic volatility function, 46
digital option, 266
discounted CF, 32, 170
J. Zhu, Applications of Fourier Transform to Smile Modeling, Springer Finance,
DOI 10.1007/978-3-642-01808-4,
c
Springer-Verlag Berlin Heidelberg 2000, 2010
327
328 Index
discrete geometric average, 245
double square root, 63
double-exponential distribution, 166
drift, 3
drift compensator, 185
dynamic hedging, 7
eigenvalues, 132
equivalent martingale measure, 12
Euler scheme, 114
exchange option, 260
exotic options, 223
exponential distribution, 163
fast Fourier transform, 78
Feller’s condition, 49, 116
Feynman-Kac theorem, 7
FFT, 78
filtration, 2
finite activity, 176
finite moment log-stable process, 196
finite variation, 176
first passage time, 181
flat volatility, 280
floorlet, 276
floors, 275
forward change, 278
forward measure, 24
forward volatility, 281
forward-starting option, 224
Fourier transform, 25
fractional FFT, 92
full rank formulations, 287
full truncation scheme, 124
futures price, 229
FX option, 17
Gamma, 81
gamma distribution, 193, 199
gamma function, 194
gamma Ornstein-Uhlenback process, 186
gamma process, 179
GARCH, 47
Garman-Kohlhagen formula, 18
Gaussian integration, 86
generalized Fourier transform, 188
generalized hyperbolic distribution, 179
geometric Brownian motion, 3
Girsanov theorem, 12
Gy
¨
ongy theorem, 108
Heston model, 48
hitting probability, 227
Hull and White’s weights, 289
implied volatility, 15
in-the-money, 17
in-the-period, 245
infinite activity, 176
infinite variation, 176
infinitely divisible, 177
interest rate derivatives, 273
inverse Gaussian distribution, 181, 200
inverse transform method, 199
inverse-transform algorithm, 114
investment certificates, 223
It
ˆ
o process, 4
It
ˆ
o’s lemma, 4
iteration algorithm, 101
ITM knock-out, 230
jump intensity, 158
jump size, 158
jump-diffusion process, 156
Kahl-J
¨
ackel scheme, 124
Kolmogorov’s backward equation, 39
Kou model, 165
kurtosis, 3, 31
L
´
evy exponent, 176
L
´
evy measure, 174
L
´
evy process, 174
L
´
evy triple, 176
L
´
evy-Khintchine theorem, 176
Laplace transform, 188
leptokurtic, 46
leverage effect, 188
leverage-neutral measure, 189
Libor, 275
Libor Market Model, 274
linear swap model, 292
Lipschitz condition, 4
LMM, 274
local volatility model, 46
local volatility process, 108
log-normal scheme, 117
lognormal jump, 160
Longstaff model, 144
lookback option, 235
main argument, 99
market completeness, 9
market price of risk, 8
Markovian projection, 108, 299
martingale, 12
mean Vega, 82
mean-reversion property, 46
Milstein scheme, 115
Index 329
mimicking process, 110
model calibration, 104
model risk, 220
modified asset process, 140
modified geometric average, 254
modular pricing, 205
modulus, 99
moment-generating function, 31, 178
moment-matching, 115
money market account, 15
moneyness, 16
multi-asset model, 130
multi-domain integration, 88
nested stochastic volatility, 306
non-central chi-square distribution, 49
normal inverse Gaussian model, 181
numeraire, 14
ordinary differential equation, 3
Ornstein-Uhlenbeck process, 56
OTC markets, 223
OTM knock-out, 227
out and in parity, 233
out-of-the-money, 17
out-of-the-period , 245
Pareto jump, 163
partial integro-differential equation, 157
payer swap, 282
Piterbarg model, 298
Poisson distribution, 198
Poisson process, 154
portfolio, 5
present value, 7
principal components, 132, 309
principle component analysis, 309
probability measure, 2
probability space, 2
product option, 264
programming codes, 96
put option, 7
put spread, 266
put-call parity, 262
PVBP, 284
QE scheme, 120
quadratic variation, 176
quotient option, 263
Radon-Nikodym derivative, 12
Rayleigh distribution, 129
Rebonato formula, 290
receiver swap, 282
reflecting principle, 228
remaining uncertainty, 32
risk sensitivity, 80
risk-neutral valuation, 7
risk-reversal volatility, 19
rotation count algorithm, 102
SABR model, 294
Sch
¨
obel-Zhu model, 56
self-financing, 9
short rate process, 22
Simpson’s rule, 92
simulation of maximum, 126
simulation of minimum, 126
skew, 16
skew function, 296
skewness, 3, 31
slope of smile, 84
smile, 16
smile surface, 16
sneer, 16
soybean complex spread, 257
spot measure, 279
spot Vega, 83
square root process, 48
stochastic clock model, 177
stochastic differential equation, 3
stochastic volatility model, 46
stop-loss strategy, 226
strike vector computation, 88
subordinator, 183
swap annuity, 284
swap contract, 282
swap market model, 284
swap rate, 284
swaption, 284
symmetric smile, 16
tenor structure, 275
term structure of interest rates, 137
terminal correlation, 288
terminal distribution, 109
terminal measure, 279
time-homogenous process, 4
trading strategy, 9
transformed volatility scheme, 118
Trapezoid rule, 92
truncation function, 175
truncation scheme, 117
two-factor model, 259
Vanna, 84
variance-gamma model, 179
Vasicek model, 142
330 Index
Vega , 8 2
volatility, 3
volatility persistence, 46
volatility smile, 16
volatility term structure, 16, 282
Vo l g a , 8 4
Wiener process, 2
Wu and Zhang model, 302
zero-coupon bond, 24, 136
Zhu model, 305