adapted process, 108, 135, 183
ADI, see alternating direction implicit
alternating direction implicit, 104, 115
analytic solution, 30, 45, 85, 86
antithetic variates, 90
Apple, 159
arbitrage, 2, 3, 8, 19–21, 34, 53, 60, 61, 86, 109, 110, 123, 137, 164, 170, 180
arbitrage-free price, 25, 35
at-the-money, 28, 51
barrier
continuous, 93, 101, 116, 125, 127
discretely monitored, 49, 93, 101, 149
knock-in, 45, 59, 101, 107, 130
knock-out, 45, 59, 107, 130
lower, 49
single, 49
upper, 49
window, 122
barrier option, see option, barrier
bisection search, 83
Black-Scholes, 4, 26, 40, 42, 43, 51, 52, 54, 59, 73, 118, 121, 136, 160, 177
Black-Scholes equation, 29–30, 36, 37, 59, 61
Black-Scholes formula, 26–29, 31, 34, 44, 51, 53, 61, 86, 112, 113, 129, 142, 148, 150, 170
Black-Scholes process, 48, 78
bond, 61
boundary condition, 30, 36, 37, 59, 68, 69, 74
Feller, 71–73
zero gamma, 96
zero price, 95
Box-Muller transform, 88
Breeden-Litzenberger, 56, 116, 137, 170
broker, 51, 109, 157
Brownian motion, 9–12, 32, 45, 65, 77, 87, 92, 108, 117, 119, 154, 163, 183
correlated, 17, 161
bump function, 58
calendar spread, 110, 111
calibration, 63, 64, 71, 124–125
call, 3
call spread, 56, 91
callable trade, 93, 101, 106
cash settlement, 2, 157
cell averaging, 101
central bank, 5
central difference, 95, 103, 104
CEV, see constant elasticity of variance
Chase Manhattan, 131
Cholesky decomposition, 90
CIR, see Cox-Ingersoll-Ross model
close out, 2, 147
CMS, see constant maturity swap
cockroach, 141
commodities, 2, 8
constant elasticity of variance, 74
constant maturity swap, 158
continuous barrier option, see option, barrier
copula, 154–155, 170
correlation, 17, 54, 64, 89
asset-asset, 153, 154
implied, 156, 157, 160–162, 175
instantaneous, 153, 166
local-stochastic, 166
spot-volatility, 65, 66, 73, 84, 113, 116, 117, 122, 127, 130, 142, 165
correlation matrix, 164
correlation product, 153
correlation smile, 155–162, 173–175
cost of carry, 8
Cox-Ingersoll-Ross process, 66, 71, 72
Craig-Sneyd scheme, 105
Crank-Nicolson scheme, 97, 99, 105
credit rating, 5
cross asset, 160
cross smile, 160, 162–164, 170, 173
cross volatility surface, see cross smile
cumulative normal function, 28, 46
bivariate, 155, 168
inverse, 86
delta, 38, 85, 138, 139, 141
Black-Scholes, 52, 112, 137, 161
forward, 52, 137
delta function, see Dirac delta function
delta hedging, see hedging, delta
derivative, 1, 3
derivatives trading, 40
digital payout, see option, digital
Dirac delta function, 56, 69, 70, 110, 115, 172
discount factor, 27, 31, 33, 55, 56, 67, 86
discounted expectation, 27, 34
dividend, 8, 19, 20, 61
domestic currency, 6
drift, 12, 16, 21, 47, 61, 121, 134, 161, 163, 183, 184
terminal, 16
Dupire local volatility model, see local volatility model
dynamic trading strategy, 36
efficient market, 2
eigenvalue, 98
energy, 2
equities, 2, 8, 51, 54, 141, 153
equivalent probability measure, see measure, equivalent
euro-dollar, 51, 53, 153
European payout, see option, European
exchange, 2
expectation, 180
expiry date, 3, 51, 82, 83
expiry time, see expiry date
explanatory variables, 94
explicit scheme, 96
exponential Ornstein-Uhlenbeck model, see Ornstein-Uhlenbeck process
factorisation method, 104
federal funds rate, 5
Feller boundary condition, see boundary condition, Feller
filtration, 15, 58, 108, 135, 147, 183
financial engineer, 183
financial mathematician, 183
finite difference method, 86, 94, 104, 106, 115, 116, 145
first generation exotic option, see option,
first generation exotic
fixed income, 6
floating smile, 112, 113
Fokker-Plank equation, see forward Kolmogorov equation
foreign exchange, 2, 8, 33, 51, 53, 63, 82, 112, 125, 140, 141, 153, 157, 160
forward contract, 1, 6–8, 19, 27, 40, 52, 53, 57, 107
forward induction, 114, 116, 124, 125, 131, 177
forward Kolmogorov equation, 58–60, 62, 71, 84, 115
forward level, 1, 8, 27, 28, 33, 35, 43, 44, 121, 140, 146, 147
forward start, see option, forward starting
forward volatility agreement, 146–151
asset, 147
cash, 147
Fourier transform, 69, 70
FTSE, 54, 63
fundamental theorem of on-smile pricing, 108–110, 114, 146
FVA, see forward volatility agreement
FX, see foreign exchange
gamma, 38–40, 85, 139
Gauss-Hermite integration, 122
Gaussian, see normal variable
Girsanov‘s theorem, 22–24, 34, 45, 47, 179–184
going short, see short position
gold, 8, 19, 107, 158, 160, 165, 172, 173
government bond, 20
Greeks, 38–39, 41, 85, 106, 123, 138, 177
Gyöngy‘s theorem, see fundamental theorem of on-smile pricing
heat kernel, 75
hedge fund, 40, 140
hedging, 1, 85, 123, 141, 178
delta, 37–38, 40, 41, 63, 133, 137, 177
dynamic, 35–37, 140
hedging portfolio, 36
hedging strategy, 38
Heston model, 65–73, 75, 78, 83, 84, 88, 106
IBM, 29, 157, 159
implicit scheme, 97, 103, 106, 115, 131
implied volatility, see volatility, implied
implied volatility dynamic, 111, 112
implied volatility smile, see smile
implied volatility surface see volatility surface, 51
in-the-money, 29, 35, 51, 141
inflation, 20
insurance company, 35
integral
Lebesgue, 181, 182
Riemann, 181
integration by parts, 149
interest rate, 5–6, 19, 51, 61, 121, 126, 140
continuously compounding, 5, 19
stochastic, 87
interfacing, 102, 145
investment bank, 35, 40, 42, 122, 133, 140, 160, 165
Ito calculus, 12
Ito process, 13, 34, 179
Ito‘s lemma, 13–14, 16–17, 36, 58, 67, 73, 74, 109, 115, 118, 135, 184
product rule, 14, 26, 184
Jacobian, 88
joint probability distribution, see probability distribution, joint, 93
JP Morgan, 131
jumps, 110, 135, 140
knock-in barrier, see barrier, knock-in
knock-out barrier, see barrier, knock-out
lambda-SABR model, 75, 84, 114, 118, 124, 126, 127, 165
Laplace transform, 71, 72
law of large numbers, 87, 88
Lehman bankruptcy, 63
Lévy‘s theorem, 184
linear congruential generator, 87
liquid asset, 3
local correlation model, 162–164, 170, 172
local stochastic correlation model, 165–166, 172
local stochastic volatility model, 54, 84, 113–116, 121, 122, 125, 127, 130, 131, 135, 143–145, 149, 165, 177
local volatility correction, 114–118, 121, 122, 124, 125, 130, 131
local volatility model, 54, 58–62, 82, 84, 86, 94, 107, 109, 111, 112, 118, 123–127, 130, 149, 172
multi-asset, 153, 155
log contract, 136, 152
log return, 134, 137, 139
log-normal, 14–15, 26, 27, 43, 44, 47, 67, 75, 87, 94, 114, 119, 120
long expiry date, 84, 140
Longstaff-Schwarz, 94
low discrepancy random numbers, see random numbers, low discrepancy
LSV, see local stochastic volatility model
LVCC, see local volatility, multi-asset
market data bump, 85
market maker, 40, 109, 160
Markov chain, 79
Markov property, 15
Martingale, 24–26, 30, 34, 61, 67, 135, 136, 184
continuous, 25
Martingale pricing, 19
Martingale pricing equation, 25, 27, 32, 86, 105, 169
matching delta, 173, 174
mean reversion, 63, 64, 66, 72, 75–77, 84, 117, 118, 121, 123–125, 127, 151, 152
mean reversion level, 64
measure, 21, 26, 34, 180
equivalent, 22, 181, 183
real world, 183
risk-neutral, 64, 66, 148, 166
measure change, 21–24, 119, 161, 166, 169, 179–184
measure theory, 179
Mersenne twister algorithm, 87
method of characteristics, 72
method of planes, 102, 106, 144, 150
mixing parameter, 114, 124–131, 144, 145, 152
mixture model, 77–79, 122, 123, 126, 127, 131, 145, 166
model uncertainty, 178
moment matching, 44, 50
money market account, see rolling money
market account
Monte Carlo, 30, 86–94, 105, 106, 144, 154, 164, 167
error, 90
Greeks, 91
multiple dimensions, 88
standard deviation, 88
variance reduction, 90
multiple assets, 16, 153
Newton-Raphson method, 86
no arbitrage principle, see arbitrage
normal process, 76, 119, 127
normal variable, 9, 15, 87, 88, 122, 154, 155
notional, 2, 35
numeraire, 21, 25, 26, 29, 31, 33, 34, 48, 49, 61, 68, 86, 161, 163, 166, 169, 172
numeraire symmetry, 172–173
numerical integration, 65, 86, 92, 122
numerical stability, 78, 83, 85
ODE, see ordinary differential equation
off-smile, 126
oil, 2
OIS, see over night indexed swap
one-sided differencing, 96
optimisation problem, 82
option
American, 93
Asian, 43–45, 50, 102, 106, 144
at-the-money, 28, 40, 54, 63, 129, 142, 147
barrier, 45–50, 59, 114, 116, 118, 119, 121, 122, 124, 125, 130, 131, 142, 151
basket, 153, 157, 171, 173–175
best-of, 34, 158, 167–169, 171
call, 19, 26, 31, 38, 48, 54, 60, 63, 68, 109, 153
composite, 159, 162, 163, 172, 173, 175
digital, 31, 43, 55–57, 68, 91, 116, 128, 129, 149
dual asset digital, 171
European, 42–43, 50, 55, 57, 61, 86, 109, 122, 131, 154, 167
exercise, 3
exotic, 2, 4, 59, 64, 107, 114, 123, 124, 126, 130, 131
first generation exotic, 116, 122, 131, 143, 144
forward starting, 123, 147
geometric basket, 159
in-the-money, 40, 54
no-touch, 130, 149, 165
one-touch, 124–128, 130
out-of-the-money, 40, 54, 63
put, 54, 55, 63
quanto, 157
short dated, 39, 52
spread, 158
vanilla, 3, 29, 38, 43, 54–57, 59, 61, 64, 65, 67, 82, 86, 107, 116, 133, 136, 140
worst-of, 158, 171
ordinary differential equation, 69
Ornstein-Uhlenbeck process, 76–77, 84, 125
OU process, see Ornstein-Uhlenbeck process
out-of-the-money, 29, 51, 141
overnight, 39
overnight indexed swap, 5
parallel shift, 151
parameter averaging, 84
partial differential equation, 29, 30, 34, 58, 59, 61, 64, 67–69, 81, 115, 124, 126, 131, 154
multiple dimensions, 103–105
numerical solution, 30, 83, 86,
94–106, 144, 166, 177
parabolic, 95
stability, 96–100
payout, 3
PDE, see partial differential equation
PDE approach, see partial differential equation, numerical solution; see also finite difference method
PDE solver, 101–103, 106, 115, 116, 131, 150, 166, 167
PDF, see probability density function
percentage notional, 29
physical settlement, 157, 159
positive definite matrix, 164
pounds sterling, 20
precious metals, 2
predictor-corrector method, 104
present value, 27, 121
probability, 181
risk-neutral, 68
probability density function, 55, 58, 59, 71, 72, 167
probability distribution, 58, 59, 108
cumulative, 55
implied, 55, 86
joint, 45, 148, 154, 169–172
marginal, 148, 154, 155
probability measure, see measure
probability space, 180, 182, 183
product rule, see Ito‘s lemma, product rule
pseudo-random numbers, see random numbers, pseudo- random
put, 3
put-call parity, 53, 147, 148, 159, 171
PV, see present value
quant, see quantitative analyst
quantitative analyst, 2, 27, 33, 150, 183
quasi-random numbers, see random
numbers, low discrepancy
Radon-Nikodým derivative, 22, 47, 48, 119, 182, 183
random numbers
low discrepancy, 92
pseudo-random, 87, 92
random variable, 180
Rannacher stepping, 100
real world probability measure, see measure, real world
real world process, 19, 21, 22, 26, 35, 37
realised variance, 133, 136, 141, 145
reflection principle, 45–48, 50, 119
regime switching model, 79–81, 145, 166
regularisation, 164
replicating portfolio, 7
replication, 7, 35, 40, 56, 57, 86, 128, 140, 141, 149, 152, 177
static, 178
Reuters, 133
rho, 38
risk, 85
risk free rate, 5, 20
risk management, 122
risk manager, 85, 176, 178
risk reversal, 82
risk-neutral expectation, 136, 147
risk-neutral measure, 27, see measure, risk-neutral
risk-neutral probability, 52, 56, 60, 61, 137
risk-neutral process, 26, 31, 42
rolling money market account, 20, 25–27, 29, 31, 42, 67
SABR formula, 74, 75, 82
SABR model, 73–75, 78, 82–84, 113, 118, 125
sample space, 179–181
SDE, see stochastic differential equation
seed, 91
self-financing portfolio, 36, 37
short expiry date, 84
short position, 35
shorting, see short position
sigma algebra, 179–182
silver, 165, 172, 173
skew, 51, 54, 63, 64, 82, 83, 125, 129, 130, 138, 142
smile, 45, 51–55, 59, 61, 63–65, 74, 77, 78, 83, 84, 107, 111, 112, 124, 126, 129, 130, 133, 137, 138, 142–144, 148, 149, 151, 160, 170, 178
convexity, 63–66, 74, 77, 78, 82–84, 118, 124, 125, 138
interpolation, 52
Sobol sequence, 92
speculation, 2
spot, 2
spot dynamic, 54, 59
spot price, see spot
spread, 40
standard deviation, 95
step function, 31, 55
sterling-dollar, 153, 157
sticky delta, 112–113, 124, 125, 138
sticky strike, 111–112, 124, 125, 138
stochastic differential equation, 12, 58, 62, 65, 73, 74, 118, 163, 173
stochastic integral, 9, 22
stochastic process, 9
stochastic variance, 71
stochastic volatility model, 54, 63–65, 73, 78, 83, 107, 111, 113, 114, 116, 118, 122–124, 127, 130, 142, 143, 149, 151
calibration, 82–84
straddle, 146, 147
strangle, 82
strike, 2, 35, 51, 55, 60, 82, 111
at-the-money, 51
supply and demand, 53–55
SV model, see stochastic volatility model
swap rate, 5
Taylor expansion, 29
term structure, 6, 16, 28, 42, 64, 118,
120–123, 126
terminal volatility, 118
theoretical value, 126, 127
theta, 38, 40
theta scheme, 97
Thomas algorithm, 97, 104
time decay, 39
time value of money, 111
tradeable asset, 20, 25, 26, 31
trader, 27, 37, 42, 85, 91, 124–126, 130, 141, 149, 151, 174, 176
trading strategy, 35
triangle rule, 160–163, 165, 166, 173
TV, see theoretical value
unconditional stability, 100, 105
United States dollar, 2, 20
vanilla, see option, vanilla
vanilla market, 59, 107–109, 113, 124
vanilla payout, see option, vanilla
variance reduction, see Monte Carlo, variance reduction
variance swap, 57, 133–141, 143, 151
vega, 38–40, 85, 118, 121, 129, 141, 150, 151, 178
vol-of-vol, see volatility of volatility
volatility, 4, 12, 16, 21, 37, 40, 47, 63, 133
at-the-money, 51, 83, 126, 129, 138, 143, 147
fair, 134, 141, 143
implied, 45, 51, 53, 54, 56, 59, 61, 63, 74, 111, 114, 130, 133, 137, 142, 146, 147, 150, 160
instantaneous, 63, 83, 84, 113, 114, 117, 120, 146
interpolation, 106, 138
local, 58–61, 108, 110, 112, 114, 117, 130, 146, 163
realised, 64, 65, 123, 133, 147
stochastic, 63, 124
terminal, 16, 121, 122, 131
volatility frown, 55
volatility of variance, 66, 72
volatility of volatility, 63, 64, 73, 74, 77, 83, 117, 118, 121–125, 131, 151
volatility process, 64
volatility smile, see smile
volatility surface, 51, 52, 54, 85, 108, 110, 116, 126, 130, 148, 165, 178
arbitrage, 109
volatility swap, 141–146, 151
instantaneous, 146
volatility trading, 40
wavelets, 106
wheat, 1, 8
winding number, 70
wingyness, 140, 143, 144
yield, 6, 19, 29
yield curve, 6, 8, 85
zero coupon bond, 20, 67, 68