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