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531

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Index

accreting Bermudan swaptions 381–382adjustment factors, Black–Scholes

paradigm 120–121advanced numerical engine objects 26–27advanced security objects 26–27aggregation grids, Monte Carlo 424American options 66, 106–110analysis

see also basket option analysisContingent Convertibles 446–448hybrid optionalities 459–466product of two calls 518–519reversion products 499–501roll-up mortgages 508–511

annuity measure, European swaptions 333arbitrage, collateral 250–251architecture, object-orientated 27–29Asian options

barrier 395basket analysis 203–205Black–Scholes paradigm 93–100conditioning geometric mean 93–98House Price Index-linked 512

Asian tails, basket options 204–205ATM Vega map, Bermudan swaptions 385at-the-money-forward (ATMF) volatility 261–262attributes tables, approach 28autocallable (kickout) products

dual-index 176equity-linked 171–175hedging 173–175multi-step copulas 210–213specific risks 173–175variations 175

automatic exercise, physical settlement 68–69automatic restriction, CoCo 438

Bachelier (normal) caplets 258–260, 272–274backward pricing PDE 334–335bail-in regime, Basel III 42–43barrier events, TARN 395

barrier optionsBlack–Scholes 110–113discrete dividends 66–67local volatility framework 131–132stochastic local volatility framework 158–159

barrier reverse convertibles 170–171, 183–186Basel III

capital requirements 43–45capital structure 41–43CoCo 435–450counterparty credit risk 45–47internal model method 48–55leverage 47liquidity 47–48principles 40–41technical requirements 41–48

basic numeric objects 26–27basis sensitivity, PRDC swaps 415–416basket asset returns, general expression 197–198basket options

Asian options 203–205autocallable 172–173, 210–213Bermudan swaptions 382Black–Scholes 100–103copula pricing models 198–213, 217–219copula terminal mapping 210–213correlation influence effects 219–222correlation skew 205–210, 223–224equity-linked 172–173, 175–177equity-linked products 197–225European options 203–204, 208–209historic volatility surfaces 213–216implied volatility surfaces 217–224integration limits 200multi-step copulas for auto-callable

baskets 210–213pricing 101–102, 373pricing model comparisons 202–205risks 197–198singular value decomposition 211–212skew and implied volatility surfaces 219–224

COPYRIG

HTED M

ATERIAL

532 INDEX

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basket risks, Black–Scholes paradigm 102–103basket skew

correlation influence 219–222correlation skew influence 223–224

BBF formula see Berestycki, Busca and Florent formula

BDT model see Black–Derman–Toy modelbenefits of wrapper categories 33–34Berestycki, Busca and Florent (BBF)

formula 140–142Bermudan swaptions 250, 303, 339–341,

381–386bespoke options, Black–Scholes 104–106bi-variate t-distribution 374BK model see Black–Karasinski modelblack caplets 259, 261–262, 271–272Black formula, CMS rates 357–359Black–Scholes paradigm 87–122

American options 106–110Asian options 93–100barrier options 110–113basic framework 87–93basket options 100–103Brownian Bridges 90–93dividend futures and options 103–106hedging 117–122hindsight options 113–117lookback options 113–117partial differential equation 89Q probability measure 87–88volatility dynamics 117–122

Black volatility 300, 309Black–Derman–Toy (BDT) model 294Black–Karasinski (BK) model 294bond option decomposition 461–462bounds, copula pricing models 199–200BREN see Buffered Return Enhanced NotesBrownian Bridges 90–93Buffered Return Enhanced Notes (BREN) 169buffers, Basel III 43bullet Bermudan swaptions 381bump-and-run Vega sensitivities 308–309butterfly, equity derivatives 76

calendar spread, equity derivatives 76calibrated marginal distributions (CDFs),

CMS 364–367calibration

Cheyette model 328CMS 350–355, 360, 361, 369–372Contingent Convertibles 446–448LGM model 297–302Libor Local volatility 341–342LMM model 307–308local volatility 317–318, 336–339, 341–342quant library objects 26–27

SABR volatility surfaces 283, 291stochastic local volatility 150–154

callable Bermudan swaptions 381callable House Price Index-linked options 512callable PRDC swaps 413callable products 387–392callable range accrual notes (CRAN) 390–392callable reverse floaters (CRF) 387–390callable snowballs 389–390callable step-up fixed-rate notes 387callable zero bonds 382call option pricing

Asian options 96automatic exercise 68CMS replication 345dividend options 103FX volatility smile boundaries 409optimum exercise 69

call spreads, equity derivatives 76–79cap/floor

see also caplets; floorletsat-the-money-forward volatility 261–263calibrating 308–309callable range accrual notes 390callable reverse floaters 388CMS 369CMS products 393exotic products 380GMAB annuities 453GMDB annuities 454LGM model 299–300log-normal, shifted log-normal and

normal 258–260tenor volatility transformation 265–274vanilla interest rate options 258–274volatility surfaces 271–274

capital, Basel III requirements 40–41capital-at-risk growth products 164–168, 180–183capital-at-risk income products 165–168, 170–171capital-protected growth products 164–170,

180–183capital-protected vanilla products, China 18capital requirements

Basel III 43–45CPPI 193

Capital Requirements Directive IV (CRD IV) 40–41Capital Requirements Regulations (CRR), Basel

III 40–41capital structures

Basel III 41–43, 436CoCo 437

caplet/floorlet pricing formula 332caplets

CMS 369–370distributions, vanilla interest rate

options 258–260

Index 533


tenor volatility transformation 265–274multi-curve 269–270single curve 266–269

vanilla interest rate options 258–260volatility stripping 263–265volatility surfaces 271–274

cash-collateralized trades, CSA 238cash dividends 61–67, 130–131cash-settled swaptions 275categories of wrappers 32–34categorization, equity-linked products 163–166CBs see Convertible BondsCCB see cross-currency basisCCP see central counterpartiesCCR see counterparty credit riskCCS see cross-currency swapsCDF see cumulative distribution functionsCDFs see calibrated marginal distributionsCDS see credit default swapcentral counterparties (CCP), Basel III

requirements 41certificates, features and benefits 33CEV model see Constant Elasticity of Variance

modelchannels for distribution 36–37Cheyette model 312–318, 327–329China 16–19, 215–216Cholesky decomposition 102, 364–365CIR see Cox–Ingersoll–Ross processesCliquet options 159–161closed-form solutions

American options 107–110barrier options 110–111product of two calls 518–519rainbow barrier options 111–112

CME-LCH basis, collateral effects 251–252CMS see Constant Maturity Swapcollateral

arbitrage 250–251discounting curves 248–252interest rate derivatives 248–252special purpose vehicles 34

La Commissione Nazionale per le Società e la Borsa (CONSOB) 13–14

complex derivative product distribution 35complex financial products, Italy 14complexity, collateral discounting 249–250components

legs 27object-orientated libraries 26–27

conditioning geometric mean (CGM) modelsAsian options 93–98, 203–205basket options 101, 203–205correlation skew 207–208European basket options 203–204smile/skew adjustment 96–98

conflicts of interest 35CONSOB see La Commissione Nazionale per le

Società e la BorsaConstant Elasticity of Variance (CEV) model 294Constant Maturity Swap (CMS)

calibration 350–355convexity 343–344marginal distributions 356–359, 362–373replication and spread options 343–374spread option pricing 356–362

Constant Proportion Portfolio Insurance (CPPI) 187–193

features 187–189, 193gap options 187–189, 190–193hedging 190–193path dependency 189–190rebalancing schemes 188–189

Contingent Convertibles (CoCo) 435–450categories 436–437comparison with Convertible Bonds 439–440credit derivative approach 439–440direct modelling approaches 442–450equity derivative approach 439–440features 435–436historic time series 443–445indirect modelling 439–442joint diffusion models 448–449jump diffusion models 446–448market calibration 446–448model analysis 446–448model criteria/formulation 445–446Monte Carlo pricing 442risk factors 438–439structural asset approach 440–442

continuous dividend yieldAmerican options 66barrier options 66–67versus discrete dividends 63–67

contractual risks, basket options 103Convertible Bonds (CBs), CoCo

comparisons 439–440convexity

CMS 343–344equity derivatives 72–74fixed-rate mortgage portfolios 473–476long-dated equity hedges 74yield curve stripping 240–241

copula pricing modelsAsian basket options 203–205auto-callable baskets 210–213basket options 198–213, 224–225CMS 359comparisons 202–205correlation skew 205–210, 223–224European basket options 203–204framework 199–201

534 INDEX


Frechet’s bounds 199–200full market marginal distributions 362–373Gaussian copulas 201, 207implied volatility surfaces 217–219multi-step 210–213power copulas 202singular value decomposition 211–212Sklar’s theorem 199Student’s t-copula 202tail dependency 201terminal mapping 210–213

copula terminal mapping (CTM) 135–138, 210–213

correlation influence, basket skew 219–222correlation sensitivity, PRDC swaps 417–418correlation skew 360

basket option analysis 102, 205–210basket skew effects 223–224CGM models 207–208European options 208–209local volatility framework 209–210

correlation structuresAsian options 94–96CMS 372–373equity-linked baskets 176–177LMM model 306–307

cost of equity, market implied 439co-tenor swaptions 321–324co-terminal swaptions 324–325counterparty credit risk (CCR) 41, 45–47coupons

callable reverse floaters 387–388callable snowball 389–390cancellation, CoCo 438target redemption note 394–395volatility bond 396zero coupon range accrual 392

Cox–Ingersoll–Ross (CIR) processes 147, 294CPPI see Constant Proportion Portfolio InsuranceCRAN see callable range accrual notesCRD IV see Capital Requirements Directive IVcredit crunch 6–7credit default swaps (CDS) 230–232, 397credit elements

LOIS 231–232tenor basis swaps 232–234

credit-linked notes 397credit skew, LOIS 231–232credit spread, equity-linked products 168Credit Support Annex (CSA) discounting 238,

248–250Credit Value Adjustment (CVA)

FVA 433–434Grid Monte Carlo 422–434implementation frameworks 420linear Gauss–Markov models 425–426

multi-factor risk-neutral techniques 420–421numerical techniques 420–422portfolios 419–434valuation implementation frameworks 420XVA 433–434

CRF see callable reverse floaterscrises 4–7criteria, Contingent Convertible models 445–446cross-currency basis (CCB), multi-curve 234–237cross-currency swaps (CCS)

collateral discounting 249–250multi-currency bootstrapping 241–242multi-curve environments 234–237par bootstrap equations 243–244

cross Gamma risks, basket options 102CRR see Capital Requirements RegulationsCSA see Credit Support AnnexCTM see copula terminal mappingcubic spline volatility versus moneyness 80cumulative distribution functions (CDF)

caplet volatility stripping 264–265implied basket volatility 217–219

cumulative prepayment distributions 477–478customer care duty, vanilla products 379

DDSV see displaced diffusion and stochastic volatility

decompositionGMIB annuities

bond options 461–462swaptions 462–464

product of two calls 515–516deferred annuity products 451–453

see also Variable Annuity productsdeflated value, PDE pricing 334degrees of freedom, stochastic local volatility

framework 150–153Delta hedging

Black–Scholes 117–122CMS calibration 352, 353–354equity-linked products 194interest rate derivatives 286–290Jacobian transformations 287–290Quanto 70–71smile dynamics 290–291

deposit instrumentsfeatures and benefits 33yield curve stripping 240–241

derivative losses, high-profile 5–6development

modelling principles 22–31product design principles 22–31

diffusion models, CoCo 446–449digital options

callable range accrual notes 390stochastic local volatility framework 156–157

Index 535


digital risksautocallable products 173–174target redemption note 395

direct modelling approaches, CoCo 442–450discount curves

collateral impacts 248–252yield curve stripping 237–248

discount factors, par bootstrap equations 238–239discrete dividends 61–67, 130–131

American options 66barrier options 66–67stochastic processes 62versus continuous yields 63–67

discretization 129–130displaced diffusion and stochastic volatility

(DDSV) 310–312“distance rule”, CMS rates 356distance to trigger, CoCo 438distribution 31–37, 193–195dividend indices, equity derivatives 62dividend options, payoffs 103dividend risks 106dividends, discrete 61–67, 130–131DNT see double-no-touch (DNT)Dodd–Frank Wall Street Reform and Consumer

Protection Act (Dodd–Frank) 8double-no-touch (DNT) barrier prices 158–159DownIn Puts 185–186DownOut Puts 184–185drifts

Libor Local Volatility Model 330–331LMM model 304

dual-index autocallable products 176Dupire local volatility models 152, 154,

318–320, 322duty of customer care 379dynamic hedges 344dynamics, Internal Model Method 51–55

Early Redemption Charges (ERC) 486–488economic value analysis, Bermudan

swaptions 382–385Edgeworth expansion models, basket options 101effective local volatility 153–154Eisenhower matrices 166–168eligible capital, Basel III 40–41EM algorithm see expectation-maximization

algorithmembedded options calls, China 18embedded risks, HPI options 513EMIR see European Market Infrastructure

RegulationENE see Expected Negative Exposureenhanced premiums, equity derivatives 164–166enhanced stochastic Alpha Beta Rho volatility

surfaces 280–282

EONIA curves 377EPE see Expected Positive Exposuree-platforms 4, 36–37equal weighted indices 62Equity Conversion Contingent Convertibles 436equity derivatives 59–225

autocallable products 171–175, 210–213automatic exercise 68–69barrier reverse convertibles 170–171, 183–186basket option analysis 197–225Black–Scholes paradigm 87–122CoCo 439–440convexity adjustments 72–74copula pricing models 198–213, 217–219,

224–225copula terminal mapping 210–213correlation skew 205–210, 223–224CPPI 187–193discrete dividends 61–67Eisenhower matrices 166–168functional form implied volatility surfaces 82–84future versus forward 72–74general non-arbitrage conditions implied

volatility surfaces 75–79growth products with tax liabilities 180–183historic volatility surfaces 213–216implied volatility surfaces 74–85, 217–224issuance risks 193–195local volatility framework 123–144major indices 62market features 61–86Monte Carlo simulation 132–138multi-step copulas 210–213non-parametric implied volatility surfaces 79–82optimum exercise 69–70option settlement delay 68–70PDE solving 127–132plain vanilla options 169–170pricing model comparisons 202–205Quanto options 70–72risk control indices 178–179stochastic local volatility framework 145–162stripping 132–137structured products 163–195target volatility 177–179tax liabilities 179–183zero coupon bonds 164–166, 179–180

equity index underlyings 61–62equity-linked structured products 163–195

autocallable 171–175barrier reverse convertibles 170–171, 183–186baskets 175–177Constant Proportion Portfolio

Insurance 187–193Eisenhower matrices 166–168features 168–183

536 INDEX


fundamental categorization 163–166general payoff category 163–168growth products with tax liabilities 180–183issuance risks 193–195plain vanilla options 169–170risk control indices 178–179soft barriers 184–186spread effects 168target volatility 177–179tax liabilities 179–183zero coupon bonds 164–166, 179–180

equity release schemes 491–492female mortality base rates 494–496House Price Index-linked options 512–514male mortality base rates 492–494mortality and pricing 492–497, 499–501,

508–511portfolios 501–507principles 491–492reversion products 497–501roll-up mortgages 507–512

Equity Variance Swap 347equivalent standard volatilities 284–285ERC see Early Redemption ChargesESMA see European Securities and Markets

Authorityessential risk measures 39–40ETF see Exchange Traded FundEURIBOR

CMS quotes 348interest rate products 377LOIS 230–232

European Market Infrastructure Regulation (EMIR) 8

European optionsbasket analysis 203–204, 208–209correlation skew methodology 208–209CPPI 192discrete versus continuous 64–66House Price Index-linked 512optimum exercise 69–70product of two calls 515–516

European Securities and Markets Authority (ESMA) 9

European sovereign-debt crisis 6–7European swaptions

CMS calibration 350collateral discounting 250key features 275LGM model 299market quotes 276–277Most-Expensive 382–383, 385non-standard 284–285pricing formula 332–334SABR volatility surfaces 279–284, 291settlement rules 275

smile adjustment 277–279vanilla interest rate derivatives 274–286volatility 277–284

European Union-11 Financial Transaction Tax 9–11

European Union (EU)Basel III implementation 40–41Key Information Documents 36product wrappers and trends 12–16regulatory environments 8–11Solvency II 41

EUROSTOCK 50, historic volatility 215–216EUR yield curves 375, 377evolution of risk systems, Internal Model

Method 50–51excess return (ER) indices 62exchange-traded certificates 33exchange-traded dividend options 104Exchange Traded Fund (ETF), China 18exchange-traded notes 33exchange-traded securities

China 18United Kingdom 15

exotic product risks 380expectation-maximization (EM) algorithm 374expected lifetimes

mortality rates 492–496reversion product analysis 499

Expected Negative Exposure (ENE) calculations 425–431

Expected Positive Exposure (EPE) calculations 420–422, 425–431

exponential terms, tenor vs. 314extrapolation

caplet volatility stripping 264FX volatility

smile to long end 407–410term structure to long end 403–407

non-arbitrage equity smile/skew 78–79

fair-value, structured products 164–166fat tails, CMS 357, 367favourable pricing, interest rate products 376features of wrapper categories 33–34female mortality base rates 494–496financial risk management 39–58

Basel III 40–48FRTB 55–58

financial rule books 39–41Financial Services Compensation Scheme (FSCS),

UK 15Financial Transaction Tax (FTT) 9–11fixed coupon products, China 16–17fixed deferred annuities, features 452fixed/fixed cross-currency swaps 235fixed indexed annuities 452

Index 537


fixed legstrade compression 8vanilla interest rate derivative

swaptions 284–286fixed maturities, CPPI 187–188fixed-rate loans 378fixed-rate mortgage portfolios 473–489

convexity 473–476cumulative prepayment 477–478Early Redemption Charges 486–488instantaneous prepayment 477interest rate risk sensitivity 481–482Monte Carlo pricing 478–479option-based prepayment 476–489prepayment optionalities 473–479prepayment profiles 479–480prepayment risks 479–486propensity risk sensitivity 484–486selling period optionalities 488–489single monthly mortality data 476–477volatility risk sensitivity 482–483

fixed-rate notes, callable 387fixed tenor rolling swaps 320–321, 325, 326,

329–331fixed terminal rolling swap 324, 326flat early redemption charge schemes

486–487floating/fixed cross-currency swaps 235floating/floating cross-currency swaps 235floating legs

CMS 347–348non-standard swaptions 284–286

floating notes, multi-curve impacts 379floorlets

LGM model 299tenor volatility transformation 265–274

multi-curve 269–270single curve 266–269

vanilla instruments 258–260, 332volatility surfaces 271–274

floors see cap/floorFokker–Plank (Kolmogorov forward)

equation 149–150foreign markets, Quanto options 70–72formulation

CoCo models 445–446PDE for local volatility framework 127–128reversion derivative models 498–499stochastic local volatility models 147–150

forward FX volatilityspot volatility relationship 403–405term structure 406–407

forward measure, Libor Local Volatility Model 330

forward rates, LMM model 304, 307–308forward risks, real estate portfolios 506–507

forward smile/skewstochastic local volatility framework 154–161

barrier option 158–159Cliquet option 159–161digital option 156–157

forward start option risks, equity-linked 194forward swap rates, dynamics 320forward variances, LGM model 301FRA instruments, yield curve stripping 240–241Frechet’s bounds 199–200FRTB see Fundamental Review of the Trading

BookFSCS see Financial Services Compensation SchemeFTSE-100, historic volatility 214–215FTT see Financial Transaction Taxfull capital protection, principles 164full market marginal distributions 362–373functional form implied volatility surfaces, equity

derivatives 82–84Fundamental Review of the Trading Book

(FRTB) 55–58funding spread, equity-linked structured

products 168Funding Valuation Adjustment (FVA) 433–434fund-linked products, Italy 14fund wrappers, features and benefits 34futures

convexity adjustment 72–74yield curve stripping 240–241

future versus forward, equity derivatives 72–74FVA see Funding Valuation AdjustmentFX correlation risks, basket options 102FX forwards/swaps

multi-currency bootstrapping 244–246multi-curve environments 237

FX forward volatilityspot volatility relationship 403–405term structure 406–407

FX self-Quanto 346–347

Gammabarrier reverse convertibles 183–186Financial Transaction Tax effects 11

“Gamma trap” 21GAO see Guaranteed Annuity Optionsgap conditions, CPPI 187–189, 190–193gap condition tables, CPPI 188–189gap multiples, Bermudan swaptions 385Gaussian copulas, basket options 201, 207Gaussian process

CMS 364–366, 374LGM model 295

general non-arbitrage conditions, equity derivatives 75–79

generation of scenarios, internal model method 51–52

538 INDEX


geographic features of product wrappers and trends 11–19

geometric average models, basket options 101Germany, products and trends 12–13global optimization, yield curve stripping 246–247GMAB see Guaranteed Minimum Accumulation

Benefit annuitiesGMC see Grid Monte CarloGMDB see Guaranteed Minimum Death Benefit

annuitiesGMIB see Guaranteed Minimum Income Benefit

annuitiesGMWB see Guaranteed Minimum Withdrawal

Benefit annuitiesgolden formulae, replication 346–347golden rules, interest rate derivatives 376Grid Monte Carlo (GMC) 422–434

computational speed 426–431FVA 433–434implementation 425–432numerical features 432–433portfolio Credit Value Adjustment 422–434in practice 432–434risk sensitivity 431–432storage optimization 424XVA extensions 433–434

growth productscapital-at-risk 164–168, 180–183capital-protected 164–170, 180–183with tax liabilities 180–183zero coupon bonds 164–165

guaranteed amounts, GMIB annuities 460Guaranteed Annuity Options (GAO) 459,

470–471see also Guaranteed Minimum Income Benefit

annuitiesguaranteed annuity rates (GAR) 454, 455, 460Guaranteed Minimum Accumulation Benefit

(GMAB) annuities 453Guaranteed Minimum Death Benefit (GMDB)

annuities 454Guaranteed Minimum Income Benefit (GMIB)

annuitiesbond option decomposition 461–462features 454guaranteed amounts 460hedging 460–466hybrid optionality analysis 459–466joint equity and swap rates 464–466multi-factor model framework 459replication pricing 460–466swaption decomposition 462–464transaction costs 466–467

Guaranteed Minimum Withdrawal Benefit (GMWB) annuities 454–455, 459

Gyöngy’s theorem 148

Heath–Jarrow–Morton model (HJM) 293–295hedging

see also Delta hedging; Vega hedgingAsian options 98–100autocallable products 173–175Black–Scholes paradigm 117–122CMS calibration 353CMS replication 344collateral discounting 249counterparty credit risk 46–47CPPI 190–193Delta adjustment 117–120equity release products 503–505Financial Transaction Tax effects 11GMIB annuities 460–466Grid Monte Carlo 434Jacobian transformations 286–290mechanisms 31–32non-arbitrage derivative pricing framework 23in practice 121–122product purposes 4purposes 4Quanto effect 70–71soft barriers 184–186target volatility products 179term structures 120–121value chains 21–22with volatility bond 396volatility dynamics 117volatility smile/skew 117–122zero-coupon interest rate swaps 380

Heston, Stein & Stein and Hull & White model 146

high-dimensional HJM models 295high-profile losses 5–6hindsight options 113–117historical correlations, CMS rates 356–357historic time series, CoCo models 443–445historic volatility surfaces, basket options 213–216HJM model see Heath–Jarrow–Morton modelHouse Price Index (HPI)

instantaneous forward rate 498options products 512–514rate and forward risks 506–507reversion products 497–501

Hull–White (HW) model 294, 297hybrid derivative products 397hybrid optionalities, GMIB annuities 459–466hybrid risks

autocallable products 174–175PRDC swaps 413–418

iCPPI see individual Constant Proportion Portfolio Insurance products

IMM see internal model methodimmediate annuity products 451

Index 539


implied basket volatility surfaces 217–224implied forward volatility smile/skew 155–156implied volatility surfaces

approximation 138–142basket option analysis 217–224BBF formula 140–142CMS 359copula technique 217–219correlation influence on basket skew 219–222correlation skew and basket skew 223–224equity derivatives 74–85from local volatility 138–142general non-arbitrage conditions 75–79log-normal local volatility 124–125Markov terminal mapping 133–135most likely path 138–139, 142non-parametric 79–82skew adjustment 219–224

independent model validation (IMV) 25–26indirect costs, Financial Transaction Tax 10–11indirect modelling approaches, CoCo 439–442individual Constant Proportion Portfolio Insurance

products (iCPPI) 193instantaneous forward rate, House Price Index 498instantaneous prepayment distributions, fixed-rate

mortgage portfolios 477Instrument objects 26–27instrument price grids 424integrals, product of two calls 516–518Integrated Product Development 22integration of copula pricing models 200interconnections, object-orientated models 27–29interest rate derivatives 227–398

Bermudan swaptions 381–386callable products 387–392cap/floor 258–274caplet volatility stripping 263–265Cheyette model 312–318, 327–329CME-LCH basis 251–252collateral arbitrage 250–251collateral impacts 248–252Constant Maturity Swaps 343–374, 392–393credit-linked 397Credit Support Annex 238cross-currency swaps 234–237, 241–244CSA & non-CSA discounting 248–250deltas incorporating smile dynamics 290–291Dupire local volatility model 318–320, 322European swaptions 274–286, 291FX forwards/swaps 244–246Hull–White model 294, 297Jacobian transformations 286–290key model characteristics 293–295Libor Local Volatility Model 329–342Libor Market Model 295, 303–312Linear Gauss–Markov model 293, 295–303

local volatility stripping 321–325, 331–332, 336–339

log-normal, shifted log-normal and normal model caplets/floorlets 258–260

marginal distributions 356–359, 362–373market-to-market cross-currency swaps

242–244Martingdale pricing theorem 255–258model choice 376multi-curve environments 229–237, 252–253non-standard swaptions 284–285OIS and LOIS 230–232Overnight Indexed Swaps 239–240par bootstrap equations 238–239practical models 293–342replication 343–374risks 375–381rolling swaps 320–321, 324–326, 329SABR volatility surfaces 279–284, 291spread option pricing 356–362stripping 263–265, 271–274swap local volatility stripping 319–329target redemption notes 394–395tenor basis swaps 232–234, 238–239tenor volatility transformation 265–274vanilla options 255–291volatility bonds 396yield curve stripping 237–248

interest rate exotics, United States 12Interest Rate Hedging Products (IRHR),

SMEs 378interest rate risk sensitivity, fixed-rate mortgage

portfolios 481–482interest rate swaps (IRS) 251–252, 343–344interfaces, models 28intermediaries, distribution 36–37internal model method (IMM) 48–55

risk infrastructure 48–49risk systems evolution 50–51scenarios generation 51–52VaR engines 49–50yield curve scenarios 52–55

interpolationcaplet volatility stripping 264K-axis 80T-axis 81–82yield curve stripping 246–247

inverse problems, copula pricing 199inverted yield curves, interest rate derivatives 377investment plans 15, 34, 36investor protections, MiFID II 9–10investors, issuances and hedge mechanisms 31–32IRHR see Interest Rate Hedging Productsissuance 31–34issuance risks 193–195Italy, products and trends 13–15

540 INDEX


Jacobian transformations 286–290Jex, Henderson and Wang model 147joint-calibration approach, callable reverse

floaters 389joint diffusion models, CoCo 448–449joint equity, GMIB annuities 464–466joint holders

reversion derivatives 498roll-up mortgages 508survival probability 496–497

jump conditionsEuropean options 64–66local volatility framework 130–131

jump diffusion modelsCoCo 446–448CPPI 192

jump risks, basket options 103

K-axisimplied volatility surfaces 82–83interpolation 80

Key Information Documents (KIDs) 36kickout see autocallable productsknock-out features, TARN 395know your customer (KYC) 35know your product providers (KYPP) 35Kolmogorov forward (Fokker–Plank)

equation 149–150KYC see know your customerKYPP see know your product providers

La Commissione Nazionale per le Società e la Borsa (CONSOB) 13–14

large-step Monte Carlo 132–138, 364–366LCR see Liquidity Coverage Ratioslegs, components 27leverage

Basel III requirements 40–41, 47debt crises 7Italy 14ratios and target volatility 178volatility 7

LGM see linear Gauss–Markov modelsLGM model see Linear Gauss–Markov

modelliabilities, annuity products 453, 466–472LIBOR

LOIS 230–232multi-curve stripping 240–241tenor basis swaps 232–234

Libor Local Volatility Model 329–342Libor Market Model (LMM) 295, 303–312

market calibration 307–308numerical implementation 305–307standard formulation 303–305

libraries, object-orientated quanting models 26–31

life assurance wrappers 34Linear Gauss–Markov (LGM) models 293,

295–303, 425–426linear skew, T-axis interpolation 81–82linear variance, T-axis interpolation 81–82linear variance of logarithmic moneyness 80linear volatility of logarithmic moneyness 80linear volatility of moneyness 80liquidity

Basel III requirements 41, 47–48LOIS 231–232tenor basis swaps 232–234

Liquidity Coverage Ratios (LCR) 41, 47–48liquid products, Italy 13–14LMM see Libor Market Modellocal volatility (LV) frameworks 123–144

Asian basket options 203–205barrier options 131–132basket options 203–205BBF formula 140–142calibration 317copula terminal mapping 135–138correlation skew methodology 209–210discrete dividends 130–131European basket options 203–204functional form 316–318implied volatility approximations 138–142into Cheyette model 327–329Libor Local Volatility Model 329–341log-normal 123–125Markov terminal mapping 133–135, 138model 318–342Monte Carlo 132–138most likely path approximations 138–139, 142multi-step copulas for auto-callable

baskets 212–213normal 125–127PDE solving 127–132practical issues 142–143stripping 123–127, 321–325, 331–332,

336–339log-normal caplets 258–260log-normal local volatility 123–125log-returns, CMS rates 356–357LOIS 230–232long-dated fund (equity) forward 467–468long-dated fund (equity) volatility 468–469long-dated FX volatility

hybrid risks 410–418PRDC swaps 411–418real life options and derivatives 401–418smile extrapolation to long end 407–410term structure extrapolation to long

end 403–407volatility surfaces 401–403

long-dated hedges, convexity 74

Index 541


long Gamma, barrier reverse convertibles 183lookback options, Black–Scholes

paradigm 113–117loss absorption mechanisms, CoCo 438loss scenarios, vanilla products 379low-dimensional HJM models 294low interest rates, multi-curve impact 379–380LSV see stochastic local volatility frameworkLV see local volatility frameworks

major equity indices 62major OIS indices 230male mortality base rates 492–494management, Basel III requirements 41marginal CMS spread options 359marginal distributions, CMS 356–359, 362–373marginals, semi-analytic copula 368market calibration

CoCo 446–448LMM model 307–308

market cap weighted indices 62market data objects 26–27market features of equity derivatives 61–86market implied cost of equity, CoCo 439marketing, value chain principle 22market-making, Financial Transaction Tax 10market marginal distributions, CMS 362–373market places, distribution 36–37market quotes

European swaptions 276–277FX volatility 401–403

market risk, VA products 456–457market scenarios, wings 351–352Markets in Financial Instruments Directive II

(MiFID II) 9–10market smile calibration 339market-to-market cross-currency swaps (resetting

CCS) 235, 242–244market value of annuity (MVA) 457–458marking to the model 23Markovian processes 293–294, 315Markovian projection 148, 153–154Markov terminal mapping (MTM) 133–135, 138Martingdale pricing theorem

basic pricing formula 256–257LMM model 304probability measure change 257–258Q-measure 88vanilla options 255–258

maturity, autocallable products 172maximum put, hindsight options 114MC see Monte Carlomean reversion, LGM model 297, 301–303medium-term notes (MTN) 375MEE swaptions see Most-Expensive European

swaptions

MiFID II 9–10mis-selling and mis-management of Guaranteed

Annuity Options 470–471MLP see most likely pathmodelling

attributes tables 28Black–Scholes paradigm 87–122CoCo 440–450development principles 22–31implementation 23–24independent validation 25–26interfaces 28object-orientated libraries 22, 26–31specification 23system testing 25testing 24–25, 30–31

model risks, basket options 103moment matching models, basket options 101money market basis 232–234, 238–241monotonic payoff curve, CMS 344–346Monte Carlo (MC) simulation

CMS 364–366Contingent Convertibles 442copula terminal mapping 135–138fixed-rate mortgage portfolios 478–479grid-type for CVA 422–434LMM model 305local volatility framework 132–138Markov terminal mapping 133–135portfolio Credit Value Adjustment 420–434

mortalityfemale base rates 494–496male base rates 492–494property-linked roll-up mortgages 508–511real estate derivatives pricing 492–497reversion products 499–501single monthly 476–477

Most-Expensive European (MEE) swaptions 382–383, 385

most likely path (MLP) 138–139, 142MTM see Markov terminal mappingMTN see medium-term notesmulti-currency bootstrap frameworks 241–246multi-curve environments

cross-currency swaps 234–237FX forwards/swaps 237interest rate derivatives 229–237, 252–253Martingdale pricing theorem 256–257OIS and LOIS 230–232tenor basis swaps 232–234tenor volatility transformation 269–270yield curve stripping 239–248

multi-curve impacts of low interest rate 379–380multi-factor hybrid modelling 301multi-factor model frameworks, VA

products 458–459

542 INDEX


multi-factor risk-neutral techniques, portfolio CVA 420–421

multi-factor stochastic processes (SLV SDEs) 148multi-step copulas, auto-callable baskets 210–213multi-underlying products see basket productsmutual fund wrappers, features and benefits 34MVA see market value of annuity

NEA see Net Exposure Amountsnegative interest rates 377–378Net Exposure Amounts (NEA) 503–505Net Stable Funding Ratio (NSFR) 41non-arbitrage conditions

equity derivatives 75–79FX volatility 408

non-arbitrage derivative pricing framework 23non-callable PRDC swaps 412–413non-CSA discounting 248–250non-deliverable cross-currency swaps 235non-linear payoff, CMS 344–346non-parametric implied volatility surfaces 79–82non-standard swaptions 284–285non-structured products, China 16–17non-uniform grids, local volatility

framework 128–130normal (Bachelier) caplets 258–260, 272–274normal local volatility, stripping 125–127normal processes

Libor Local Volatility Model 335–336swap local volatility stripping 325–327

notes, features and benefits 33notional risks, retail equity-linked

products 194–195NSFR see Net Stable Funding Rationumerical features of Grid Monte Carlo 432–433numerical procedures

discrete dividends 61–67portfolio CVA 419–434stochastic local volatility framework 150–154yield curve stripping 246–248

object-orientated libraries 26–31architecture 27–29attributes tables 28documentation 29–31interfaces 28orthogonal combination 26technical documentation 29–30testing documentation 30–31

OBPT see option-based prepayment techniqueOEB see Optimal Exercise BoundaryOIS see Overnight Indexed SwapsOpen Ended Investment Companies (OEIC) 187operator objects, quant libraries 27Optimal Exercise Boundary (OEB), American

options 107

optimum exercise, physical settlement 69–70option-based prepayment technique

(OBPT) 476–489application 488–489Early Redemption Charges 486–488interest rate risk sensitivity 481–482prepayment optionalities 476–486prepayment profiles 479–480propensity risk sensitivity 484–486volatility risk sensitivity 482–483

option settlement delay 68–70Ornstein–Uhlenbeck (OU) processes 147orthogonal combination 26OTC see over-the-counter productsOU see Ornstein–Uhlenbeck processesOvernight Indexed Swaps (OIS) 230–232,

238–240over-the-counter (OTC) products

distribution 35–37Key Information Documents 36

Packaged Retail and Insurance-Based Investment Products (PRIIPs) 36

Packaged Retail Investment Products (PRIPs) 10par bootstrap equations 238–239, 243–244Partial Differential Equations (PDE)

American options 107, 108–110Black–Scholes paradigm 89discrete dividends 63–66local volatility 127–132, 322pricing 302–303, 334–335, 340–341smile model formulation 316

path-dependent options 302CPPI 189–190discrete dividends 66–67PDE for local volatility 128–130

payoffCMS 344–347foreign currencies 346–347

PCA see principal component analysisPDE see Partial Differential EquationsPDF see probability density functionsPhoenix autocallable products 175physical settlement

automatic exercise 68–69equity derivatives 68–70European swaptions 275optimum exercise 69–70

pillars 21–38distribution 35–37issuance and wrappers 31–34model and product development 22–31value chains 21–22

plain vanilla optionssee also vanilla optionsfeatures 169–170

Index 543


Point of Non-Viability (PoNV) triggers 438Poisson jumps, CPPI 192portfolio Credit Value Adjustment 419–434

Expected Positive Exposure calculations 420–422Funding Valuation Adjustment 433–434Grid Monte Carlo 422–434hedging 434implementation frameworks 420, 425–432linear Gauss–Markov models 425–426multi-factor risk-neutral techniques 420–421XVA 433–434

portfolio real estate products 501–507equity release products 503–505forward risks 506–507rate risks 506–507risk analysis 505–506

power copulas 202, 364Power Reverse Dual Currency (PRDC)

swaps 411–418basis sensitivity 415–416callable optionalities 413correlation sensitivity 417–418hybrid risks 413–418interest rates 415–416non-callable optionalities 412–413spot sensitivity 413–414swaption volatility 416–417volatility sensitivity 414–415

PRDC see Power Reverse Dual Currency swapsprecipice bonds 170–171, 183–186prepayment optionalities

convexity 473–476fixed-rate mortgage portfolios 473–479profiles 479–480risks 473–479single monthly mortality data 476–477

price comparisons, CMS 369–370price ladders, Asian options 99–100price weighted indices 62pricing

see also replicationAsian options 93–96basket options 101–102, 373Bermudan swaptions 341, 383callable reverse floaters 388CMS spread options 356–362derivatives with smile 339–341Financial Transaction Tax effects 11Guaranteed Minimum Income Benefit

annuities 460–466interest rate derivative products 376marking to the model 23Martingdale theorem 256–257mortality 492–497, 499–501, 508–511PDEs 302–303, 334–335vanilla formulae 332–334

PRIIPs see Packaged Retail and Insurance-Based Investment Products

principal component analysis (PCA) 53–55PRIPs see Packaged Retail Investment Productsprobability density functions (PDF)

CMS calibration 353, 355–358, 363, 367, 374copula pricing models 200historic basket volatility surfaces 213–216log-normal local volatility 123–124

probability measure change 257–258product distribution 31–37, 193–195product issuance 31–34, 193–195product objects 26–27products

development principles 22–31geographic features 11–19

product of two calls 515–519product wrappers

geographic features 11–19principles 31–34

propensity risk sensitivity 484–486property-linked roll-up mortgages 507–512public distribution 37put option pricing

Asian options 96automatic exercise 69dividend options 103FX volatility smile boundaries 409optimum exercise 69–70

puts, CMS replication 346PVBP cash-settled swaptions 275

QDII see qualified domestic institutional investors

Q probability measure 87–88, 255–258quadratic fitting 128–130qualified domestic institutional investors (QDII),

China 17quantitative documentation, object-orientated

libraries 29–31quant libraries

see also object-orientated librariesprinciples 26–31

Quanto 376Quanto adjustment 71–72, 256–257, 464Quanto hedging 70–71Quanto options, equity derivatives 70–72quants

object-orientated model libraries 26–31responsibilities and procedures 22–26value chain principle 22

Radon–Nikodym theorem 257–258, 348rainbow barrier options 111–112range accrual notes, callable 390–392RDR see Retail Distribution Review, UK

544 INDEX


real estate derivatives 491–514equity release scheme principles 491–492female mortality base rates 494–496House Price Index-linked options 512–514male mortality base rates 492–494mortality and pricing 492–497, 499–501,

508–511Net Exposure Amounts 503–505portfolio products 501–507reversion products 497–501roll-up mortgages 507–512survival probability 494–497

real life options and derivatives (RLOD) 399–514Contingent Convertibles 435–450equity release scheme principles 491–492Expected Positive Exposure calculations

420–422fixed-rate mortgage portfolios 473–489Funding Valuation Adjustment 433–434Grid Monte Carlo 422–434House Price Index-linked options 512–514long-dated FX volatility 401–418mortality and real estate pricing 492–497,

499–501, 508–511portfolio Credit Value Adjustment 419–434portfolio real estate products 501–507PRDC swaps 411–418prepayment optionalities 473–479real estate 491–514replication pricing 460–466reversion products 497–501roll-up mortgages 507–512selling period optionalities 488–489Variable Annuity products 451–472

rebalancing schemes, CPPI 188–189reciprocal Gamma models, basket options 101recursive bisection, Brownian Bridges 92–93“redeemable” notes see callable step-up

fixed-rate notesregulatory environments

China 17–18European Union 8–11, 13–16general 7–11Germany 13Italy 13–14United Kingdom 10, 15–16United States 8

relative entropy, historic basket volatility surfaces 214

release change control procedures 25Ren, Madden and Qian model, stochastic local

volatility 147repayment (upon death), roll-up

mortgages 507–512replication

see also pricing

CMS 343–374examples 346–350Guaranteed Minimum Income Benefit

annuities 460–466principles 344–346

resetting CCS see market-to-market cross-currency swaps

Retail Distribution Review, UK (RDR) 10retail products

distribution 35–37issuance risks 193–195Key Information Documents 36

reverse convertibles see barrier reverse convertiblesreverse floaters 387–390, 394reversion products 497–501risk analysis

lookback and hindsight options 115–117property-linked roll-up mortgages 508–511real estate portfolios 505–506

risk controlindices 62, 178–179model specification 23value chain principle 22

risk engine objects 27risk factors

CoCo 438–439VA products 456–458, 469–472

risk-free savings products 164risk infrastructure, internal model method 48–49risk management

see also financial risk managementCMS spread options 360principles 21–22retail equity-linked products 193–195target volatility products 177–179

risk measures 39–41risk-neutral (RN) multi-factor techniques, portfolio

CVA 420–421risk-neutral (RN) probability density

functions 214risks

autocallable products 173–175basket options 102–103dividend options 106exotic products 380interest rate derivative products 375–378Quanto adjustment 71–72target redemption note 395vanilla products 378–379

risk scenarios, House Price Index-linked options 513–514

risk sensitivityAsian options 98–100CMS 348Grid Monte Carlo calculations 431–432Jacobian transformations 286–290

Index 545


property-linked roll-up mortgages 511–512vanilla interest rate derivatives 286–291

risk systems evolution, internal model method 50–51

Risk-Weighted Assets (RWA) 40, 43–45RLOD see real life options and derivativesrogue trading 5–6rolling swaps

fixed tenor 320–321, 324–325, 326, 329fixed terminal 324, 326

roll-up mortgages 507–512running realized volatility 178–179RWA see Risk-Weighted Assets

S&P index, historic volatility 215–216SABR see stochastic Alpha Beta Rhoscaling, LGM model 297–298scenario generation, IMM 51–52SDEs see stochastic processesself-consistent pricing 360–362selling period optionalities 488–489semi-analytical copulas 366–373settlement rules, European swaptions 275Shibor-linked derivatives 18–19shifted log-normal caplets 258–260, 272–273shifting, LGM model 297short Gamma, barrier reverse convertibles 183short rate models 293–294single currency bootstrap framework 238–241single curve environments, tenor volatility

transformations 266–269single holders

property-linked roll-up mortgages 508reversion derivative products 497–498survival probability 496

single monthly mortality (SMM) data 476–477singular value decomposition (SVD), basket

options 102, 211–212skew adjustment

Asian options 96–98barrier options 112–113implied volatility surfaces 219–224lookback and hindsight options 114–115stochastic local volatility framework 154–161target volatility products 179

skew boundaries, equity derivatives 76–79skew decay, equity derivatives 77–78skewed t-distribution 366–367, 373Sklar’s theorem 199, 363SLV SDEs see multi-factor stochastic processessmall and medium-sized enterprises (SMEs), IRHR

risks 378smile adjustment

Asian options 96–98barrier options 112–113European swaptions 277–279

FX volatility 407–410lookback and hindsight options 114–115pricing 339–341stochastic local volatility framework 154–161vanilla interest rate derivatives 277–279,

290–291smile boundaries

equity derivatives 76–79FX volatility 408–410

smile calibration 312–313, 318, 336–339, 341–342smile decay, equity derivatives 77–78smile model formulation 315–317, 319SMM see single monthly mortality data; Swap

Market Modelsmooth monotone mapping 154snowball see callable snowballSobol, bias 91soft barriers 184–186solvency II 41, 193sovereign-debt crisis 6–7special purpose vehicles (SPVs) 34spline interpolation 80–82spot FX volatility 403–406spot processes

local volatility formula 331–332PDE pricing 334–335, 340–341vanilla pricing formulae 332–334

spot sensitivity, PRDC swaps 413–414spread options, CMS 343–374, 393SPVs see special purpose vehiclessquare terminal payoff 346stamp duty, United Kingdom 16standard swaps, par bootstrap equations 238–239state variable grids 424static hedges 344static replication see replicationsteepening, CMS products 392–393step early redemption charge schemes 487–488step-up fixed-rate notes 387sticky delta, concepts 118–122sticky local volatility, concepts 118–122sticky strikes, concepts 118–122stitching, volatility wings 351–355stochastic Alpha Beta Rho (SABR) framework

279–284, 291, 350–353, 355, 366stochastic local volatility framework

(LSV) 145–162alternative calibration technique 153–154barrier option 158–159calibration 150–154Cliquet option 159–161digital option 156–157Fokker–Plank equation 149–150Gyöngy’s theorem 148implied forward volatility smile/skew 155–156Markovian projection 148, 153–154

546 INDEX


model formulation 147–150numerical implementation 150–154numerical results 154–161in practice 161–162stochastic volatility models 145–146

stochastic processes (SDEs)Black–Scholes paradigm 90–93Cheyette model 313–314CMS 374discrete dividends 62local volatility formulation 145–150Markovian projection 148, 153–154Variable Annuity product models 458–459

stochastic volatility, LMM model 310–311storage optimization, Grid Monte Carlo 424strategy indices, equity derivatives 62strikes

Bermudan swaptions 384CMS products 393

strippingcaplet volatility 263–265local volatility 123–127, 331–332, 336–339swap local volatility 319–329yield curves 237–248

structural asset approach, CoCo 440–442structured certificates 13–15, 33–34“structured” collars 378–379structured deposits 15, 17structured notes 13, 33–34structured products

ESMA governance 9geographic features 11–19purposes 4

Student’s t-copulas 202Student’s t-distribution 366supervision, Basel III 41surrender charges 453survival probability 494–497swap curves, CMS 349swap local volatility stripping 319–329Swap Market Model (SMM) 295swaps

see also interest rate swapsLGM model 298–299

swaptionssee also Bermudan swaptions; co-tenor

swaptions; co-terminal swaptions; European swaptions

calibrating to 309–310callable range accrual notes 390–391callable reverse floaters 388–389Cheyette model calibration 328CMS calibration 350–352CMS rates 358collateral discounting 250exotic products 380

GMIB annuities 462–464LGM model 299market quotes 276–277Most-Expensive European 382–383, 385non-standard 284–285SABR volatility surfaces 279–284, 291smile adjustment 277–279vanilla interest rate derivatives 274–286volatility 277–284, 416–417

synthetic fixed legs, trade compression 8systemic risks, Basel III requirements 41system testing, models 25

tables, gap conditions 188–189tail dependency, copula pricing models 201tail risks, Variable Annuity products 470–472tails, “Gamma trap” 21Tanh-Cosh volatility functional form 83tapered surrender charges 453target redemption notes (TARN) 394–395target volatility products 177–179target weighting, CPPI 188–189TARN see target redemption notestax efficient scheme wrappers 34T-axis

implied volatility surfaces 84interpolation 81–82

tax liabilitiesequity-linked products 179–183growth products 180–183zero coupon bonds 179–180

Taylor expansion models 101t-distribution 374technical documentation template 29–30technical requirements, Basel III 41–48tenor basis swaps 232–234, 238–241tenor terms

see also fixed tenor rolling swapsexponential vs. 314

tenor volatility transformation 265–274multi-curve environments 269–270single curve environments 266–269

terminal payoff, replication 346term structures

adjustment factors 120–121basket skew 222cap/floor implied volatilities 260–263FX volatility 403–407

testing, models 24–25Tier 1/2 capital, Basel III 42–43time-bucketed Vegas, autocallable products 173time changes, CMS spread options 360total assets, Basel III 40–41total return (TR) indices, equity derivatives 62trade compression, EMIR 8trading, value chains 21–22

Index 547


transaction costsFinancial Transaction Tax 10Variable Annuity products 466–467

transparency, MiFID II 9–10trigger levels, Contingent Convertibles 437–438trust companies, China 17

UCITS see Undertakings for Collective Investments in Transferable Securities

underlyingsautocallable products 172–173equity indexes 61–62special purpose vehicles 34

Undertakings for Collective Investments in Transferable Securities (UCITS) 10, 34

undiscounted Bachelier calls 126–127United Kingdom (UK)

FTSE-100 214–215product wrappers and trends 10, 15–16Retail Distribution Review 10stamp duty 16

United States of America (USA)Dodd–Frank 8product wrappers and trends 11–12

utility objects 27

VA see Variable Annuity productsValue at Risk engines (VaR engines) 49–50value chains 21–22vanilla options

CMS 360, 361, 374interest rate derivatives 255–291

cap/floor 258–274caplet volatility stripping 263–265deltas incorporating smile dynamics 290–291European swaptions 274–286Jacobian transformations 286–290log-normal, shifted log-normal and normal

model caplets/floorlets 258–260Martingdale pricing theorem 255–258non-standard swaptions 284–285risk sensitivities 286–291SABR volatility surfaces 279–284, 291tenor volatility transformation 265–274volatility term structures 260–263

LGM model 298–300log-normal local volatility 124plain equity-linked 169–170pricing formulae under spot process 332–334

vanilla products, risks 378–379VaR engines see Value at Risk enginesVariable Annuity (VA) products 451–472

bond options decomposition 461–462features 452–453general multi-factor model framework

458–459

GMIB annuities 454, 459–466GMWB annuities 454–455, 459hedging 460–466hybrid pricing models 458–466joint equity and swap rates 464–466key types 453–455liabilities 453, 466–472long-dated fund (equity) forward 467–468long-dated fund (equity) volatility 468–469market risk 456–457market value of annuity 457–458practicalities 466–469Quanto adjustment 464replication pricing 460–466risk factors 456–458, 469–472swaption decomposition 462–464tail risks 470–472transaction costs 466–467

varianceBrownian Bridges 90–93equity derivatives 76

Vasicek model 294Vega hedging

autocallable products 173Bermudan swaptions 385–386Black–Scholes paradigm 122CMS calibration 352–355retail equity-linked products 194–195target volatility products 179

volatilityAsian option smile/skew 96–98barrier options 112–113basket skew and correlation effects 219–224caplet stripping 263–265, 271–274discrete dividends 65Eisenhower matrices 166–168European swaptions 277–284leverage 7local stripping 123–127lookback and hindsight options 114–115non-arbitrage conditions of implied volatility

surfaces 76–79stochastic models 145–146tenor transformations 265–274

volatility bonds 396volatility dynamics, Black–Scholes paradigm

117volatility exposure, Bermudan swaptions

385–386volatility functions, low-dimensional HJM

models 294volatility indices, equity derivatives 62volatility note see volatility bondvolatility sensitivity

PRDC swaps 414–415real estate portfolios 482–483, 505–506

548 INDEX


volatility separability 314–315volatility smile

extensions 310–312high-dimensional HJM models 295

volatility surfacessee also implied volatility surfacesblack caplets 271–272cap/floor, vanilla interest rate options 271–274real life long-dated FX options 401–403

volatility term structures, vanilla options 260–263volatility wings

CMS calibration 350–352SABR volatility surfaces 283stitching 351–355

Volcker rule 8

warrants, features and benefits 33wealth management products (WMPs) 16–17weighting, local volatility framework 128–129Wiener process 365wings see volatility wingswithdrawal guarantees, GMIB annuities 455withdrawal process optimization, GMIB

annuities 455WMPs see wealth management productswrappers

categories 32–34features and benefits 33–34principles 31–34

Write-down Contingent Convertibles 436Write-down/Write-up Contingent Convertibles 436

XVA, Grid Monte Carlo 433–434

yield, discrete versus continuous dividends 63–64

yield curvesBermudan swaptions 381CMS 348, 350, 392–393interest rate derivatives 375–378internal model method 52–55stripping

cross-currency swaps 241–242FX forwards/swaps 244–246interest rate derivatives 237–248key steps 240market-to-market cross-currency

swaps 242–244multi-curve environments 239–248numerical procedures 246–248single currency bootstrapping 238–241single-curve environments 238–239

zero Bermudan swaptions 381–382zero-coupon bonds 304

capital-at-risk growth products 164–165capital-at-risk income products 165–166capital-protected growth products 164–165Martingdale pricing theorem 255–258with tax liabilities 179–180

zero-coupon interest rate swaps (zIRSs) 380zero-coupon range accrual 392zero sum game 376zIRSs see zero-coupon interest rate swaps