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Copyrighted Material Index AA estimate, 68–69 At-the-money (ATM) SPX variance Affine jump diffusion (AJD), 15–16 levels/skews, 39f AJD. See Affine jump diffusion Avellaneda, Marco, 114, 163 Alfonsi, Aurelien,´ 163 American Airlines (AMR), negative book Bakshi, Gurdip, 66, 163 value, 84 Bakshi-Cao-Chen (BCC) parameters, 40, 66, American implied volatilities, 82 67f, 70f, 146, 152, 154f American options, 82 Barrier level, Amortizing options, 135 distribution, 86 Andersen, Leif, 24, 67, 68, 163 equal to strike, 108–109 Andreasen, Jesper, 67, 68, 163 Barrier options, 107, 114. See also Annualized Heston convexity adjustment, Out-of-the-money barrier options 145f applications, 120 Annualized Heston VXB convexity barrier window, 120 adjustment, 160f definitions, 107–108 Ansatz, 32–33 discrete monitoring, adjustment, 117–119 application, 34 knock-in options, 107 Arbitrage, 78–79. See also Capital structure knock-out options, 107, 108 arbitrage limiting cases, 108–109 avoidance, 26 live-out options, 116, 117f calendar spread arbitrage, 26 one-touch options, 110, 111f, 112f, 115 vertical spread arbitrage, 26, 78 out-of-the-money barrier, 114–115 Arrow-Debreu prices, 8–9 Parisian options, 120 Asymptotics, summary, 100 rebate, 108 Benaim, Shalom, 98, 163 At-the-money (ATM) implied volatility (or Berestycki, Henri, 26, 163 variance), 34, 37, 39, 79, 104 Bessel functions, 23, 151. See also Modified structure, computation, 60 Bessel function At-the-money (ATM) lookback (hindsight) weights, 149 option, 119 Bid/offer spread, 26 At-the-money (ATM) option, 70, 78, 126, minimization, 114 149, 151 COPYRIGHTEDBid/offer MATERIAL volatilities, graphs, 38f, 83f At-the-money (ATM) volatility (or variance) Black, Fisher, 84, 163 skew, 37, 62, 66 Black-Scholes (BS) equation 27, 55 computation, 60–61 Black-Scholes (BS) (flat) volatility smile, 99 decay, 65 Black-Scholes (BS) formula, 4, 17, 79, 113 effect of jumps, 61, 65, 94 derivatives, 12, 150 effect of stochastic volatility, 35, 94 generalization, 9 term structure, 64t Black-Scholes (BS) forward implied variance zero expiration, at-the-money strike limit, function, defining, 27 90 Black-Scholes (BS) gamma, defining, 27 169 170 THE VOLATILITY SURFACE Black-Scholes (BS) implied total variance, 11 Carr-Lee result, 151, 155 Black-Scholes (BS) implied volatility (or Cauchy Residue Theorem, 59 variance), 29, 34, 61, 88, 95, 99. See CBOE. See Chicago Board Options Exchange also At-the-money implied volatility CDS. See Credit default swap computation, 76 Characteristic function implicit expression, derivation, 60 definition, 142 path-integral representation, 26 for exponential Brownian motion, 57 process, 57 for a Levy´ process, 56 skew, 89, 95, 102 for Heston, 21, 57 parametrization, 140 for jump-diffusion, 57 term structure. See Heston model implied volatility from, 60 Black-Scholes (BS) model, success of, 1 methods, 56–65 Bloomberg recovery rate, 83 option prices from, 58–60 Bonds, 85 volatility skew from, 60 recovery, 94 Chen, Zhiwu, 66, 163 Brace, Alan, 103, 163 Chicago Board Options Exchange (CBOE), Breeden, Douglas, 8, 133, 163 156–157 Brigo, Damiano, 25, 163 VIX index, 156–158 Broadie, Mark, 23, 118, 119, 163, 164 VXB futures, 158–160 Broadie-Kaya simulation procedure, 23 Chou, Andrew, 107, 164 Brotherton-Ratcliffe, Rupert, 24, 163 Chriss, Neil, 26, 137, 164 Brownian Bridge density, 33–34. See also Clark, Peter K., 1, 164 Stock price Cliquets. See Digital cliquets; Exotic cliquets; maximization, 31 Locally capped globally floored cliquet; Brownian Bridge process, 32–33 Reverse cliquet Brownian motion, 1, 56, 85 See also deals, 130 Exponential Brownian motion definition, 105 BS. See Black-Scholes payoff, illustration, 105f Busca, Jer´ ome,ˆ 26, 163 Butterfly. See Negative butterflies Compound Poisson process, 141 ratio. See Calendar spread Constant volatility assumption, 112 Cont, Rama, 103, 164 Cadl` ag` stochastic process, 56 Convergence of Heston simulation, 21 Calculation Agent, 124 Convexity adjustment, Calendar spread. See Negative calendar between volatility and variance, 143–146, spreads 149, 152. See Heston model arbitrage, elimination, 37 graphs, 145f butterfly, ratio, 45 VXB futures, 158–160 Call option, 104 graph, 160f Black-Scholes formula, 80 Coupon period, 130f hedging, 134 Cox, John, 15, 144, 164 expected realized profit, 28 Cox Ingersoll Ross (CIR) model, 144 value on default, 77 Credit default swap (CDS), 84 Call price, characteristic function prices, 85 representation, 59 Credit spreads, 76f, 78 Call spreads, capped and floored cliquet as relationship to volatility skew. See strip of, 123 Volatility skew Cao, Charles, 66, 163 term structure, 86 Capital structure arbitrage, 77–79 CreditGrades model, 74, 84–86 Capped and floored cliquets, 122–125 calibration, 86 Carr, Peter, 58, 107, 133, 147, 164 setup, 84 Index 171 Curvature, dependence of variance swap Euro annual swap rate, 125, 130 value on, 138–140 EURO STOXX 50 index, 123, 128 European binary call da Fonseca, Jose,´ 103, 164 model independence, 111 Default valuation, 104 CreditGrades model, 74, 84–86 value under stochastic/local volatility boundary condition, 85 assumptions, 111f effect on option prices, 82–84 European capped calls, 109 Merton (jump-to-ruin) model, 74–76 European options, constraint on volatility option values, 153 Default-free counterparty, 77 extraction of risk-neutral pdf from, 29 Delta function peak, 30 Heston solution for, 16–20 Delta hedging, 6, 7, 28, 137, 138, 154 put-call symmetry, 113 Delta-function weight, 149 realized profit on sale, 28 Demeterfi, Kresimir, 137, 164 valuation under local volatility, 25 Derman, Emanuel, 8, 26, 137, 164 weights in strip for, Diffusion coefficient, 80. See also general European payoff, 134 State-dependent diffusion coefficient log contract, 136 Diffusion processes, 8 variance swaps, 137 impact on short dated skew, 51 volatility swap (zero correlation case), independence from jumps, 61 149 Digital cliquets, 103–106 European payoffs, spanning. See Generalized Digital options, 103–106 European payoffs valuation, 104 Exact transition law, sampling from, 23–24 Dirac delta function, 14 Exercise, pseudo-probability of, 17 Discrete monitoring, adjustment. See Barrier Exotic cliquets, 122 options Exotic option traders, 131, 132 Discretely monitored lookback options, 119 Expected instantaneous variance, 14, 28, 138 Discretization of Heston process, 21–23 Expected quadratic variation, See Variance Dividend yield, 9, 10 swaps; Expected total variance Dow Jones EURO STOXX 50 index, 122 Expected total variance, 32 Down-and-out call option, 114 Exponential Brownian motion, 57 static hedge, 113 Exponential quadratic-variation payoff, fair Downside skew, 76f value, 148 Dragulescu, Adrian, 99, 164 Extended transform, computation, 16 Duffie, Darrell, 17, 67, 74, 164 Extreme strikes, 97–99 Dumas, Bernard, 8, 164 Extreme tails, 3f Dupire, Bruno, 8, 9, 10, 11, 13, 26, 45, 137, 154, 155, 160, 165 Fat tails, 2 Dupire equation Finger, Christopher, 164 derivation, 9–11 Finkelstein, Vladimir, 74, 84, 165 applied to Heston-Nandi model, 45 Five-cent bids, 50–51 Durrleman, Valdo, 103, 165 Fleming, Jeff, 8, 164 Florent, Igor, 26, 163 Effective theory, 8 Fokker-Planck equation, 9 Empirical SPX implied volatility surface, 72f Forde, Martin, 155, 165 Equity as call on value of company, 84 Forward BS implied variance, 13, 29 Equity volatility in the CreditGrades model, Forward volatility surface, under local 86 volatility, 102–103 Equity-linked investments, guaranteed, 138 Forward-starting options, 106, 122 Euler discretization, 22 cliquet as strips of, 131 172 THE VOLATILITY SURFACE Fouque, Jean-Pierre, 95–96, 165 Heston European option valuation formula, Fourier transform, See also Characteristic 15 function integration, complex logarithm, 19–20 of covered call position, 58–59 Heston model, 15, 89 of probability of exercise, 17 Black-Scholes implied volatility inversion 19, 59 skew, 35–36 Friedman, Craig, 26, 163 term structure, 34 Friz, Peter, 33, 98, 146, 149, 163, 165 convexity adjustment, 144–146 example. See Lognormal model independence of volatility level and skew, Gastineau, Gary, 165 35 Gatheral, Jim, 26, 37, 146, 149, 165 local variances, 33 Gatto, Mary-Ann, 165 computation, 32 Generalized European payoffs, spanning, pdf of instantaneous variance, 159 133–136 popularity, 24 Glasserman, Paul, 23, 118, 119, 163, 165 variance swaps, 138 Global floor, 126 VXB convexity adjustment, 158–159 Goldman, Barry, 112, 165 Heston process, 15–16 Goldys, Ben, 103, 163 characteristic function, 21, 57 Goodyear Tire and Rubber (GT) simulation, 21–24 Alfonsi discretization, 22 credit spread, 83 Euler discretization, 21 implied volatilities, 82t exact, 23 –24 Milstein discretization, 22 Heston-Nandi (HN) density, 45 Hagan, Patrick, 91, 165 Heston-Nandi (HN) model, 43 Hazard rate, 56, 80, 83. See also Poisson local variance, 43–44, 46 process numerical example, 44–48 Heaviside function, 14, 106 probability density, 45f Hedge funds and variance swaps, 137 results, discussion, 49 Hedging. See also Quasistatic hedging Heston-Nandi (HN) parameters, 44, complexity of barrier option hedging, 121 115–116, 123, 126, 146 difficulty of hedging Napoleons, 131 convexity adjustment with, 144 Heston, Steven, 15, 43, 165 Holmes, Richard, 26, 163 Heston Hull-White (H-W) model, 74 assumptions, valuation under, 123–124, 126, 128–129 IBM, volatility distribution, 4 characteristic function, 45 Implied volatility (or variance), 12, 36–37, derivation, 20–21 90. See also Heston model; inversion, 153 Jump-to-ruin model convexity adjustments, 159 computation, 46–48, 60 European-style
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