Index

AA estimate, 68–69 At-the-money (ATM) SPX variance Affine (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 arbitrage, 26 one-touch options, 110, 111f, 112f, 115 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 (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 , 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 , 35, 94 generalization, 9 term structure, 64t Black-Scholes (BS) forward implied variance zero , 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) (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 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 , 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 (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 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/ 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 (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 , pseudo-probability of, 17 Discrete monitoring, adjustment. See Barrier Exotic cliquets, 122 options 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 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 , 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 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 option, value, 16 in the Heston model, fit, 41f term structure, 34 local variance, numerical computation, 47f volatility skew, 35 local volatility approximation, 123 in the Heston-Nandi model, 48f Napoleon valuation, 129 interpolation/extrapolation, 26 parameters, 23, 43, 44, 71, 152. See also Merton model, 76f Standard & Poor’s 500 models. See Stochastic implied volatility SDE, integration, 144 models skew, 95 representation in terms of local volatility, solution. See European options 26–31 Index 173

Brownian bridge density, 30, 31f fit to GT option prices, 82t, 83f path integral representation, 28 implied volatility, 79–82 skew, 35. See also Heston model local variance (volatility) surface, 79–82 empirical, 38f, 39t, 39f option payoffs, 77 long expiration limit, 95 option valuation, 80 short expiration limit, 94 Jumps. See also Lognormally distributed small volatility of volatility limit, 97 jumps structure, computation. See At-the-money adding jumps, 50. See also Stochastic implied volatility volatility plus jumps surface, 25. See also Standard & Poor’s characteristic functions, 57–58 500; Volatility surface contribution to skew, 62, 93, 94 dynamics, 8 compensator, 94 fit to observed, 36 decay of skew, 63 shape, 25, 103 frequency, 63 term structure, 35 impact on valuation of variance swaps, understanding, 26–31 140–143. See also Skew; Volatility upper bound for extreme strikes, 97 impact on terminal return distribution, 64, Ingersoll, Jonathan, 15, 144, 164 65f Instantaneous variance (or volatility), 34, 43, necessity for explaining short-dated smile, 90 50–52 conditional expectation of, 13–14, 25 SVJ (jumps in stock price only), 65–68 jumps in, 68 SVJJ (jumps in stock price and volatility), pdf in Heston model, 159 68–71 risk-neutral expectation, 14 SDE under stochastic volatility, 6 Kahl, Christian, 20, 166 unconditional expectation, 32 Kamal, Michael, 137, 164 Intrinsic value, Kani, Iraj, 8, 26, 164 of European capped call, 109 Kaya, Ozg¨ ur,¨ 23, 164 Inverse Fourier transform, 17–18, 19 Klebaner, Fima, 103, 163 Investor motivation. See Napoleon Ito’sˆ lemma, Kloeden, Peter, 166 application of, 6–7, 14, 93 Knock-in options, 107 Knock-on benefits of VXB futures, 160 Knock-out option 107, 108 Jackel,¨ Peter, 20, 165 case of no optionality, 108 Jarrow, Robert, 164 closed-form formula, 113 JD. See Jump diffusion values, 115f, 116f Jeffery, Christopher, 132, 166 limiting cases, Jump compensator, 62, 63, 65, 70, 93, 94 Jump diffusion (JD), 52–56, 62, 66 limit orders, 108 at-the-money variance skews, 66 European capped calls, 109 inability to fit the volatility surface, 68 model sensitivity, 109 inconsistency with mean reversion of Kou, Steven, 118, 119, 163 volatility, 2 Kritzman, Mark, 165 models, fits. See Standard & Poor’s 500 Kumar, Deep, 91, 165 parameters, volatility smile, 63t Kurtosis. See Risk-neutral density process, 15 skew behavior, 61–63 Ladders, 120 Jump size, Lagrangian Uncertain Volatility Model, 114 known jump size, 51–53 Laplace transform of quadratic variation, uncertain jump size, 54–56 147. See also Quadratic variation Jump-to-ruin model, 75 Lardy, Jean-Pierre, 74, 84, 166 174 THE VOLATILITY SURFACE

Ledoit, Olivier, 103, 166 Local volatility model, Lee, Roger, 29, 88, 97–100, 147, 164, 166 as single-factor, 49 moment formula, 97–98 dymamics of the volatility skew, 102–103 Lesniewski, Andrew, 91, 165 Locally capped globally floored cliquet, Levy, Arnon, 114, 163 122–125 Levy´ processes, 56 performance, 124–125, 127 Levy-Khintchine´ representation, 56 Log contract, 135–136 applications, 57–58 Lognormal model for volatility, 151–154 Lewis, Alan, 58, 96, 166 Heston model, example, 152–154 Limit orders, 108–109 Lognormal SABR formula, 92 with guaranteed execution, 108 Lognormal volatility dynamics, 152 Listed quadratic-variation based securities, Lognormally distributed jumps, 143 156–160 Log-OU model, 95 Litzenberger, Robert, 8, 133, 163 Log-strike, definition, 147 Live-out options, 107, 116 Log-strip, 141 intuition, 109 value, 142 values as function of barrier level, 117f Long expirations, Long-dated skew, Local stochastic volatility models, 155 Longer-dated volatility skews, 35–36, 71, Local variance (or volatility), 1, 7–14, 44 See 95–96 Long-Term Capital Management (LTCM) also Heston model; Jump-to-ruin model meltdown, 137 approximate formula in Heston model, 33 Lookback options, 112, 117, 118f approximate formula in Heston-Nandi discrete monitoring, 119 model, 44 valuation adjustment, 119 as a conditional expectation of Lookback hedging argument, 112–113 instantaneous variance, 13–14 Lower bound on volatility skew, 79 Brigo-Mercurio parameterization, 25 Luo, Chiyan, 139 computation in Heston-Nandi model, 45–46, 47f Madan, Dilip, 58, 133, 164 definition, 14 Market price of volatility risk, 6–7 Dupire equation for, 9. See also Implied Martingale, 14, 28 volatility; Volatility skew option prices as, 103 exact formula in jump-to-ruin model, 80 Matytsin, Andrew, 40, 68, 166 flattening of volatility surface over time, Mean jump arrival rate, 56 102–103 Mean reverting random variable, 4 formula in terms of implied, 11–13, 102 Mediobanca Bond Protection 2002–2005, history, 7–8 122–125 in jump-to-ruin model, 81f estimated coupons, 125t minimizing value of options on volatility, historical performance, 124, 125f 155 intuition, 123 short-dated skews, 88 payoff, 122 valuation of valuation, 123, 124f knock-out calls, 115f, 116f Mediobanca Reverse Cliquet European binary, 111f Telecommunicazioni, 125–127 live-out calls, 117f historical performance, 127, 128f locally capped locally floored cliquet, intuition, 126 123–124 payoff, 126 lookback options, 118f valuation, 126, 127f Napoleon 128–129, 129f Mediobanca S.p.A., 122 one-touch, 111f, 112f Mediobanca 2002–2005 World Indices Euro reverse cliquet 126, 127f Note Serie 46, 127–131 Index 175

estimated coupons, 131t NIKKEI 225, 128 general comments, 131 Nogel,¨ Ulrich, 25, 166 historical performance, 130, 130f Nonlinear fit, 36 intuition (or lack thereof), 129 Numerical PDE investor motivation, 130–131 computation of option prices, 46, 48f payoff, 127–128 valuation, 128–129, 129f One-touch double barrier construction, 120 Medvedev, Alexey, 89, 93, 166 One-touch options, 109, 113, 115–116 Medvedev-Scaillet formula, 89–93 bid-ask spread in FX market, 155 Mercurio, Fabio, 25, 163 intuition, 110 Merton, Robert, 75, 84, 166 ratio to European binary, 111f jump diffusion (JD) model, 57–58, 61, value, 112f 142 Options. See American options; Amortizing model of default, 75–76. See also Default Options; Barrier options; European default probability, 78 options; Digital options; Lookback Merton-style lognormally distributed jump options; Parisian options, Range process, 65–66 options; Variance options; Volatility Mikhailov, Sergei, 25, 166 options; Mikosch, Thomas, 166 Out-of-the-money barrier options, 114–115 Milstein discretization, 22–23 Out-of-the-money puts close to expiration, recommendation, 23 51 Model calibration, 25 Overshoot of discrete barrier, 119 Model independence, of generalized European payoff formula, Pan, Jun, 17, 67, 164 134 Papanicolaou, George, 95–96, 165 of variance swap replication, 136 Paras,´ Antonio, 114, 163 options on volatility, 154–156 Parisian options, 120 Model-independent upper/lower bounds for Parisian-style features, 120 volatility options, 155 Partial differential equation (PDE), 24, 56 Modeling assumptions, numerical solution. See Numerical PDE sensitivity of price of one-touch to, 110 Partial integro-differential equation (PIDE), Modified Bessel function, 149 56 Moment generating function, 147 Path-integral representation of implied Monte Carlo methods, 23 volatility, 26–31 Monte Carlo simulation, 15, 24 Payoff, 114 Morokoff, William, 137, 164 generation, 102 hedging, 101 Nandi, Saikat, 43, 165 PDE. See Partial differential equation Napoleon, 122, 127–132. See also PDF. See Probability density function Mediobanca 2002–2005 World Indices Perturbation expansion, 89 Euro Note Serie 46 PIDE. See Partial integro-differential history, 131 equation intuition (or lack thereof), 129 Platen, Eckhard, 166 investor motivation, 130–131 Poisson intensity, 56 payoff, 127–128 Poisson process, 66, 93, 140 valuation, 128–129, 129f hazard rate, 54 Neftci, Salih, 166 Portfolio. See Quasistatic hedge portfolio Negative butterflies, 26 hedge, 5, 53 Negative calendar spreads, 26 of stock with knock-out call, 108 Negative variances, 22 Power payoff, fair value, 146–147 176 THE VOLATILITY SURFACE

Price of volatility risk, 6–7 Riemann sheet, 20 Principal guarantee, 122, 126, 127 Risk Probability density function (PDF), market price of volatility risk, 6, 16 conditional on final stock price, 31f Risk-free hedge portfolio, 5, 53 of final stock price, 10 Risk-neutral process. See Stock price process; of realized variance, 149 Volatility process Pseudo-probability, Risky bonds, 75, 77 of default, 78 Risky options, 77 of a jump, 54 arbitrage bounds, 79 of option exercise, 17 Risky rate, 77 Put spread combination, payoff, 79f Ross, Steven, 15, 144, 164 Put-call parity, with risk-free and risky Rubinstein, Mark, 26, 166 bonds, 77–78 Put-call symmetry, 113–114 SABR. See Stochastic alpha beta rho Samperi, Dominick, 26, 163 Q-Q plot, 2. See also Standard & Poor’s Santa-Clara, Pedro, 103, 166 Scaillet, Olivier, 89, 93, 166. See also 500 Medvedev-Scaillet formula Quadratic variation, See also Expected Scholes, Myron, 84, 163 quadratic variation; Variance swaps Schonbucher,¨ Philipp, 103, 166 definition, 136 SDE. See Stochastic differential equation law of, 155 Self-financing hedge, 4–5, 55, 75, 133–134 options, 154. See also Variance, options on Shimko, David, 26, 166 Laplace transform under zero correlation, Short-dated ATM skew, 35, 62, 71f, 87–89, 147–149 89f, 91, 94 probability density, Short-dated implied volatilities, 51t, 52f, 90, lognormal approximation vs exact 91, 93 Heston, 154f Short-dated smile, 51–52, 51t, 52f Qualitative valuation, 114–117 Single-stock variance swaps, 160 Quasistatic hedge portfolio, 114 Singleton, Kenneth, 17, 67, 74, 164 Quasistatic hedging, 114–117 Sircar, K. Ronnie, 95–96, 165 Skew. See At-the-money skew; Long-dated Radon-Nikodym , 29 skew; Short-dated skew; Variance skew; Randall, Curt, 46, 120, 167 Volatility skew Range options, 120 behavior under jump-diffusion, 61 Realized variance, 149. See also Quadratic computation from characteristic function, variation; Total variance 60 Realized volatility, 143 impact on digital option value, 104 Rebalancing decay of due to jumps, 63–65 of , 112–113 impact on variance swaps, 138–140 of variance swap, 136 generated by Jump-to-ruin (Merton) Rebate, 108 model, 83 Recovery rate, 84 Smile,1,2,37See also Volatility surface Reduced form models of default, 74 empirical, 36f, 38f, 41, 51t, 52f Reflection principle, 109–110 SVI parameterization, 37 Reflected path, 110f under jump-diffusion, 61–63, 63f Return distribution, Sosin, Howard, 112, 165 empirical distribution, 2 Spanning generalized European payoffs, with jumps, 64, 65f 133–134 Reverse cliquet, 122, 125–127 SPX. See Standard & Poor’s 500 Revuz, Daniel, 120, 166 Square root process, 101 Index 177

Standard & Poor’s 500 (SPX), 104, 128 valuation equation, 6 ATM skew, graph, 39f derivation, 4–6 ATM variance, graph, 40f Stochastic volatility inspired (SVI) cliquet, resetting, 105f parameterization, 37 daily log returns, 2f, 3f plots, 38f fitted parameters, Stochastic volatility plus jumps (SVJ), 65–73 Heston, 40t See also Jumps SVJ, 71t approximate additivity of short-dated frequency distribution, 3f skews, 66, 67f futures, 158 at-the-money variance skew, 67f, 70f implied volatility surface, 36–42 empirical fit to SPX, 71t, 72f See also comparison with Heston fit, 41f Standard & Poor’s 500 comparison with SVJ fit, 72f fits from the literature, 69t graphs, 36f, 38f process. 65 model fits, 66–68, 69t model, 66 option prices (as of 15-Sep-2005), 51t variance swap valuation, 143 percentage of variance overnight, 50 why SVJ wins, 73 principal component analysis of variance, Stochastic volatility with simultaneous jumps 155 in stock price and volatility (SVJJ), probability of large move, 50 68–71. Q-Q plot, 3f at-the-money variance skew, 70f, 71f State vector, 15–16 characteristic function, 68 State-dependent diffusion coefficient, 8 short-dated limit, 70 Static hedge, 114 Stock market crash (October 1987), 1 Stineman, Russell, 37, 166 Stock price, 57 Stineman monotonic spline interpolation, Brownian Bridge density, 30 37 process, 4, 9, 43, 52, 57, 65 Stochastic alpha beta rho (SABR) Stock as call on value of company, 84 implied volatility formula, 91 See also Stork, Jabairu, 74 Lognormal SABR formula Structural models of default, 74 model, 91–93 Surface. See Volatility surface process, 91 Survival probability, 85–86 short-dated skew, 92 SV. See Stochastic volatility volatility process, 93 SVI. See Stochastic volatility inspired Stochastic differential equation, 7, 26, 43, SVJ. See Stochastic volatility plus jumps 44, 52, 57, 87, 91 SVJJ. See Stochastic volatility with Stochastic implied volatility models, 103 simultaneous jumps in stock price and martingale contstraint, 103 volatility Stochastic volatility (SV), 1–7. See also Swaps. See Variance swaps; Volatility swaps Heston model contrast with local volatility, 49 Taleb, Nassim, 107, 110, 120, 167 calibration, 25–26 Tavella, Domingo, 46, 120, 167 characteristic shape of volatility surface, Terminal return distributions with jumps, 42 64, 65f characteristic skew behavior, 96 Time zero skew, 62, 89, 91, 94 computational complexity, 7–8 Total variance, See also Quadratic variation process (SDE), 4, 87 expected, 32, 159 short-dated skew, 88, 91 implied 11, 88, 102 SV models in general, 24, 39, 40, 42, 99, realized 136–138, 144, 148 100, 101, 116 Trading time, 1 time homogeneity, 103 Triple witching day, 36, 36f 178 THE VOLATILITY SURFACE

Up-barrier, 118 perturbation expansion, 96–97 Upper bound on option price, 79f Volatility skew See also At-the-money Upper bound on implied variance skew, volatility skew 87–89 credit spreads, relationship, 86 dynamics, 101, 152 Valuation equations, under local volatility, 102–103 jump-diffusion, under stochastic volatility, 101–102 known jump size, 54 effect of default risk, 76 uncertain jump size, 55 empirical levels as of 15-Sep-2005, 39f jump-to-ruin model, 75 extreme strikes, 97–99 stochastic volatility, 6 impact of jumps, 61–65 Variance (or volatility), options on, long-dated expansion, 95–96 152–156 short-dated, 50–52 call value, 153f inconsistency with diffusion, 102 lognormal approximation, 151–152 short-dated expansion, 87–94 Variance swaps, 136–146 under stochastic volatility, 87–91 dependence on skew and curvature, 139 effect of jumps, 139–143 with jumps, 93–94 fair value, 137 small volatility of volatility, 96–97 in the Heston model, 138 Volatility surface, in terms of implied volatility, 139 asymptotics, 87 under diffusion, 137 dynamics, 101 hedging strategy, 136 empirically lognormal, 101 history, 137–138 flattening under local volatility, lognormal approximation, 151–152 102–103 Variance vs volatility empirical. See also Standard & Poor’s 500 convexity adjustment, 143 SPX as of 15-Sep-2005, 38f, 51t, 52f in the Heston model, 144–146 empirical fit, 36f, graphs, 145f comparison with Heston fit, 41f VIX index, 156–158 comparison with SVJ fit, 72f calculation, 157–158 fitting, 7,16, 21, 25, 49, 52, 153, 154 Volatility. See also Implied volatility; CreditGrades, 86–87 Instantaneous volatility Heston fit, 35–42, 40t, 41f as a random variable, 2 jump-to-ruin model, 82–83 clustering, 2 model fits from the literature, 67–68, derivatives, 133, 161 69t valuation, 146–156 SABR, 91 jumps in 68–71 SVJ fit, 65–73, 71t, 72f SVJJ, 68–70 parameterization (SVI), 37 mean reversion, 2 Volatility swaps, 136, 143 near constancy for short times under diffusion, 51–52 convexity adjustment, 143 options, 154–156 in the Heston model, 144–146 process, 4, 65, 68, 97 graphs, 145f risk lognormal approximation, 152 extra return per unit, 7 fair value under zero-correlation, market price of, 6–7, 16 149–151 Volatility index. See VIX index proof, 150–151 Volatility of volatility, 35, 51, 97 replication, 151 fair value of variance independent of, VXB futures, 158–160 138 contract settlement, 158 Index 179

convexity adjustment, 158 Yakovenko, Victor, 99, 164 in the Heston model, 159 Yan, Shu, 103, 166 Dupire’s valuation method, 160 Yor, Marc, 120, 166

Whaley, Robert, 8, 164 Zero correlation assumption between Wilmott, Paul, 107, 167 volatility and stock price, 146 market price of volatility risk, 6 fair value of implied volatility, 149 valuation under stochastic volatility, 4–6 Laplace transform of quadratic valuation under jump-diffusion, 52 variation, 147. See also Quadratic Womersley, Robert, 103, 163 variation Woodward, Diana, 91, 165 Zou, Joseph, 137, 164