60–3central banks, interest rates 28Central Limit Theorem 46chooser optionsconcepts 49–51definition 49example 50–1strategy rationale 51terms 49–50cliquet optionssee also ratchet optionsconcepts 83–5definition 83forward starting options 83–5risk 85collarssee also risk reversal strategiesconcepts 39–40definition 39composite optionsconcepts 125–8, 129–30, 167–8definition 125delta hedging 126–8, 132–3example 125–6
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FX exposure 126–8implied volatility 126outperformance options 129, 130–1pricing 127–8quanto options 125variance 126, 167–8computer programmes, Monte Carlosimulations 48conditional coupons, autocallable notes94–7conditional double barrier optionssee also double barrier optionsconcepts 68–9continuous dividends 10conventional optionssee also Greeksconcepts 1, 3–21, 157–65financially engineered structures157–65forward prices 10forward starting options 71–2interest rate exposures 21pricing 1, 6–10, 12–14, 34, 45–8,71–2, 105–7strategies 35–44variance swap replication 140–9,169–74convexity principle, vega 148–9correlationbest of options risk 135–7composite options risk 125–6, 131–2dispersion 151–6exotic options 1interest/equity correlation factors 98outperformance options 131–2, 136–7quanto options risk 119–22worst of options risk 135, 137–8cost of carry, definition 9coupons 87–91, 93–8, 99–104, 157–8,160–3autocallable notes 93–8, 99–104interest rates 90–1reverse convertibles 87–91, 93–6,99–104, 160–3covariance 152curvature parameter, implied volatility31–2deltasee also dynamic hedgingarithmetic Asian out options 109–10
180
Index
delta (Continued)Asian options 109–12barrier options 58–65, 161–3Black–Scholes options pricing formula12–14collars 40composite options 126–8, 132–3concepts 12–14, 21, 25–6, 29–32, 40,58–63, 90–1, 109–15, 122–3,126–8, 132–3, 156definition 12, 25dispersion 156down-and-in put barrier options 58–63,161–3extreme values 12forward starting options 72–3gamma 14–16, 23–4, 25–6, 27, 29, 54,60, 75, 79–80, 110, 113–15,142–5interest rates exposures 21outperformance options 132–3quanto options 122–3skew effects 29–32, 147–8, 156variance swaps 141–5, 147–8, 156delta cashconcepts 25–6, 144–5definition 25example 26digital optionscall spread option strategy 53–5, 95–6concepts 1, 53–5, 62, 95–8definition 53payoff profiles 53–4strike-selection criteria 55DIP see down-and-in put barrier optionsdiscounts 84–5, 89–90dispersionsee also basket optionsconcepts 151–6correlation 151–6definition 151delta 156ratio’s vega 155skew delta 156trading possibilities 153–5variance swaps 153, 155vega 153–5volatility 151–5dividends 6–12, 39–40, 120–2, 159Black–Scholes options pricing formula6–7, 9–10, 120–1
collars 40concepts 9–12, 39–40, 159continuous dividends 10forward starting options 72forwards 8–9, 10, 39–40premiums 159double barrier optionssee also barrier optionsconcepts 57–8, 68–9conditional double barrier options68–9definition 68examples 68–9risk 68–9types 68–9unconditional double barrier options69down-and-in call barrier options,concepts 58, 163down-and-in put barrier optionssee also barrier optionsairbags 160–3autocallable notes 93–6, 101–4barrier shifts 60–3, 67–8, 96, 104,161–3concepts 57–63, 66, 87–90, 93–6,101–4, 160–3definition 58delta 58–63, 161–3exposure equity products for cautiousinvestors 160–3gamma 60maturity dates 62, 66payoff profiles 58–60, 66–7pricing 58–63, 66, 89–90, 95–8,101–4, 162–3reverse convertibles 87–90, 93–6,101–4risk 58–60share-buying situations 63skew 58, 66–8time to maturity 62volatility factors 60–2down-and-out call barrier options,concepts 58, 68, 103–4down-and-out put barrier options,concepts 58, 103–4, 161–3downside options 27–34, 39–40, 67–8,80–1duration, arithmetic Asian out options108–10
Index
DVO1 91dynamic hedgingsee also deltaconcepts 14, 21, 23–4, 26, 27, 30–2,90–1, 109–15, 122–3, 132–3definition 14dynamically short skew, forward startingoptions 74–5early redemption features, autocallablenotes 95–6, 97–8, 101–2equity/FX contrasts, skew 32–4EUR/BRL exchange rate, skew 33–4EUR/USD exchange rate, skew 32–4European options 3–4, 11–18, 47–8,49–51, 53–5, 58–69, 95–8definition 3–4, 49put–call parity 11–12event risk, variance swaps 146–7exerciseAmerican options 4, 46–8definition 3–4exotic optionscorrelation components 1definition 1, 53, 87full spectrum 1gamma sign changes 16pricing 1, 34, 45, 49risks 1skew 34, 35, 58–60vanna/vomma 19–21vega 16–17expected fair value of the stock, forwards4–6expiration dates see maturity datesexposure equity products for cautiousinvestors 157, 160–3falling markets, volatility 27–8financially engineered structuressee also retail investorscapital guaranteed products for riskaverse investors 157–9concepts 157–65exposure equity products for cautiousinvestors 157, 160–3leveraged products for risk-seekinginvestors 157, 163–5fixed strike lookback optionssee also lookback optionsconcepts 79–80
181
floating strike lookback optionssee also lookback optionsconcepts 79, 80–1foreign exchange see FXforward starting optionsanalytically short skew 74–5cliquet options 83–5concepts 71–5, 80–1, 83–5, 111conventional options 71–2definition 71delta 72–3dividends 72dynamically short skew 74–5examples 71–5the Greeks 72–3, 75, 111hedging 72–3, 74–5implied volatility 72–4lookback options 80–1maturity dates 75pricing 71–5, 80–1, 83–5, 111risk 71, 72–5, 85skew risk 71, 72–5, 85strike prices 71–5, 111term structure of the skew 73–4vega 74–5, 80–1forwardsAsian in forwards 112–18Black–Scholes options pricing formula10concepts 4–21, 51cost of carry 9definition 4–6, 8dividends 8–9, 10, 39–40expected fair value of the stock 4–6interest rates 10, 39–40payoff profiles 5–6pricing 4–5, 8–10strike prices 6, 8–9, 12FXcomposite options 125–8, 167–8quanto options 119–23skew equity contrasts 32–4volatility 120–3, 126–8gamma 1, 14–16, 18, 23–9, 54, 60, 75,79–80, 110, 113–15, 142–5, 146–7arithmetic Asian out options 110Asian in forwards 113–15at the money options 16concepts 1, 14–16, 18, 23–9, 54, 60,75, 79–80, 110, 113–15
182
Index
gamma (Continued)definition 14delta 14–16, 23–4, 25–6, 27, 29, 54, 60,75, 79–80, 110, 113–15, 142–5down-and-in put barrier options 60examples 23–4, 26exotic options 16hidden gamma 113–14importance 16, 23–4lookback options 79–80profit-determination role 1, 16, 23–4,27, 29, 54, 113–14, 143–4, 146–7theta 23–4variance swaps 142–5, 146–7vega comparisons 18gamma cashconcepts 25–6, 143–6definition 25gearing 39–40, 53–5, 66, 95–6, 164–5see also leveraged products . . .geometric Asian out optionssee also Asian . . .concepts 105–7implied volatility 107, 108geometric averages, arithmetic Asian outoptions 108, 110geometric basket options 167–8Girsanov theorem, concepts 7the Greekssee also delta; gamma; theta; vegaarithmetic Asian out options 107–10Asian in options 111barrier options 57–65concepts 1, 3, 12–21, 57, 63forward starting options 72–3, 75, 111variance swaps 140–7GSK shares 125–8hedging 8–9, 11–12, 14, 21, 23–4, 26, 27,30–2, 53–5, 58–60, 63–4, 72–5,90–1, 105, 109–10, 113–15, 122–3,126–8, 132–3hidden gamma, Asian in forwards 113–14higher order derivatives, concepts 19–21hybrids, concepts 138implied volatilitysee also vega; volatility; vommaarithmetic Asian out options 108, 110Asian options 107, 108basket options 152–3
Black–Scholes options pricing formula8, 45, 107call spread option strategy 36–7composite options 126concepts 8, 16–18, 19–21, 27–34,36–7, 64–5curvature parameter 31–2definition 8forward starting options 72–4geometric Asian out options 107,108premiums 159reverse convertibles 90skew 27–34, 35, 36–7, 58, 66–8, 71,72–5, 85, 141–2, 147–8, 156smiles 29–34strike prices 27–34, 35, 36–7, 58, 66–8,71, 72–5, 85, 141–2, 147–8, 156up-and-out call barrier options 16–17,64–5, 67–8variance swaps 141–9in the money options 12–14, 16, 19, 27,30–2, 50–1, 59–61, 63–4, 69,110–12indicesautocallable notes 93–8options 28skew 156interest rate hedge, reverse convertibles90–1interest rates 6–7, 11–12, 18–19, 21,27–8, 39–40, 90–1, 106–7, 126,167see also risk free . . .central banks 28concepts 6–7, 11–12, 18–19, 21, 27–8,39–40, 90–1, 106–7, 126, 167coupons 90–1delta hedging 21forwards 10, 39–40future rates 27–8liquid markets 7reverse convertibles 90–1zero coupon bonds 159interest/equity correlation factors,autocallable notes 98investment banks 85, 113–15investorsbarrier options 57–8capital guaranteed products for riskaverse investors 157–9
Index
cautious equity investors 157, 160–3exposure equity products for cautiousinvestors 157, 160–3fears 28, 157–65financially engineered structures157–65leveraged products for risk-seekinginvestors 157, 163–5risk attitudes 157–65risk averse investors 157–9risk-seeking investors 157, 163–5types 157–65Ito’s lemma, concepts 7Jensen’s inequality 108knock-in reverse convertiblessee also reverse convertiblesconcepts 87–90knock-out/knock-in barrier optionssee also barrier optionsbarrier parity 65ladder optionsbarrier options 78concepts 77–8, 79–80definition 77example 77lookback options 79–80pricing 78, 79–80leveraged products for risk-seekinginvestors 157, 163–5see also gearingliquid markets, interest rates 7lognormal returns, standard deviation145–6lookback optionsbarrier options 79–80concepts 79–81definition 79fixed strike lookback options79–80floating strike lookback options 79,80–1forward starting options 80–1gamma 79–80ladder options 79–80pricing 79–81risk 80–1skew 81types 79
183
maturity dates 3–21, 62, 65–6, 75see also time to maturityautocallable notes 94barrier options 62, 65–6definition 3–4down-and-in put barrier options 62,66forward starting options 75reverse convertibles 88skew 28–30Monte Carlo simulations 1, 45–8, 57,59–60, 64, 66, 68–9, 71–3, 79–80,107, 114–18, 135, 151arithmetic Asian out options 107Asian in forwards 114–18Asian options 107, 114–18barrier options 57, 58, 59–60, 64, 66,68–9basket options 151binomial trees 46–7, 48computer programmes 48concepts 45–8exotic options 45pricing 45–8, 57, 58, 59–60, 64, 66,68–9, 71–3, 79–80, 107, 114–18,135, 151principle 45–6workings 48Newton–Raphson process, concepts 102,117Normal distributions 7, 139–40, 141–2,145–6notional, definition 119optionssee also call . . . ; conventional . . . ;exotic . . . ; Greeks; pricing;put . . .concepts 1, 3–21, 157–65financially engineered structures157–65forward prices 10indices 28interest rate exposures 21premiums 21, 31–2, 90, 123, 127–8,132–3, 158–9strategies 35–44, 51–69, 85,157–65out of the money options 12–14,27
184
Index
outperformance optionscomposite options 129, 130–1concepts 129–33, 136–7, 167–8correlation position 131–2, 136–7definition 129delta hedging 132–3example 129–33hybrids 138payoff profiles 129, 130–1pricing 129–31variance 131, 167–8payoff profilesbest of options 135–6call options 4–5call spread option strategy 36–7capital guaranteed products 158collars 39–40concepts 4–8digital options 53–4down-and-in put barrier options 58–60,66–7forwards 5–6outperformance options 129, 130–1put options 4–5put spread option strategy 37–9straddles 42, 44strangles 43–4supertrackers 164–5variance swaps 139–40, 142worst of options 137performance comparisonsbest of options 135–8outperformance options 129–33,167–8worst of options 135, 137–8Philips shares 23–4, 26pin risk, digital options 54price earnings ratios 27–8pricingsee also Greeksarithmetic Asian out options 107–8Asian in forwards 114–18Asian in options 111Asian options 105–18Asian out options 105–8autocallable notes 95–8, 99–104barrier options 57–69, 77–8, 89–90,95–8, 162–3basket options 151best of options 135
binomial trees 46–8Black–Scholes options pricing formula6–10, 12–14, 84–5, 107, 120–1,131, 151call options 6–8callable reverse convertibles 99–102chooser options 49–51cliquet options 83–5composite options 127–8concepts 1, 4–10, 12–14, 34, 45–8, 71conventional options 1, 6–10, 12–14,34, 45–8, 71–2, 105–7dividends 6–9, 10–12down-and-in put barrier options 58–63,66, 89–90, 95–8, 101–4,162–3exotic options 1, 34, 45, 49forward starting options 71–5, 80–1,83–5, 111forwards 4–5, 8–10geometric Asian out options 105–7ladder options 78, 79–80lookback options 79–81methods 1, 6–8Monte Carlo simulations 45–8, 57, 58,59–60, 64, 66, 68–9, 71–3, 79–80,107, 114–18, 135, 151outperformance options 129–31put options 6–8, 10put–call parity 11–12puttable reverse convertibles 102–4quanto options 119–22reverse convertibles 89–90, 95–8,99–104, 162–3skew uses 34up-and-out call barrier options 64–5variance swaps 140–2worst of options 135profit-determination role, gamma 1, 16,23–4, 27, 29, 54, 113–14, 143–4,146–7put optionssee also Greekschooser options 49–51concepts 1–8, 10, 12–21, 23–44,158–65definition 3down-and-in put barrier options 57–63,66, 87–90, 93–6, 160–3forward prices 10payoff profiles 4–5
Index
pricing 6–8, 10up-and-out put barrier options 58, 65,68put spread option strategy, concepts 37–9put–call parity, concepts 11–12puttable reverse convertiblesconcepts 99, 102–4definition 99quanto optionscomposite options 125concepts 119–23correlation risk 119–22definition 119delta hedging 122–3FX volatility 120–3pricing 119–22ratchet optionssee also cliquet optionsconcepts 83–5definition 83investment bank losses 85ratio’s vega dispersion 155see also dispersionrealised-variance/standard-deviationcontrasts, variance swaps 139–40,145–6rebates, barrier options 64, 77–8, 97–8,101–2, 160–3recessions 27–8references 175replication, variance swaps 140–9,169–74retail investorsbarrier options 57–8capital guaranteed products for riskaverse investors 157–9cautious equity investors 157, 160–3exposure equity products for cautiousinvestors 157, 160–3financially engineered structures157–65leveraged products for risk-seekinginvestors 157, 163–5risk averse investors 157–9risk-seeking investors 157, 163–5types 157–65reverse convertiblesautocallable notes 93–6, 99–104barrier options 87–90, 93–6, 99–104
185
callable reverse convertibles 99–102concepts 87–91, 93–6, 99–104,160–3conditional coupons 161–3coupon-magnitude market conditions89–90definition 87demand determinants 89–90down-and-in put barrier options 87–90,93–6, 101–4examples 87–9expiry scenarios 88exposure equity products for cautiousinvestors 160–3hedging 90–1implied volatility 90–1interest rate hedge 90–1pricing 89–90, 95–8, 99–104,162–3puttable reverse convertibles 99,102–4risk 88, 90–1volatility hedge 90–1riskattitudes 157–65concepts 1, 35–44, 51–69, 157–65exotic options 1risk free interest rates 6–7, 11–12, 39–40,126, 167Black–Scholes options pricing formula6–7, 121concepts 6–7, 11–12risk managementbarrier options 57–69strategies 35–44, 51–69risk neutral portfolios, concepts 6–8risk reversal strategiessee also collarsconcepts 39–40risk averse investors, capital guaranteedproducts 157–9risk-seeking investors, leveragedproducts 157, 163–5Royal Dutch/Shell shares 8–9settlement pricessee also strike pricesAsian options 105–18definition 105short players, falling markets 28simple option strategies 35–44
186
Index
single barrier optionssee also barrier optionsconcepts 57–68retail demand 57–8skewsee also implied volatilitybarrier options 57, 58, 66–8call spread option strategy 36–7, 85concepts 27–34, 35, 58, 66–8, 71,72–5, 85, 141–2, 147–8, 156definition 27delta effects 29–32, 147–8, 156dispersion 156down-and-in put barrier options 58,66–8exotic options 34, 35, 58–60forward starting options 71, 72–5, 85FX/equity contrasts 32–4indices 156lookback options 81maturity dates 28–30options on indices 28pricing uses 34term structure of the skew 28–30,73–4up-and-out call barrier options 67–8up-and-out put barrier options 68variance swaps 141–2, 147–8, 156smiles 29–34standard deviationlognormal returns 145–6realised variance 139–40, 145–6variance swaps 141–2, 145–6standard options see conventional optionsstochastic processes 121see also Brownian motionstraddleschooser contrasts 51concepts 40–2, 44, 51definition 40–1example 41–2, 44payoff profiles 42, 44stranglesconcepts 42–4definition 42payoff profiles 43–4strategies 35–44, 51–69, 85, 157–65bearish strategies 37–9, 63, 157–9,164–5bullish strategies 35–6, 39–40, 63,157–9, 163–5
call spread option strategy 35–7, 53–5,85capital guaranteed products for riskaverse investors 157–9cautious equity investors 157, 160–3chooser options 51collars 39–40exposure equity products for cautiousinvestors 157, 160–3financially engineered structures157–65leveraged products for risk-seekinginvestors 157, 163–5put spread option strategy 37–9retail investors 157–65risk averse investors 157–9simple strategies 35–44straddles 40–2, 51strangles 42–4strike pricesAsian options