P1: JYS ind JWBT294-Sinclair May 7, 2010 17:11 Printer: Courier Westford, Westford, MA

Index

Bold number refers to definition

A arbitrage bounds for prices boxes, 38–39 adjust position, 230 the “butterfly” relationship, 32 ADR. See American Depository , maximum value of a, Receipt (ADR) 25–26 all or nothing (AON), 271 call option, minimum value for a, alpha, 87 26–27 as function of call delta, 888 calls with different expirations, as function of underlying price, 87 relationship between, 34 American Depository Receipt (ADR), calls with different strikes, 128, 271 29–30 American option, 271 calls with different strikes, Black-Scholes-Merton (BSM) model, comparing three, 33 63, 68 options, American compared to “Butterfly” relationship, 32 European, 25 calls with different expirations, option values, 21–24 34–35 portfolio 1, 36 calls with different strikes, 29–30 portfolio 2: no , 37 Euopean option vs., 23, 28, 31, 39 pseudoarbitrage, 24–25 exercise style, 10 put, maximum value of a, 26 implied curve, 138 put call parity, 35–38 intrinsic value of, 28 put call parity relationship, 35 nondividend paying stocks, 38 , minimum value for a, options, early exercise of, 243 27–29 option strategies, 92 puts with different expirations, put call paristy,COPYRIGHTED 36 MATERIALrelationships between, 34–35 short option position, 26 puts with different strikes, 30–31, Anscombe’s Quartet, 267 31 AON. See all or nothing (AON) summary, 39–40 arbitrage, defined, 271 values, absolute maximum and arbitrage, structured product, 13–14 minimum, 25–39

285 P1: JYS ind JWBT294-Sinclair May 7, 2010 17:11 Printer: Courier Westford, Westford, MA

286 INDEX

arbitrage relationships at-the-money (ATM) call about, xiii, xvi, 23 delta as a function of time until dependent variables, 43 expiry parameters are those in surface, 183 figure 5.6, 69 put call parity, 35 as a function of time until expiry, , 271, 275 66 asset, underlying, 10 at-the-money (ATM) forward, 271 “Asset Allocation” (Sharpe), 117 at-the-money (ATM) put asymptotic solution of Whalley and delta as a function of time until Wilmott, 198–200 expiry parameters are the same at-the-money (ATM) as those for figure 5.6, 70 call, 205 as a function of time until expiry, call delta vs. time to , 66 68–69 auction, pre-opening, 167–69 call options, 49, 63, 66, 75 automatic exercise, 271 defined, 271 delta of the, 68, 71, 74 B implied volatility, 147 implied volatility curve, 151, back months, 271 187 back office, 233, 271 implied volatility curve as a function back spread, 271 of expiration, 145 backwardation, 271 option, with volatility will lower the bad things happen, 135–36 gamma, 199 Barings Bank, 233 options, gamma is maximized for basis point, 84, 271 short-dated, 88 basis risk, 242, 272 options, out-of-the-money options , 240, 272, 275 relative to, 140 Bayes’s theorem, 177–81 options, theta is maximized for bear, 272 short-dated, 88 Bermudan option, 10, 272 options, value of, 66–67, 71 bias, 123 options, vega is short, 111 efficiency and, 123–25 options, volatility value of, 77 bid and ask option with high gamma, 72, 74, 75, implied volatility for Google 80 February option December 21, put options, 63, 66 2009, 186 , 150 spread in practice, varying the, strike, 77, 104, 109, 186, 220, 229 179–80 strike, upside strikes trade at bid/ask spread, 59, 123, 228, 272 premium vs., 140 binomial model, 48–55 strike, volatility of, 142–43 about, 42, 272, 282 below, 138 binomial tree volatility curve, 138 European call option, pricing of, 55 volatility index of CBOE, model-free one-step, evolution of the stock implied, 147 price in, 49 P1: JYS ind JWBT294-Sinclair May 7, 2010 17:11 Printer: Courier Westford, Westford, MA

Index 287

single node, pricing of a call option , second-order, 84 at, 54 put delta as function of the three-step, evolution of the stock underlying price, 69 price in, 53 put value as function of the Black-Scholes-Merton (BSM) model underlying price, 65 about, xi, xv–xvi, 5, 42–43, 55–61, rho, 82–83 63–67, 272 summary, 88 alpha, 87 theta, 76–78 alpha as a function of the underlying vega, 80–82 price, 87 vega as function of the underlying alpha as function of the call delta, price, 81 88 vega as function of time and expiry ATM call as a function of time until for an at-the-moment option, expiry, 66 81 ATM call delta as function of time block trade, 272 until expiry parameters are Boston Options Exchange, Inc. (BOX), those in figure 5.6, 69 15 ATM put as a function of time until boxes, 38–40, 272 expiry, 66 option, 59 ATM put delta as function of time breakeven, concept of, 78–80 until expiry parameters are brokerage firm, xii, 15–18, 272 those for figure 5.6, 70 BSM model. See Black-Scholes-Merton breakeven, concept of, 78–80 (BSM) model call delta as a function of the bucket shop, 4, 272 underlying price, 68 Buffett, Warren, 1 call rho as a function of the bull, 272 underlying price, 83 butterfly, 93, 272 call theta as a function of the condor vs., 113 underlying price, 77 iron, 111–12, 277 call value as a function of the long, 110–11, 201 underlying price, 65, 68 with missing wing, 114–15 carry rho, 83–84 no-arbitrage condition, 144 DdeIT, 85–86 relationship, 32 DdeIV, 85 short straddle vs., 219 DdeIV as function of the underlying static hedging and, 201 price, 85 buying DdelT for calls as function of the call as a hedge, 12–13 underlying price, 86 put (call) to speculate on a fall delta, 67–71 (rise), 12 gamma, 72–76 buy-in risk, 242–43 gamma as a function of the underlying price, 73 C gamma as a function of time until expiry for an out-of-the-money , 24, 115, 272, option, 74, 75 282 P1: JYS ind JWBT294-Sinclair May 7, 2010 17:11 Printer: Courier Westford, Westford, MA

288 INDEX

call closing price, 7, 118, 120, 123, 126–29, delta as function of the underlying 133, 269, 273 price, 68 closing trade, 273 on a future, exercising a, 246 CME. See Chicago Mercantile rho as a function of the underlying Exchange (CME) price, 83 COMEX, 273 on a stock, exercising a, commodity, 60, 273 246–47 markets, 271 theta as function of the underlying option expirations, 240 price, 77 products, 109 value as function of the underlying trader, 154 price, 65, 68 Commodity Futures Trading call-around market, 16, 272 Commission (CFTC), 273 call option(s) commodity trading advisor, 273 American vs. European, 25 , 273 with different strikes, 29–30, contingent order, 273, 278 33 contract month, 273, 276 expirations, relationship between contract size, 274 different, 34 contract unit, 10–11 maximum value of, 25–26 convergence, 54, 124, 274 minimum value for, 26–27 conversion, 113, 121, 221, 243, 272, values, 21–24 274 values, absolute maximum and correlation, 240–41 minimum, 25–39 Anscombe’s Quartet, 267–69 call spread, 29–31, 38, 40, 95, 103–6, correlation does not tell us “things 272, 277 are moving together,” 266 carry rho, 83–84 correlation is not causation, cash settlement, 273 265 Cauchy distribution, 259 correlation measures degree of CBOE. See Chicago Board Options linear association, 266 Exchange (CBOE) data that has a clear relationship CBOT. See Chicago Board of Trade and a low correlation, 266 (CBOT) fitted regression line, 264 center, quantifying the, 254 highly correlated, divergent series, CFTC. See Commodity Futures 266 Trading Commission (CFTC) for a population, 263 Chicago Board of Trade (CBOT), positively correlated returns and 273 negatively correlated Chicago Board Options Exchange volatilities, 267 (CBOE), 5, 15–16, 118, 147, 273, scatter graph of two variables, 264 275–77, 283 Simpson’s Paradox, 270 Chicago Mercantile Exchange (CME), coupon, 14, 242–43, 274 273, 276 , 19, 100, 274 Christmas tree, 273, 282 credit risk, 4, 17, 274 clearinghouse, 16–17, 273, 279 critical strike for put exercise as a close-to-close estimator, 128 function of time, 245 P1: JYS ind JWBT294-Sinclair May 7, 2010 17:11 Printer: Courier Westford, Westford, MA

Index 289

D variance (of a sample), 257 width, quantifying the, 256–61 day order, 274 dividend changes, 47–48 DdeIT, 85–86 dividends for calls as a function of the absence of, 59–60 underlying price, 86 buy-in risk and, 242–43 DdeIV, 85 dividend spread, 247–48 as a function of the underlying price, dividend yield 85 ATM call as function of time until deep (in-the-money), 28, 38, 42, 64, 74, expiry, 66 221, 246–47, 274 Black-Scholes-Merton (BSM) model, delivery date, 274 64–65 delivery options, 274 call delta as function of the delivery price, 274 underlying price, 68 delta, 67–71, 234 call value as function of the , 75, 192–93, 229, 274 underlying price, 65 dependent variables, 45–48 carry rates and, 83 designated primary market maker defined, 39, 47, 59–60, 64, 275 (DPM), 275 dividend changes and, 47 , 274 dividends, absence of, 59–60 direct method, 197 gamma as function of the underlying dispersion, 204, 274 price, 73 dispersion trading, 240, 274 gamma as function of time until distribution of returns, 60–61 expiry for out-of-the-money distributions option, 74 Cauchy distribution, 259 interest rates and, 39 center, quantifying the, 254 normal, 243 distribution A, 261 par value as function of underlying distribution B, 261 price, 65 distribution of height, repro rates and forward prices, corresponding, 255 84 histogram of random numbers vega as a function of time until drawn from a uniform expiry for at-the-money option, distribution, 260 81 histogram of random numbers from do nothing, 229 a normal distribution, 258 double-asymptotic method of kurtosis, 260–61 Zakamouline, 200–201 mean absolute deviation (MAD), 257 DPM. See designated primary market measured heights of 11 professional maker (DPM) athletes, 255 dynamic hedge, 223, 275 moments and the “shape” of distributions, 254–62 E normal distribution, 253–54 robustness of efficiency, 260 ECN. See electronic communication skewness, 256 network (ECN) tails, quantifying, 260–62 edge, 154–56 P1: JYS ind JWBT294-Sinclair May 7, 2010 17:11 Printer: Courier Westford, Westford, MA

290 INDEX

efficiency, 123 theta as we approach expiry, bias and, 123–25 224 EFT. See exchange-traded fund (ETF) wrong options, exercising the, Eisenhower, General Dwight D., 227 221–23 elasticity, 275 exponential utility function, 195 electronic communication network (ECN), 275 F electronic exchange, 15, 114, 184, 275, 279 fast market, 275 estimator, close-to-close, 128 FIFO. See first-in, first-out (FIFO) EUREX, 275 fill or kill (FOK), 275 European call option, 55 final settlement price, 276 European option, 275 first-in, first-out (FIFO), 181, 275 American option vs., 10, 25, 28, 39, fitted regression line, 264 243, 272 flipping, 183–85 Bermudan option vs., 272 FOK. See fill or kill (FOK) Black-Scholes-Merton (BSM) model, forward risk, 221 63 FXI, 7, 8 boxes and, 39 FXP, 7, 8 BSM partial differential equation, 42 call option, minimum value for, G 27–28 convergence and, 54 gambler’s ruin, 158–62 option nomenclature, 92 gamma, 72–76, 234–38 options, early exercise of, 243 as a function of the underlying price, puts with different expirations, 35 73 wrong options, exercising, 222 as a function of time until expiry for exchange-traded fund (ETF), 275 an out-of-the-money option, 74, exercise and assignment, 17–18 75 expiration date, 10 general principles of trading and expiration trading hedging forward risk, 221 about, 153–54 Google price on October 16, 2009, edge, 154–56 218 gambler’s ruin, 158–62 Greeks, irrelevance of the, 223–25 hedging, 156–58 implied volatility on expiration day Kelly sizing, example of, 162–63 for Google, October 16, 2009, probability distribution of result of 218 St. Petersburg game, 160 nonexpiration days in American probability of going bankrupt with stocks, 216 55 percent chance of winning option value as we approach expiry, and betting our entire bankroll, 224 159 pinning, 215–19 scalability and breadth, 163–64 pinning in American stocks, 216 scalability within products, 163–64 , 219–21 summary, 164 summary, 225 trade sizing and leverage, 158–63 P1: JYS ind JWBT294-Sinclair May 7, 2010 17:11 Printer: Courier Westford, Westford, MA

Index 291

Google price on October 16, 2009, implied method, 198 218 implied volatility Greeks about, 137 irrelevance of the, 223–25 bid and ask, for the Google February second-order, 84 option expiration at 2 p.m. on December 21, 2009, 186 H curve, 236 dynamic hedging, 141 Halmos, Paul, 227 dynamics, 146–51 hedge frequency, 206 expiration day hedging, 156–58, 192–93 for Google, October 16, 2009, based on underlying price changes, 218 194 implied correlation, 141–42 to a delta band, 194 implied volatility curve, 138–42 error as a function of the number of for an index, typical, 139 rebalances, distribution of the, as function of expiration, 207 145–46 a long position with a put, 11–12 parameterizing and measuring in practice, 193–203 the, 142–45 at regular time intervals, 193 for a stock, typical, 139 static, 201–3 implied volatility dynamics, 146–51 strategy for the one-year option, model misspecification, 138 realizations of the, 204 raw and scaled, of the QQQQ volatility, choosing a, 208–13 Options as a function of the hedging bands out-of-the-money delta as functions of the BSM delta which September 13, 2007, 143 is shown as a dashed line, summary, 151 optimal, 196 supply and demand, 140–41 long call, 196 surface, trading correlated assets, short call, 196 185–88 from the WW Asymptotic method as VIX and the S&P 500, 150 functions of the BSM delta, VIX implied volatility index, 148 199 VIX volatility of the, 148 from the Zakamouline Asymptotic volatility curves method as functions of the BSM that differ by a shift in the level, delta, 202 two, 146 histogram that differ by a twist, two, 146 of random numbers drawn from a that differ in convexity, two, uniform distribution, 260 147 of random numbers from a normal volatility curves that differ by a shift distribution, 258 in the level, two, 146 volatility curves that differ by a I twist, two, 146 volatility curves that differ in ICE. See International Commodities convexity, two, 147 Exchange (ICE) initial risk slide, 239 P1: JYS ind JWBT294-Sinclair May 7, 2010 17:11 Printer: Courier Westford, Westford, MA

292 INDEX

interest rate order-driven, 166–67 changes, dependence on the, 46–47 structure, 14–19, 166–69 existence of a single constant, 59 market maker International Commodities Exchange profit and loss distribution for the, (ICE), 276 174 inventory, 232–34 profit and loss distribution when inventory-based market-making, orders are unbalanced, 175 172–76 simulation, underlying price paths in I Want to be a Mathematician our, 173 (Halmos), 227 market-making, inventory-based, 172–76 K market making techniques about, 165–66 Kelly sizing, 162–63 auction, pre-opening, 167–69 kurtosis, 260–61 Bayes’s theorem, 177–81 bid and ask implied volatility for the L Google February option expiration at 2 p.m. on LEAP. See long-term equity December 21, 2009, 186 anticipation product (LEAP) bid and ask spread in practice, leverage and trade sizing, 158–63 varying the, 179–80 LIFFE. See London International flipping, 183–85 Financial Futures Exchange inventory-based market-making, (LIFFE) 172–76 log returns, why use?, 119–23 market, order-driven, 166–67 London International Financial market depth for QQQQ at 12:52 Futures Exchange (LIFFE), CST on February 25, 2009, 277 170 long call hedging bands, 196, 202 market making, 169–76 long calls (puts), 19 market-making, information-based, long stock (covered call) 176 short calls, 19 market structure, 166–69 long-term equity anticipation product mimicry: the ultimate method, (LEAP), 277 180–81 order book, pre-open, 168 M order book for QQQQ at 12:52 CST on February 24, 2009, 167 Maginn, John L., 117 order book immediately after the Managing Investment Portfolios: A open, 169 Dynamic Process (Maginn and price paths in our market maker Tuttle), 117 simulation, underlying, 173 , strategy-based, 18–19 profit and loss distribution for the market market maker, 174 depth for QQQQ at 12:52 CST on profit and loss distribution for the February 25, 2009, 170 market maker when orders are execution, 15–16 unbalanced, 175 P1: JYS ind JWBT294-Sinclair May 7, 2010 17:11 Printer: Courier Westford, Westford, MA

Index 293

profit as a function for the market OPM. See “Other people’s money” maker, 174 (OPM) “ratio” trade, 181–83 Option Clearing Corporation (OCC), stop hunting, 185 14, 16–18, 278 summary, 189 option contract, specifications for an, supply and demand curves implied 9–11 by the pre-open order book, option order, 14–15 168 option position, 235 trading based on order-book options information, 181–89 about, 9 trading correlated assets: the buying a call as a hedge, 12–13 implied volatility surface, buying a put (call) to speculate on a 185–88 fall (rise), 12 trading day, 169 clearing and the Option Clearing trading game, practicing, 188–89 Corporation (OCC), 16–17 mathematics, role of, xiv–xvi contract unit, 10–11 mean absolute deviation (MAD), 257 early exercise of, 243–48 measured heights of 11 professional exercise and assignment, 17–18 athletes, 255 exercising wrong, 221–23 median of the deviations from the expiration date, 10 median MAD, 257 FXI and FXP, 8 mimicry, 180–81 hedging a long position with a put, modeling principles, 42–43 11–12 MSFT prices and returns, 121 long calls (puts), 19 margin, 18 N market execution, 15–16 market structure, 14–19 The New York Mercantile Exchange option contract, specifications for (NYMEX), 273, 278 an, 9–11 The New York Stock Exchange option order, 14–15 (NYSE), 13, 278 option type, 9 Nixon, Richard, 227 over-the-counter trading (OTC), 17 nonexpiration days portfolio margin, 19 in American stocks, 216 short calls (puts), 19 normal distribution, 253–54 short calls and long stock (covered NYMEX. See The New York Mercantile call), 19 Exchange (NYMEX) strategy-based margin, 18–19 NYSE. See The New York Stock strike price, 10 Exchange (NYSE) structured product arbitrage, 13–14 O structured products, creation of, 13 summary, 20 OCC. See Option Clearing Corporation trading costs, 18 (OCC) underlying asset, 10 OCO. See one cancels other (OCO) uses of, 11–14 one cancels other (OCO), 278 volatility trading, 13 P1: JYS ind JWBT294-Sinclair May 7, 2010 17:11 Printer: Courier Westford, Westford, MA

294 INDEX

option strategies profit and loss of the 95 105 , bearish underlying forecast, 90 110 bullish underlying forecast, 89 profit and loss of the 100 105 110 call calendar spread, 115 ladder, 114 call spread, 103–5 profit and loss of the 100 105 one by conversions and reversals, two call spread, 113 115–16 profit and loss of the 100 straddle, dependence of the straddle value on 108 time, 109 put spread, 106 forecasting and strategy selection, , 113–14 89–91 realized volatility, decreasing, implied volatility, decreasing, 91 90–91 implied volatility, increasing, 90 realized volatility, increasing, 90 iron butterfly, 111–12 , 106–8 ladder, 114–15 risk tolerance, 91 long butterfly, 110–11 short call, 98–100 long call, 96–98 short put, 100–102 long condor, 112–13 strategies, 93–115 long put, 102–3 summary, 116 long straddle, 108–9 theta of the 95 105 call spread, 104 long strangle, 109–10 underlying price, 97 neutral underlying forecast, 90 value of a call spread position as it neutral volatility, 91 approaches expiry, 105 option nomenclature, 91–93 vega of the 95 105 call spread, 105 payoff of a short call position, volatility forecast, 90 99 option trading profit and loss diagram for 95 100 conclusion, 249–51 call spread, 95 history, 1–6 profit and loss diagram for a more stock options, total annual volume complex strategy, 96 in U.S., 5 profit and loss graph, constructing, option type, 9 93–96 option value profit and loss of a long put position, as we approach expiry, 224 102 order book profit and loss of a short put immediately after the open, position, 101 169 profit and loss of the 94 98 102 106 pre-open, 168 call condor, 112 for QQQQ at 12:52 CST on February profit and loss of the 95 100 24, 2009, 167 butterfly, 111 supply and demand curves implied profit and loss of the 95 105 call by the pre-open, 168 spread, 103 order-driven market, 166–67 profit and loss of the 95 105 put OTC. See over-the-counter (OTC) spread, 106 “Other people’s money” (OPM), profit and loss of the 95 105 risk 279 reversal, 107 over-the-counter (OTC), 17, 279 P1: JYS ind JWBT294-Sinclair May 7, 2010 17:11 Printer: Courier Westford, Westford, MA

Index 295

P single node in the binomial tree, pricing of a call option at, 54 Parkinson estimator, 129 strike, dependence on the, 45–46 Parkinson variance, sampling error in, summary, 61 127 three-step binomial tree, evolution Path 1, 204 of the stock price in, 53 Path 2, 205 time changes, dependence on the, pinning, 215–19 47 in American stocks, 216 tools, choosing the, 43 pin risk, 219–21 tradable underlying, existence of a, population variance as function of 59 sample size, convergence of underlying, dependence on the, 45 variance to true, 124 variable, choosing suitable, 43–44 portfolio 1, 36 volatility, dependence on the, portfolio 2: no exercise, 37 47 portfolio margin, 19 volatility is constant, 60 position repair, 228–32 pricing variables, other, 48 pre-opening auction, 167–69 primary risks, 231 price paths in our market maker probability simulation, underlying, distribution of result of the St. 173 Petersburg game, 160 pricing models of going bankrupt with 55 percent about, 41–42 chance of winning and betting binomial model, 48–55 our entire bankroll, 159 Black-Scholes-Merton (BSM) model, professional trading, xii–xiv 55–61 profit dependent variables, choice of, as a function for the market maker, 45–48 174 distribution of returns, assumption as a function of time for a short about, 60–61 position hedged at the realized dividend changes, dependence on volatility, 211 the, 47–48 as a function of time for a short dividends, absence of, 59–60 position hedged at implied European call option in the binomial volatility, 212 tree, pricing of, 55 profit and loss (P/L) distribution interest rate, existence of a single for $1,000 vega of initially constant, 59 at-the-money options when interest-rate changes, dependence hedged once a day, 207 on the, 46–47 for $1,000 vega of initially modeling principles, general, at-the-money options when 42–43 hedged once a week, 206 one-step binomial tree, evolution of of hedged option positions, 203–13 the stock price in, 49 for the market maker, 174 pricing variables, other, 48 for the market maker when orders results, interpretation of, 44 are unbalanced, 175 short the underlying, ability to, 60 pseudoarbitrage, 24–25 P1: JYS ind JWBT294-Sinclair May 7, 2010 17:11 Printer: Courier Westford, Westford, MA

296 INDEX

put position repair, example of, delta as a function of the underlying 228–32 price, 69 primary risks, 231 on a stock, exercising a, 244–46 put on a stock, exercising a, 244–46 value as a function of the underlying rho, 242 price, 65 risk side, 235 put call parity, 35–38 risk side after buying 10 80 puts, relationship, 35 237 put exercise as a function of time, risk side after selling 200 shares, 237 critical strike for, 245 risk side when using a floating put option(s) implied volatility, 236 with different expirations, roll the position, completely, 229 relationships between, 34–35 secondary risks, 231 with different strikes, 30–31 Societ´ eG´ en´ erale´ example, 233 maximum value of a, 26 stock risk: dividends and buy-in risk, minimum value for a, 27–29 242–43 Sumitomo example, 233 R summary, 248 tertiary risks, 231 “ratio” trade, 181–83 vega, 238–40 results, interpretation of, 44 risks returns and MSFT prices, 121 buy-in, 242–43 rho, 82–83, 242 forward, 221 risk-aversion parameter, 197 pin, 219–21 risk management primary, 231 about, 227–28 secondary, 231 adjust position, 230 stock, 242–43 Barings Bank example, 233 tertiary, 231 call on a future, exercising a, risk side, 235 246 after buying 10 80 puts, 237 call on a stock, exercising a, after selling 200 shares, 237 246–47 when using a floating implied correlation, 240–41 volatility, 236 critical strike for put exercise as a robustness of efficiency, 260 function of time, 245 roll the position, 229 delta, 234 dividend spread, 247–48 S do nothing, 229 exit the trade, 230 scalability, 163–64 gamma, 234–38 scatter graph of two variables, 264 implied volatility curve for the SEC. See Securities and Exchange example, 236 Commission (SEC) initial risk slide, 239 secondary risks, 231 inventory, 232–34 Securities and Exchange Commission option position, 235 (SEC), 154, 280 options, early exercise of, Sharpe, William, 117 243–48 short call hedging bands, 196, 202 P1: JYS ind JWBT294-Sinclair May 7, 2010 17:11 Printer: Courier Westford, Westford, MA

Index 297

short calls (puts), 19 U long stock (covered call), 19 short covering, 281 underlying, dependence on the, 45 short interest, 281 underlying asset, 10 short selling, 281 utility-based methods, 194–97 short squeeze, 281 short the underlying, ability to, 60 V Simpson’s Paradox, 270 skewness, 256 variable, choosing suitable, Societ´ eG´ en´ erale,´ 233 43–44 S&P volatility, 130 variance (of a sample), 257 static hedging, 201–3 variance convergence to true stock options population variance as function total annual volume in U.S., 5 of sample size, 124 stock price vega, 80–82, 238–40 in three-step binomial tree, 53 as a function of the underlying price, stock risk: dividends and buy-in risk, 81 242–43 as a function of time and expiry for stop hunting, 185 an at-the-moment option, 81 strategy-based margin, 18–19 volatility strike, dependence on the, 45–46 cone for MSFT, from two years of strike price, 10 closing prices ending December strip in Europe, 281 31, 2008, 133, 134 structured product arbitrage, 13–14 constant, 60 structured products, creation of, 13 in context, 132–35 Sumitomo, 233 dependence on the, 47 for different option positions as a T function of market direction, 209 tails, quantifying, 260–62 estimator, “quick and dirty,” tertiary risks, 231 125–29 theta, 76–78 forecasting, 129–32 as we approach expiry, 224 S&P, 130 time changes, dependence on the, for True Religion Apparel, Inc., 131, 47 132 tools, choosing the, 43 two price paths with identical, tradable underlying, existence of, 59 209 trade sizing, 158–63 volatility estimation trading about, 117–19 based on order-book information, bad things happen, 135–36 181–89 bias, 123 correlated assets, 185–88 bias and efficiency, 123–25 day, 169 close-to-close estimator, 128 game, 188–89 convergence of variance to true trading, volatility, 13 population variance, 124 trading costs, 18 efficiency, 123 Tuttle, Donald L., 117 log returns, why use?, 119–23 P1: JYS ind JWBT294-Sinclair May 7, 2010 17:11 Printer: Courier Westford, Westford, MA

298 INDEX

volatility estimation (Continued) hedging in practice, 193–203 MSFT prices and returns, 121 hedging strategy for one-year Parkinson estimator, 129 option, 204 Parkinson variance, sampling error hedging to a delta band, 194 in, 127 hedging volatility, choosing a, S&P volatility, 130 208–13 summary, 136 implied method, 198 volatility, forecasting, 129–32 optimal hedging bands as functions volatility cone for MSFT, from two of BSM delta, 196 years of closing prices ending Path 1, 204 December 31, 2008, 133, 134 Path 2, 205 volatility estimator, “quick and P/L distribution for $1,000 vega of dirty,” 125–29 initially ATM options when volatility for True Religion Apparel, hedged once a day, 207 Inc., 30-day moving window P/L distribution for $1,000 vega of close-to-close, 131 initially ATM options when volatility for True Religion Apparel, hedged once a week, 206 Inc., 30-day moving window P/L distribution of hedged option “jump excluded” close-to-close positions, 203–13 volatility and EWMA, 132 profit as a function of time for a volatility in context, 132–35 short position hedged a the volatility trading realized volatility, 211 about, 70, 80, 145, 181, 250, 283 profit as a function of time for a asymptotic solution of Whalley and short position hedged at Wilmott, 198–200 implied volatility, 212 direct method, 197 risk-aversion parameter, choosing, double-asymptotic method of 197 Zakamouline, 200–201 short call hedging bands, 196, 202 exponential utility function, 195 static hedging, 201–3 hedge frequency, 206 summary, 213 hedging, 192–93 utility-based methods, 194–97 hedging at regular time intervals, volatility, two price paths with 193 identical, 209 hedging bands, long call, 196, 202 volatility as a function of market hedging bands from the WW direction, 209 Asymptotic method as functions of the BSM delta, W approximate, 199 hedging bands from the width, quantifying the, 256–61 Zakamouline Asymptotic “with a tick,” 283 method as functions of the BSM wrong options, exercising the, 221–23 delta, 202 hedging based on underlying price Z changes, 194 hedging error as function of a Zakamouline’s double-asymptotic number of rebalances, 207 method, 200–201 P1: JYS ind JWBT294-Sinclair May 7, 2010 17:11 Printer: Courier Westford, Westford, MA

299 P1: JYS ind JWBT294-Sinclair May 7, 2010 17:11 Printer: Courier Westford, Westford, MA

300