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D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 7: 1 Chapter 7: Advanced Option Strategies “It takes two things to make a good trader,” Struve advises Norman. “You have to understand the mathematics, and you need street smarts. You don’t want to be the guy with thick glasses who is reading the sheet just when the freight train is about to roll over on you. The street-smart guy will pick up a couple of quarters and get out of the way.” Thomas A. Bass The Predictors, p. 1999, pp

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D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 7: 2 Important Concepts in Chapter 7 n Profit equations and graphs for option spread strategies, including money spreads, collars, calendar spreads and ratio spreads n Profit equations and graphs for option combination strategies including straddles and box spreads

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D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 7: 3 Option Spreads: Basic Concepts u Definitions F spread vertical, strike, money spreadvertical, strike, money spread horizontal, time, calendar spreadhorizontal, time, calendar spread F spread notation June 120/125June 120/125 June/July 120June/July 120 F long or short long, buying, debit spreadlong, buying, debit spread short, selling, credit spreadshort, selling, credit spread

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D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 7: 4 Option Spreads: Basic Concepts (continued) n Why Investors Use Option Spreads u Risk reduction u To lower the cost of a long position u Types of spreads F bull spread F bear spread F time spread is based on volatility

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D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 7: 5 Option Spreads: Basic Concepts (continued) n Notation u For money spreads F X 1 < X 2 < X 3 F C 1, C 2, C 3 F N 1, N 2, N 3 u For time spreads F T 1 < T 2 F C 1, C 2 F N 1, N 2 u See Table 7.1, p. 236 for AOL option data Table 7.1, p. 236Table 7.1, p. 236

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D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 7: 6 Money Spreads n Bull Spreads u Buy call with strike X 1, sell call with strike X 2. Let N 1 = 1, N 2 = -1 Profit equation: = Max(0,S T - X 1 ) - C 1 - Max(0,S T - X 2 ) + C 2 = -C 1 + C 2 if S T X 1 < X 2 = -C 1 + C 2 if S T X 1 < X 2 = S T - X 1 - C 1 + C 2 if X 1 < S T X 2 = S T - X 1 - C 1 + C 2 if X 1 < S T X 2 = X 2 - X 1 - C 1 + C 2 if X 1 < X 2 < S T F See Figure 7.1, p. 237 for AOL June 125/130, C 1 = $13.50, C 2 = $ Figure 7.1, p. 237Figure 7.1, p. 237 u Maximum profit = X 2 - X 1 - C 1 + C 2, Minimum = - C 1 + C 2 u Breakeven: S T * = X 1 + C 1 - C 2

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D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 7: 7 Money Spreads (continued) n Bull Spreads (continued) u For different holding periods, compute profit for range of stock prices at T 1, T 2, and T using Black-Scholes model. See Figure 7.2, p Figure 7.2, p. 239Figure 7.2, p. 239 u Note how time value decay affects profit for given holding period. u Early exercise not a problem.

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D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 7: 8 Money Spreads (continued) n Bear Spreads u Buy put with strike X 2, sell put with strike X 1. Let N 1 = -1, N 2 = 1 Profit equation: = -Max(0,X 1 - S T ) + P 1 + Max(0,X 2 - S T ) - P 2 = X 2 - X 1 + P 1 - P 2 if S T X 1 < X 2 = X 2 - X 1 + P 1 - P 2 if S T X 1 < X 2 = P 1 + X 2 - S T - P 2 if X 1 < S T < X 2 = P 1 - P 2 if X 1 < X 2 S T = P 1 - P 2 if X 1 < X 2 S T F See Figure 7.3, p. 240 for AOL June 130/125, P 1 = $11.50, P 2 = $ Figure 7.3, p. 240Figure 7.3, p. 240 F Maximum profit = X 2 - X 1 + P 1 - P 2. Minimum = P 1 - P 2. F Breakeven: S T * = X 2 + P 1 - P 2.

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D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 7: 9 Money Spreads (continued) n Bear Spreads (continued) u For different holding periods, compute profit for range of stock prices at T 1, T 2, and T using Black-Scholes model. See Figure 7.4, p Figure 7.4, p. 242Figure 7.4, p. 242 u Note how time value decay affects profit for given holding period. u Note early exercise problem. n A Note About Put Money Spreads u Can construct call bear and put bull spreads.

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D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 7: 10 Money Spreads (continued) n Collars u Buy stock, buy put with strike X 1, sell call with strike X 2. N S = 1, N P = 1, N C = -1. Profit equation: = S T - S 0 + Max(0,X 1 - S T ) - P 1 - Max(0,S T - X 2 ) + C 2 = X 1 - S 0 - P 1 + C 2 if S T X 1 < X 2 = X 1 - S 0 - P 1 + C 2 if S T X 1 < X 2 = S T - S 0 - P 1 + C 2 if X 1 < S T < X 2 = X 2 - S 0 - P 1 + C 2 if X 1 < X 2 S T = X 2 - S 0 - P 1 + C 2 if X 1 < X 2 S T u A common type of collar is what is often referred to as a zero-cost collar. The call strike is set such that the call premium offsets the put premium so that there is no initial outlay for the options.

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D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 7: 11 Money Spreads (continued) n Collars (continued) F See Figure 7.5, p. 244 for AOL July 120/136.23, P 1 = $13.625, C 2 = $ That is, a call strike of generates the same premium as a put with strike of 120. This result can be obtained only by using an option pricing model and plugging in exercise prices until you find the one that makes the call premium the same as the put premium. Figure 7.5, p. 244Figure 7.5, p. 244 F This will nearly always require the use of OTC options. F Maximum profit = X 2 - S 0. Minimum = X 1 - S 0. F Breakeven: S T * = S 0.

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D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 7: 12 Money Spreads (continued) n Collars (continued) u The collar is a lot like a bull spread (compare Figure 7.5 to Figure 7.1). Figure 7.5Figure 7.1Figure 7.5Figure 7.1 F The collar payoff exceeds the bull spread payoff by the difference between X 1 and the interest on X 1. F Thus, the collar is equivalent to a bull spread plus a risk-free bond paying X 1 at expiration. u For different holding periods, compute profit for range of stock prices at T 1, T 2, and T using Black-Scholes model. See Figure 7.6, p Figure 7.6, p. 248Figure 7.6, p. 248

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D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 7: 13 Money Spreads (continued) n Butterfly Spreads u Buy call with strike X 1, buy call with strike X 3, sell two calls with strike X 2. Let N 1 = 1, N 2 = -2, N 3 = 1. Profit equation: = Max(0,S T - X 1 ) - C 1 - 2Max(0,S T - X 2 ) + 2C 2 + Max(0,S T - X 3 ) - C 3 = -C 1 + 2C 2 - C 3 if S T X 1 < X 2 < X 3 = -C 1 + 2C 2 - C 3 if S T X 1 < X 2 < X 3 = S T - X 1 - C 1 + 2C 2 - C 3 if X 1 < S T X 2 < X 3 = S T - X 1 - C 1 + 2C 2 - C 3 if X 1 < S T X 2 < X 3 = -S T +2X 2 - X 1 - C 1 + 2C 2 - C 3 if X 1 < X 2 < S T X 3 = -S T +2X 2 - X 1 - C 1 + 2C 2 - C 3 if X 1 < X 2 < S T X 3 = -X 1 + 2X 2 - X 3 - C 1 + 2C 2 - C 3 if X 1 < X 2 < X 3 < S T F See Figure 7.7, p. 250 for AOL July 120/125/130, C 1 = $16.00, C 2 = $13.50, C 3 = $ Figure 7.7, p. 250Figure 7.7, p. 250

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D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 7: 14 Money Spreads (continued) n Butterfly Spreads (continued) u Maximum profit = X 2 - X 1 - C 1 + 2C 2 - C 3, minimum = -C 1 + 2C 2 - C 3 u Breakeven: S T * = X 1 + C 1 - 2C 2 + C 3 and S T * = 2X 2 - X 1 - C 1 + 2C 2 - C 3 u For different holding periods, compute profit for range of stock prices at T 1, T 2, and T using Black-Scholes model. See Figure 7.8, p Figure 7.8, p. 251Figure 7.8, p. 251 u Note how time value decay affects profit for given holding period. u Note early exercise problem.

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D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 7: 15 Calendar Spreads u Buy call with longer time to expiration, sell call with shorter time to expiration. u Note how this strategy cannot be held to expiration because there are two different expirations. u Profitability depends on volatility and time value decay. u Use Black-Scholes model to value options at end of holding period if prior to expiration. u See Figure 7.9, p Figure 7.9, p. 253Figure 7.9, p. 253 u Note time value decay. See Table 7.2, p. 254 and Figure 7.10, p Table 7.2, p. 254 Figure 7.10, p. 255Table 7.2, p. 254 Figure 7.10, p. 255 u Early exercise can be problem. u Can be constructed with puts as well.

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D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 7: 16 Ratio Spreads u Long one option, short another based on deltas of two options. Designed to be delta-neutral. Can use any two options on same stock. u Portfolio value F V = N 1 C 1 + N 2 C 2 Set to zero and solve for N 1 /N 2 = - 2 / 1, which is ratio of their deltas (recall that Set to zero and solve for N 1 /N 2 = - 2 / 1, which is ratio of their deltas (recall that = N(d 1 ) from Black-Scholes model). u Buy June 120s, sell June 125s. Delta of 120 is.630; delta of 125 is.569. Ratio is –(.569/.630) = For example, buy 903 June 120s, sell 1,000 June 125s u Note why this works and that delta will change. u Why do this? Hedging mispriced option

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D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 7: 17 Straddles u Straddle: long an equal number of puts and calls Profit equation: Profit equation: = Max(0,S T - X) - C + Max(0,X - S T ) - P (assuming N c = 1, N p = 1) = S T - X - C - P if S T X = X - S T - C - P if S T < X u Either call or put will be exercised (unless S T = X). u See Figure 7.11, p. 258 for AOL June 125, C = $13.50, P = $ Figure 7.11, p. 258Figure 7.11, p. 258 u Breakeven: S T * = X - C - P and S T * = X + C + P u Maximum profit: , minimum = - C - P u See Figure 7.12, p. 261 for different holding periods. Note time value decay. Figure 7.12, p. 261Figure 7.12, p. 261

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D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 7: 18 Straddles (continued) n Applications of Straddles u Based on perception of volatility greater than priced by market n A Short Straddle u Unlimited loss potential u Based on perception of volatility less than priced by market

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D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 7: 19 Box Spreads u Definition: bull call money spread plus bear put money spread. Risk-free payoff if options are European u Construction: F Buy call with strike X 1, sell call with strike X 2 F Buy put with strike X 2, sell put with strike X 1 Profit equation: = Max(0,S T - X 1 ) - C 1 - Max(0,S T - X 2 ) + C 2 + Max(0,X 2 - S T ) - P 2 - Max(0,X 1 - S T ) + P 1 = X 2 - X 1 - C 1 + C 2 - P 2 + P 1 if S T X 1 < X 2 = X 2 - X 1 - C 1 + C 2 - P 2 + P 1 if X 1 < S T X 2 = X 2 - X 1 - C 1 + C 2 - P 2 + P 1 if X 1 < X 2 S T

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D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 7: 20 Box Spreads (continued) u Evaluate by determining net present value (NPV) F NPV = (X 2 - X 1 )(1 + r) -T - C 1 + C 2 - P 2 + P 1 F This determines whether present value of risk-free payoff exceeds initial value of transaction. F If NPV > 0, do it. If NPV 0, do it. If NPV < 0, do the reverse. u See Figure 7.13, p Figure 7.13, p. 264Figure 7.13, p. 264 u Box spread is also difference between two put-call parities.

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D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 7: 21 Box Spreads (continued) u Evaluate June 125/130 box spread F Buy 125 call at $13.50, sell 130 call at $ F Buy 130 put at $14.25, sell 125 put at $11.50 F Initial outlay = $4.875, $ for 100 each F NPV = 100[( )(1.0456) ] = F NPV > 0 so do it u Early exercise a problem only on short box spread u Transaction costs high

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D. M. ChanceAn Introduction to Derivatives and Risk Management, 6th ed.Ch. 7: 22 Summary

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