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## Chapter 15

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**Chapter15**Options Markets**Option Terminology**• Buy - Long • Sell - Short • Call Option: gives its holder the right to purchase an asset for a specified price before or on a specified expiration date. • Put Option: gives its holder the right to sell an asset at a specified price before or on a specified expiration date. • Key Elements • Exercise or Strike Price • Premium or Price • Maturity or Expiration**Market and Exercise Price Relationships**In the Money - exercise of the option would be profitable Call: market price>exercise price Put: exercise price>market price Out of the Money - exercise of the option would not be profitable Call: market price<exercise price Put: exercise price<market price At the Money - exercise price and asset price are equal**American vs European Options**American - the option can be exercised at any time before expiration or maturity European - the option can only be exercised on the expiration or maturity date**Different Types of Options**• Stock Options • Index Options • Futures Options • Foreign Currency Options • Interest Rate Options**Payoffs and Profits on Options at Expiration - Calls**Notation Stock Price = ST Exercise Price = X Payoff to Call Holder (ST - X) if ST >X 0 if ST < X or Max {ST – X, 0} Profit to Call Holder Payoff - Purchase Price**Payoffs and Profits on Options at Expiration - Calls**Payoff to Call Writer - (ST - X) if ST >X 0 if ST < X or Min {X – ST, 0} Profit to Call Writer Payoff + Premium**Profit Profiles for Calls**Profit Call Holder 0 Call Writer Stock Price**Payoffs and Profits at Expiration - Puts**Payoffs to Put Holder 0 if ST> X (X - ST) if ST < X or Max {X-ST, 0} Profit to Put Holder Payoff - Premium**Payoffs and Profits at Expiration - Puts**Payoffs to Put Writer 0 if ST > X -(X - ST) if ST < X or Min {ST - X, 0} Profits to Put Writer Payoff + Premium**Profit Profiles for Puts**Profits Put Writer 0 Put Holder Stock Price**Exercise in class**1. You purchase one IBM July 120 call contract for a premium of $5. You hold the option until the expiration date when IBM stock sells for $123 per share. You will realize a ______ on the investment. A) $200 profit B) $200 loss C) $300 profit D) $300 loss 2. You purchase one IBM July 120 put contract for a premium of $5. You hold the option until the expiration date when IBM stock sells for $123 per share. You will realize a ______ on the investment. A) $300 profit B) $200 loss C) $500 loss D) $200 profit**Exercise in class**A call option on Brocklehurst Corp. has an exercise price of $30. The current stock price of Brocklehurst Corp. is $32. The call option is __________. A) at the money B) in the money C) out of the money D) none of the above According to the Wall Street Journal on May 27, 1994, the price quotations for a Boeing call option with a strike price of $45 due to expire in August was 2 1/2 while the stock price of Boeing is $47. The premium on one Boeing August 90 call contract is __________. A) $0.50 B) $2.50 C) $50.00 D) $250.00**Equity, Options & Leveraged Equity - Text Example**Investment Strategy Investment Equity only Buy stock @ 80 100 shares $8,000 Options only Buy 80 calls @ 10 800 options $8,000 Leveraged Buy 80 calls @ 10 100 options $1,000 equity Buy T-bills @ 2% $7,000 Yield**Equity, Options & Leveraged Equity - Payoffs**Microsoft Stock Price $75 $80 $100 All Stock $7,500 $8,000 $10,000 All Options $0 $0 $16,000 Lev Equity $7,140 $7,140 $9,140**Equity, Options & Leveraged Equity - Rates of Return**Microsoft Stock Price $75 $80 $100 All Stock -6.25% 0% 25% All Options -100% -100% 100% Lev Equity -10.75% -10.75% 14.25%**Put-Call Parity Relationship**ST< X ST > X Payoff for Call Owned 0 ST - X Payoff for Put Written -( X -ST) 0 Total Payoff ST - XST - X**Payoff of Long Call & Short Put**Payoff Long Call Combined = Leveraged Equity Stock Price Short Put**Arbitrage & Put Call Parity**Since the payoff on a combination of a long call and a short put are equivalent to leveraged equity, the prices must be equal. C - P = S0 - X / (1 + rf)T If the prices are not equal arbitrage will be possible**Put Call Parity - Disequilibrium Example**Stock Price = 110 Call Price = 17 Put Price = 5 Risk Free = 10.25% Maturity = .5 yr X = 105 C - P > S0 - X / (1 + rf)T 17- 5 > 110 - (105/1.05) 12 > 10 Since the leveraged equity is less expensive, acquire the low cost alternative and sell the high cost alternative**Put-Call Parity Arbitrage**ImmediateCashflow in Six Months Position Cashflow ST<105 ST> 105 Buy Stock -110 ST ST Borrow X/(1+r)T = 100 +100 -105 -105 Sell Call +17 0 -(ST-105) Buy Put -5 105-ST 0 Total 2 0 0**Option Strategies**Protective Put Long Stock Long Put Covered Call Long Stock Short Call Straddle (Same Exercise Price) Long Call Long Put**Exercise in class**1 You buy one Chrysler August 50 call contract and one Chrysler August 50 put contract. The call premium is $4.25 and the put premium is $5.00. Your highest potential loss from this position is __________. A) $75 B) $925 C) $5,000 D) unlimited 2 An investor purchases a long call at a price of $2.50. The expiration price is $35.00. If the current stock price is $35.10, what is the break even point for the investor? A) $32.50 B) $35.00 C) $37.50 D) $37.60**Option Strategies**Spreads - A combination of two or more call options or put options on the same asset with differing exercise prices or times to expiration Vertical or money spread Same maturity Different exercise price Horizontal or time spread Different maturity dates**Exercise in class**You buy one Chrysler August 50 call contract and one Chrysler August 50 put contract. The call premium is $4.25 and the put premium is $4.50. Your strategy is useful if you believe that the stock price __________. A) will be lower than $41.25 in August B) will be between $41.25 and $58.75 in August C) will be higher than $58.75 in August D) either a or c**Chapter16**Option Valuation**Option Values**• Intrinsic value - payoff that could be made if the option was immediately exercised • Call: stock price - exercise price • Put: exercise price - stock price • Time value - the difference between the option price and the intrinsic value**Time Value of Options: Call**Option value Value of Call Intrinsic Value Time value X Stock Price**Factors Influencing Option Values: Calls**FactorEffect on value Stock price increases Exercise price decreases Volatility of stock price increases Time to expiration increases Interest rate increases Dividend yield decreases**A Simple Binomial Model**• A stock price is currently $20 • In three months it will be either $22 or $18 Stock Price = $22 Stock price = $20 Stock Price = $18**A Call Option**A 3-month call option on the stock has a strike price of 21. Stock Price = $22 Option Price = $1 Stock price = $20 Option Price=? Stock Price = $18 Option Price = $0**22D – 1**18D Setting Up a Riskless Portfolio • Consider the Portfolio: long D shares short 1 call option • Portfolio is riskless when 22D – 1 = 18D or D = 0.25**Valuing the Portfolio(Risk-Free Rate is 12%)**• The riskless portfolio is: long 0.25 shares short 1 call option • The value of the portfolio in 3 months is 22´0.25 – 1 = 4.50 • The value of the portfolio today is 4.5e– 0.12´0.25 = 4.3670**Valuing the Option**• The portfolio that is long 0.25 shares short 1 option is worth 4.367 • The value of the shares is 5.000 (= 0.25´20 ) • The value of the option is therefore 0.633 (= 5.000 – 4.367 )**Example:**• Suppose the stock now sells at $100, and the price will either double to $200 or fall in half to $50 by the year-end. A call option on the stock might specify an exercise price of $125 and a time to expiration of one year. The interest rate is 8%. What is the option price today?**Black-Scholes Option Valuation**Co= Soe-dTN(d1) - Xe-rTN(d2) d1 = [ln(So/X) + (r – d + s2/2)T] / (s T1/2) d2 = d1 - (s T1/2) where Co = Current call option value. So= Current stock price N(d) = probability that a random draw from a normal dist. will be less than d.**Black-Scholes Option Valuation**X = Exercise price. d = Annual dividend yield of underlying stock e = 2.71828, the base of the nat. log. r = Risk-free interest rate (annualizes continuously compounded with the same maturity as the option. T = time to maturity of the option in years. ln = Natural log function s = Standard deviation of annualized cont. compounded rate of return on the stock**Call Option Example**So = 100 X = 95 r = .10 T = .25 (quarter) s = .50 d = 0 d1 = [ln(100/95)+(.10-0+(.5 2/2))]/(.5.251/2) = .43 d2 = .43 - ((.5)( .251/2) = .18**Probabilities from Normal Dist.**N (.43) = .6664 Table 17.2 d N(d) .42 .6628 .43 .6664 Interpolation .44 .6700**Probabilities from Normal Dist.**N (.18) = .5714 Table 17.2 d N(d) .16 .5636 .18 .5714 .20 .5793**Call Option Value**Co= Soe-dTN(d1) - Xe-rTN(d2) Co = 100 X .6664 - 95 e- .10 X .25 X .5714 Co = 13.70 Implied Volatility Using Black-Scholes and the actual price of the option, solve for volatility. Is the implied volatility consistent with the stock?**Put Option Value: Black-Scholes**P=Xe-rT [1-N(d2)] - S0e-dT [1-N(d1)] Using the sample data P = $95e(-.10X.25)(1-.5714) - $100 (1-.6664) P = $6.35**Put Option Valuation: Using Put-Call Parity**P = C + PV (X) - So = C + Xe-rT - So Using the example data C = 13.70 X = 95 S = 100 r = .10 T = .25 P = 13.70 + 95 e -.10 X .25 - 100 P = 6.35**75**C 0 Call Option Value X = 125 Binomial Option Pricing:Text Example 200 100 50 Stock Price**Binomial Option Pricing:Text Example**Alternative Portfolio (assume discrete discount) Buy 1 share of stock at $100 Borrow $46.30 (8% Rate):=50/(1+0.08) Net outlay $53.70 Payoff Value of Stock 50 200 Repay loan - 50 -50 Net Payoff 0 150 150 53.70 0 Payoff Structure is exactly 2 times the Call**75**C 0 Binomial Option Pricing:Text Example 150 53.70 0 2C = $53.70 C = $26.85**Another View of Replication of Payoffs and Option Values**Alternative Portfolio - one share of stock and 2 calls written (X = 125) Portfolio is perfectly hedged Stock Value 50 200 Call Obligation 0-150 Net payoff 50 50 Hence 100 - 2C = 46.30 or C = 26.85**Exercise in class**The stock price of Ajax Inc. is currently $105. The stock price a year from now will be either $130 or $90 with equal probabilities. The interest rate at which investors can borrow is 10%. Using the binomial OPM, the value of a call option with an exercise price of $110 and an expiration date one year from now should be worth __________ today. A) $11.60 B) $15.00 C) $20.00 D) $40.00 The stock price of Bravo Corp. is currently $100. The stock price a year from now will be either $160 or $60 with equal probabilities. The interest rate at which investors invest in riskless assets at is 6%. Using the binomial OPM, the value of a put option with an exercise price of $135 and an expiration date one year from now should be worth __________ today. A) $34.09 B) $37.50 C) $38.21 D) $45.45**CHAPTER 18**Performance Evaluation and Active Portfolio Management**Introduction**• Complicated subject • Theoretically correct measures are difficult to construct • Different statistics or measures are appropriate for different types of investment decisions or portfolios • Many industry and academic measures are different • The nature of active managements leads to measurement problems