Fin 5633: Investment Theory and Problems: Chapter#20 Solutions

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1 Fin 5633: Investment Theory and Problems: Chapter#2 Solutions 1. Cost Payoff Profit a. Call option, X = $22.5 $2.7 $2.5 -$.2 b. Put option, X = $22.5 $1.9 $. -$1.9 c. Call option, X = $25. $1.5 $. -$1.5 d. Put option, X = $25. $3.1 $. -$3.1 e. Call option, X = $27.5 $.75 $. -$.75 f. Put option, X = $27.5 $4.8 $2.5 -$ In terms of dollar returns, based on a $1, investment: Price of Stock 6 Months from Now Stock Price $8 $1 $11 $12 All stocks (1 shares) $8, $1, $11, $12, All options (1, options) $ $ $1, $2, Bills + 1 options $9,36 $9,36 $1,36 $11,36 In terms of rate of return, based on a $1, investment: Price of Stock 6 Months from Now Stock Price $8 $1 $11 $12 All stocks (1 shares) -2% % 1% 2% All options (1, options) -1% -1% % 1% Bills + 1 options -6.4% -6.4% 3.6% 13.6% Rate of return (%) All options 11 All stocks Bills plus options S T 1 Page1

2 3. a. From put-call parity: P = C S + [X/(1 + r f ) T ] = [1/(1.1) 1/4 ] = $7.65 b. Purchase a straddle, i.e., both a put and a call on the stock. The total cost of the straddle is: $1 + $7.65 = $17.65 This is the amount by which the stock would have to move in either direction for the profit on the call or put to cover the investment cost (not including time value of money considerations). Accounting for time value, the stock price would have to move in either direction by: $ /4 = $ a. From put-call parity: C = P + S [X/(l + r f ) T ] [5/(1.1) 1/4 ] = $5.18 b. Sell a straddle, i.e., sell a call and a put to realize premium income of: $ $4 = $9.18 If the stock ends up at $5, both of the options will be worthless and your profit will be $9.18. This is your maximum possible profit since, at any other stock price, you will have to pay off on either the call or the put. The stock price can move by $9.18 in either direction before your profits become negative. c. Buy the call, sell (write) the put, lend: $5/(1.1) 1/4 The payoff is as follows: Position Immediate CF CF in 3 months S T X S T > X Call (long) C = 5.18 S T 5 Put (short) P = 4. (5 S T) Lending position 5 1 / = Total 5 C P + 1 / = 5. S T S T By the put-call parity theorem, the initial outlay equals the stock price: S = $5 In either scenario, you end up with the same payoff as you would if you bought the stock itself. 5. a. Outcome S T X S T > X Stock Put S T + D X S T S T + D Total X + D S T + D Page2

3 b. Outcome S T X S T > X Call S T X Zeros X + D X + D Total X + D S T + D The total payoffs for the two strategies are equal regardless of whether S T exceeds X. c. The cost of establishing the stock-plus-put portfolio is: S + P The cost of establishing the call-plus-zero portfolio is: C + PV(X + D) Therefore: S + P = C + PV(X + D) This result is identical to equation According to put-call parity (assuming no dividends), the present value of a payment of $23.75 can be calculated using the options with July maturity and exercise price of $ PV(X) = S + P C PV($23.75) = $ $2.5 $2.5 =$ The following payoff table shows that the portfolio is riskless with time-t value equal to $1: Position S T 1 S T > 1 Buy stock S T S T Write call, X = $1 (S T 1) Buy put, X = $1 1 S T Total 1 1 Therefore, the risk-free rate is: ($1/$9.5) 1 =.526 = 5.26% 2. From put-call parity: C P = S X/(l + r f ) T If the options are at the money, then S = X and: Page3

4 21. a., b. C P = X X/(l + r f ) T The right-hand side of the equation is positive, and we conclude that C > P. Position S T < 1 1 S T 11 S T > 11 Buy put, X = $11 11 S T 11 S T Write put, X = $1 (1 S T) Total 1 11 S T The net outlay to establish this position is positive. The put you buy has a higher exercise price than the put you write, and therefore must cost more than the put that you write. Therefore, net profits will be less than the payoff at time T. 1 Payoff 1 11 Profit S T c. The value of this portfolio generally decreases with the stock price. Therefore, its beta is negative. 22 a. Joe s strategy Position Cost Payoff S T 4 S T > 4 Stock index 4 S T S T Put option, X = $4 2 4 S T Total 42 4 S T Profit = payoff $42 2 S T 42 Sally s strategy Position Cost Payoff S T 39 S T > 39 Stock index 4 S T S T Put option, X = $ S T Page4

5 Total S T Profit = payoff $ S T 415 Profit Sally Joe 39 4 S T b. Sally does better when the stock price is high, but worse when the stock price is low. The break-even point occurs at S T = $395, when both positions provide losses of $2. c. Sally s strategy has greater systematic risk. Profits are more sensitive to the value of the stock index. 23. a., b. (See graph on next page) This strategy is a bear spread. Initial proceeds = $9 $3 = $6 The payoff is either negative or zero: Position S T < 5 5 S T 6 S T > 6 Buy call, X = $6 S T 6 Write call, X = $5 (S T 5) (S T 5) Total (S T 5) 1 c. Breakeven occurs when the payoff offsets the initial proceeds of $6, which occurs at stock price S T = $56. The investor must be bearish: the position does worse when the stock price increases. Page5

6 Profit S T -1 Payoff Page6