Fin 3710 Investment Analysis Professor Rui Yao CHAPTER 14: OPTIONS MARKETS

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1 HW 6 Fin 3710 Investment Analysis Professor Rui Yao CHAPTER 14: OPTIONS MARKETS 4. Cost Payoff Profit Call option, X = Put option, X = Call option, X = Put option, X = Call option, X = Put option, X = In terms of dollar returns: Price of Stock Six Months From Now Stock price: All stocks (100 shares) 8,000 10,000 11,000 12,000 All options (1,000) shares ,000 20,000 Bills options 9,360 9,360 10,360 11,360 In terms of rate of return, based on a $10,000 investment: Price of Stock Six Months From Now Stock price: All stocks (100 shares) -20% 0% 10% 20% All options (1,000) shares -100% -100% 0% 100% Bills options -6.4% -6.4% 3.6% 13.6% Rate of return (%) All options 110 All stocks Bills plus options S T 100

2 6. a. Purchase a straddle, i.e., both a put and a call on the stock. The total cost of the straddle would be: ($10 + $7) = $17 b. Since the straddle costs $17, this is the amount by which the stock would have to move in either direction for the profit on either the call or put to cover the investment cost (not including time value of money considerations). 7. a. Sell a straddle, i.e., sell a call and a put to realize premium income of: ($4 + $7) = $11 b. If the stock ends up at $50, both of the options will be worthless and your profit will be $11. 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 $11 (your initial revenue from writing the two at-themoney options) in either direction before your profits become negative. c. Buy the call, sell (write) the put, lend the present value of $50. The payoff is as follows: Final Payoff Position Initial Outlay S T < X Long call C = 7 0 S T > X S T 50 Short put -P = -4 -(50 S T ) 0 Lending 50/(1 + r) (1/4) Total [50/(1 + r) (1/4) ] S T S T The initial outlay equals: [(the present value of $50) + $3]. In either scenario, you end up with the same payoff as you would if you bought the stock itself. 8. a. By writing covered call options, Jones receives premium income of $30,000. If, in January, the price of the stock is less than or equal to $45, he will keep the stock plus the premium income. But the most he can have is $450,000 + $30,000 because the stock will be called away from him if its price exceeds $45. (We are ignoring interest earned on the premium income from writing the option over this short time period.) The payoff structure is: Less than $45 (10,000 times stock price) + $30,000 Greater than $45 $450,000 + $30,000 = $480,000 This strategy offers some premium income but leaves the investor with substantial downside risk. At the extreme, if the stock price falls to zero, Jones would be left with only $30,000. This strategy also puts a cap on the final value at $480,000, but this is more than sufficient to purchase the house.

3 b. By buying put options with a $35 strike price, Jones will be paying $30,000 in premiums in order to insure a minimum level for the final value of his position. That minimum value is: [($35 10,000) $30,000] = $320,000. This strategy allows for upside gain, but exposes Jones to the possibility of a moderate loss equal to the cost of the puts. The payoff structure is: Less than $35 $350,000 $30,000 = $320,000 Greater than $35 (10,000 times stock price) $30,000 c. The net cost of the collar is zero. The value of the portfolio will be as follows: Less than $35 $350,000 Between $35 and $45 10,000 times stock price Greater than $45 $450,000 If the stock price is less than or equal to $35, then the collar preserves the $350,000 in principal. If the price exceeds $45, then Jones gains up to a cap of $450,000. In between $35 and $45, his proceeds equal 10,000 times the stock price. The best strategy in this case is (c) since it satisfies the two requirements of preserving the $350,000 in principal while offering a chance of getting $450,000. Strategy (a) should be ruled out because it leaves Jones exposed to the risk of substantial loss of principal. Our ranking is: (1) c (2) b (3) a CHAPTER 15: OPTION VALUATION 2. a. Put A must be written on the lower-priced stock. Otherwise, given the lower volatility of stock A, put A would sell for less than put B. b. Put B must be written on the stock with lower price. This would explain its higher value. c. Call B. Despite the higher price of stock B, call B is cheaper than call A. This can be explained by a lower time to expiration. d. Call B. This would explain its higher price. e. Not enough information. The call with the lower exercise price sells for more than the call with the higher exercise price. The values given are consistent with either stock having higher volatility.

4 3. Note that, as the option becomes progressively more in the money, its hedge ratio increases to a maximum of 1.0: X Hedge ratio X Hedge ratio /150 = /150 = /150 = /150 = /150 = /150 = a. When S = 130, then P = 0. When S = 80, then P = 30. The hedge ratio is: [(P + P )/(S + S ) = [(0 30)/(130 80)] = 3/5 b. Riskless portfolio S =80 S = shares puts Total Present value = ($390/1.10) = c. Portfolio cost = 3S + 5P = $ P = $ Therefore 5P = $ P = $54.545/5 = $ The hedge ratio for the call is [[(C + C - )/(S + S - ) = [(20 0)/(130 80)] = 2/5 Riskless portfolio S =80 S = shares Short 5 calls Total C = (160/1.10) = C = Put-call parity relationship: P = C S 0 + PV(X) = (110/1.10) 100 = Step 1: Calculate the option values at expiration. The two possible stock prices are: S + = $120 and S = $80. Therefore, since the exercise price is $100, the corresponding two possible call values are: C + = $20 and C = $0. Step 2: Calculate the hedge ratio: (C + C )/(S + S ) = (20 0)/(120 80) = 0.5

5 Step 3: Form a riskless portfolio made up of one share of stock and two written calls. The cost of the riskless portfolio is: (S 0 2C 0 ) = 100 2C 0 and the certain end-of-year value is $80. Step 4: Calculate the present value of $80 with a one-year interest rate of 10% = $72.73 Step 5: Set the value of the hedged position equal to the present value of the certain payoff: $100 2C 0 = $72.73 Step 6: Solve for the value of the call: C 0 = $13.64 Notice that we never use the probabilities of a stock price increase or decrease. These are not needed to value the call option. 30. Step 1: Calculate the option values at expiration. The two possible stock prices are: S + = $130 and S = $70. Therefore, since the exercise price is $100, the corresponding two possible call values are: C + = $30 and C = $0. Step 2: Calculate the hedge ratio: (C + C )/(S + S ) = (30 0)/(130 70) = 0.5 Step 3: Form a riskless portfolio made up of one share of stock and two written calls. The cost of the riskless portfolio is: (S 0 2C 0 ) = 100 2C 0 and the certain end-of-year value is $70. Step 4: Calculate the present value of $70 with a one-year interest rate of 10% = $63.64 Step 5: Set the value of the hedged position equal to the present value of the certain payoff: $100 2C 0 = $63.64 Step 6: Solve for the value of the call: C 0 = $18.18 Here, the value of the call is greater than the value of the call in the lower-volatility scenario.

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