FINC 381 Test 3 Examples

The Following price quotations are for exchange listed option on a company's common stock Strike 55 Exp Feb Call 7.25 Put .48 With transaction costs ignored how much would a buyer have to pay for one call option contact. Assume each contract is for 100 shares

$725 7.25 * 100 = 725

Consider a 1-year option with exercise price $120 on a stock with annual standard deviation 15%. The T-bill rate is 3% per year. Find N(d1) for stock prices $115, $120, and $125.

115: .4965 120: .6083 125: .7079 We first calculate d1 = formula35.mml, and then find N(d1), which is the Black Scholes hedge ratio for the call. We can observe from the following that when the stock price increases, N(d1) increases as well. X 120 r 3 % σ 15 % T 1 S d1 N(d1) $115 −0.0087 0.4965 $120 0.2750 0.6083 $125 0.5471 0.7079

A. If, in a two-state model, a stock can take a price of 132 or 99, what would be the hedge ratio for each of the following prices: $132, $125, $115, $99? B. What do you conclude about the hedge ratio as the option becomes progressively more in the money?

A. 132: 0 125: .21 115: .52 99: 1 H = Cu − Cd uS0 − dS0 uS0 − dS0 = 132 − 99 = 33 X Cu − Cd Hedge Ratio 132 0 − 0 0/33 = 0.00 125 7 − 0 7/33 = 0.21 115 17 − 0 17/33 = 0.52 $9 33 − 0 33/33 = 1.00 B: Increase to 1.0

You establish a straddle on Walmart using September call and put options with a strike price of $55. The call premium is $4.5 and the put premium is $5.25 A. what is the most you can lose on this position B. What will your profit or loss if Walmart is selling for $63 in September? C. At what stock prices will you break even on the straddle

A. Maximum Loss: 9.75 Loss 4.50 + 5.25 = 9.75 B. Loss: Final Value - Original Investment (St - X) - (C + P) 8 - 9.75 = -1.75 C. St>X (St - X) - (C+P) (St - 55) - 9.75 = 0 St= 64.75 St<X (X - St) - (C + P) (55- St) - 9.75 = 0 St = 45.25

You purchase one IBM July 120 call contract (equaling 100 shares) 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.

Answer: 200 Loss Long call profit = Max [0, ($123 - $120)(100)] - $500 = -$200

An investor is bearish on a particular stock and decided to buy a put with a strike price of $25. Ignoring commissions, if the option was purchased for a price of $.85, what is the break-even point for the investor?

Answer: 24.15 Breakeven = 25 - .85 = 24.15

Suppose a stock is currently selling for $62 and can have a price of $70 or $56 in one period. There is a call option with a strike price of $65. The risk-free rate is 5%. Estimate fair price of the call today?

Answer: 3.09 Hedge ratio = 5/14 = 0.3571 call = 0.3571*62 - (0.3571*56)/1.05 = $3.09

You are considering purchasing a put option on a stock with a current price of $33. The exercise price is $35, and the price of the corresponding call option is $2.25. According to the put-call parity theorem, if the risk-free rate of interest is 4% and there are 90 days until expiration, the value of the put should be ____________.

Answer: 3.91 P = C - S0 + Xe-rT or P = 2.25 - 33 + (35)e−(.04)(90/365) = 3.91

You sell one IBM July 90 call contract for a premium of $4 and two puts for a premium of $3 each. You hold the position until the expiration date, when IBM stock sells for $95 per share. You will realize a ______ on this strip.

Answer: 500 Profit Selling an IBM July 90 strip entails selling two IBM July 90 puts and one IBM July 90 call. Initial income = C90 + 2P90 = [4 + 2(3)](100) = $1,000. If the final stock price is $95, the position value is found as: Profit = [-Max ($0, $95 - 90) + 2Max ($0, $90 - $95)](100) + $1,000 = -$500 + $1,000 = $500

You find digital option quotes on jobless claims. You can buy a call option with a strike price of 300,000 jobless claims. This option pays $100 if actual claims exceed the strike price and pays zero otherwise. The option costs $68. A second digital call with a strike price of 305,000 jobless claims is available at a cost of $53. Suppose you buy the option with the 300,000 strike and sell the option with the 305,000 strike and jobless claims actually wind up at 303,000. Your net profit on the position is ______.

Answer: 85$ Initial cost = -C300 + C305 = -$68 + $53 = -$15 At actual jobless claims of 303,000, at contract maturity the C300 call is worth $100 and the C305 call is worthless. Profit = +$100 - $0 - $15 = $85

You purchase a call option on a stock. The profit at contract maturity of the option position is ___________, where X equals the option's strike price, ST is the stock price at contract expiration, and C0 is the original purchase price of the option.

Answer: Max (-Co, St - C - Co)

An investor buys a call at a price of $4.50 with an exercise price of $40. At what stock price will the investor break even on the purchase of the Call?

Break even price: $44.50 40 +4.5 = 44.50

A call option with a strike price of $56 on a stock selling at $64 costs $8.7. What are the call option's intrinsic and time values?

Intrinsic Value: 8 IV= Present Stock- Strike Price 64 - 56 = 8 Time Value: .7 Time Value= Costs - IV 8.7 - 8 = .7

At contract maturity the value of a CALL option is ___________, where X equals the option's strike price and ST is the stock price at contract expiration.

MAX(0, St- X)

An investor purchases a stock for $56 and a put for .8 with a strike price of 66$. The investor sells a call for .8 with a strike price of 66$. What is the maximum profit and loss for this position?

Maximum Profit: 10 66-56 = 10 Maximum Loss: -6 50-56=-6 initial outlay is 56 pay off will be between 50 and 66

A call option on Jupiter Motors stock with an exercise price of $50 and one-year expiration is selling at $4. A put option on Jupiter stock with an exercise price of $50 and one-year expiration is selling at $3.5. If the risk-free rate is 11% and Jupiter pays no dividends, what should the stock price be?

Stock Price: $45.55 Using put-call parity: Put = C − S0 + PV(X) +PV(Dividends) $3.5 = $4 − S0 + $50/(1 + 0.11) + 0 --> S0 = $45.55

You buy a share of stock, write a one-year call option with X = $22, and buy a one-year put option with X = $22. Your net outlay to establish the entire portfolio is $20.60. What must be the risk-free interest rate? The stock pays no dividends.

The following payoff table shows that the portfolio is riskless with time-T value equal to $22. Therefore, the risk-free rate is: ($22/$20.60) - 1 = 0.0680 = 6.80%***

Suppose a stock is currently selling for $100. It can go up by 15% or down by 10%. There is a call option with a strike price of $97.50. The risk-free rate is 10%. Will there be any arbitrage profit or loss if call is selling for $14.50?

Answer: Yes Arbitrage Profit is 1.95 Hedge ratio = 17.5/25 = 0.70 call = 0.70*100 - (0.70*90)/1.10 = $12.73 If c= $14.50 then buying 0.70 share at 100 and selling call at 14.50 and financing 55.50 for 10% will give $1.95 whether price in the future is 90 or 115.

You are attempting to value a call option with an exercise price of $130 and one year to expiration. The underlying stock pays no dividends, its current price is $130, and you believe it has a 50% chance of increasing to $155 and a 50% chance of decreasing to $105. The risk-free rate of interest is 8%. Based upon your assumptions, calculate your estimate of the the call option's value using the two-state stock price model.

Value of the Call: 16.39 Step 1: Calculate the option value at expiration based upon your assumption of a 50% chance of increasing to $155 and a 50% chance of decreasing to $105. The two possible stock prices are: S+ = $155 and S- = $105. Therefore, since the exercise price is $130, the corresponding two possible call values are: Cu = $25 and Cd = $0. Step 2: Calculate the hedge ratio: (Cu - Cd)/(uS0 - dS0) = (25 - 0)/(155 - 105) = 0.50 Step 3: Form a riskless portfolio made up of one share of stock and two written calls. The cost of the riskless portfolio is: (S0 - 2C0) = 130 - 2C0 and the certain end-of-year value is $105. Step 4: Calculate the present value of $105 with a one-year interest rate of 8%: $105/1.08 = $97.22 Step 5: Set the value of the hedged position equal to the present value of the certain payoff: $130 - 2C0 = $97.22 Step 6: Solve for the value of the call: C0 = $16.39

A put option on a stock with a current price of $45 has an exercise price of $47. The price of the corresponding call option is $4.05. According to put-call parity, if the effective annual risk-free rate of interest is 6% and there are four months until expiration, what should be the value of the put?

Value of the Put: $5.15 Put = 4.05 - 45 + 47/(1 +.06)^(4/12) + 0 = 5.15 Put = C − S0 + PV(X) +PV(Dividends)

We will derive a two-state call option value in this problem. Data: S0 = 280; X = 290; 1 + r = 1.1. The two possibilities for ST are 320 and 200. a. The range of S is 120 while that of C is 30 across the two states. What is the hedge ratio of the call? B. Calculate the value of a call option on the stock with an exercise price of 290. (Do not use continuous compounding to calculate the present value of X in this example, because the interest rate is quoted as an effective per-period rate.)

A: Hedge Ratio: .25 hedge ratio for the call is: H = Cu - Cd = 30 - 0 = 0.25 uS0 - dS0 320 -200 B: Call Value: 24.55 Riskless Portfolio S = 200 S = 320 1 shares 200 320 Short 4 calls 0 -120 Total 200 200 Present value = $200/1.10 = $181.818 Portfolio cost = 1S - 4C = 280 - 4C = $181.818 formula5.mml C = $24.55 Put-call parity relationship:P = C - S0 + PV(X) $24.55 = $24.55 - $280 + ($290/1.10) = $24.55

A stock priced at $65 has a standard deviation of 30%. Three-month calls and puts with an exercise price of $60 are available. The calls have a premium of $7.27, and the puts cost $1.10. The risk-free rate is 5%. Since the theoretical value of the put is $1.525, you believe the puts are undervalued. If you want to construct a riskless arbitrage to exploit the mispriced puts, you should ____________.

Answer: Write the call and buy the put and buy the stock and borrow the present value of the exercise price For parity to hold, the following condition must be met: C - P = S0 - Xe−rT C-P= 6.17 So-Xe^(rT)= 5.75 Correct Price for put = 1.52 Buy what's underpriced (the left side of the equation) and sell what's overpriced (the right side of the equation)

Imagine that you are holding 5,300 shares of stock, currently selling at $40 per share. You are ready to sell the shares but would prefer to put off the sale until next year due to tax reasons. If you continue to hold the shares until January, however, you face the risk that the stock will drop in value before year-end. You decide to use a collar to limit downside risk without laying out a good deal of additional funds. January call options with a strike price of $45 are selling at $3, and January puts with a strike price of $35 are selling at $4. What will be the value of your portfolio in January (net of the proceeds from the options) if the stock price ends up at $28, $40, $48? What will the value of your portfolio be if you simply continued to hold the shares? Portfolio Value 28 40 48 If collar is used? if you continued to hold the shares?

The collar involves purchasing a put for $4 and selling a call for $3. The initial outlay is $1. ST = $28: Value at expiration = Value of call + Value of put + Value of stock = $0 + ($35 - $28) + $28 = $35 *** Given 5,300 shares, the total net proceeds will be: (Final value - Original investment) × # of shares = ($35 - $1) × 5,300 = $180,200**** ***Net proceeds without using collar = ST × # of shares = $28 × 5,300 = $148,400*** ST = $40: Value at expiration = Value of call + Value of put + Value of stock = 0 + 0 + $40 = $40 ***Given 5,300 shares, the total net proceeds will be: (Final value - Original investment) × # of shares = ($40 - $1) × 5,300 = $206,700*** ***Net proceeds without using collar = ST × # of shares = $40 × 5,300 = $212,000*** ST = $48: Value at expiration = Value of call + Value of put + Value of stock = ($45 - $48) + 0 + $48 = $45 ***Given 5,300 shares, the total net proceeds will be: (Final value - Original investment) × # of shares = ($45 - $1) × 5,300 = $233,200*** ***Net proceeds without using collar = ST × # of shares = $48 × 5,300 = $254,400***