Question

2. A stock has two possible ending prices six months from now: $120 or $90. A call option written on this stock has an exercise price of $110. The option expires in six months. The risk-free rate is 6% per year. The current price of the stock is $100. a. Show how you can create a hedge portfolio using a combination of the stock and call option on this stock. b. What is the equilibrium price of the call option on this stock? 3. A stock has two possible ending prices six months from now: $ 45 or $60. A call option written on this stock has an exercise price of $48. The option expires in six months. The risk-free rate is 4% per year. The current price of the stock is $50. What is the equilibrium price of the call option on this stock? Suppose you find this call option trading at $3.00, describe an arbitrage strategy you can use to take advantage of the mispricing and calculate your arbitrage profit per share used in the strategy. 4. Assume the following information for a stock and a call option written on the stock: Exercise price = $45 Black-Scholes OPM: Current stock price = $30 C=S[N(di)] - Xer [N(D2)] 0²= 0.25 di = In(S/X) + (r + 1/202)t Time to expiration, t = 0.25 Risk-free rate, r = 0.05 dz = d.- o t0.5 o to.5 a. Use the Black-Scholes formula to determine the value of the call option. b. What is the value of the corresponding put option on the same stock with the same exercise price and time to maturity? c. Repeat a and b when the time to expiration is 0.5. d. Repeat a and b when the exercise price = $35.

Answer 1

2) Using Binomial Model:

Facts:

Spot price = $100, term = 6 months, Exercise price = $110, FSP = $120 or $90, interest rate=6%

Step 1:

situation FSP Value of call

1 120 -10

2 90 0

change Δ 30 10

Step 2: Calculation of Δ shares

Δ shares = change in value of call/change in FSP = 10/30 = 0.3333 shares

step 3: future value of portfolio   

Situation FSP share value value of call portfolio value

1 120 (120*0.333)= 40 -10 30

2 90 (90*0.333)= 30 0 30

Step 4: P.V of portfolio:

present value = 30*1/1.03 = 29.126

Step 5: P.V of  Δ shares

= Δ shares*spot price

= 0.3333*$100

= $33.33

Step 6: Value of call

value of portfolio = value of call + value of Δ shares

29.126 = value of call + 33.33

value of call = $4.206

The strategy is write 1 call option and hold 0.333 shares

3) Calculation of value of call option:

Step 1:

situation FSP Value of call

1   60 -12

2 45 0

change Δ 15 12

Δ shares = change in value of call/change in FSP = 12/15 =0.8 shares

step 2: future value of portfolio   

Situation FSP share value value of call portfolio value

1 60 (60*0.8)= 48 -12 36

2 45 (45*0.8)=36 0 36

Step 3: P.V of portfolio:

present value = 36*1/1.02 = 35.29

Step 4: P.V of  Δ shares

= Δ shares*spot price

= 0.80*$50

=$40

Step 5: Value of call

value of portfolio = value of call + value of Δ shares

35.29 = value of call + 40

value of call = $4.71

strategy is write 1 call option and hold 0.8 shares

If the call is trading at $3 then hold a call as it was less than equilibrium price

4) Black Schole's model:

Given exercise price = $45 current stock price $30 r² = 0.25 t = 0.25 r=0.05 stepir Ln (s/x) = Ln(30/45) = Ln(0.6667) = -0,4054 Stepar a di = Ln(s)x)+(0/+rf]xt rXVE = -0.4054 + [ 0. 25/2 + 0.05] x 0.25 10.25% (0.25 -0.3616 Step 3: da de = d.- [ox NE] -0.3616 - (0.5% 10:25] = -0.6116 Steph's Nelay means noinal table value of at, Molde = N (di) = N(-0.3616) = 0.3594 Stepsr N(42) = 0.2709 Step 6: - 0.250.05 -0.0125 - e -e 0.9876 stepts value of call option Ne [sxn(d,)] - [ax e tyNede)] -the

9393 = (30x 0.3594) - [45x 0.9876x0.2709] = 10.782 - 12-039 = $ 1,2573_fie, - $1.2573] Steps & Nl-de) N-di) = 1- Nld,) = 1-0.3594 = 0.6406 Stepa. Nl-da) = 1-N(d) - 1-0.2709 - 0.7291 Step 10 value of put option elp = [axetx Nl-dz] - [sxNl-d,)] - [ H5x0.9876X0.7291] - [30x 0.6406) = 32.40-19.218 13.184