ECN603 Asset Pricing/Workshop 4
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ECN603 Asset Pricing/Workshop 4
(2) Give two reasons that the early exercise of an American call option on a non-dividend-paying stock is not optimal. The first reason should involve the time value of money. The second reason should apply even if interest rates are zero.
(3) A one-month European put option on a non-dividend-paying stock is currently selling for $2.5. The stock price is $47, the strike price is $50, and the risk-free interest rate is 6% per annum . What opportunities are there for an arbitrageur?
(4) The prices of European call and put options on a non-dividend-paying stock with 12 months to maturity, a strike price of $120, and an expiration date in 12 months are $20 and $5, respectively. The current stock price is $130. What is the implied risk-free rate?
(5) You are the manager and sole owner of a highly leveraged company. All the debt will mature in one year. If at that time the value of the company is greater than the face value of the debt, you will pay off the debt. If the value of the company is less than the face value of the debt, you will declare bankruptcy and the debt holders will own the company.
a. Express your position as an option on the value of the company.
b. Express the position of the debt holders in terms of options on the value of the company.
c. What can you do to increase the value of your position?
(6) What is meant by a protective put? What position in call options is equivalent to a protective put?
(7) Call options on a stock are available with strike prices of $15,$17.5 , and $20 and expiration dates in three months. Their prices are $4, $2, and,$0.5 respectively. Explain how the options can be used to create a butterfly spread. Construct a table showing how
profit varies with stock price for the butterfly spread.
(8) A call option with a strike price of $50 costs $2. A put option with a strike price of $45 costs $3. Explain how a strangle can be created from these two options. What is the pattern of profits from the strangle?
(9) Use put–call parity to relate the initial investment for a bull spread created using calls to the initial investment for a bull spread created using puts.
(10) “A box spread comprises four options. Two can be combined to create a long forward position and two can be combined to create a short forward position.” Explain this statement.
(11) A stock price is currently $40. It is known that at the end of one month it will be either $42 or $38. The risk-free interest rate is 8% per annum with continuous compounding. (a) Calculate the risk-neutral probabilities in this case, and determine the value of a one-month European call option with a strike price of $39. (b) Calculate how many shares you need to construct a risk-free portfolio of shares and one European call option with a strike price of $39, determine the value of that portfolio and show how this value corresponds to your answer in (a).
(12) A stock index is currently 1,500. Its volatility is 18%. The risk-free rate is 4% per annum (continuously compounded) for all maturities and the dividend yield on the index is 2.5%. Calculate values for u, d, and p when a six-month time step is used. What is the value a 12-month American put option with a strike price of 1,480 given by a two-step binomial tree.
(13) For the following notation for a one period binomial model of a stock price:
S0: current share price
1−d: percentage decrease in stock price when there is a down movement
u−1: percentage increase in stock price when there is a up movement
fu: value of option for up movement in the underlying share price
fd: value of option for down movement in the underlying share price
T: length of time period (in years)
r: risk free rate per annum (continuously compounded)
Construct a risk free portfolio consisting of Δ shares and short one option, and show that the current value of the option is
f=exp(−rT)*[pfu+(1−p)fd] ,
where
p=(exp(rT)−d) / (u−d)
2023-05-22