# 辅导 EC333 Topics in Financial Economics: Theories and International Finance PROBLEM SET 4 - OPTIONS讲解

EC333 Topics in Financial Economics: Theories and International Finance

PROBLEM SET 4 - OPTIONS

1)   Put-Call Parity

a)  Suppose that the stock price and the prices of at-the-money out and call options are  given by S = \$100, P = \$10, and C =\$15. What must be the one-year interest rate that would prevent an arbitrage?

b)  If the one year, risk-free interest rate is 4%, is there an arbitrage opportunity? If so, what would you do to obtain risk-less profit?

2)   Binomial Model - Equity Option

This question asks you to use the binomial model to compute the price of a European put option on Meta. stock with a strike price of \$60. The current price of Meta stock is \$60 per share, and the price will either increase by 100% or decrease by 50% on the    exercise date of the option (that is, u = 2 and d = 0.5). The interest rate is 10%. Meta.   stock does not pay any dividends.

a)  Using the replicating portfolio approach, what portfolio of Meta. stock and borrowing  or lending will replicate the change in the value of the option? What is the theoretical value of the put option described above?

b)  Using the risk-neutral probability approach, what is the theoretical value of the put option described above?

Now, there are two periods and each period the price will either increase by 100% or decrease by 50% (that is, u = 2 and d = 0.5) . The interest rate is 10% per period.

If there are two periods then there are 3 dates; call the three dates times 0, 1, and 2. The put option expires at time 2, and the payoff of the put option is based on the stock price  at time 2. The strike price of the option is \$60.

c)  Using the replicating portfolio approach;

. If the Meta. stock’s price increases by 100% in the first period, what portfolio of Meta. stock and borrowing or lending at time 1 will replicate the change in the   value of the option from time 1 to time 2? (That is, what is the “replicating portfolio?”) What is the theoretical value of the put option at time 1?

. If the Meta. stock’s price decreases by 50% in the first period, what portfolio of Meta. stock and borrowing or lending at time 1 will replicate the change in the  value of the option from time 1 to time 2? (That is, what is the “replicating portfolio?”) What is the theoretical value of the put option at time 1?

. What portfolio of Meta stock and borrowing or lending at time 0 will replicate the change in the value of the option from time 0 to time 1? What is the theoretical value of the put option at time 0?

d)  Using the risk-neutral probability approach, what is the theoretical value of the put option at time 1 if the Meta. stock’s price increases by 100% in the first period and   what is the value if the price decreases by 50% in the first period? What is the theoretical value of the put option at time 0?

3)   Binomial Model - Currency Option

The Australian dollar is currently worth 0.6100 US dollars. The Australian risk-free rate is 7% and the US risk-free rate 5%.What is the value of one-year call option with a strike of 0.60. For next year, the Australian dollar will either appreciate by 150% or depreciate by 60%.