Chapter 20 Finance

20.27 Consider the payoff structures of the following two portfolios:

a. Buying a call option on one share in one month at a strike price of $50 and saving the present value of $50 (so that at expiration it will have grown to $50 with interest).

b. Buying a put option on one share in one month at a strike price of $50 and buying one share of stock.

What conclusion can you draw about the relation between call prices and put prices?

Solution:

The payoff of these two portfolios is identical. In the first case, if the stock price is below $50 at expiration, you will not exercise and you will be left with $50. If it is above $50, you will exercise and you will have a share of stock (whatever it is worth). In the second case, if the price is below $50, you will exercise your put option and get $50 for your share of stock. If it is above $50, you will not exercise and you will retain your share of stock (whatever it is worth).

From this, we can see that the value of the two portfolios today must be the same. That is:

C + K / (1 + Rrf) = P + S

where C is the value of a call, P is the value of a put, both options have a strike price K and the same expiration, S is the current value of the stock and Rrf is the risk-free interest rate. If we know the value of a call (and the current stock price and interest rate), we can calculate the value of a put. This relation is call put-call parity. Note that this relation is not dependent on any option pricing model.

20.28 One way to extend the binomial pricing model is by including multiple time periods. Suppose Splittime, Inc., is currently trading for $100 per share. In one month the price will either increase by $10 (to $110) or decrease by $10 (to $90). The following month will be the same. The price will either increase by $10 or decrease by $10. Note that in two months, the price could be $120, $100, or $80. The risk-free rate is 1 percent per month. Find the value today of an option to buy one share of...