Homework 4(For Practice)Problem 1The price of a stock is $40. The price of a one-year European put option on the stock with a strike price of $30 is quoted as $7 and the price of a one-year European call option on the stock with a strike price of $50 is quoted as $5. Suppose that an investor buys 100 shares, shorts 100 call options, and buys 100 put options. Draw a diagram illustrating how the investor’s profit or loss varies with the stock price over the next year. How does your answer change if the investor buys 100 shares, shorts 200 call options, and buys 200 put options?

Solution:Figure 1.1 shows the way in which the investor’s profit varies with the stock price in the first case. For stock prices less than $30 there is a loss of $1,200. As the stock price increases from $30 to $50 the profit increases from –$1,200 to $800. Above $50 the profit is $800. Students may express surprise that a call which is $10 out of the money is less expensive than a put which is $10 out of the money. Figure 1.2 shows the way in which the profit varies with stock price in the second case. In this case the profit pattern has a zigzag shape. The problem illustrates how many different patterns can be obtained by including calls, puts, and the underlying asset in a portfolio.

Figure 1.1 Profit in first case considered Problem 1

Figure 1.2 Profit for the second case considered Problem 1

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0 20 40 60 80

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Long Stock

Long Put

Short Call

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Problem 2A European call option and put option on a stock both have a strike price of $20 and an expiration date in three months. Both sell for $3. The risk-free interest rate is 10% per annum, the current stock price is $19, and a $1 dividend is expected in one month. Identify the arbitrage opportunity open to a trader.

If the call is worth $3, put-call parity shows that the put should be worth

This is greater than $3. The put is therefore undervalued relative to the call. The correct arbitrage strategy is to buy the put, buy the stock, and short the call. This costs $19. If the stock price in three months is greater than $20, the call is exercised. If it is less than $20, the put is exercised. In either case the arbitrageur sells the stock for $20 and collects the $1 dividend in one month. The present value of the gain to the arbitrageur is

Problem 3A diagonal spread is created by buying a call with strike price and exercise date

and selling a call with strike price and exercise date . Draw a

diagram showing the profit when (a) and (b) .

There are two alternative profit patterns for part (a). These are shown in Figures 3.1 and 3.2. In Figure 3.1 the long maturity (high strike price) option is worth more than the short maturity (low strike price) option. In Figure 3.2 the reverse is true. There is no ambiguity about the profit pattern for part (b). This is shown in Figure 3.3.

K1 K2

Profit

ST

Figure 3.1: Investor’s Profit/Loss in Problem 3a when long maturity call is worth more than short maturity call

K1 K2

Profit

ST

Figure 3.2 Investor’s Profit/Loss in Problem 3b when short maturity call is worth more than long maturity call

K2 K1

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ST

Figure 3.3 Investor’s Profit/Loss in Problem 3b

Problem 4A stock price is currently $40. Over each of the next two three-month periods it is expected to go up by 10% or down by 10%. The risk-free interest rate is 12% per annum with continuous compounding.

a. What is the value of a six-month European put option with a strike price of $42?

b. What is the value of a six-month American put option with a strike price of $42?

a. A tree describing the behavior of the stock price is shown in Figure 4.1. The risk-neutral probability of an up move, , is given by

Calculating the expected payoff and discounting, we obtain the value of the option as

The value of the European option is 2.118. This can also be calculated by working back through the tree as shown in Figure 4.1. The second number at each node is the value of the European option.

b. The value of the American option is shown as the third number at each node on the tree. It is 2.537. This is greater than the value of the European option because it is optimal to exercise early at node C.

40.0002.1182.537

44.000 0.8100.810

36.0004.7596.000

48.4000.0000.000

39.6002.4002.400

32.4009.6009.600

A

B

C

Figure 4.1 Tree to evaluate European and American put options in Problem 4. At each node, upper number is the stock price, the next number is the European put price, and the final number is the American put price