The Economics of DerivativesKen Danger

Problem Set 4 Solutions

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1. A trader buys a European call for $1. The strike price is $30. Draw a diagram that shows the trader’s variation in profit as a function of the stock price at expiration.

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profit from call

profit from call

2. A trader buys a European call with a strike price of K and a maturity date of T and at the same time writes a put with the same price and maturity date. What is the investor’s position? Draw a graph showing the profit for the put, call and combined position, assuming the strike price for the options are $40. Assume that the price of the call and the put are both $2.

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Profit from PutProfit from CallCombined Profit

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3. A trader sells a call option with a strike price of $50 and a put option with a strike price of $35. The options have the same maturity date. Suppose the premium for the call is $6 and for the put it is $5. Draw a diagram showing the variation in the trader’s profit as a function of the stock price for the put, call and combined position.

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Put ProfitCall ProfitCombined

4. Why is an American option always worth at least as much as a European option assuming both have the same strike price and expiration date?

The holder of an American option has all of the same attributes as a European option and more. For example, a European option can only be exercised on option expiration day while an American option can be exercised at any time. If an American option were priced less than a European option, an arbitrageur should buy the American option and sell the European option. In doing so, the arbitrageur would earn a riskless profit.

5. Why is an American option always worth at least as much as its intrinsic value?

When a call option is in the money, the intrinsic value of a call is the difference between the price of the underlying (e.g., the stock) and the strike price (i.e., S – K). For a put option that is in the money, the intrinsic value is the difference between the price of the strike less the value of the underlying (i.e., K – S). The intrinsic value of an out of the money option is zero. When value of an in the money option is less than its intrinsic value, a trader should buy the option and exercise it immediately.

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6. What is the lower bound price of a 6-month European call option on a non-dividend paying stock? Assume the call has a strike price of $50, the price is $52, and the risk-free rate is 3%.

Lower bound = 52 - 50e−. 03*. 5= 2 . 7444

7. What is the lower bound price for a 3-month European put option on a non-dividend paying stock? Assume the stock price is $30, the strike price is $25, and the risk free rate is 2%.

Lower bound = 25e−. 02*.25 - 30 = -5 . 1247

Because the value produced by the equation is negative, the lower bound price for the put is zero. The minimum value for an option is zero (i.e., it is worthless).

8. A one-month European put option on a non-dividend paying stock is currently selling for $1.50. The stock price is $52, the strike price is $55 and the risk free rate is 2%. Is there an arbitrage opportunity? If so, what should the trader do to take advantage of it?

Time =112

= .08333 years

Lower bound = 55e−. 02*.08333 - 52 = 2 . 908413

To execute an arbitrage trade, an arbitrageur should borrow money to buy the underlying stock and a put.

The cost of buying the stock and the put = $52 +$1.50 = $53.50. Thus, the arbitrageur should borrow $53.50 for one month.

The cost of repayment = $53 .50e . 02*. 08333= $53.58924

If the stock is below $55, the arbitrageur exercises the put .

The gain at expiration = $55 - $53 .58924 = $1 . 410763

If the stock is above $55, the put option expires worthless . In that event, profit is at least $1. 410763 . For example, if the stock price was $60, the arbitrageur would earn $6 . 410743 .

These values would need to be discounted to determine the persent value of the profit intoday's terms .

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9. What is the change in the lower bound price of a 3-month European put option if the stock price moves from $30 to $29. Assume the strike price is $35 and the risk free rate is 3%.

Lower bound when stock price equal 30 = 35e−. 03*.25 - 30 = 4 . 738482Lower bound when stock price equal 29 = 35e−. 03*.25 - 29 = 5 . 738482

Thus, the lower bound price of the put has increased by $1.

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