Chapter 2
An Introduction to Forwards and Options
古代婚礼Question 2.1
The payoff diagram of the stock is just a graph of the stock price as a function of the stock price:
In order to obtain the profit diagram at expiration, we have to incorporate the initial costs of the stock, i.e., how we finance the initial investment. We do this by assuming we borrow $50 at 10% interest. Note that even if we u our own funds to finance the purcha, we want to incorporate the opportunity costs of our investment (i.e., the interest we could have earned). By borrowing $50, after one year we have to pay back: $50(10.1)$55.
×+= The cond figure shows the graph of the stock, the $55 we have to pay back, and of the sum of the two positions, which is the profit graph. The arrows show that at a stock price of $55, the profit at expiration is indeed zero. When the stock price is $55, our stock is worth exactly what we owe and, hence, we break even.
10 McDonald • Fundamentals of Derivatives Markets
Question 2.2
Since we shorted the stock initially, our payoff at expiration is negative and equal—the stock price. This is the amount we have to spend in order to replace the share of stock we borrowed.
计划生育管理条例
Chapter 2 An Introduction to Forwards and Options 11 In order to obtain the profit diagram at expiration, we have to lend out the money we received from the short sale of the stock. We do so by buying a bond for $50. After one year we receive from the investment in the bond: $50(10.1)$55.
不问青红皂白×+= The cond figure shows the graph of the payoff of the shorted stock, the money we receive from the investment in the bond, and the sum of the two positions, which is the profit graph. The arrows show that at a stock price of $55, the profit at expiration is indeed zero. At $55, the amount of money we receive from our bond investment is exactly offt by the amount of money we need to buy back the one share of stock we borrowed.
Question 2.3
The position that is the opposite of a purchad call is a written call. A ller of a call option is said to be the option writer, or to have a short position in the call option. The call option writer is the counterparty to the option buyer, and his payoffs and profits are just the opposite of tho of the call option buyer. Similarly, the position that is the opposite of a purchad put option is a written put option. Again, the payoff and profit for a written put are just the opposite of tho of the purchad put.
It is important to note that the opposite of a purchad call is NOT the purchad put. If you do not e why, plea draw a payoff diagram with a purchad call and a purchad put.
Question 2.4
京剧常识1. The payoff to a long forward at expiration is equal to:
=−
Payoff to long forward Spot price at expiration forward price
12 McDonald • Fundamentals of Derivatives Markets
Therefore, we can construct the following table:
Price of ast in 6 months Agreed forward price Payoff to the long forward
40 50 −10
45 50−5
50 500
55 505
60 5010
2. The payoff to a purchad call option at expiration is:
=−
师德考核总结Payoff to call option max[0,spot price at expiration strike price] The strike price given is equal to $50. Therefore, we can construct the following table:
Price of ast in 6 months Strike price Payoff to the call option
40 50
45 500
50 500
55 505
60 5010
3. If we compare the two contracts, we immediately e that the call option has a protection for adver
movements in the price of the ast.
If the spot price is below $50, the buyer of the call option is able to walk away and does not incur a loss; whereas, the holder of the long forward position incurs a loss since he is obligated to buy the ast for $50.
If the spot price is above $50, the holder of the call option and the holder of the long forward position
have identical payoffs. Therefore, the call option should be more expensive due to this attractive walk away feature.
Question 2.5
1. The payoff to a short forward at expiration is equal to:
Payoff to short forward forward price spot price at expiration
=−
Therefore, we can construct the following table:
Price of ast in 6 months Agreed forward price Payoff to the short forward
40 50 10
45 505
50 500
55 50−5
先进制造技术60 50−10
Chapter 2 An Introduction to Forwards and Options 13
2. The payoff to a purchad put option at expiration is:
=−
Payoff to put option max[0,strike price spot price at expiration] The strike price given is equal to $50. Therefore, we can construct the following table:
Price of ast in 6 months Strike price Payoff to the put option
40 50 10
和珅如何玩黑玫瑰45 505
50 500
55 500
60 500
3. The same logic as in Question 2.4(3) applies. If we compare the two contracts, we e that the put
option holder is protected from increas in the price of the ast.
If the spot price is above $50, the buyer of the put option is able to walk away and does not incur a loss; whereas, the holder of the short forward position incurs a loss since he is obligated to ll the ast for $50.
If the spot price is above $50, the holder of the put option and the holder of the short forward position have identical payoffs. Therefore, the put option should be more expensive due to this walk away feature.
Question 2.6
We need to solve the following equation to determine the effective annual interest rate: $91(1)$100.
r
×+=
We obtain 100/9110.0989,
问卷调查网
r=−= which means that the effective annual interest rate is approximately 9.9%. Remember that when we drew profit diagrams for the forward or call option, we drew the payoff on the vertical axis, and the index price at the expiration of the contract on the horizontal axis. Since the default-free zero-coupon bond pays $100 regardless of the index price, the payoff diagram is just a horizontal line equal to $100 (on the y-axis).
The textbook provides the answer to the question concerning the profit diagram in the ction “Zero-Coupon Bonds in Payoff and Profit Diagrams.” When we were calculating profits, we saw that we had to find the future value of the initial investment. In this ca, our initial investment is $91. How do we find the future value? We u the current risk-free interest rate and multiply the initial investment by it. However, as our bond is default-free, and does not bear coupons, the effective annual interest rate is exactly the 9.9% we have calculated before. Therefore, the future value of $91 is $91(10.0989)$100,
×+= and our profit in six months is zero! In general, by incorporating the time value of money in our profit diagrams, all risk free borrowing and lending has no affect on the profit diagrams for any investment strategy.
