An Introduction to Forwards and Options

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 ﬁnance the initial investment. We do so by selling a bond for $50. After one year we have to pay back: $50 × (1 + 0.1) = $55. The second ﬁgure shows the graph of the stock, of the bond to be repaid, 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.

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Part 1 Insurance, Hedging, and Simple Strategies

Question 2.2.

Since we sold the stock initially, our payoff at expiration from being short the stock is negative.

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Chapter 2 An Introduction to Forwards and Options

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 × (1 + 0.1) = $55. The second ﬁgure shows the graph of the sold stock, of the money we receive from the investment in the bond, 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.

Question 2.3.

The position that is the opposite of a purchased call is a written call. A seller of a call option is said to be the option writer, or to have a short position. The call option writer is the counterparty to the option buyer, and his payoffs and proﬁts are just the opposite of those of the call option buyer.

Similarly, the position that is the opposite of a purchased put option is a written put option. Again, the payoff and proﬁt for a written put are just the opposite of those of the purchased put.

It is important to note that the opposite of a purchased call is NOT the purchased put. If you do not see why, please draw a payoff diagram with a purchased call and a purchased put.

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Part 1 Insurance, Hedging, and Simple Strategies

Question 2.4.

a)

The payoff to a long forward at expiration is equal to:

Payoff to long forward = Spot price at expiration – forward price

Therefore, we can construct the following table:

Price of asset in 6 months Agreed forward price

40

50

45

50

50

50

55

50

60

50

b)

Payoff to the long forward

−10

−5

0

5

10

The payoff to a purchased call option at expiration is:

Payoff to call option = max[0, spot price at expiration – strike price]

The strike is given: It is $50. Therefore, we can construct the following table:

Price of asset in 6 months

40

45

50

55

60

Strike price

50

50

50

50

50

Payoff to the call option

0

0

0

5

10

c)

If we compare the two contracts, we immediately see that the call option has a protection for adverse movements in the price of the asset: If the spot price is below $50, the buyer of the call option can walk away, and need not incur a loss. The buyer of the long forward incurs a loss, while he has the same payoff as the buyer of the call option if the spot price is above $50. Therefore, the call option should be more expensive. It is this attractive option to walk away that we have to pay for. Question 2.5.

a)

The payoff to a short forward at expiration is equal to:

Payoff to short forward = forward price – spot price at expiration

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Chapter 2 An Introduction to Forwards and Options

Therefore, we can construct the following table:

Price of asset in 6 months Agreed forward price

40

50

45

50

50

50

55

50

60

50

b)

Payoff to the short forward

10

5

0

−5

−10

The payoff to a purchased put option at expiration is:

Payoff to put option = max[0, strike price – spot price at expiration]

The strike is given: It is $50. Therefore, we can construct the following table:

Price of asset in 6 months

40

45

50

55

60

Strike price

50

50

50

50

50