Arbitrage - options contracts part b

Question 1

The option holder (long position) will only decide to exercise the option

Answer 1

when it benefits them
•Given this, the gain (payoff) the holder makes from the option can be expressed as follows:
–Call Option: payoff(long) = max (S_T-X,0); and,
Put Option: payoff(long) = max(X-S_T,0)

Question 2

option holder (long position) will only decide to exercise the option when it benefits them

The payoff for the writer of the option (short position) is exactly the opposite of the payoff for the holder

Answer 2

Call Option: payoff(short) = min(-[S_T-X],0) = min(X-S_T,0).
Put Option: payoff(short) = min(-[X-S_T],0) = min(S_T-X,0)

Question 3

payoff from a long and short position in an European call option

Answer 3

Question 4

payoff from a long and short position in a European put option

Answer 4

Question 5

Given the holder of the option will only exercise the option if it benefits them (i.e. their minimum payoff is 0) and the payoff for the writer is the exact opposite of the holder’s payoff (i.e. their maximum payoff is 0), why would anyone want to be the writer of an option?

Answer 5

People write options (i.e. take the short position) in light of the following 2 factors:

1. They have the opposite expectation about future price movements as the holder, therefore believing option exercise is unlikely; and,
2. In order to write the option, they receive a premium from the holder:
   •Call Option: We denote the call option premium as c; and,
   •Put Option: We denote the put option premium as p.

Question 6

the profit for a party to an option is not only the function of the option payoff, but also of the premium

what does this mean?

Answer 6

The holder of the option must also take into account the fact that they have had an additional outflow, namely the option premium; and,

The writer of the option must also take into account the fact that they have had an additional inflow, namely the option premium

Question 7

what do profit diagrams show?

Answer 7

profit for the option holder is the exact opposite of the profit for the option writer

Question 8

profit from a long position in a European call option

Answer 8

Question 9

profit from a short position in a European call option

Answer 9

Question 10

profit from a long position in a European PUT option

Answer 10

Question 11

profit from a short position in a European PUT option

Answer 11

Question 12

Suppose that on March 20, you purchased one contract (for 100 shares) of September $10 BHP call options. At that time, the price of a BHP share was $10.50 and the price of the call options on the Australian Stock Exchange (ASX) was $1.28. That is, at the time of entering the contract, you paid 100 * $1.28 = $128 for the right to purchase 100 BHP shares for $10 each at any time before the contract matures.

It is now September 20, which is the maturity date for September options, and the BHP stock price is $12. Will you exercise the option and, regardless, what is the payoff and profit from the option contract?

Answer 12

In this case, you will want to exercise the option: you pay $10 * 100 = $1,000 and receive 100 BHP shares which are worth $1,200 in total. Hence, the payoff from the option contract is $200 and the profit (net of the initial cost) is $72 ($200-$128)

Question 13

Suppose that on March 20, you purchased one contract (for 100 shares) of September $10 BHP call options. At that time, the price of a BHP share was $10.50 and the price of the call options on the Australian Stock Exchange (ASX) was $1.28. That is, at the time of entering the contract, you paid 100 * $1.28 = $128 for the right to purchase 100 BHP shares for $10 each at any time before the contract matures.

It is now September 20, which is the maturity date for September options, and the BHP stock price is $9. Will you exercise the option and, regardless, what is the payoff and profit from the option contract?

Answer 13

In this case, you would let the option lapse and no funds would change hands. You would clearly be unwilling to pay $10 per share by exercising the option when the stock is only worth $9. That is, the payoff is zero and the profit is –$128 (0-$128)

Question 14

Exercising a Put Option

Suppose that on March 20, you purchased one contract (for 100 shares) of September $10 BHP put options. At that time, the price of a BHP share was $10.50 and the price of the put options on the Australian Stock Exchange (ASX) was $0.53. That is, at the time of entering the contract, you paid 100 * $0.53 = $53 for the right to sell 100 BHP shares for $10 each at any time before the contract matures.

It is now September 20, which is the maturity date for September options, and the BHP stock price is $12. Will you exercise the option and, regardless, what is the payoff and profit from the option contract?

Answer 14

In this case, you will not want to exercise the option: You would clearly be unwilling to sell BHP shares for $10 per share by exercising the option when the stock is actually worth $12. That is, the payoff is zero and the profit is –$53 ($0-$53)

Question 15

Exercising a Put Option

Suppose that on March 20, you purchased one contract (for 100 shares) of September $10 BHP put options. At that time, the price of a BHP share was $10.50 and the price of the put options on the Australian Stock Exchange (ASX) was $0.53. That is, at the time of entering the contract, you paid 100 * $0.53 = $53 for the right to sell 100 BHP shares for $10 each at any time before the contract matures.

It is now September 20, which is the maturity date for September options, and the BHP stock price is $9. Will you exercise the option and, regardless, what is the payoff and profit from the option contract?

Answer 15

you would exercise the option. You receive $10 * 100 = $1,000 in return for 100 BHP shares (which are worth only $900)

Hence, the payoff from the option contract is $100 and the profit (net of the initial cost) is $47 ($100-$53)

Question 16

Selling to Close

options over shares are traded on

Answer 16

the Australian Stock Exchange (ASX) (note that options can be exchange traded, as in the case of share options, but, in some other instances, are traded over the counter)

Question 17

Selling to Close

All stock options on the ASX are

Answer 17

of the American type. Therefore, if the holder of the option so wishes, they may exercise the options at any time before maturity

Question 18

Selling to close

An alternative way for the holder of the option to liquidate their option position is

Answer 18

An alternative way for the holder of the option to liquidate their option position is by issuing a sell to close order.

Question 19

what is a sell to close order?

Answer 19

This procedure simply involves an option buyer selling their rights under the option contract to a third party.

Question 20

Suppose, that on March 20, you purchased one contract (100 shares) of September $10 BHP call options when the BHP stock price was $10.50 and September $10 call price was $1.28. Also suppose that it is now August 20, still one month before maturity and that you want to cash out of your position. The BHP stock price is now $10.70 and the call option price is $0.84. There are two ways to cash out of this contract

Answer 20

–First, you may exercise the option. This will generate a benefit of $0.70 per option (you pay $10 for a share that is worth $10.70); or,

–Second, you may sell the option as a live contract by issuing a sell to close order.This will generate a benefit of $0.84. Issuing a sell to close order results in youroption position being extinguished and all of the rightsyou had under the option contract arepassed on to the subsequent buyer.

Question 21

As with forward and futures contracts, option contracts can also be valued by

Answer 21

arbitrage

Question 22

option valuation

what is Put-call parity

Answer 22

is a pricing relation between the underlying stock and put and call options written on the stock.

Question 23

option valuation

For European put and call options with the same exercise price and maturity date put-call parity is given by:

Answer 23

is a pricing relation between the underlying stock and put and call options written on the stock.

Question 24

Suppose that European call and put options on Brambles with exercise prices of $20 and six months to maturity are selling for $2.50 and $1.50 respectively, that the Brambles stock price is $20, and that the riskless rate of interest is 8% p.a.

is put-call parity violated? If so, tabulate the strategy you would employ to make an arbitrage profit.

Answer 24

put-call parity is violated since c = 2.50 > p + S0 – X(1 + r_f)^–T = 2.25. The violation occurs because of any one or all of the following possibilities:

–The call is overpriced;
–The put is underpriced; and / or
–The stock is underpriced.

Question 25

Suppose that European call and put options on Brambles with exercise prices of $20 and six months to maturity are selling for $2.50 and $1.50 respectively, that the Brambles stock price is $20, and that the riskless rate of interest is 8% p.a.

is put-call parity violated? If so, tabulate the strategy you would employ to make an arbitrage profit.

Answer 25

The arbitrage portfolio, therefore, must account for all three of these possibilities simultaneously (remembering we always “buy low and sell high”). Consider the following portfolio:

Question 26

Suppose that European call and put options on Brambles with exercise prices of $20 and six months to maturity are selling for $2.50 and $1.50 respectively, that the Brambles stock price is $20, and that the riskless rate of interest is 8% p.a.

is put-call parity violated? If so, tabulate the strategy you would employ to make an arbitrage profit.

this case, put-call parity is violated since c = 2.50 > p + S₀ – X(1 + r_f)^–T = 2.25. The violation occurs because of any one or all of the following possibilities:

–The call is overpriced;
–The put is underpriced; and / or
- The stock is underpriced

Answer 26

In forming the portfolio a certain arbitrage profit of $0.25 is earned immediately. Note that:

–We sell the call (the left hand side of the put call parity equation) since it is relatively overvalued;
–We buy the portfolio on the right hand side of the equation since it is relatively undervalued. This involves buying the stock and the put;
–Since the portfolio must be self-financing, we sell the bond (which is equivalent to borrowing to finance the purchase of the stock and the put); and,
–If the call was relatively undervalued, all of these transactions would be reversed.

Question 27

feature that distinguishes American options from European options is that

Answer 27

American options can be exercised prior to expiration. Since an American option conveys all of the rights of a European option plus more (i.e., the added right to exercise early) an American option must be at least as valuable as its European counterpart