Option Pricing Unit 1 - Self Check
Explain why the market maker's bid-offer spread represents a real cost to options investors.
A "fair" price for the option can reasonably be assumed to be half way between the bid and the offer price quoted by a market maker. An investor typically buys at the market maker's offer and sells at the market maker's bid. Each time he or she does this there is a hidden cost equal to half the bid-offer spread.
Explain carefully the difference between hedging, speculation, and arbitrage.
A trader is hedging when she has an exposure to the price of an asset and takes a position in a derivative to offset the exposure. In a speculation the trader has no exposure to offset. She is betting on the future movements in the price of the asset. Arbitrage involves taking a position in two or more different markets to lock in a profit.
Why is an American call option on a dividend-paying stock always worth at least as much as its intrinsic value? Is the same true of a European call option? Explain your answer.
An American call option can be exercised at any time. If it is exercised its holder gets the intrinsic value. It follows that an American call option must be worth at least its intrinsic value. A European call option can be worth less than its intrinsic value. Consider, for example, the situation where a stock is expected to provide a very high dividend during the life of an option. The price of the stock will decline as a result of the dividend. Because the European option can be exercised only after the dividend has been paid, its value may be less than the intrinsic value today
"The early exercise of an American put is a trade-off between the time value of money and the insurance value of a put." Explain this statement.
An American put when held in conjunction with the underlying stock provides insurance. It guarantees that the stock can be sold for the strike price, K. If the put is exercised early, the insurance ceases. However, the option holder receives the strike price immediately and is able to earn interest on it between the time of the early exercise and the expiration date.
What is the effect of an unexpected cash dividend on (a) a call option price and (b) a put option price?
An unexpected cash dividend would reduce the stock price on the ex-dividend date. This stock price reduction would not be anticipated by option holders. As a result there would be a reduction in the value of a call option and an increase the value of a put option.
What is arbitrage? Explain the arbitrage opportunity when the price of a dually listed mining company stock is $50 (USD) on the New York Stock Exchange and $52 (CAD) on the Toronto Stock Exchange. Assume that the exchange rate is such that 1 USD equals 1.01 CAD. Explain what is likely to happen to prices as traders take advantage of this opportunity.
Arbitrage involves carrying out two or more different trades to lock in a profit. In this case, traders can buy shares on the NYSE and sell them on the TSX to lock in a USD profit of 52/1.01−50=1.485 per share. As they do this the NYSE price will rise and the TSX price will fall so that the arbitrage opportunity disappears.
Give two reasons that the early exercise of an American call option on a non-dividend-paying stock is not optimal. The first reason should involve the time value of money. The second reason should apply even if interest rates are zero.
Delaying exercise delays the payment of the strike price. This means that the option holder is able to earn interest on the strike price for a longer period of time. Delaying exercise also provides insurance against the stock price falling below the strike price by the expiration date. Assume that the option holder has an amount of cash K and that interest rates are zero. When the option is exercised early it is worth ST at expiration. Delaying exercise means that it will be worth max (K,ST) at expiration.
Calculate the intrinsic value and time value from the mid-market (average of bid and offer) prices of the September 2013 call options in the table below. Do the same for the September 2013 put options. Assume in each case that the current mid-market stock price is $871.30. (an attached graph of Prices of Call Options)
For strike prices of 820, 840, 860, 880, 900, and 920 the intrinsic values of call options are 51.30, 31.30, 11.30, 0, 0, and 0. The mid-market values of the options are 76.90, 63.40, 51.75, 41.30, 32.45 and 25.20. The time values of the options are given by what is left from the mid-market value after the intrinsic value has been subtracted. They are 25.60, 32.10, 40.45, 41.30, 32.45, and 25.20, respectively. For strike prices of 820, 840, 860, 880, 900, and 920, the intrinsic values of put options are 0, 0, 0, 8.70, 28.70, and 48.70. The mid-market values of the options are 24.55, 31.40, 39.65, 49.30, 60.05, and 72.55. The time values of the options are given by what is left from the mid-market value after the intrinsic value has been subtracted. They are 24.55, 31.40, 39.65, 40.60, 31.35 and 23.85, respectively. Note that for both puts and calls the time value is greatest when the option is close to the money.
Explain why a futures contract can be used for either speculation or hedging.
If an investor has an exposure to the price of an asset, he or she can hedge with futures contracts. If the investor will gain when the price decreases and lose when the price increases, a long futures position will hedge the risk. If the investor will lose when the price decreases and gain when the price increases, a short futures position will hedge the risk. Thus either a long or a short futures position can be entered into for hedging purposes. If the investor has no exposure to the price of the underlying asset, entering into a futures contract is speculation. If the investor takes a long position, he or she gains when the asset's price increases and loses when it decreases. If the investor takes a short position, he or she loses when the asset's price increases and gains when it decreases.
"Buying a put option on a stock when the stock is owned is a form of insurance.'' Explain this statement.
If the stock price declines below the strike price of the put option, the stock can be sold for the strike price.
A trader buys a call option with a strike price of $30 for $3. Does the trader ever exercise the option and lose money on the trade? Explain.
If the stock price is between $30 and $33 at option maturity the trader will exercise the option, but lose money on the trade. Consider the situation where the stock price is $31. If the trader exercises, she loses $2 on the trade. If she does not exercise she loses $3 on the trade. It is clearly better to exercise than not exercise.
Suppose that a European call option to buy a share for $100.00 costs $5.00 and is held until maturity. Under what circumstances will the holder of the option make a profit? Under what circumstances will the option be exercised? Draw a diagram illustrating how the profit from a long position in the option depends on the stock price at maturity of the option.
Ignoring the time value of money, the holder of the option will make a profit if the stock price at maturity of the option is greater than $105. This is because the payoff to the holder of the option is, in these circumstances, greater than the $5 paid for the option. The option will be exercised if the stock price at maturity is greater than $100. Note that if the stock price is between $100 and $105 the option is exercised, but the holder of the option takes a loss overall.
Suppose that a European put option to sell a share for $60 costs $8 and is held until maturity. Under what circumstances will the seller of the option (the party with the short position) make a profit? Under what circumstances will the option be exercised? Draw a diagram illustrating how the profit from a short position in the option depends on the stock price at maturity of the option.
Ignoring the time value of money, the seller of the option will make a profit if the stock price at maturity is greater than $52.00. This is because the cost to the seller of the option is in these circumstances less than the price received for the option. The option will be exercised if the stock price at maturity is less than $60.00. Note that if the stock price is between $52.00 and $60.00 the seller of the option makes a profit even though the option is exercised.
What is the difference between entering into a long forward contract when the forward price is $50 and taking a long position in a call option with a strike price of $50?
In the first case the trader is obligated to buy the asset for $50. (The trader does not have a choice.) In the second case the trader has an option to buy the asset for $50. (The trader does not have to exercise the option.)
You would like to speculate on a rise in the price of a certain stock. The current stock price is $29, and a three-month call with a strike of $30 costs $2.90. You have $5,800 to invest. Identify two alternative strategies, one involving an investment in the stock and the other involving investment in the option. What are the potential gains and losses from each?
One strategy would be to buy 200 shares. Another would be to buy 2,000 options (or 20 option contracts). If the share price does well, the second strategy will give rise to greater gains. For example, if the share price goes up to $40 you gain [2,000 × ($40 −$30)] − $5,800 = $14,200 from the second strategy and only 200 × ($40 − $29) = $2,200 from the first strategy. However, if the share price does badly, the second strategy gives greater losses. For example, if the share price goes down to $25, the first strategy leads to a loss of 200 × ($29 - $25) = $800, whereas the second strategy leads to a loss of the whole $5,800 investment. This example shows that options contain built in leverage.
Explain carefully the difference between selling a call option and buying a put option.
Selling a call option involves giving someone else the right to buy an asset from you. It gives you a payoff of: −max(ST − K, 0) = min (K − ST, 0) Buying a put option involves buying an option from someone else. It gives a payoff of: max(K − ST, 0) In both cases the potential payoff is K−ST. When you write a call option, the payoff is negative or zero. (This is because the counterparty chooses whether to exercise.) When you buy a put option, the payoff is zero or positive. (This is because you choose whether to exercise.)
Explain why an American option is always worth at least as much as a European option on the same asset with the same strike price and exercise date.
The holder of an American option has all the same rights as the holder of a European option and more. It must therefore be worth at least as much. If it were not, an arbitrageur could short the European option and take a long position in the American option.
Explain why an American option is always worth at least as much as its intrinsic value.
The holder of an American option has the right to exercise it immediately. The American option must therefore be worth at least as much as its intrinsic value. If it were not an arbitrageur could lock in a sure profit by buying the option and exercising it immediately
Suppose that a March call option to buy a share for $50 costs $2.50 and is held until March. Under what circumstances will the holder of the option make a profit? Under what circumstances will the option be exercised? Draw a diagram showing how the profit on a long position in the option depends on the stock price at the maturity of the option.
The holder of the option will gain if the price of the stock is above $52.50 in March. (This ignores the time value of money.) The option will be exercised if the price of the stock is above $50.00 in March. The profit as a function of the stock price is shown here: **Graph depicts profit on Y axis (-4, -2, 0, 2, 4, 6, 8) Stock Price on X axis (40, 45, 50, 55, 60) The line is straight horizonal at just below -2 (2.50) and then increases straight up at the 50 mark
The current price of a stock is $94, and three-month call options with a strike price of $95 currently sell for $4.70. An investor who feels that the price of the stock will increase is trying to decide between buying 100 shares and buying 2,000 call options (20 contracts). Both strategies involve an investment of $9,400. What advice would you give? How high does the stock price have to rise for the option strategy to be more profitable?
The investment in call options entials higher risks but can lead to higher returns. If the stock price stays at $94, an investor who buys call options loses $9,400 whereas an investor who buys shares doesn't gain or lose anything.
In March, a US investor instructs a broker to sell one July put option contract on a stock. The stock price is $42 and the strike price is $40. The option price is $3. Explain what the investor has agreed to. Under what circumstances will the trade prove to be profitable? What are the risks?
The investor has agreed to buy 100 shares of the stock for $40 in July (or earlier) if the party on the other side of the transaction chooses to sell. The trade will prove profitable if the option is not exercised or if the stock price is above $37 at the time of exercise. The risk to the investor is that the stock price plunges to a low level. For example, if the stock price drops to $1 by July, the investor loses $3,600. This is because the put options are exercised and $40 is paid for 100 shares when the value per share is $1. This leads to a loss of $3,900 which is only a little offset by the premium of $300 received for the options
An investor sells a European call on a share for $4. The stock price is $47 and the strike price is $50. Under what circumstances does the investor make a profit? Under what circumstances will the option be exercised? Draw a diagram showing the variation of the investor's profit with the stock price at the maturity of the option.
The investor makes a profit if the price of the stock is below $54 on the expiration date. If the stock price is below $50, the option will not be exercised, and the investor makes a profit of $4. If the stock price is between $50 and $54, the option is exercised and the investor makes a profit between $0 and $4.
An investor buys a European put on a share for $3. The stock price is $42 and the strike price is $40. Under what circumstances does the investor make a profit? Under what circumstances will the option be exercised? Draw a diagram showing the variation of the investor's profit with the stock price at the maturity of the option.
The investor makes a profit if the price of the stock on the expiration date is less than $37. In these circumstances the gain from exercising the option is greater than $3. The option will be exercised if the stock price is less than $40 at the maturity of the option. The variation of the investor's profit with the stock price is shown below:
What is the difference between the over-the-counter market and the exchange-traded market?
The over-the-counter market is a telephone- and computer-linked network of financial institutions, fund managers, and corporate treasurers where two participants can enter into any mutually acceptable contract. An exchange-traded market is a market organized by an exchange where the contracts that can be traded have been defined by the exchange.
On May 8, 2013, an investor owns 100 Google shares. As indicated in Table 1.3, the share price is about $871 and a December put option with a strike price $820 costs $37.50. The investor is comparing two alternatives to limit downside risk. The first involves buying one December put option contract with a strike price of $820. The second involves instructing a broker to sell the 100 shares as soon as Google's price reaches $820. Discuss the advantages and disadvantages of the two strategies
The second alternative involves what is known as a stop or stop-loss order. It costs nothing and ensures that $82,000, or close to $82,000, is realized for the holding in the event the stock price ever falls to $820. The put option costs $3,750 and guarantees that the holding can be sold for $8,200 any time up to December. If the stock price falls marginally below $820 and then rises the option will not be exercised, but the stop-loss order will lead to the holding being liquidated. There are some circumstances where the put option alternative leads to a better outcome and some circumstances where the stop-loss order leads to a better outcome. If the stock price ends up below $820, the stop-loss order alternative leads to a better outcome because the cost of the option is avoided. If the stock price falls to $800 in November and then rises to $850 by December, the put option alternative leads to a better outcome. The investor is paying $3,750 for the chance to benefit from this second type of outcome.
Suppose that a June put option to sell a share for $60 costs $4 and is held until June. Under what circumstances will the seller of the option (i.e., the party with a short position) make a profit? Under what circumstances will the option be exercised? Draw a diagram showing how the profit from a short position in the option depends on the stock price at the maturity of the option.
The seller of the option will lose money if the price of the stock is below $56.00 in June. (This ignores the time value of money.) The option will be exercised if the price of the stock is below $60.00 in June. The profit as a function of the stock price is shown in the diagram here:
List the six factors affecting stock option prices.
The six factors affecting stock option prices are the stock price, strike price, risk-free interest rate, volatility, time to maturity, and dividends.
"Options and futures are zero-sum games." What do you think is meant by this statement?
The statement means that the gain (loss) to the party with the short position is equal to the loss (gain) to the party with the long position. In aggregate, the net gain to all parties is zero.
A company declares a 2-for-1 stock split. Explain how the terms change for a call option with a strike price of $60.
The strike price is reduced to $30, and the option gives the holder the right to purchase twice as many shares.
It is May and a trader writes a September call option with a strike price of $20. The stock price is $18, and the option price is $2. Describe the investor's cash flows if the option is held until September and the stock price is $25 at this time.
The trader has an inflow of $2 in May and an outflow of $5 in September. The $2 is the cash received from the sale of the option. The $5 is the result of the option being exercised. The investor has to buy the stock for $25 in September and sell it to the purchaser of the option for $20.
A trader writes a December put option with a strike price of $30. The price of the option is $4. Under what circumstances does the trader make a gain?
The trader makes a gain if the price of the stock is above $26 at the time of exercise. (This ignores the time value of money.)
A trader sells a put option with a strike price of $40 for $5. What is the trader's maximum gain and maximum loss? How does your answer change if it is a call option?
The trader's maximum gain from the put option is $5. The maximum loss is $35, corresponding to the situation where the option is exercised and the price of the underlying asset is zero. If the option were a call, the trader's maximum gain would still be $5, but there would be no bound to the loss as there is in theory no limit to how high the asset price could rise.
A United States investor writes five naked call option contracts. The option price is $3.50, the strike price is $60.00, and the stock price is $57.00. What is the initial margin requirement, using the requirement explained in the Hull text?
The two calculations are necessary to determine the initial margin. The first gives: 500 × (3.5 + 0.2 × 57 − 3) = 5,950 The second gives: 500 × (3.5 + 0.1 × 57) = 4,600 The initial margin is the greater of these, or $5,950. Part of this can be provided by the initial amount of 500 × 3.5 = $1,750 received for the options.
What are the bid and offer quotes of a market maker in the over-the-counter market?
When a market maker quotes a bid and an offer, the bid is the price at which the market maker is prepared to buy and the offer is the price at which the market maker is prepared to sell.
Explain why margin accounts are required when clients write options but not when they buy options.
When an investor buys an option, cash must be paid up front. There is no possibility of future liabilities and therefore no need for a margin account. When an investor sells an option, there are potential future liabilities. To protect against the risk of a default, margins are required.
Suppose you own 5,000 shares that are worth $25 each. How can put options be used to provide you with insurance against a decline in the value of your holding over the next four months?
You could buy 50 put option contracts (each on 100 shares) with a strike price of $25 and an expiration date in four months. If at the end of four months the stock price proves to be less than $25, you can exercise the options and sell the shares for $25 each.
Suppose that you write a put contract with a strike price of $40 and an expiration date in three months. The current stock price is $41 and the contract is on 100 shares. What have you committed yourself to? How much could you gain or lose?
You have sold a put option. You have agreed to buy 100 shares for $40 per share if the party on the other side of the contract chooses to exercise the right to sell for this price. The option will be exercised only when the price of stock is below $40. Suppose, for example, that the option is exercised when the price is $30. You have to buy at $40 shares that are worth $30; you lose $10 per share, or $1,000 in total. If the option is exercised when the price is $20, you lose $20 per share, or $2,000 in total. The worst that can happen is that the price of the stock declines to almost zero during the three-month period. This highly unlikely event would cost you $4,000. In return for the possible future losses, you receive the price of the option from the purchaser.
A trader has a put option contract to sell 100 shares of a stock for a strike price of $60. What is the effect on the terms of the contract of: a) A $2 dividend being declared b) A $2 dividend being paid c) A 5-for-2 stock split d) A 5% stock dividend being paid.
a) No effect b) No effect c) The put option contract gives the right to sell 250 shares for $24 each d) The put option contract gives the right to sell 105 shares for 60/1.05 = $57.14
A trader writes five naked put option contracts, with each contract being on 100 shares. The option price is $10, the time to maturity is six months, and the strike price is $64. a) What is the margin requirement if the stock price is $58, as described in the Hull text? b) How would the answer to (a) change if the stock price were $70? c) How would the answer to (a) change if the trader is buying instead of selling the options?
a) The margin requirement is the greater of 500×(10 + 0.2×58) = 10,800 and 500×(10+0.1×64) = 8,200. It is $10,800. b) The margin requirement is the greater of 500×(10+0.2×70-6) = 9,000 and 500×(10+0.1×64) = 8,200. It is $9,000. c) No margin is required if the trader is buying.
Consider an exchange-traded call option contract to buy 500 shares with a strike price of $40 and maturity in four months. Explain how the terms of the option contract change when there is: a) A 10% stock dividend b) A 10% cash dividend c) A 4-for-1 stock split
a) The option contract becomes one to buy 500 × 1.1 = 550 shares with an exercise price of 40/1.1 = 36.36. b) There is no effect. The terms of an options contract are not normally adjusted for cash dividends. c) The option contract becomes one to buy 500 × 4 = 2,000 shares with an exercise price of 40/4 = $10.
