8. OPTION PRICING THEORY
Objectives: After reading this chapter, you will
1. Understand the role of options in financial markets and the terminology used.
2. Calculate the value of an option using the Black-Scholes model.
3. Apply the put-call parity theorem.
4. Use options in portfolio management and the valuation of risky securities.
8.1 Options
Suppose you believe that the price of gold is going to increase in the near future and you want to buy some gold in anticipation of its price rise. However, you do not have enough capital to finance your purchase and you do not want to take the risk of a major loss in the event of a sharp drop in the price of gold. You can overcome both these problems by buying a "call option" on gold. If the gold rises in price you can exercise your option to buy gold at a preset price and resell it in the market for an immediate profit. If the price drops, you have to do nothing, and your loss will be limited to the premium paid for the call option. The call option gives you the right but not the obligation to buy an asset at a previously agreed upon price.
There are several elements in a call option:
1. A call option is a contract between a buyer of the call option and a seller of the option. The buyer and seller enter into the contract by mutual agreement.
2. The buyer of the call pays a certain amount of money to the seller of the call to initiate this contract. This amount is non-refundable, and is called the call price or call premium.
3. This contract gives the buyer of the call the right but not the obligation to buy a certain asset. The asset may be a piece of land, an ounce of gold, or 100 shares of Home Depot stock. The buyer of the call exercises the call option if he buys the assets. Of course, he may not exercise the option at all. If the option is exercised, the seller of the call is obligated to sell the asset. It is an asymmetric contract. The buyer of the call must compensate the seller of the call for this disadvantage by paying a premium for the call, C.
4. There is a strict time limit for this contract, T. When this time has elapsed, the call expires and the contract becomes void.
5. There is a certain exercise price, X, which is the purchase price of the asset. This is the price that buyer of the call option must pay to the seller of the call if he (the buyer of the call) decides to buy the asset by exercising the call during the life of the option.
The buyer of a call will exercise the call only if it gives him some financial advantage. For instance, if the exercise price of a call is $40 and the stock is trading at $43 per share
just before the expiration of the call, then the owner of a call will exercise it and buy the stock by paying only $40 per share for the stock. This gives him an advantage of $3 per share.
The buyer of a call believes that the price of the asset will rise above the exercise price during the life of the contract and that he will be able to buy the asset at less than its market value. In case of a large drop in the value of the asset, his loss is limited to the premium paid for the call.
The seller of the call believes that the price of the asset will remain the same, perhaps drop a little. He expects that the call will not be exercised against him and that he will keep the asset and pocket the premium. When the call expires at time T, which was not exercised, he may want to sell another call.
If you own a call option, you may take any one of these actions:
1. Exercise your call and buy the asset, by paying the exercise price;
2. Sell the call to another investor before expiration, who may be interested in its profit potential; or,
3. Do nothing, and let the option expire. After expiration, the value of a call is zero.
Another example of an option is the ticket to a sports event. If you buy a basketball ticket for $5 from University of Scranton, you can do any of the three things: You can exercise the option by watching the game, or, you can sell the ticket to a friend, or, you may let the option expire by not attending the game. The University keeps the $5 in any case.
When you buy a put option, it gives you the right but not the obligation to sell an asset at a certain exercise price within a given time. The buyer of a put believes that the value of the underlying asset will fall in the near future and that he will be able to sell it at a fixed price by exercising his put and thus make a profit. The seller of a put believes that the value of the asset will actually rise and that he will keep the put premium.
The most important form of puts and calls are those on common stocks. For example you can buy a call option on Boeing stock that will expire after 3 months. These options are traded on well organized options exchanges. One can see real-time option prices on the Internet. A good website for financial information is www.yahoo.com and its financial section.
We make the following observations from the table.
(1) The call price decreases as the exercise price rises, for the same expiration time.
(2) For the same exercise price, the call price rises as the time to expiration increases.
(3) For the same time to maturity, the put price rises as the exercise price increases.
(4) For the same exercise price, the put price increases as the time to maturity increases.
Microsoft Corp. (MSFT) 30.45 ê 0.64 (2.06%)January 25, 2007, at 4:00 PM ET
CALL OPTIONS Expire at close Fri, Mar 16, 2007
Strike Last Chg Bid Ask Vol Open Int
25.00 5.80 ê0.44 5.50 5.70 85 290
27.50 3.30 ê0.56 3.20 3.40 1,428 601
30.00 1.40 ê0.30 1.35 1.40 6,127 1,414
32.50 0.40 ê0.04 0.40 0.45 7,046 5,515
35.00 0.15 é0.05 0.10 0.15 761 53
PUT OPTIONS Expire at close Fri, Mar 16, 2007
Strike Last Chg Bid Ask Vol Open Int
27.50 0.20 é0.10 0.15 0.20 1,069 642
30.00 0.80 é0.30 0.75 0.80 7,987 3,294
32.50 2.27 é0.45 2.25 2.35 722 127
35.00 4.18 ê0.12 4.50 4.60 5 780
37.50 6.64 0.00 6.90 7.10 10 10
CALL OPTIONS Expire at close Fri, Jan 18, 2008
Strike Last Chg Bid Ask Vol Open Int
20.00 11.30 ê0.51 11.10 11.30 732 68,052
22.50 8.90 ê0.60 8.80 9.10 97 56,618
25.00 6.90 ê0.40 6.70 6.90 369 186,080
27.50 4.80 ê0.40 4.80 5.00 127 113,036
30.00 3.30 ê0.20 3.20 3.30 1,420 365,211
32.50 2.00 ê0.11 1.95 2.00 1,358 483
35.00 1.05 ê0.10 1.00 1.10 2,726 105,582
40.00 0.25 ê0.05 0.25 0.30 450 59,188
PUT OPTIONS Expire at close Fri, Jan 18, 2008
Strike Last Chg Bid Ask Vol Open Int
20.00 0.15 0.00 0.10 0.15 4 227,277
22.50 0.23 é0.03 0.20 0.30 478 86,706
25.00 0.45 é0.05 0.45 0.50 467 145,824
27.50 0.90 é0.20 0.85 1.00 743 112,323
30.00 1.65 é0.25 1.65 1.70 774 84,062
32.50 2.70 é0.16 2.85 3.00 93 1,768
35.00 4.70 é0.50 4.60 4.80 261 8,781
40.00 9.08 0.00 9.40 9.60 2 155
Table 8.1: Option data for January 25, 2007
When the call options expire, they are in the money if the stock price is higher than the exercise price. They are at the money if the strike price is just equal to the stock price. They are out of the money, hence worthless, if the stock price is less than the exercise price. An option that is in the money has some value. You can unlock this value by exercising it, and buying the stock at the exercise price, which is less than the current stock price. For instance, at expiration, when the stock is selling at $50 and the exercise price is $45, then the value of a call is just $5. We may generalize this result by the equations
CT = ST − X, if ST X (8.1a)
= 0, if ST ≤ X (8.1b)
where CT = call price at time T, that is, at expiration. Also, ST is the stock price at time T, and X is the exercise price. The may write (8.1a) and (8.1b) as
CT = max(ST − X, 0) (8.2)
Here "max" means the greater of the two quantities in the parenthesis.
Before expiration, the value of a call option is the sum of its intrinsic value and its time value.
Total value of an option = Intrinsic value + Time value
The intrinsic value is the value of the option if it is exercised immediately. If the stock price is less than or equal to the exercise price then you do not want to exercise the option. In that case the intrinsic value is zero.
Consider the Microsoft options of the Table 8.1. The stock is priced at $30.45. The March30 is selling for $1.40. If we buy one of these calls and exercise it immediately, it will give us a benefit of 45¢ per share, because we are able to buy the $30.45 stock for only $30. The intrinsic value of this option is thus 45¢. Subtracting it from the total value of the option, we find the time value of the call to be 1.40 − .45 = $0.95.
Next we consider the January35 call option that is selling for $1.05. Its entire value is its time value, and it has no intrinsic value at all. The time value of an option is always positive and it gradually becomes zero as the time to expiration dissipates.
8.2 Black-Scholes Option Pricing Model
We have already seen that the value of a call depends upon the stock price, the exercise price, and the time to maturity. Its value at maturity is given by (8.2). Calculating its value prior to maturity is a much more difficult problem. Further analysis reveals that it depends upon two more factors, the riskless interest rate r and the volatility of the stock measured by its σ.
Define the following:
S = Market or current price of the underlying asset. This asset could be an ounce of gold, a share of IBM stock, a piece of land, or any other suitable asset. The price of this asset is a stochastic variable: it may go up or down in price in a random manner.
X = Exercise price of the option. This is a fixed price, agreed upon by the buyer and the seller, at which the option holder has a right to buy the asset. The exercise price of the option can be above or below the market value of the asset.
T = The time period during which the option is viable. An "American" option can be exercised at any time during this period whereas a "European" option can be exercised only at the end of this period. The life of an option can be anywhere from one day to several years.
σ = The standard deviation of the continuously compounded rate of return due to price changes of the underlying asset. This is the volatility of the asset. As noted earlier, the price S of the asset is a variable. If it is changing rapidly and by large amounts then σ is large. If the price of an asset is not changing at all then its sigma is obviously zero.
r = Riskless rate of interest. One can determine this quantity by using the yield of Treasury securities.
C = Price of a call option prior to maturity.
Fischer Black 1938-1995
Myron Scholes 1941-
Robert Merton 1944-
The relationship between the call price of an option and the other five parameters was first discovered by Fischer Black and Myron Scholes in 1973, and independently by Robert Merton. This remarkable result can be expressed as
C = S N(d1) − X e−rT N(d2) (8.3)
where d1 =
ln(S/X) + (r + σ2/2) T
(8.4)
o T
and d2 =
ln(S/X) + (r − σ2/2) T
= d1 − σ T (8.5)
o T
and N(d) is the cumulative normal density function, which is equal to the area under the normal probability distribution curve from minus infinity to the point d. The table at the end of this book give the numerical values to find out N(d). We may also express N(d) as a definite integral as