Finance Assignment - Financial Management

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

138

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.

Analytical Techniques

8. Option Pricing Theory

	Microsoft Corp. (MSFT) 30.45 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 0.44 5.50 5.70 85 290 27.50 3.30 0.56 3.20 3.40 1,428 601 30.00 1.40 0.30 1.35 1.40 6,127 1,414 32.50 0.40 0.04 0.40 0.45 7,046 5,515 35.00 0.15 0.05 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 0.10 0.15 0.20 1,069 642 30.00 0.80 0.30 0.75 0.80 7,987 3,294 32.50 2.27 0.45 2.25 2.35 722 127 35.00 4.18 0.12 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 0.51 11.10 11.30 732 68,052 22.50 8.90 0.60 8.80 9.10 97 56,618 25.00 6.90 0.40 6.70 6.90 369 186,080 27.50 4.80 0.40 4.80 5.00 127 113,036 30.00 3.30 0.20 3.20 3.30 1,420 365,211 32.50 2.00 0.11 1.95 2.00 1,358 483 35.00 1.05 0.10 1.00 1.10 2,726 105,582 40.00 0.25 0.05 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 0.03 0.20 0.30 478 86,706 25.00 0.45 0.05 0.45 0.50 467 145,824 27.50 0.90 0.20 0.85 1.00 743 112,323 30.00 1.65 0.25 1.65 1.70 774 84,062 32.50 2.70 0.16 2.85 3.00 93 1,768 35.00 4.70 0.50 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)

· T

and d2 =

ln(S/X) + (r − σ2/2) T

= d1 − σ T (8.5)

· 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

N(d) =

1 d

ex2/2 dx (8.6)



2π

−∞

Example (8.1) gives the Maple code to find the price of a call option using equations (8.3) - (8.6).

It is possible to show that the call price is positively correlated with the asset price, time to maturity, riskless rate, and variability of price returns, but it is negatively correlated to the exercise price. We can express it as

C = f(S +, X −, T +, r +, σ +)

Black-Scholes formula gives the price of a European call option of a non-dividend paying stock or some other asset. It also assumes that people are rational investors, that r and σ remain constant, that there are no taxes or transaction costs, and that the capital markets are efficient. Despite all these restrictions it is a remarkably accurate and practical formula for options valuation.

Fig. 8.1: The value of a call, X = 100, T = .25, r = .05, σ = .3, for varying stock price.

Fig. 8.1 shows the value of a call with time to maturity T = 6 months, exercise price X =

$100, riskless rate r = 6%, volatility σ = .3 as the stock price S changes from $80 to $120.

Hans Stoll (1969) discovered a very important relationship between the value of a call and the value of a put. We can write the relationship, known as the put-call parity theorem, as

P + S = C + X e−rT (8.7)

Substituting the value of C from (8.3), we get

	P + S = S N(d1) − X e−rT N(d2) + X e−rT
Or,	P = S N(d1) − S − X e−rT N(d2) + X e−rT
Or,	P = S [N(d1) – 1] − X e−rT [N(d2) −1]	(8.8)

With the help of (8.8), we can find the value of a European put on a stock.

By judicious use of put or call options, it is possible to manage the risk inherent in the investment process. For example if you own the stock of a corporation, you may wish to sell call options on your stock. In case of a drop in the price of the stock the options will expire worthless. The premium you have already collected on the options will be yours to keep and it will offset some of your loss in the value of the stock. What you have done is to "hedge" your exposure to risk. In fact, it is possible to eliminate risk altogether by setting up a riskless hedge. This can be done as follows.

Suppose you buy h shares of a stock and sell one call option. Here h is unknown but it is the proper number of shares to set up the riskless hedge. The total money invested in the hedge, or the value V of the hedge is

V = h S − C

where S is the price of the stock and C the price of the call option. The value of the hedge should not vary as a result of variation in the price of the stock, and therefore the partial derivative of V with respect to S should be zero.

∂V

∂S = 0

Or,

∂

∂S (h S − C) = 0

∂C ∂C

Or, h − ∂S = 0, or h = ∂S (8.9)

Since C = S N(d1) − X e−rT N(d2) (8.3)

Differentiating the above expression with respect to S gives us, after considerable algebra,

∂C

∂S = N(d1) (8.10)

Comparing (8.9) and (8.10) we note that

h = N(d1) (8.11)

The number of shares of stock, h, that one should buy for each option sold is called the "hedge ratio" and it is just equal to N(d1). Hedging is also used to take advantage of any temporary mispricing of the options. If the call options happen to be selling at a price which is more than their theoretical value, one can sell them and buy an appropriate number of shares. Likewise, if the options are relatively underpriced one can buy them and sell the stock, using the same hedge ratio.

8.3 Options, Stockholders, and Bondholders

Consider a company that is financed partly by stockholders and partly by bondholders. They are all stakeholders in the company. Because of the provisions of the indenture, the bondholders have a stronger claim on the company. If the company is liquidated, the bondholders will get their money first and then the stockholders. In other words, the stockholders will get the leftover amount, after the bondholders are satisfied. This is also the case if the bonds reach maturity and the bondholders are ready to receive the face value of the bonds.

Consider a company with zero-coupon bonds with face value $25 million, which will mature after 10 years. The bondholders are not getting any interest and they will have to wait for 10 years before they receive their share. Suppose the value of the company is V after 10 years, consider the following three possibilities for V and the division of that amount. The amount received by each stakeholder will depend on the final value of the firm.

Firm value

Share of Bondholders

Share of Stockholders

Explanation

V > 25 25 V − 25 Suppose the total value of the firm is $35 million.

Bondholders will receive their share first, which is $25 million, and the stockholders will get the remaining value of the firm, which is $10 million

V = 25 25 0 If the value of the firm is $25 million, the

bondholders will liquidate the firm and get the face value of the bonds, which is also $25 million. The stockholders will get nothing.

V < 25	V	0	Suppose the value of the firm after 10 years is
			only $20 million. The bondholders will not be able to get the face value of their bonds. They will just have to settle for $20 million. Each bond will

be worth only 20/25*1000 = $800. The

bondholders will get 80¢ on the dollar. The stockholders will get nothing.

Now consider a call option and its payoff at maturity. Recall equation (8.2)

CT = max(ST − X, 0) (8.2)

This equation implies that the value of the call, at maturity, is the difference between the stock price and the exercise price provided the stock price is higher than the exercise price, otherwise it is zero.

Comparing the payoff of a call option and the relationship between bondholders and stockholders, we reach a very important conclusion.

The stockholders of a corporation hold a call option on the assets of the firm, with an exercise price equal to the face value of the zero-coupon bonds,

and time to maturity equal to the maturity of the bonds.

Examples 8.8-8.10 illustrate this relationship.

Examples

8.1. Anglia Corporation stock price is $40 a share. The risk-free rate is 6%, and the volatility of the stock, σ is .4. Find the price of a call option that will expire after 6 months, with the exercise price $35. What is the price of the corresponding put option?

First, we write the information in symbolic form as follows: S = 40, X = 35, r = .06, T =

.5, and σ = .4. Substitute these numbers in (8.4)

ln(S/X) + (r + σ2/2) T

which gives

d1 =

(8.4)

· T

d1 =

ln(40/35) + [0.06 + (0.4)2/2] (0.5)

= .7196

0.4 0.5

ln(S/X) + (r − σ2/2) T

Similarly, d2 =

gives

= d1 − σ T (8.5)

· T

d2 =

ln(40/35) + [0.06 − (0.4)2/2] (0.5)

= .4367

0.4 0.5

Draw a normal probability distribution curve, with 0 at the center and stretching up to ∞ on both sides. First take d1 = .7196, which lies to the right of center. N(d1) is defined as the area under the curve from −∞ to d1, which will be somewhat more than .5. To find its value, check the tables for d1 = .7196. This comes out as

Similarly,

N(d1) = .5 + .2611 + .96(.2642 − .2611)] = 0.7641

N(d2) = .5 + .1664 + .67(.1700 − .1664)] = 0.6689

Put this in (8.3),

C = S N(d1) − X e−rT N(d2) (8.3)

which gives C = (40)(.7641) − (35)(e−.06(.5))(.6689) = $7.85 ♥

To find the value of the put option, use (8.8),

P = S [N(d1) – 1] − X e−rT [N(d2) −1] (8.8) This gives P = 40[.7641 – 1] − 35 e−.06(.5) [.6689 −1] = $1.81 ♥

For WolframAlpha, the procedure is as follows.

For Enter Result

0.719592

0.436749

0.764112

0.668853

d1	(LN(40/35)+(.06+.4^2/2)*.5)/(.4*SQRT(.5))
d2	(LN(40/35)+(.06-.4^2/2)*.5)/(.4*SQRT(.5))
N(d1)	1/SQRT(2*Pi)*INTEGRATE[EXP[-x^2/2],{x,-infinity,.719592}]
N(d2)	1/SQRT(2*Pi)*INTEGRATE[EXP[-x^2/2],{x,-infinity,.436749}]
C	40*.764112-35*EXP(-.06*.5)*.668853	7.84649
P	40*(.764112-1)-35*EXP(-.06*.5)*(.668853-1)	1.81208

To do it as a shortcut on WolframAlpha, enter Black Scholes in the input space, then the following information about the option itself.

option name: European

option type: call strike price:

$35

6 mo

$40

40 %

0 %

6 %

time to expiration: underlying price: volatility: dividend yield:

risk‐free interest rate:

To get the value of the put option, change “call” into “put”.

The Maple code for the problem is as follows:

restart;assign(S=40,X=35,T=.5,r=.06,sigma=.4); Nd:=1/sqrt(2*Pi)*int(exp(-x^2/2),x=-infinity..d): d1:=evalf((ln(S/X)+(r+sigma^2/2)*T)/sigma/sqrt(T)); d2:=evalf(d1-sigma*sqrt(T)); Nd1:=evalf(subs(d=d1,Nd)); Nd2:=evalf(subs(d=d2,Nd));

C:=evalf(S*Nd1-X*exp(-r*T)*Nd2); P:=evalf(S*(Nd1-1)-X*exp(-r*T)*(Nd2-1));

The result comes out to be call = $7.85 and put = $1.81. ♥

To do the problem in Excel, proceed as follows.

	A	B	C
1	Stock price, S =	40	dollars
2	Exercise price, X =	35	dollars
3	Riskfree rate, r =	0.06	per year
4	Time to maturity, T =	0.5	year
5	Volatility, σ =	0.4	per √(year)
6	d1 =	=(LN(B1/B2)+(B3+B5^2/2)*B4)/B5/SQRT(B4)
7	d2 =	=B6-B5*SQRT(B4)
8	N(d1) =	=NORMDIST(B6,0,1,true)
9	N(d2) =	=NORMDIST(B7,0,1,true)
10	Call =	=B1*B8-B2*EXP(-B3*B4)*B9	dollars
11	Put =	=B1*(B8-1)-B2*EXP(-B3*B4)*(B9-1)	dollars

One can write Black-Scholes model and its components using WolframAlpha as follows.

For Enter Result

AAA

BBB

FFF

GGG

d1	(LN(S/X)+(r+σ^2/2)*T)/(r*SQRT(T))
d2	(LN(S/X)+(r-σ^2/2)*T)/(r*SQRT(T))
N(d1)	1/SQRT(2*Pi)*INTEGRATE[EXP[-x^2/2],{x,-infinity,AAA}]
N(d2)	1/SQRT(2*Pi)*INTEGRATE[EXP[-x^2/2],{x,-infinity,BBB}]
C	S*FFF-X*EXP(-r*T)*GGG	Call ♥
P	S*(FFF-1)-X*EXP(-r*T)*(GGG-1)	Put ♥

8.2. (A) Uriah Heep has just bought 100 oz of gold at $350 per ounce. He has calculated that the standard deviation of returns in gold investment is 0.243, and that the riskless rate is 11%. He would like to sell call options on gold at an exercise price of $375 per ounce, with a maturity of three months. What is the correct value of these options?

(B) Suppose Uriah was able to sell options on 10 oz of gold. The price of gold at the end of 3 months is $400 per ounce. Now he liquidates all his gold and settles the options, what is his total profit?

(C) Using a discount rate of 15%, find the NPV of this investment.

(A) Here we are given that:

S = current price of the underlying asset = $350

X = exercise or the striking price of the option = $375

r = riskless rate of interest = 0.11 per year

T = time to maturity of the option = 0.25 years

σ = standard deviation of the continuously compounded rate of return from the price fluctuations of the underlying asset = 0.243

The price of the option is calculated by using the Black-Scholes model as shown below: ln(350/375) + [0.11 + 0.5 (0.243)2] (0.25)

d1 =

= − 0.2808

0.243 0.25

d2 = − 0.280755 − 0.243 0.25 = − 0.4023

Draw normal probability distribution curve, with 0 at the center and stretching up to ∞ on both sides. First take d1 = − 0.2808, which will lie slightly left of center. N(d1) is defined as the area under the curve from −∞ to d1, which will be somewhat less than .5. To find its value, check the tables for .2808. This comes out as

N(d1) = .5 – [.1103 + .08(.1141 − .1103)] = 0.3894

Figure 8.2. N(d1) is defined as the area under the normal probability distribution fumction curve from −∞ to

d1. This is the shaded area in the diagram.

Similarly, N(d2) = .5 – [.1554 + .2256(.1591 − .1554)] = 0.3438

C = 350 (0.3894) − 375e−0.11(0.25) (0.3438) = $10.86 ♥

(B) Since Heep sold options on only 10 oz of gold, he was able to sell 90 oz of gold at a profit of $50 per oz. The options ended up in the money, that is, the final price of gold was higher than the exercise price. As a result the buyers of the options exercised their options by forcing Heep to sell the gold to them at the rate of $375 per oz. On these ten ounces of gold he made $25 per ounce besides collecting the $10.86 option premium

calculated in part (A). Let us define profit as the difference between the final payoff and the initial investment, without regard to the risk involved, and without the time value of money. The total profit works out as follows:

Initial investment = cost of buying gold − cash received by selling the options

= 100(350) – 10(10.86) = $34,891.40

Final payoff = money received by selling 90 oz of gold in the open market at $400 an oz

+ money received by selling 10 oz of gold to the option holders at $375 an oz

= 90(400) + 10(375) = $39,750

Profit = 39750 − 34891.40 = $4858.60 ♥

(C) To calculate the NPV, we have to subtract the initial investment from the present value of the future payoff, using a discount rate that includes the risk of the investment. Using the continuously compounded discount rate, as we used it in the calculation of the option price, NPV comes out as

NPV = − 34,891.40 + 39,750e−(.15)(.25) = $3395.58 ♥

Note that NPV is less than the profit and it is a more conservative measure of the profitability of an investment.

8.3. William Horner bought 100 oz of gold at $1663 an oz. Then he sold call options on 25 oz of gold, exercise price $1680, for $100 each; and options on 35 oz of gold, exercise price $1700, for $80 each. The cost of capital for Horner is 9%. All the options will expire after 6 months and then Horner will liquidate his position. Use continuous discounting, to calculate the NPV of this hedge if the price of gold after 6 months is expected to be $1700 an oz.

Initial investment = (value of 100 oz of gold at $1663 per oz)

· (value of 25 options sold, at $100 each, with X = 1680)

· (value of 35 options sold, at $80 each, with X = 1700)

= 100*1663 – 25*100 – 35*80 = $161,000

If the expected final price of gold is $1700, options with X = 1680 will be exercised, and he will deliver 25 oz of gold and receive $1680 per oz. The options with X = 1700 will expire worthless because when the stock price is exactly equal to the exercise price, at expiration, then the value of the option is zero. Therefore, he will sell the remaining 75 oz of gold in the market at $1700 per oz. Thus

Final payoff = money received because some of the options have been exercised + money received by selling the rest of gold in open market = 25*1680 + 75*1700 = $169,500.

To summarize, his initial invest was $161,000 and the final payoff was $169,500. With these two numbers, we can find the following on this investment.

Profit = 169,500 − 161,000 = $8,500

The above value of the profit is misleading because we did not consider the time value of money and we did not take into account the risk involved. To compensate for these factors, we should find the NPV of the investment. We can do it in two ways, in discrete time and in continuous time. The results are as follows.

Discrete time, NPV = – 161,000 + 169,500(1.09)−.5 = $1351.56

With continuous discounting, NPV = – 161,000 + 169,500e−.09*.5 = $1041.57 ♥

8.4. Adam Diller bought 100 shares of Apple stock at $580.32 per share. He also sold 1 call option on the stock, at 41.66, with exercise price 590, and with 132 days till expiration. Using a discount rate of 12%, continuously compounded, find the stock price where Adam will just break even in this investment. Neglect transaction costs.

At the break-even point, the NPV of the investment will be zero. By selling the call, the net cost of stock is reduced by $41.66 per share. The break-even price of the stock should be around 580.32 − 41.66 = $538.66. The buyer of the option will not exercise the option because the final stock price, $538.66 is much less than the exercise price of the call option. Let us find the answer more accurately including the time value of money.

The initial cost of stock = 100(580.22) = $58,022

Cash received by selling the call option = 100(41.66) = $4166 Net cost of this hedge = 58,022 − 4166 = $53,856

Suppose the final stock price is x. This is around $540, as seen by the approximate calculation. The final payoff from selling the stock at x per share will be 100x. Its present value, using 12% continuously compounded discount rate and 132 days to maturity, will be 100xe−.12(132/365). Setting NPV = 0, we get

− 53,856 + 100xe−.12(132/365) = 0

You may solve it at WolframAlpha by using the instruction

-53856+100*x*exp(-.12*132/365)=0

The result is x = $562.45. ♥

8.5. You own 1,000 shares of GM stock which is currently selling for $75 a share, and your estimate of its sigma is 0.225. The riskless rate is 11.2%. What is the price of three month call options at an exercise price of $80? How many call options should you sell in order to set up a perfect hedge?

In this problem, S = 75, X = 80, T = .25, r = .112, σ = .225

ln(75/80) + (.112 + .2252/2)(.25)

d1 =

d2 =

= −.2685368546

.225 .25

ln(75/80) + (.112 − .2252/2)(.25)

= −.3810368545

.225 .25

N(d1) = =NORMDIST(-.2685368546,0,1,TRUE) = .3941430552 N(d2) = =NORMDIST(-.3810368545,0,1,TRUE) = .3515879509

C = 75*.3941430552 – 80*exp(−.112*.25)*(.3515879509) = 2.21032647

The call price is $2.21.

One can set up a hedge by buying the shares of GM and selling call options on them. In order to set up a riskless hedge, one has to buy N(d1) shares of stock and sell one option. With the proper hedge ratio as N(d1) = 0.3941, one should buy 0.3941 shares per call, or 1/0.3941 calls per share. But we already have 1,000 shares, therefore, we have to sell 1,000/0.3941 calls altogether. This works out to be 2537 calls.♥

As the time to maturity changes and the stock price fluctuates, the hedge ratio N(d1) also changes. To keep the hedge riskless, we have to recalculate N(d1) periodically and adjust the hedge accordingly.

8.6. (A) Stanley Corporation stock is currently selling for $76 a share, riskless rate is 12%, and the sigma for Stanley is 0.25. Find the price of a nine-month Stanley call option with an exercise price of $70.

(B) Suppose the options in part (A) are selling for $10 each. Explain how you would set up a hedge to take advantage of the mispricing.

(A) From the option pricing formula we get: d1 = 0.9038, d2 = 0.6873, N(d1) = 0.8169, N(d2) = 0.7540, and call price = $13.85.

(B) Since options are selling for $10 apiece, which is substantially less than their theoretical value of $13.85, you must buy them. To set up the hedge you buy the calls and sell the stock short in the proper hedge ratio of 0.8169. The overall size of this hedging operation depends upon the amount of available funds. For example, you may buy 1,000 options and sell 817 shares short, maintaining the hedge ratio of 0.817. When the price of the options reach an equilibrium, you can take your guaranteed profits. The cost of buying calls is 1,000(10) = $10,000. The proceeds from the short sale of stock is 817(76)

= $62,092. The net proceeds are 62,092 − 10,000 = $52,092. You should invest this amount in riskless government bonds while waiting for the profits to occur.

Here we are assuming that our estimate of the σ of Stanley is absolutely correct and the rest of the market does not know it yet. We also assume that there are no transaction costs, that is, no brokerage commissions. We also assume that there are no restrictions against short selling, and that we are continuously adjusting the hedge ratio while the stock and option prices are changing. In practice this is very difficult to do. ♥

8.7. (A) Denver Corporation stock is currently selling for $100, riskless rate is 12%, and the sigma for Denver is .25. Find the price of a nine month Denver call option with an exercise price of $100.

(B) Suppose the options in the last problem are selling for $15 each. Explain how you would set up a hedge to take advantage of the mispricing.

In part (A) we get: d1 = 0.5239, d2 = 0.3074, N(d1) = 0.6998, N(d2) = 0.6207, and call price = $13.25.

In part (B) we notice that the calls are overpriced at $15 each compared to their theoretical value of $13.25, and we should sell them. Because the hedge ratio N(d1) is roughly 0.7, we should buy 0.7 shares of stock for each option sold. For example we can sell 1,000 options but buy only 700 share of Denver. This will require a cash outlay of 700(100) − 1000(15) = $55,000. If we could borrow that money at a rate equal to the riskless interest rate, and if we wait until the prices regain equilibrium then we should make a profit of 1,000 (15 − 13.25) = $1,750. ♥

8.8. Glenn Corporation has an overall market value of $40 million. The firm has zero- coupon bonds outstanding, maturing in 5 years, with the face value $25 million. The σ for this company is 0.25, and the riskless rate is 8%. Glenn has one million shares of common stock. What is the market value per share of its common stock?

The stockholders of a company hold a call option on the assets of the company after the bondholders are satisfied. The bondholders have a senior claim on the assets of the firm in the case of liquidation of the firm. The stockholders share whatever is left over, after all the other claims are satisfied.

The value of the underlying asset is the current market value of the firm, namely, $40 million. The exercise price is the face value, not the market value, of the zero coupon bonds, $25 million. The value of the option is the total market value of the common stock of the firm. Substituting S = 40, X = 25, T = 5, r = .08, and σ = .25, in the Black-Scholes formula, we find: d1 = 1.8358, d2 = 1.2768, N(d1) = 0.9668, N(d2) = 0.8992, and call price

= 23.60. This means that the value of 1 million shares of common stock is $23,600,000.

The price of the stock per share comes to $23.60. ♥

8.9. Fischer Black is the sole stockholder of Black Belt Co., which has an overall value of $50,000. The company has borrowed some money from an investor, Myron Scholes, and has promised to pay him back the entire amount as a lump sum of $30,000 after 5

years. The σ of Black Belt is 0.25, and the riskless rate is 6%. Find the market value of the holdings of Black and Scholes individually.

Being a stockholder, Black holds an option on the assets of the firm after Scholes has been satisfied. The value of the option can be found by the option pricing formula with S

= 50,000 X = 30,000 which gives C = $28,369. This represents Black's portion of the assets. The total market value of a firm is equal to the market value of its common stock plus the market value of its debt. The value of the debt in this case is thus 50,000 – 28,369 = $21,631. This is the value of Scholes' claims on the company.

If we evaluate his claim as if it were riskless, its value is 30,000 e−0.06(5) = $22,225. Because the debt is not riskless, its value is somewhat less. The difference between

$22,225 and $21,631 is $594, which is about 2.67% of $22,225. The debt is quite safe because there is a good possibility that the $50,000 firm will have a terminal value of

$30,000 after 5 years. ♥

8.10. Carolina Inc has a total value of $5 million. It has zero coupon bonds maturing in 10 years with a face value of $4 million. The riskless rate is 10%, and σ of Carolina is

.25. Using Black-Scholes model, find the market value of a single $1,000 bond.

Use the following in the Black-Scholes formula: S = 5, X = 4, T = 10, r = .1, and σ = .25. This gives us d1 = 1.9425, d2 = 1.1519, N(d1) = .9740, N(d2) = .8753, C = 3.582.

This means that the market value of the stock of Carolina is $3,582,000, and that of the bonds 5,000,000 − 3,582,000 = $1,418,000. Since the face value of the bonds is

$4,000,000, each $1,000-bond is selling for 1,000*(1,418,000/4,000,000) = $354.50 ♥

If the bonds were riskless they would be selling for 1,000 e−0.1(10) = $367.88 each. This agrees with the price of the risky bonds found above, because the riskless bonds are somewhat more valuable.

8.11. Enceladus Corporation has a total value of $5 million. It has $2 million of zero- coupon bonds maturing in 12 years. The sigma of Enceladus is .4 and riskless bonds with 12-year maturity have a yield of 9%. Find the market value of a $1,000 Enceladus bond.

Using Black-Scholes formula with S = 5, X = 2, r = 0.09, σ = 0.4, and T = 12, we get d1 = 2.1335, d2 = 0.74788, N(d1) = 0.98356, N(d2) = 0.77273, and C = 4.393. This means that

the value of equity is $4.393 million, and the value of debt is 5 − 4.393 = $0.607 million.

A thousand dollar bond sells for (1000)(0.607/2) = $303.50. ♥

8.12. Calhoun's Saloon is run jointly by Calhoun and his brother-in-law Breckinridge. Calhoun is the sole stockholder of the company, but the company owes Breckinridge

$10,000 which will be paid as a lump sum after 5 years. Considering the income generated by the business, it is estimated that the value of the business is $20,000. The risk of the business is measured by its σ which is estimated to be 0.5. The riskless rate is

10%. Calhoun wants to pay a fair price to Breckinridge for his loan and thus become the sole proprietor of the business. How much should Calhoun pay Breckinridge now?

Here we have to use the Black-Scholes formula with the following values: S = 20,000, X

= 10,000, T = 5, σ = 0.5, r = 0.1. This gives: d1 = 1.626, d2 = 0.5082, N(d1) = 0.9481,

N(d2) = 0.6943, C = 14,750. In general, the stockholders of a firm hold a call option on

the assets of a firm, with an exercise price equal to the face value of the bonds of the

firm. Calhoun is the stockholder and he holds a call option on the assets of the firm after the bondholder, Breckenridge, is satisfied. Thus the value of Calhoun's investment is

$14,750. The total value of a firm equals the value of stock plus the value of the debt. The present value of Breckinridge's loan, the value of debt, is thus 20,000 − 14,750 =

$5,250. Therefore Calhoun should pay Breckinridge $5,250 to buy him out. ♥