MBA Finance

Chapter

Tool Kit			Chapter 8				11/20/18
Financial Options and Applications in Corporate Finance
8-1 Overview of Financial Options
An option is a contract which gives its holder the right to buy (or sell) an asset at a predetermined price within a specified period of time. Option contracts, though often quoted in terms of single shares, usually are contracts for a 100 shares. A call option describes a situation in which one investor may sell to someone the right to buy his/her shares of a stock over some interval of time. In this scenario, the writer of the call option (the party that surrenders the right to exercise) is said to hold a short position on the option. Meanwhile, the party that has purchased this right to buy is said to hold a long position on the option. The predetermined price that the stock may be purchased for is called the strike, or exercise, price.
When an investor "writes" call options against stock held in his/her portfolio, this is called a "covered call". When the call options are written without the stock to back them up, they are they are called "naked calls". When the strike price is below the current market price, the call option is said to be "in-the-money". Likewise, when the strike price exceeds the current market price, the call option is said to be "out-of-the-money". For instance, if you believed that the price of stock was primed to rise, a call option would allow you to capture a profit off of the rise in price.
A put option allows you to buy the right to sell a stock at a specified price within some future period. If you happened to believe that the price of a stock was ready to fall, a put option would allow you to turn a profit out of that decline. In the cases of both call and put options, the profit or loss made on an options transaction is determined by the value of the underlying asset, the strike price of the option, and the price of the option.
FOR A CALL, AT EXPIRATION
If the value of the underlying asset exceeds the strike price, the profit/loss from the call transaction would be equal to the difference between the value of the asset and the strike price less the price of the call. In this case there could be either a net profit or loss depending upon the exercise value and the price of the call.
If the value of the underlying asset equals the strike price, the profit/loss from the call transaction would be equal to the price of the call, because whether exercised or unexercised the call value would be zero. In this case there is a loss equal to the price of the call.
If the value of the underlying asset is less than that of the strike price, the profit/loss from the call transaction would be equal to the price of the call, because the option would not be exercised if the strike price was greater than the market price. In this case there is a loss equal to the price of the call.
FOR A PUT, AT EXPIRATION
If the value of the underlying asset is less than the strike price, the profit/loss from the put transaction would be equal to the difference between the strike price and value of the asset less the price of the put. In this case there could be either a profit or loss depending upon the exercise value and the price of the put.
If the value of the underlying asset equals that of the strike price, the profit/loss from the put transaction would be equal to the price of the call, because whether exercised or unexercised the put value would be zero. In this case there is a loss equal to the price of the put.
If the value of the underlying asset exceeds that of the strike price, the profit/loss from the put transaction would be equal to the price of the put, because the option would not be exercised if the market price was greater than the strike price. In this case there is a loss equal to the price of the put.
Table 8-1
January 8, 2019, Listed Options Quotations
		CALLS—LAST QUOTE			PUTS—LAST QUOTE
Closing Price	Strike Price	February	March	May	February	March	May
General Computer Corporation (GCC)
53.50	50	4.25	4.75	5.50	0.65	1.40	2.20
53.50	55	1.30	2.05	3.15	2.65	r	4.50
53.50	60	0.30	0.70	1.50	6.65	r	8.00
U.S. Biotec
56.65	55	5.25	6.10	8.00	2.25	3.75	r
Food World
56.65	55	3.50	4.10	r	0.70	r	r
Note: r means not traded on this date.
Suppose you purchase GCC's May call option with a strike price of $50 and the stock price goes to $60. What is the rate of return on the stock? What is the rate of return on the option?
Stock Return
Intital stock price		$53.50
Final stock price		$60.00
Rate of return on stock		12.1%
Call Option Return
Intital cost of option		$5.50
Market price of stock		$60.00
Strike price		$50.00
Profit from exercise		$10.00
Rate of Return		81.8%
Suppose you purchase GCC's May put option with a strike price of $50 and the stock price goes to $45. What is the rate of return on the stock? What is the rate of return on the option?
Stock Return
Intital stock price		$53.50
Final stock price		$45.00
Rate of return on stock		-15.9%
Put Option Return
Intital cost of option		$2.20
Market price of stock		$45.00
Strike price		$50.00
Profit from exercise		$5.00
Rate of Return		127.3%
What is the exercise value of GCC's May call option with a strike price of $50? What is the exercise value of GCC's May call option with a strike price of $55?
Exercise Value
Stock price		$53.50
Strike price		$50.00
Exercise value		$3.50
Stock price		$53.50
Strike price		$55.00
Exercise value		$0.00
8-2 The Single-Period Binomial Option Pricing Approach
Consider a call option on a stock. The stock's current price, denoted by P, is $40 and the strike price, denoted by X, is $35. The option expires in 6 months. The nominal annual risk-free rate is 8%.
PAYOFFS IN A SINGLE-PERIOD BINOMIAL MODEL
At expiration, the stock can take on only one of two possible values. It can either go up in price by a factor of 1.25, or down in price by a factor of 0.80.
Inputs:					Key output:
		Current stock price, P =	$40.00		VC =	$7.71
		Nominal annual risk-free rate, rRF =	8%
		Strike price, X =	$35.00
		Up factor for stock price, u =	1.25
		Down factor for stock price, d =	0.80
		Years to expiration, t =	0.50
		Number of periods until expiration, n =	1 Mike Ehrhardt: Do not change this input.
Consider the value of the stock and the payoff of the option.
Figure 8-1
Binomial Payoffs
		Strike price: X =	$35.00
		Current stock price: P =	$40.00
		Up factor for stock price: u =	1.25
		Down factor for stock price: d =	0.80
					Cu,
		Ending up			ending up
		stock price			option payoff
		P (u) =		MAX[P(u) − X, 0] =
		=A146*D135 =	$50.00	=MAX[D141− D133,0] =		$15.00
P,			VC,
current			current
stock price			option price
$40			?
					Cd,
		Ending down			ending down
		stock price			option payoff
		P (d) =			MAX[P(d) − X, 0] =
		=A146*D136 =	$32.00		=MAX[D151− D133,0] =	$0.00
THE HEDGE PORTFOLIO APPROACH
We can form a portfolio by writing 1 call option and purchasing Ns shares of stock. We want to choose Ns such that the payoff of the portfolio if the stock price goes up is the same as if the stock price goes down. This is a hedge portfolio because it has a riskless payoff.
Step 1. Find the number of shares of stock in the hedge portfolio.
Ns =	Cu - Cd	=	0.83333
	P(u - d)
Step 2. Find the hedge portfolio’s payoff.
If the stock price goes up:
	Portoflio payoff =	Ns (P)(u) - Cu
	=	$26.6667
If the stock price goes down:
	Portoflio payoff =	Ns (P)(d) - Cd
	=	$26.6667
Figure 8-2
The Hedge Portfolio with Riskless Payoffs
				Strike price: X =	$35.00
				Current stock price: P =	$40.00
				Up factor for stock price: u =	1.25
				Down factor for stock price: d =	0.80
				Up option payoff: Cu = MAX[0,P(u)-X] =	$15.00
				Down option payoff: Cd =MAX[0,P(d)-X] =	$0.00
				Number of shares of stock in portfolio: Ns = (Cu - Cd) / P(u-d) =	0.83333
			Stock price = P (u) =	$50.00
P,				Portfolio's stock payoff: = P(u)(Ns) =	$41.67
current				Subtract option's payoff: Cu =	$15.00
stock price				Portfolio's net payoff = P(u)Ns - Cu =	$26.67
$40
			Stock price = P (d) =	$32.00
				Portfolio's stock payoff: = P(d)(Ns) =	$26.67
				Subtract option's payoff: Cd =	$0.00
				Portfolio's net payoff = P(d)Ns - Cd =	$26.67
Step 3. Find the present value of the hedge portfolio's riskless payoff.
The present value of the riskless payoff disounted at the risk-free rate (we assume daily compounding) is:
		Frequency compounded in year:	365
		Nominal risk-free rate, rRF:	8%
		Years to end of binomial period:	0.5
		Payoff of hedged portfolio:	$26.6667
	Inputs to PV function:
		N =	182.5	= t(Frequency)
		I/YR =	0.02192%	= Rrf / Frequency
		PMT =	0
		FV =	-$26.6667	Negative because the PV is the amount we want in exchange for payoff of hedged portfolio
		PV of payoff =	$25.6212	Use the PV function: =PV(I/YR,N,PMT,FV)
Alternatively, you can use the present value equation and adjust for the frequency of compounding:
	Pv of payoff =	Payoff		=	$26.6667	=	$25.6212
		(1 + rRF/365)365*(t/n)			1.04081
Step 4. Find the option's current value.
The current value of the hedge portolio is the the stock value (Ns x P) less the call value (VC). But the hedge portfolio has a riskless payoff, so the hedge portfolio's value must also be equal to the present value of the riskless payoff disounted at the risk-free rate (we assume daily compounding). With a little algebra, we get:
	VC =	Ns (P) - Present value of riskless payoff
	VC =	$7.71
THE REPLICATING PORTFOLIO
If a portfolio can be formed such that is has the same cash flows as an option, the the option value must equal the value of this replicating portfolio. It is possible to replicate an option's cash flows with a portfolio of stock and risk-free bonds, as we show in the next section.
Suppose we form a portfolio with Ns shares of stock (as determined by the formula for the number of shares of stock in the hedge portfolio). How much could we borrow so that the net payoff from the stock and the repayment of the loan (and its interest) has the same payoff as the option?
Inputs:
		Current stock price, P =	$40.00
		Risk-free rate, rRF =	8%
		Strike price, X =	$35.00
		Up factor for stock price, u =	1.25
		Down factor for stock price, d =	0.80
		Years to expiration, t =	0.50
		Number of periods until expiration, n =	1 Mike Ehrhardt: Do not change this input.
Intermediate calculations:
		Up payoff for stock, Pu =	$50.00
		Down payoff for stock, Pd =	$32.00
		Cu =	$15.00
		Cd =	$0.00
	Ns =	Cu - Cd	=	0.8333
		P(u - d)
If we form a portfolio with Ns shares of stock, how much can we afford to borrow so that the portfolio's net payoff is equal to the option's payoff?
			Value of stock in portfolio if up =	Ns P u
			=	$41.67
			Cu =	$15.00
			Amount of borrowing (plus interest) that can be repaid =	$26.6667
			Value of stock in portfolio if down =	Ns P d
			=	$26.67
			Cd =	$0.00
			Amount of borrowing (plus interest) that can be repaid =	$26.6667
Notice that the amount of borrowing (plus interest) that we can afford to repay is the same whether the stock goes up or down. To find the amount we can borrow, we find the present value fo the amount we can repay. Option pricing assumes that interest rates are compounded very frequently. We will assume daily compounding (which is a good approximation for continuous compounding).
	Amount borrowed =	Amount repaid		=	25.6212
		(1 + rRF/365)365*(t/n)
A summary of the replicating portfolio value and payoff's is shown below:
Replicating Portfolio Payoffs
			Number of shares of stock: Ns =	0.8333
			Current stock price: P =	$40.00
			Up factor for stock price: u =	1.2500
			Up stock price: P(u) =	$50.00
			Down factor for stock price: d =	0.8000
			Down stock price: P(d) =	$32.00
			Risk-free rate: rRF =	8.00%
			Years to expiration: t =	0.50
			Number of periods until expiration: n =	1
			Amount of principal and interest repaid =	$26.67
			Amount borrowed =	$25.62
					(Ns) x (Pu) =	$41.67
					Loan repayment =	$26.67
					Net portfolio payoff =	$15.00
	Current value of portfolio:
	(Ns) x (P) =	$33.33
	Amount borrowed =	$25.62
	Total portfolio net cost =	$7.71
					(Ns) x (Pd) =	$26.67
					Loan repayment =	$26.67
					Net portfolio payoff =	$0.00
The call option has the same cash flows as the replicating portfolio, so the call's price must be equal to the value of the replicating portfolio:
		VC = Total portfolio value =	$7.71
8.3 The Single-Period Binomial Option Pricing Formula
The step-by-step hedge portfolio approach works fine, but for problems in which you want to change the inputs, it is easier to use the binomial option pricing formula shown below.
Inputs:
		P =	$40.00
		X =	$35.00
		u =	1.25
		d =	0.80
		Cu =	$15.00
		Cd =	$0.00
		Risk-free rate, rRF =	8%
		Years to expiration, t =	0.50
		Number of periods until expiration, n =	1 Mike Ehrhardt: Do not change this input.
		VC =	$7.7123
The Simplified Binomial Option Pricing Formula
		P =	$40.00
		X =	$35.00
		u =	1.25
		d =	0.80
		Cu =	$15.00
		Cd =	$0.00
		Risk-free rate, rRF =	8%
		Years to expiration, t =	0.50
		Number of periods until expiration, n =	1 Mike Ehrhardt: Do not change this input.
We can simplify the model by defining pu and pd as:
The binomial option pricing model then simplifies to:
	VC =	Cu pu + Cd pd
For Western's 6-month options, we have:
	pu =	0.5142
	pd =	0.4466
We can find the value of Western's 6-month call with a $35 strike price:
VC =	Cu	x	pu	+	Cd	x	pd
VC =	$15.00	x	0.5142	+	$0.00	x	0.4466
VC =	$7.71
Find the value of a 6-month call option with a $30 strike price:
	x =	$30.00
	Cu =	MAX[0,Pu-X] =		$20.00
	Cd =	MAX[0,Pd-X] =		$2.00
VC =	Cu	x	pu	+	Cd	x	pd
VC =	$20.00	x	0.5142	+	$2.00	x	0.4466
VC =	$11.18
In fact, we can use the p's to find the value of any security with payoffs that depend on Western's 6-month stock price.
8-4 The Multi-Period Binomial Option Pricing Model
Suppose we divide the year into two 6-month periods. We will allow the stock to only go up or down each period, but because there are more periods there will be more possible stock prices.
The key is to keep the standard deviation of the stock's return the same as we divide the year into smaller periods. If we know the standard deviation of the stock's return and the number of periods, there is a formula that will show us what u and d must be.
s is the standard deviation of stock return. Here are the formulas relating s to to u and d:
The standard deviation of Western's stock return is shown below. Notice that this provides the values for u and d that we used in the single-period model.
				Multi-period	Single-period
			Annual standard deviation of stock return, σ =	0.315573	0.315573
			Years to expiration, t =	0.5	0.5
			Number of periods prior to expiration, n =	2 Mike Ehrhardt: The number of periods per year may not be changed by the user.	Mike Ehrhardt: Do not change this input.		1 Mike Ehrhardt: The number of periods per year may not be changed by the user.
			Mike Ehrhardt: Do not change this input.	u =	1.1709	1.250
			d =	0.8540	0.8000
Here are the other data for Western, taken from the original problem:
			Current stock price, P =	$40.00
			Risk-free rate, rRF =	8%
			Strike price, X =	$35.00
Because we are going to solve a binomial problem repeatedly, it will be easier if we go ahead and calculate the p's now.
			pu =	0.51400
			pd =	0.46620
Applying these values of u and d to the intital stock price gives the possible stock prices after 3 months. We can then apply u and d to these 3-month values to get the stock values at the end of 6 months, as shown below. Notice that because d = 1/u, the "middle" stock value at the end of the year is the same whether the stock initially went up and then went down, or whether it went down and then went up.
Notice that the range of final outcomes at 6 months is wider than the previous problem. However, the standard deviation of stock returns is the same as before, because most of the time the stock price will end up at the middle outcome rather than at the top or bottom outcomes.
Figure 8-3
The 2-Period Binomial Lattice and Option Valuation
			Standard deviation of stock return: σ =	0.315573
			Current stock price: P =	$40.00
			Up factor for stock price: u =	1.1709
			Down factor for stock price: d =	0.8540
			Strike price: X =	$35.00
			Risk-free rate: rRF =	8.00%
			Years to expiration: t =	0.50
			Number of periods until expiration: n =	2
			Price of $1 payoff if stock goes up: πu =	0.51400
			Price of $1 payoff if stock goes down: πd =	0.46620
Now			3 months			6 months
							Stock = P (u) (u) =	$54.84
						Cuu = Max[P(u)(u) − X, 0]
							Cuu =	$19.84
			Stock = P (u) =	$46.84
			Cu = Cuuπu + Cudπd
			Cu =	$12.53		Stock = P (u) (d) = P (d) (u)
P =	$40.00						Stock =	$40.00
VC = Cuπu + Cdπd						Cud = Cdu = Max[P(u)(d) − X, 0]
VC =	$7.64						Cud =	$5.00
			Stock = P (d) =	$34.16
			Cd = Cduπu + Cddπd
			Cd =	$2.57
							Stock = P (d) (d) =	$29.17
								Cdd = Max[P(d)(d) − X, 0]
							Cdd =	$0.00
To find the current value of the option, we can break the binomial lattice into three problems. Problem #1 is to find the option value at the end of six months, given that the stock moved upward from its initial value. Problem #2 is to find the option value at the end of six months, given that the stock moved downward from its intitial value. Finally, problem #3 is to find the current value of the option, given its two possible values at the end of six months.
In this example, we divided time into two periods. If we were to divide time into more periods, we would get a distribution of stock prices in the last period that would be very realistic, which would give a very accurate option price. It is true that dividing time into more periods would create more binominal problems to solve, but each problem is very easy and computers can solve them very quickly.
8-5 The Black-Scholes Option Pricing Model (OPM)
In deriving this option pricing model, Black and Scholes made the following assumptions:
	1. The stock underlying the call option provides no dividends or other distributions during the
	life of the option.
	2. There are no transaction costs for buying or selling either the stock or the option.
	3. The short-term, risk-free interest rate is known and is constant during the life of the option.
	4. Any purchaser of a security may borrow any fraction of the purchase price at the short-term,
	risk-free interest rate.
	5. Short selling is permitted, and the short seller will receive immediately the full cash proceeds
	of today's price for a security sold short.
	6. The call option can be exercised only on its expiration date.
	7. Trading in all securities takes place continuously, and the stock price moves randomly.
The derivation of the Black-Scholes model rests on the concept of a riskless hedge. By buying shares of a stock and simultaneously selling call options on that stock, an investor can create a risk-free investment position, where gains on the stock are exactly offset by losses on the option. Ultimately, the Black-Scholes model utilizes these three formulas:
	VC =	P[ N (d1) ] - X e-r t [ N (d2) ]			Note: r is the risk free rate, rRF.
	d1 =	{ ln (P/X) + [rRF + σ2 /2) ] t } / (σ t1/2)
	d2 =	d1 - σ (t 1 / 2)
In these equations, V is the value of the option. P is the current price of the stock. N(d1) is the area beneath the standard normal distribution corresponding to the value of d1. X is the strike price. rRF is the risk-free rate. t is the time to maturity. N(d2) is the area beneath the standard normal distribution corresponding to the value of d2. σ is the volatility of the stock price, as measured by the standard deviation. Looking at these equations we see that you must first solve d1 and d2 before you can proceed to value the option.
First, we will lay out the input data given earlier for Western Cellular's call option.
Inputs:			Key Output:
P =	$40		VC =	$7.39
X =	$35
rRF =	8.00%
t =	0.5
s =	0.31557
Now, we will use the formula from above to solve for d1.
	d1	=	0.8892	=(LN(B527/B528)+(B529+((B531^2)/2))*B530)/((B531)*(B530^0.5))
Having solved for d1, we will now use this value to find d2.
	d2	=	0.6661	=D535-(B531*(B530^0.5))
At this point, we have all of the necessary inputs for solving for the value of the call option. We will use the formula for V from above to find the value. The only complication arises when entering N(d1) and N(d2). Remember, these are the areas under the normal distribution. Luckily, Excel is equipped with a function that can determine cumulative probabilities of the standard normal distribution. This function is located in the list of statistical functions, as "NORMSDIST". For both N(d1) and N(d2), we will follow the same procedure of using this function in the value formula. The data entries for N(d1) are shown below.
	N(d1)	=	0.8131
	N(d2)	=	0.7473	=NORMSDIST(D539)
By applying this method for cumulative distributions, we can solve for the option value using the formula above.
	VC	=	$7.39	=(B527*D563)-(B528*EXP(-B529*B530))*D566
EFFECTS OF OPM FACTORS ON THE VALUE OF A CALL OPTION
The figure below shows 3 of Westerns's call options, each with a $35 strike price. One option has 1 year until expiration, 1 has 6 months (0.5 years), and 1 has 3 months (0.25 years).
Figure 5-5									Data for the figure.		Time until expiration
Western Cellular’s Call Options with a Strike Price of $35											1	0.5	0.25
									Stock Price	$5	$9.37	$7.39	$6.20
									$0.00	$0.00	$0.00	$0.00	$0.00
									$2.50	$0.00	$0.00	$0.00	$0.00
									$5.00	$0.00	$0.00	$0.00	$0.00
									$7.50	$0.00	$0.00	$0.00	$0.00
									$10.00	$0.00	$0.00	$0.00	$0.00
									$12.50	$0.00	$0.00	$0.00	$0.00
									$15.00	$0.00	$0.02	$0.00	$0.00
									$17.50	$0.00	$0.07	$0.00	$0.00
									$20.00	$0.00	$0.22	$0.02	$0.00
									$22.50	$0.00	$0.53	$0.09	$0.01
									$25.00	$0.00	$1.05	$0.28	$0.04
									$27.50	$0.00	$1.82	$0.68	$0.18
									$30.00	$0.00	$2.86	$1.37	$0.56
									$32.50	$0.00	$4.16	$2.41	$1.32
									$35.00	$0.00	$5.70	$3.78	$2.54
									$37.50	$2.50	$7.45	$5.46	$4.20
									$40.00	$5.00	$9.37	$7.39	$6.20
									$42.50	$7.50	$11.42	$9.51	$8.44
									$45.00	$10.00	$13.59	$11.76	$10.80
									$47.50	$12.50	$15.85	$14.11	$13.24
									$50.00	$15.00	$18.17	$16.51	$15.71
									$52.50	$17.50	$20.54	$18.95	$18.20
									$55.00	$20.00	$22.94	$21.42	$20.70
									$57.50	$22.50	$25.37	$23.90	$23.19
The figure shows that:
1.	Option prices increase as the stock price increases relative to the strike price.
2.	Option prices increase as time to expiration increases.
3.	Obviousy, an increase in the strike price will cause the option price to fall.
The impact of changes in σ.
We keep all inputs constant except the standard deviation:
	Standard deviation	Call option price
	0.00001	$6.37
	0.10000	$6.38
	0.31557	$7.39
	0.40000	$8.07
	0.60000	$9.87
	0.90000	$12.70
The impact of changes in the risk-free rate.
We keep all inputs constant except the risk-free rate:
	Risk-free rate (rRF)	Call option price
	0%	$6.41
	4%	$6.89
	8%	$7.39
	12%	$7.90
	20%	$8.93
8-6 The Valuation of Put Options
Consider two portfolios. The first has a put option and a share of stock. The second has a call option and cash equal to the present value of the strike price (discounted with continuous compounding; see Chapter 04 Web Extension 04C). What are the payoffs at expiration date T of the two portfolios if the stock price is less than the strike price at expiration? If it is above the strike price at expiration?
	PT<X	PT>=X
Put	X-PT	0
Stock	PT	PT
Portfolio 1:	X	PT
Call	0	PT - X
Cash	X	X
Portfolio 2:	X	PT
As the table shows, the two porfolios have the same payoffs. Therefore, they must have the same value today. This is called put-call parity.
	Put + Stock = Call + PV of strike price
	VP = VC - P + X exp(-rRF t)
Suppose you have the following information. What is the value of the put?
	P =	$40
	X =	$35
	rRF =	8%
	t =	0.50
	V (call price) =	$7.39
Put =	VC	-	P	+	X exp(-rRF t)
=	$7.39	-	$40	+	33.63
=	$1.02
If you do not already have the value of the call option, you can use the following formula to directly calculate the value fo the put.
	Put =	P[ N (d1) - 1 ] - X e-r t [ N (d2) -1 ]			Note: r is the risk free rate, rRF.
The formulas for d1 and d2 are the same as for the Black-Scholes call option model. In fact, the only differences between the two models is that the formula for puts subtracts 1 from N(d1) and N(d2).
	Put =	$1.02

4.8999999999999998E-3 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30 32.5 35 37.5 40 42.5 45 47.5 50 52.5 55 57.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 4.8999999999999998E-3 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30 32.5 35 37.5 40 42.5 45 47.5 50 52.5 55 57.5 1.5150622246796405E-173 8.551037223004708E-17 1.0707391850392869E-9 1.7888330135090295E-6 1.3585249859497206E-4 2.3209063311913428E-3 1.7007818860363233E-2 7.3417460290248315E-2 0.22265267412135037 0.52767094353637489 1.0464025940164552 1.8180793522198648 2.8578489451071842 4.1588837090380917 5.6985569916309835 7.4454485292264891 9.3652391884590784 11.424793940210201 13.594503070659179 15.849283904615337 18.16870054198052 20.536584798365151 22.940430900902154 25.370735065231614 4.8999999999999998E-3 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30 32.5 35 37.5 40 42.5 45 47.5 50 52.5 55 57.5 0 2.034371665111521E-32 2.1793899956117436E-18 4.4422907445989101E-12 1.9320356049049042E-8 4.3342499037726459E-6 1.8125098221685731E-4 2.6564218893350283E-3 1.93802973823062 05E-2 8.7036692966296569E-2 0.27589396880531591 0.67705362154287307 1.3723892350350759 2.4075803490956478 3.7831612533390917 5.4631471584671552 7.3917425251585094 9.5090391158147796 11.761291275124691 14.105483202163523 16.509863251030701 18.952374964508827 21.418401108009341 23.898608547159135 4.8999999999999998E-3 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30 32.5 35 37.5 40 42.5 45 47.5 50 52.5 55 57.5 0 3.1006815734593928E-63 2.4111167130696169E-35 7.2193048440695481E-23 1.0108472623198529E-15 3.8160640339364244E-11 5.0384429453763521E-8 8.1939514959356247E-6 3.3051895956191443E-4 5.0541777747451638E-3 3.8545487259123323E-2 0.17667576564411558 0.55543116117534286 1.3190314214951382 2.5430641729822128 4.2030784897650015 6.2045576034450711 8.435082930148436 10.800892256507687 13.238645143905515 15.711472657439714 18.200209025287549 20.695740198428446 23.194031602069494

Stock Price ($)

$

Exercise Value

VC if t = 1

VC if t = 0.5

VC if t = 0.25

8-1

SECTION 8-1
SOLUTIONS TO SELF-TEST
Brighton Memory's stock is currently trading at $50 a share. A call option on the stock with a $35 strike price currently sells for $21. What is the exercise value of the call option? What is the time value?
Stock price		$50
Strike price		$35
Market price of option		$21
Exercise value of option		$15.00
Time value of option		$6.00

8-2

SECTION 8-2
SOLUTIONS TO SELF-TEST
Lett Incorporated's stock price is now $50 but it is expected to either go up by a factor of 1.5 or down by a factor of 0.7 by the end of the year. There is a call option on Lett's stock with a strike price of $55 and an expiration date one year from now. What are the stock's possible prices at the end of the year? What are the call option's payoffs if the stock price goes up? If the stock price goes down? If we sell one call option, how many shares of Lett's stock must we buy to create a riskless hedged portfolio consisting of the option position and the stock? What is the payoff of this portfolio? If the annual risk free rate is 6 percent, how much is the riskless portfolio worth today (assuming daily compounding)? What is the current value of the call option?
Inputs:
Current stock price		$50
Strike price		$55
u		1.50
d		0.70
Risk-free rate		6%
Time to exercise		1.00
Stock price if u		$75.00
Stock price if d		$35.00
Option payoff if u		$20.00
Option payoff if d		$0.00
N		0.50
Hedge portfolio payoff if u		$17.50
Hedge portfolio payoff if d		$17.50
Portfolio value today (PV of payoff)		$16.48
Current option value		$8.52

8-3

SECTION 8-3
SOLUTIONS TO SELF-TEST
Yegi's Fine Phones has a current stock price of $30. You need to find the value of a call option with a strike price of $32 that expires in 3 months. The annual risk-free rate is 6%. Use the binomial model with 1 period until expiration. The factor for an increase in stock price is u = 1.15; the factor for a downward movement is d = 0.85. What are the possible stock prices at expiration? What are the option's possible payoffs at expiration? What are pu and pd? What is the current value of the option (assume each month is 1/12 of a year)?
Inputs for Problem
Current stock price		$30.00
Strike price		$32.00
u =		1.15
d =		0.85
Time in years to expiration		0.25
Number of periods until expiration		1 Mike Ehrhardt: The number of periods per year may not be changed by the user.
Risk-free rate, rRF =		6%
Binomial lattice of stock prices:
		P(u) =	$34.50
P =	$30.00
		P(d) =	$25.50
Cu =	$2.50
Cd =	$0.00
pu =	0.5422
pd =	0.4429
Current value of option:
VC =	$1.36

8-4

SECTION 8-4
SOLUTIONS TO SELF-TEST
Ringling Cycle’s stock price is now $20. You need to find the value of a call option with an strike price of $22 that expires in 2 months. You want to use the binomial model with 2 periods (each period is a month). Your assistant has calculated that u = 1.1553, d = 0.8656, pu = 0.4838, and pd = 0.5095. Draw the binomial lattice for stock prices. What are the possible prices after 1 month? After 2 months? What are the option's possible payoffs at expiration? What will the option's value be in 1 month if the stock goes up? What will the option's value be in 1 month if the stock price goes down? What is the current value of the option (assume each month is 1/12 of a year)?
Previous work done by your assistant:
		Annual standard deviation of stock return, s =	0.500
		Years to expiration, t =	0.1667
		Number of periods per year, n =	2 Mike Ehrhardt: The number of periods per year may not be changed by the user.
		u =	1.1553
		d =	0.8656
		Risk-free rate, rRF =	8%
		pu =	0.4838
		pd =	0.5095
Inputs for Problem
Current stock price		$20.00
Strike price		$22.00
u =		1.1553
d =		0.8656
Time in years to expiration		0.1667
Number of periods until expiration		2
pu =		0.4838
pd =		0.5095
Binomial lattice of stock prices:
					P(u)(u) =	$26.69
		P(u) =	$23.11
P =	$20.00				P(u)(d) = P(d)(u) =	$20.00
		P(d) =	$17.31
					P(d)(d) =	$14.99
Option payoffs at expiration
Cuu =	$4.69
Cud =	$0.00
Cdd =	$0.00
Value of option in 1 month if stock goes up:
Cu =	$2.27
Value of option in 1 month if stock goes up:
Cd =	$0.00
Current value of option:
VC =	$1.10

8-5

SECTION 8-5
SOLUTIONS TO SELF-TEST
What is the value of a call option with these data: P = $35, X = $25, rRF = 6%, t = 0.5 (6 months), and σ = 0.6?
P	$35
X	$25
rRF	6.0%
t	0.50
σ	0.6
d1	1.076
d2	0.652
N(d1)	0.8590
N(d2)	0.7427
V =	$12.05

8-6

SECTION 8-6
SOLUTIONS TO SELF-TEST
A put option written on the stock of Taylor Enterprises (TE) has an exercise price of $25 and six months remaining until expiration. The risk-free rate is 6 percent. A call option written on TE has the same exercise price and expiration date as the put option. TE's stock price is $35. If the call option has a price of $12.05, what is the price (i.e., value) of the put option?
P =	$35.00
X =	$25.00
rRF =	6.00%
t =	0.50
V (call price) =	$12.05
Put =	$1.31