OPTION CONTRACTS

1. A long straddle consists of a long call and a long put on the same stock and profits from dramatic price movement by the stock. A short straddle involves the sale of a call and a put on the same stock and profits from little or no stock price change. Investors going long would anticipate volatility in excess of that discounted by the options’ prices while investors going short would expect volatility below that already discounted. Since volatility enhances option prices, long straddles would tend to pay higher premiums for more volatile options, whereas short straddles must accept lower premiums for less volatile options.

2. A range forward is actually an option strategy that combines a long call and a short put (or vice versa) through a costless transaction. Because the options will not have the same striking price, the combination is classified as a range forward as opposed to an actual forward, created by combining long and short options with the same striking price. It is fair to view actual forwards as a special case (or zero-cost version) of range forwards.

3. CFA Examination II (1993)

Call options give the owner the right, but not the obligation, to purchase SFr’s for a pre-specified amount of domestic currency. Purchasing an at-the-money call option would guarantee the current exchange rate over the life of the option. If the SFr declines in value, the call will not be exercised since francs can be purchased more cheaply in the open market and redeeming the bond issue will be less costly.

Contrasting characteristics: (1) Currency options are traded worldwide and enjoy a liquid market. (2) Exchange-traded currency option contracts have standard amounts, maturities, etc. (3) Over-the-counter options could be tailored to meet Michelle’s needs. (4) The initial cash outflow would be the premium. (5) The use of options preserves the ability to profit. (6) No counterparty credit risk. (7) Must roll to match year obligation.

Currency forward contracts commit the seller to deliver the specified amount of currency to the buyer on a specified future date at a fixed price. A short position in a forward contract requires delivery.

Contrasting characteristics: (1) The market for forward contracts is over-the-counter and sometimes may not be as liquid as option or futures market. (2) Forward contracts may be custom designed for specific applications. (3) Cash does not change hands until a forward contract is settled. (4) Counterparty credit risk. (5) Can best match 5-year obligation.

Currency futures are like forward contracts except the gain or loss on the contract is settled daily under the supervision of an organized exchange. A short position in the futures requires either offset or delivery at expiration.

Contrasting characteristics: (1) Futures are traded in standardized contracts and are highly liquid. (2) Cash is required for daily settlement. (3) A margin account is required. (4) Management and administration costs are higher than with a forward or option contract. (5) No counterparty credit risk. (6) Must roll to match year obligation.

4. CFA Examination III (1991)

The other three factors affecting the value of call options and the ways that changes in them affect value are:

(1). Increases in underlying stock volatility. A call cannot be worth less than zero no matter how far the stock price falls, but rising stock prices can increase the call’s value without limit. Therefore, the wider the range within which a stock’s price can fluctuate (i.e. the greater its volatility), the greater the chance that the option will expire in-the-money, the higher the expected payoff from owning it, and the higher its value. A wider range of probable future prices on the underlying stock increases the probability of higher payoffs in general but because the call’s value cannot decline below zero does not symmetrically increase the probability of lower payoffs.

(2). The risk-free interest rate. Call value increase with increases in interest rates (given constant stock prices) because higher interest rates make the ownership of call options more attractive. The call owner does not pay for the stock until the option is exercised; its owner can, therefore, take advantage of the time value of money by investing free interest rate increases the time value benefit to the call owner, increasing the value of the call option.

(3). The exercise price of the option. Call values decrease with increases in the exercise price. When a call option is exercised, the payoff is the difference between the stock price at the time of exercise and the exercise (or strike) price. A higher exercise price decreases the expected payoff from the call, thus reducing the option’s value.

5. Put-call parity indicates that a long position in a stock combined with being short a call and long a put (with the same strike price) is a risk-free investment. In other words, no matter what the stock price at expiration, the payoff will be the same. Consequently, any investment in this portfolio should earn the risk-free return. The three-step process for valuing options is to

(1). Determine a distribution of future stock prices,

(2). Calculate the cash flows from the option at the future prices, and

(3). Discount these expected cash flows to the present at the risk-free rate.

It is this final step that is relevant. Cash flows can be discounted at the risk-free rate because of the riskless replicating portfolio strategy.

6. The Black-Scholes model is derived by showing how a portfolio of the underlying asset and risk-free bonds can be created that exactly mimics the price of an option. This involves taking a long position in the underlying asset to replicate a call option. For currency options, the underlying asset can be thought of as risk-free deposits (bonds) in the foreign currency. So, just as stocks pay dividends, foreign deposits pay interest. Therefore, we can just substitute the foreign risk-free rate for the dividend yield when pricing options on currency.

7. If there are no transaction costs, it is only rational to exercise a call option early, immediately before a dividend payment. This is because it will always be more profitable to sell the option and buy the stock in the open market rather than exercise the option, unless there is a dividend. When a firm pays a dividend, the price of the stock usually declines by approximately the amount of the dividend. This has two impacts on the option holder. First, the decrease in the stock price decreases the value of his option. Second, the investor does not get the dividend payment unless he actually has exercised the option and owns the stock. Consequently, if the value of the dividend is greater than C+K-S, it will be beneficial to exercise early. Note, this is most likely to happen for every in-the-money options.

For put options, it may be the case that the interest that could be earned on the proceeds from early exercise is greater than the intrinsic option value (P+S-K), so early proceeds from early exercise is optimal. Using similar logic as above, puts should always be exercised immediately after a dividend (on a dividend paying stock) because the stock’s price declines after a dividend, thus making the put more valuable.

8. In the Black-Scholes model, the expected future value of a stock is a function of the risk-free interest rate and the dividend yield. As long as the risk-free rate is greater than the dividend yield, the future expected value will be greater than today’s price. The longer the time period, the higher the expected price. So, as time to expiration increases, there are two opposing forces on the value of a European put. First, the increased time to expiration increases the chances of the option being more in-the-money. This increases put value. Second, the higher expected price at expiration decreases the expected value of the put’s payoff at expiration and, therefore, decreases the put value. Depending on which of these two effects is larger, the put may increase or decrease in price with an increase in time to expiration.

For a European call option, these two effects work in the same direction, since an increase in expected future price increases the value of a call. Hence, an increase in the time to expiration always increases the value of a European call.

9. Since the price of an option is positively related to volatility, “buying low vol and selling high vol” is the same as the idea of “buy low, sell high” for any risky asset if the other parameters that affect option prices are fixed. The other factors that affect option prices can be effectively neutralized by holding the appropriate portfolio of options and the underlying asset (beyond the scope of this question). If this is the case, then the risky asset is volatile itself, not the underlying foreign currency.

10. A decrease in security volatility will cause an increase in both the call and put option values. For example, when the volatility of the underlying asset’s price decreases, the call option becomes less valuable since this decreases the probability that the option will be deeper in the money at expiration (a similar scenario is also true for the put option).

11. On October 19, 1987, implied volatilities sky-rocketed. The jump in implied volatility increased the value of call options more than enough to offset the negative impact of the in the index level.

. CHAPTER 23

Answers to Problems

1. CFA Examination III (1987)

1(a). Cu – Cd 20 – 0
Hedge ratio = = = 0.5
uS – dS 120 – 80

(1 + r) – d (1 + 0.1) – 0.0
Implied probability (p) = = = 0.75
u – d 1.2 – 0.8

pCu + (1 – p)Cd .75(20)+.25(0)
Call value = = = 13.6
1 + r 1 + .01
OR

Step 1

Set up binomial tree and calculate the option values at expiration for each ending stock price.

Step 2

Solve for the amount to invest in the stock and the amount to borrow in order to replicate the option given its value in the up state and its value in the down state. Solve these equations simultaneously.

Step 3

Use the values derived in Step 2 to solve for the value of the option at the beginning of the period.

OR

Step 1
120
(C=20)

100

80
(C=0)

Step 2

20 = 120 x D – 1.1 x B
0 = 80 x D – 1.1 x B

-1.1(20) – (-1.1)(0) -22
D = = = 0.5
120(-1.1) – 80(-1.1) -44

C = 80(.5) – 1.1 x B
0 = 80(.5) – 1.1 x B

B = 40/1.1 36.4

Step 3

C = 50 – 36.4 = 13.6

1(b). The binomial option pricing model is a discrete version of the continuous time Black-Scholes option pricing model. As the number of intervals in the binomial model approaches infinity, the option value derived from this model approaches the option value derived from the Black-Scholes model.

The binomial model is more flexible than the Black-Scholes model because it does not require one to assume constant interest rates and constant variance throughout the horizon. These values can be changed at any of the nodes in the binomial tree. However, the binomial model is more cumbersome to use since accuracy requires that the tree include many nodes.

2. CFA Examination III (1992)

2(a). An individual put will hedge an amount of the underlying stock index equal to the underlying value of the put, which in each case here is the respective stock index times $100.

Also, due to the differing betas, the portfolio and the stock indexes are expected to produce gains or losses in proportion to their respective betas. Thus the number of puts required to hedge the portfolio must be adjusted for the betas of the respective stock indexes relative to the beta of the portfolio.

The number and cost of protective puts could be calculated indirectly or directly.

Indirect method:

(port)(Vport ) = (GAC)(VGAC) = (opt)(Vopt)

The indirect method is based on the fact that an overall portfolio’s beta is equal to the weighted average of its component betas. Since we are seeking a zero beta portfolio, the left-hand side of the equation can be set to zero.

For the S&P 100:
0 = -(1.05)(7,761,700) = (0.95)(Vopt)
Vopt = -(1.05/0.95)(7,761,700)
Vopt = -8,578,721
Number of puts = 8,578,721/(365 x 100) = 235
Cost of puts = 235 x $10.25 x 100 = $240,875
For the S&P 500:
0 = -(1.05)(7,761,700) = (1.00)(Vopt)
Vopt = -(1.05/1.00)(7,761,700)
Vopt = -8,149,785
Number of puts = 8,149,785/(390 x 100) = 209
Cost of puts = 209 x $11.00 x 100 = $229,900
For the NYSE:
0 = (1.05)(7,761,700) = (1.03)(Vopt)
Vopt = -(1.05/1.03)(7,761,700)
Vopt = -7,912,413
Number of puts = 7,912,413/(215 x 100) = 368
Cost of puts = 368 x $6.25 x 100 = $230,000
Direct method:
The direct method calculates the number of puts by utilizing the hedge ratio (HR).

HR = volatility of hedged security / volatility of hedging instrument

Beta, which measures the movement in a security relative to the S&P 500, serves as a measure of volatility.

For the S&P 100:

Number of puts = (1.05/0.95)(7,761,700)/(365 x 100) = 235

Cost of puts = 235 x $10.25 x 100 = $240,875

For the S&P 500:

Number of puts = (1.05/1.00)(7,761,700)/(390 x 100) = 209

Cost of puts = 209 x $11.00 x 100 = $229,900

For the NYSE:

Number of puts = (1.05/1.03)(7,761,700)/(215 x 100) = 368

Cost of puts = 368 x $6.25 x 100 = $230,000

2(b). The cost of the S&P 500 and NYSE puts are essentially the same and less than the cost of the S&P 100 puts; other relevant factors in the decision are correlation and liquidity.

While the hedge calculated in Part A is intended to protect GAC’s portfolio from a decline, the portfolio does not replicate any of the indices. The hedge could be less than perfect if the price movement of GAC’s portfolio does not track the index movement. In this context, the index which has the highest correlation with GAC’s portfolio (the S&P 500, 0.95) would be most preferred, while the index exhibiting the lowest correlation with GAC (the S&P 100, 0.86) would be least preferred.

The hedge should be implemented with a minimum of market impact thus another consideration is liquidity. If the hedge transaction represents a disproportionate share of average daily trading volume, the market could move adversely, increasing the cost of the hedge. The required put position represent 2.4% (235/10,000) of average daily trading volume for the S&P 100, 5.2% (209/4,000) for the S&P 500, and 36.8% (368/1,000) for the NYSE. Thus based on trading volume and likely market impact, the S&P 100 would be most preferred, with the S&P 500 also acceptable; the NYSE transaction would clearly be least desirable.

The S&P 500 options would appear to be the preferred protective position, offering the best combination of cost, hedging ability (correlation), and liquidity; the S&P 100 is a relatively costly position and the NYS position is relatively illiquid.

2(c). The scenario in Part A used options with strike prices equal to the current index value (i.e. “at the money”). Such options have no intrinsic value (the difference between the index value and the strike price) and contain only a time premium. Buying the put at-the-money will enable a hedge of the portfolio with no loss from current levels assuming that the betas used in the GAC hedge are stationary; then decreases in the portfolio value will be offset exactly by increases in the put option value.

As the relationship between the strike price and the index value changes, however, a trade off enters into the investment decision. If the strike price is below the current index value, then the puts are “out of the money.” These options will be less expensive than “at the money” or “in the money” puts; however they expose the portfolio to potential loss, with the difference between the index value and strike price being analogous to the deductible on an insurance policy. For example, if the S&P 500 index is at 400 and the strike price at 390, then GAC’s portfolio would be unprotected for the 10 point difference. The portfolio value will decline by 2.5 (10/400) if the index drops to 390 or lower. However, an index value between 390 and 400 at option expiration will not provide any offset to a portfolio decline. The portfolio would have incurred the cost of the option, only to have it expire worthless at expiration.

If the strike price is above the current index value, then the puts are “in the money.” These puts will be most expensive, since they include both intrinsic value and a time premium. Buying “in the money” puts is analogous to selling the portfolio at the strike price and owning a call option. If the S&P 500 index is 390 and the strike price is 400, then the portfolio is assured of a 2.6% (10/390) gain (ignoring the cost of the option).

In summary, the protective put strategy will provide a floor value for the portfolio. However, the various index options will each have strike prices different from their respective current index values. Thus the portfolio manager must weigh the trade off between the cost of the put option (the higher the strike price, the higher the put premium) and the degree of protection sought (level of assured portfolio value).

2(d). An option pricing model, such as the Black-Scholes or binomial approaches, can be used to establish the fair value of an option and to facilitate value comparisons across different strike prices, different times to expiration, and different stock index options. To the extent that mispricings can be identified, arbitrage opportunities may be exploited.

The Black-Scholes formula utilizes the current index price and strike price, in conjunction with the risk-free interest rate, the time to expiration, and return volatility, to establish a fair option price. Alternatively, prevailing option prices may used in the model to calculate implied volatility and to facilitate volatility comparisons across strike prices and times to expiration; this also enables assessment of option premiums against implied risk levels. The model can also be used to calculate hedge ratios, the change in portfolio value for a $1 change in index price.

In summary, option pricing models allow the portfolio manager to quantify the relative pricing of options to determine which option gives the most efficient portfolio protection per dollar of cost.

3. CFA Examination II (1998)

3(a). Critique of Belief
Joel Franklin’s belief is incorrect. There are two fundamental kinds of options: American style and European style. An American option permits the owner to exercise the option at
any time before or at expiration. The owner of a European option may exercise it only at expiration. If an option is at expiration, it will have the same value whether it is American or European.

The owner of an American option can treat the option as a European option simply by postponing the decision to exercise until expiration. Therefore, the American option cannot be worth less than the European option. However, the American option can be worth more. The American option will be worth more if circumstances make exercise of the option before its expiration desirable. So, it may have a higher premium.

3(b). European-Style Option’s Value
The formula to calculate a call option using put-call parity is c = S + p – Xe-rt

where c = the price of a European call option at time t
S = the price of the underlying stock at time t
p = the price of a European put at time t
X = the exercise price for the option
t = time to expiration = one year

Therefore, from the information given,

Call option = $4.408 = $43 + 4.00 – 45.00e-.055

3(c). Effect of Variables
Effect on Call Option’s Value
i. An increase in short-term interest rate Positive
ii. An increase in stock price volatility Positive
iii. A decrease in time to option expiration Negative

4(a). Calculate the following parameters, option values, and hedge ratios at each node:

U = 42/40=1.05
D = 38.4/40=.96
R = (1.06)1/3 = 1.01961
rf = 1.961% per period
Pu = (R D)/(U D) = (1.01961 .96)/(1.05 .96) = .05961/.09 = .662

4(a)(i). S = 40.32

If the stock moves up the option will be worth $4.34 (Cudu); if the stock moves down the option will be worth $.71 (Cudd). The value of the option and hedge ratio at this node is

Cud = .662(4.34) + (1 .662)(.71) = 2.501/1.01961 = 2.45
1.01961

HRud = 4.34 – .72 = 1.00
(1.05 – .96) x 40.32

4(a)(ii). S = 42

If the stock moves up the option will be worth $6.62 (Cuu); if the stock moves down the option will be worth $2.86 (Cud). The value of the option and hedge ratio at this node is

Cu = .662(6.62) + (1 .662)(2.86) = 5.34912/1.01961 = 5.246
1.01961

HRu = 6.62 – 2.86 = 1.00
(1.05 – .96) x 42.00
Note:
Cuu = .662(8.31) + (1 .662)(4.34) = 6.96814/1.01961 = 6.834
1.01961

4(a)(iii). S = 40

If the stock moves up the option will be worth $5.05 (Cuu); if the stock moves down the option will be worth $1.87. The value of the option and hedge ratio at this node is

C = .662(5.05) + (1 .662)(1.87) = 3.97516/1.01961 = 3.8987
1.01961

HRu = 5.05 – 1.87 = 0.8833
(1.05 – .96) x 40.00
Note:
Cd = (.662)2(4.34) + (2)(1 .662)(0.71) = 2.38194/1.0396 = 2.29
(1.01961)2

So the riskless portfolio will initially contain 1 call option and be short 0.8833 shares of stock. Since after an initial up move to 42.00 the option can only finish in-the-money, the hedge ratio is 1.00.

4(b). Ending Price Number of Paths Path Probability Total Probability
$46.31 1 .60753=.2242 .2242
$42.34 3 .60752 x .39251 = .1449 .4346
$38.71 3 .60751 x .39252 =.1449 .2808
$35.39 1 .39253=.0604 .0604

4(c).
C = .6623 (8.31) + (3)(.6622)(1 .662)(4.34) +(3)(.662)(1 .662)2 (.71) = 4.50/1.06 = 4.24
1.06

4(d). P = (1 .662)3 (2.61) = 0.1008/1.06 = .095
1.06

Check with put call parity. Does the following hold true:

C – P = S – PV(K)
4.24 – .095 = 40 – 38/1.06
4.15 = 4.15
Put-call parity does hold exactly.

5. CFA Examination II (2000)

5(a). Over two periods, the stock price must follow one of four patterns: up-up, up-down, down-down, or down-up. To construct a 2-period price lattice for a two-year option, each period consists of 365 days for a total of 730 days.

U = 1.2214

D=1/U D = 1/1.2214 D = 0.8187

The binomial parameters are:

U = 1 + percentage increase in a period if the stock price rises 1.2214
D = 1 + percentage decrease in a period if the stock price falls 0.8187
R = 1 + Risk-free rate = 1.06184

$74.59

$61.07
$50.00
$50.00
$50.00

$40.94

$33.52

Period 1 Period 2 Period 3

Period 1:
Up: $50 x 1.2214 = $61.07
Down: $50 x 0.8187 = $40.94

Period 2:
Up: $61.07 x 1.2214 = $74.59
Down: $61.07 x 0.8187 = $50.00
Up: $40.94 x 1.2214 = $50.00
Down: $40.94 x 0.8187 = $33.52

5(b). For each terminal stock price, a call option has a specific value.Because the company does not pay a dividend,  = 0. With a stock paying a dividend, the dividend yield would be subtracted from the risk-free rate.

The probability of an upward stock price movement in any period is:

0.6038 = (1.06184 – 0.08287)/(1.2214 – 0.08187)

The probability of a downward stock price movement in any period is:

1 – 0.6038 = 0.3962

The vaue of a call option at expiration must be :

Max(0, S – X)

Where:
S = current price of stock index
X= exercise price of the call option

$14.59

$8.30
$0.00
$4.72
$0.00

$0.00

$0.00

Period 0 Period 1 Period 2

Period 2:
Up-up: max(0, S-X) = max(0, $74.59 – $60.00) = max(0, $14.59) = $14.59
Up-down: max(0, S-X) = max(0, $50.00 – $60.00) = max(0, -$10.00) = $0.00
Down-up: max(0, S-X) = max(0, $50.00 – $60.00) = max(0, -$10.00) = $0.00
Down-down: max(0, S-X) = max(0, $33.52 – $60.00) = max(0, -$26.48) = $0.00

Period 1:
$14.59 x 0.6038 (probability of an upward stock movement) = $8.81
$ 0.00 x 0.3962 (probability of a downward stock movement) = $0.00
$ 0.00 x 0.6038 (probability of an upward stock movement) = $0.00
$ 0.00 x 0.3962 (probability of a downward stock movement) = $0.00

Period 0:
$8.30 x 0.6038 (probability of an upward stock movement) = $5.01
$0.00 x 0.3962 (probability ofa downward stock movement) = $0.00

With a discount factor of 0.94176, the value of the call in period 0 is:

($5.01 x 0.94176) + ($0.00 x 0.94176) = $4.72

Alternatively:

C = (UUCUU +PUDCUD + DUCDU + DDCDD)/R2

C = (0.6038 x 0.6038 x 14.59 +0.6038 x 0.3962 x 0.00 +0.3962 x 0.6038 x 0.00
+ 0.3962 x 0.3962 x 0.00)/(1.06184)2

C = $4.72

5(c). The put-call parity formula is:

pt = ct – St + Xe-r (T-t)

where:
c = call option price = $4.72
S = current stock or index pice = $50.00
X = exercise price = $60.00
e-r (T-t) = discount factor = 0.88692
T = time to expiration in periods = 2
t = current time = 0
r = risk-free rate = 0.06

p = $4.72 – $50.00 + ($60 x 0.88692)

p = $7.94

6(a). u = 33.00/30.00 = 36.00/33.00 = 1.10 d = 27.00/30.00 = 24.30/27.00 = 0.90

r = (1 + .05)1/2 = 1.0247

r – d 1.0247 – 0.90
p = = = .1247/.20 = .6235 $8.30
u – d 1.10 – 0.90 $5.68

Cuu = Max [0, 36.30 – 28] = 8.30
Cud = Max [0, 29.70 – 28] = 1.70 3.84 $1.70
Cdd = Max [0, 24.30 – 28] = 0
$1.03

$0.00
(.6235)(8.30) + (.3765)(1.70) 5.17 + 0.64
CU = = = 5.815/1.025 = $5.68
1.024695 1.024695

(.6235)(1.70) + (.3765)(0.00) 1.05995
Cd = = = $1.03
1.024695 1.024695

(.6235)(5.675) + (.3765)(1.0344) 3.538 + .3895
Co = = = $3.84
1.024695 1.024695

h = (5.675 – 1.0344)/(36.30 – 24.30) = 4.64/12.00 = 0.3867

6(b)(i). Asian-style ($33.00 + $36.30)/2 – $28.00 = $6.65

$5.28
($33.00 + $29.70)/2 – $28.00 = $3.35
($27.00 + $29.70)/2 – $28.00 = $0.35
$3.29
$0.21

($27.00 + $24.30)/2 – $28.00 = 0

(.6235)(6.65) + (.3765)(3.35) 4.146 + 1.261
CU = = = $5.28
1.024695 1.024695

(.6235)(0.35) + (.3765)(0.00)
Cd = = $0.21
1.024695

(.6235)(5.28) + (.3765)(0.21)
Co = = $3.29
1.024695

6(b)(ii). Lookback
max ($33.00, $36.30)- $28.00 = $8.30

$6.89
max ($33.00, $29.70) – $28.00 = $5.00
max ($27.00, $29.70) – $28.00 = $1.70
$4.57
$1.03

max ($27.00, $24.30) – $28.00 = 0

(.6235)(8.30) + (.3765)(5.00) 5.175 + 1.8825
CU = = = $6.89
1.024695 1.024695

(.6235)(1.70) + (.3765)(0.00)
Cd = = $1.03
1.024695

(.6235)(6.89) + (.3765)(1.03)
Co = = $4.57
1.024695
6(b)(iii). Digital
$10.00

$9.76
$10.00
$10.00
$8.17
$6.08

$0.00

(.6235)(10.00) + (.3765)(10.00) 6.235 + 3.765
CU = = = $9.76
1.024695 1.024695

(.6235)(10.00) + (.3765)(0.00)
Cd = = $6.08
1.024695

(.6235)(9.76) + (.3765)(6.08)
Co = = $8.17
1.024695

7(a). One way is to calculate approximate dividend yield: approximate annual dividend yield = 8/75 = .1067

Asset Price Strike Volatility T R(f) Div. Yield Put/Call
75.00 70.00 0.2000 0.25 0.0900 0.1067 1
B S Value Delta Gamma ~X(1) ~X(2)
5.68424 0.6798 0.0406 0.6982 0.5982

Or we could calculate the value ex-dividend: 75 2 = 73

Asset Price Strike Volatility T R(f) Div. Yield Put/Call
73.00 70.00 0.2000 0.25 0.0900 0.000 1
B S Value Delta Gamma X(1) X(2)
5.66469 0.6946 0.0429 0.6946 0.5946

The difference in price is less than 2 cents, much less than the bid ask spread.

7(b). Using put call parity

P = C S + PV(D) + PV(K)
P = 5.68 75 + (70+2)*exp(-.09*.25)
P= 1.08

7(c).
Asset Price Strike Volatility T R(f) Div. Yield Put/Call
75.00 70.00 0.2000 0.25 0.0900 0.1067 1
B S Value Delta Gamma X(1) X(2)
7.25695 0.9649 0.0334 0.9649 0.8649

So the price would increase: $1.57

7(d). An increase in the volatility to 30% would increase the call’s value. A decrease in the risk free rate to 8% would decrease the call’s value.

8(a). Stock Price Call Value Hedge Ratio
$25 $0.02 0.0104
$30 0.26 0.1003
$35 1.32 0.3409
$40 3.62 0.6331
$45 7.51 0.8438
$50 12.00 0.9463
$55 16.80 0.9838

X = $40; r = .09; T = 6 months (0.5);  = 0.25

ln (S/X) + [r + (2/2)]T
d1 = d2 = d1 –  (T)1/2
 (T)1/2

For example, if S = 25

ln(25/40) + [.09 + (.252/2)](0.5)
d1 =
(.25)(0.5)1/2

-.47 + [.09 + .03125](0.5) -.47 + .060625
= = = -.409375/.17678 = -2.31
(.25)(.7071) .17678

d2 = -2.31 – (.25)(0.5)½ = -2.31 – .17678 = -2.49

N(d1) = 1 – .9896 = 0.0104 N(d2) = 1 – .9936 = 0.0064

C = S(N(d1)) – Xe-rT(N(d2)) = 25(0.0104) – 40e-(.09)(.5)(.0064) = 0.26 – 0.24 = $0.02

8(b). The call value for each level of stock price is lower than those shown in Exhibit 23.11 because:
(1) A decrease in time to expiration (one year to six months) causes a decrease in the call value.
(2) A decrease in security volatility () causes a decrease in the call value.

8(c). For S = 40
P = Xe-rT(N(-d2)) – S(N(-d1)) = 40e-(.09)(.5) (.4325) – 40(.3669) = 16.539 – 14.676 = $1.86
where N(-d1) = 1 – .6331 = .3669 and N(-d2) = 1 – .5675 = .4325

9(a). Call:

Future Price Strike Volatility T R(f) Div. Yield Put/Call
46.50 46.50 0.2300 0.50 0.0545 0.000 1
Future Option Delta Gamma d(l) d(2)
4.14286 0.1150 0.0524 0.1150 0.1150

Put:

Future Price Strike Volatility T R(f) Div. Yield Put/Call
46.50 46.50 0.2300 0.50 0.0545 0.000 1
Future Option Delta Gamma d(l) d(2)
4.14286 0.1150 0.0524 0.1150 0.1150

9(b). You have created a long forward on a future. An investor that is bullish on the futures price would take a long position in this synthetic forward.

10(a). Asset Price Strike Volatility T R(f) Div. Yield Put/Call
653.50 670.00 0.1600 0.25 0.0650 0.0280 1
B S Value Delta Gamma X(1) X(2)
15.48 0.1860 0.0074 0.1873 0.2673

10(b).
Asset Price Strike Volatility T R(f) Div. Yield Put/Call
653.50 670.00 0.1751 0.25 0.0650 0.0280 1
B S Value Market Price Difference Delta Gamma X(1) X(2)
17.40 17.40 0.00 0.1628 0.0068 0.1640 0.2515

So, the option has an implied volatility of 17.51%.

10(c). The market price may differ from the estimated price for the following reasons:

1. The quote may be stale. That is the market maker may not have adjusted the price to reflect new market conditions.
2. Is this a bid or ask price? The ask price win have a higher implied volatility than the bid price or the midpoint.
3. We are making the assumption that dividends are paid continuously, when in fact they are clumped together. So our estimate of the effective dividend yield may be incorrect.
4. Black-Scholes model may not be the correct pricing model.

11(a). The volatility estimates are calculated as:

Period A 0.2708
Period B 0.2570

11(b).
Asset Price Strike Volatility T R(f) Div. Yield Put/Call
120.63 115.00 0.2570 0.17 0.0742 0.0365 1
B S Value Market Price Difference Delta Gamma X(1) X(2)
8.72 12.25 3.53 0.5608 0.0265 0.5643 0.4583

The market price is much higher than the calculated BS price. This implies that if all of the other parameters of the model are correct, the implied BS volatility is much higher than the historical volatility. This could be because of recent developments in the stock. Another more controversial explanation is that implied volatilities have a risk-premium built into them such that implied volatilities are on average higher than realized historical volatilities.

12(a).
Cost/Contract x 31,250
long 1.44 Call 0.0422 $1,318.75
short 1.48 Call 0.0255 $ 796.88
long 1.40 Put 0.0260 $ 812.50
short 1.44 Put 0.0422 $1,318.75

June Net Initial Long Call Short Call Long Put Short Put Total Net
USD/GBP 1.44 Profit 1.48 Profit 1.40 Profit 1.40 Profit 1.44 Profit Profit
1.36 $15.63 0 0 1250 -2500 ($1,234.38)
1.40 $15.63 0 0 0 -1250 ($1,234.38)
1.44 $15.63 0 0 0 0 $15.63
1.48 $15.63 1250 0 0 0 $1,265.63
1.52 $15.63 2500 -1250 0 0 $1,265.63

12(b).
$1,000.00

$500.00

$0.00
1.36 1.40 1.44 1.48 1.52
($500.00)

($1,000.00)

This position resembles a bull vertical spread. The purchaser of this portfolio is probably moderately bullish on the exchange rate. We see this from the willingness to give up the extreme upside in exchange for limiting the downside.

12(c). This is a simple application of put call parity
C – P = FX * exp( rf * dt) K *exp( rd *dt)
where FX is the exchange rate,
rd (rf) is the domestic (foreign) risk-free rate,
dt is the time to expiration.
C – P = 1.385*exp(-.07*.1667) – 1.44*exp(-.05*.1667) = ($0.0591)

13. CFA Examination II (1999)
13(a). The maximum loss at expiration for the straddle buyer takes place at the exercise price and is $7. This is the total cost to purchase the long call and the long put that are combined to create the straddle. If the stock price exceeds the exercise price, the straddle owner will exercise the call; if the stock price is less than the exercise price, the straddle owner will exercise the put. The maximum loss shown in the graph below and is computed as follows:

CT + PT = Max{$0, ST – X} – Ct + Max{$0, X – ST} – Pt
CT + PT = Max{$0, $50 – $50} – $4 + Max{$0, $50 -$50} – $3
CT + PT = Max{$0, $0} – $4 + Max{$0, $0} – $3
CT + PT = (-$4) + (-$3)
CT + PT = -$7

Profits will be realized for the straddle if the stock price moves above $57 or below 43 (the breakeven points.) The breakeven points for the straddle buyer are the exercise price of the options plus and minus the cost of the straddle. Stock prices, at expiration, tht are lower than $43.00 or higher than $57.00 will produce a gain for this position. These gains are shown as the intersection between the payoff profit and the zero profit line in the graph for this section, given bellow. The breakeven points are those stock prices that result in zero profit:

CT + PT = Max{$0, ST – X} – Ct + Max{$0, X – ST} – Pt
CT + PT = Max{$0, $43 – $50} – $4 + Max{$0, $50 – $43} -$3
CT + PT = Max{$0, -$7} – $4 + Max{$0, $7} -$3
CT + PT = $0 + (-$4) + $7 + (-$3)
CT + PT = $0

CT + PT = Max{$0, ST – X} – Ct + Max{$0, X – ST} – Pt
CT + PT = Max($0, $57 -$50} – $4 + Max{$0, $50 -$57} – $3
CT + PT = Max{$0, $7} – $4 + Max{$0, -$7} – $3
CT + PT = $0 + $7 – $4 – $3
CT + PT = $0

13(b). The alternative option combination, consisting of buying a put and a call with the same expiration dates and the same underlying stock but different strike prices is called a strangle. In a strangle the call has an exercise price above the current stock price and the put has an exercise price below the current stock price.

The maximum loss at expiration is $4.50, the cost of the two options (i.e., the long call and the long put). The maximum loss is computed as follows:

CT(ST,X1,T) + PT(ST,X2,T) = Max{$0, ST – X1} – Ct + Max{$0, X2 – ST} – Pt
CT(ST,X1,T) + PT(ST,X2,T) = Max{$0, $50 – $55} – $2.50 + Max{$0, $45 – $50) – $2
CT(ST,X1,T) + PT(ST,X2,T) = Max{$0 – $5} – $2.50 + Max{$0, -$5} – $2
CT(ST,X1,T) + PT(ST,X2,T) = $0 – $2.50 + $0 – $2
CT(ST,X1,T) + PT(ST,X2,T) = -$4.50

The breakeven points for the position are $40.40 and $59.50. The total cost for the two options is $4.50. The breakeven points are calculated by taking the put option strike minus the combined option premium ($45 – $4.50 = $40.50), and taking the call option strike plus the combined option premiums ($55 + $4.50 = $59.50). These are shown as the two intersections between the payoff profile and the zero profit line in the graph below.

CT(ST,X1,T) + PT(ST,X2,T) = Max{$0,ST – X1} – Ct + Max{$0,X2 – ST} – Pt
= Max{$0, $40.50 – $55} – $2.50 + Max{$0,$45 – $40.50} – $2.00
= Max{$0, -$14.50} – $2.50 + Max{$45 – $40.50} – $2.00
= $0 – $2.50 + $4.50 – $2.00
= $0

CT(ST,X1,T) + PT(ST,X2,T) = Max{$0, ST – X1} – Ct + Max{$0, X2 –ST} – Pt
= Max{$0, $59.50 – $55} – $2.50 + max{$0, $45 – $59.50} – $2.00
= Max{$0, $4.50} – $2.50 + Max{$0, -$14.50} – $2.00
= $4.50 – $2.50 + $0 – $2.00
= $0

Stock prices at expiration that are lower than $40.50 or higher than $59.50 will produce a gain for this position.

14(a) Price of ARB Profit on Profit on Net Profit on
Stock at Expiration Initial Cost Call #1 Position Call #2 Position Total Position
40 ($1.72) $0.00 $0.00 ($1.72)
45 ($1.72) $0.00 $0.00 ($1.72)
50 ($1.72) $0.00 $0.00 ($1.72)
55 ($1.72) $5.00 $0.00 $3.28
60 ($1.72) $10.00 $0.00 $8.28
65 ($1.72) $15.00 ($10.00) $3.28
70 ($1.72) $20.00 ($20.00) ($1.72)

14(b).
$8.00

$4.00

$0.00
40 45 50 55 60 65 70 75
($4.00)

($8.00)

Breakeven points are $51.72 and $68.28. Maximum profit occurs at $60.

14(c). The user of this position is betting on low volatility (that prices will stay between breakeven points). The holder has limited liability for substantial price declines and unlimited liability for substantial price increases.

15. Consider the following put option transactions: (i) long one put #1; (ii) short two puts #2, and (iii) long one put #3. The net initial cost of this set of transactions is $0.83 (= 1.70 + 6.47 – 3.67 – 3.67) and its terminal payoffs for a variety of potential expiration date SAS stock prices are:

SAS Price Value of Value of Value of Cost of
(Expiration) Put #1 Put #2 Put #3 Options Net Profit
20 15.00 40.00 25.00 0.83 -0.83
25 10.00 30.00 20.00 0.83 -0.83
30 5.00 20.00 15.00 0.83 -0.83
36 0.00 10.00 10.00 0.83 -0.83
40 0.00 0.00 5.00 0.83 4.17
45 0.00 0.00 0.00 0.83 -0.83
50 0.00 0.00 0.00 0.83 -0.83
55 0.00 0.00 0.00 0.83 -0.83
60 0.00 0.00 0.00 0.83 -0.83
Graphing these payoffs would lead to the same illustration as shown in Exhibit 23.31.

16. CFA Examination III (1997)

The suggested strategy is essentially a zero-premium collar, which is constructed by selling a call option with enough premium to fund the purchase of the put option. In this case, at-the-money protection (provided by the put) is more expensive than the out-of-money call. Therefore, the strategy requires the sale of two calls to finance one put. The call is not struck (sold) at the money because the fund manager, Ken Webster, does not want to incur the opportunity cost for a 1.05 percent rise above the S&P 500 Index’s current 668 level. By selling two 675 calls, the manager would expose the portfolio to incurring losses (from the calls) at twice the rate of the underlying portfolio’s appreciation. The portfolio would experience full market participation within the 665-675 range (plus the premium received from selling the collar). The strategy will underperform if the index has appreciated beyond 675 at expiration date and outperform if the index has declined below 665 at expiration.

16(a). Potential Returns
i. Index rises to 701. This rise would cause a substantial loss to the combined portfolio. An increase above the 675 call’s strike price would require an increase above 1.05 percent (668 to 675) in the S&P 500, at which time the call would be in the money. Because two calls were sold for every put bought in the collar, the collar would lose at twice the rate of the underlying portion of the portfolio against which it was written for any appreciation of the S&P 500 above 675. (The portfolio would realize a slight increment in return because of the premium [per collar traded] generated by the collar [2 x 4.30 – 8.05 = 0.55 per collar].)

ii. No change. The combined portfolio would experience only a slight gain as a result of the premium (per collar traded) generated by the collar (2 x 4.30 – 8.05 = 0.55 per contract). Both the put and the call would expire worthless. Therefore, the change in the portfolio would mirror the S&P 500 at 668.

iii. Index declines to 635. This decline would cause a slight loss to the combined portfolio because of the unhedged portion (668 to 675); then, the put hedge would take effect. Below 675.55, the put option guards against losses on a one-for-one basis. (The portfolio would realize a slight increment in return because of the premium [per collar traded] generated by the collar [2 x 4.30 – 8.05 = 0.55 per collar].)

16(b). Effect of Scenarios on Delta
i. Index rises to 701. The initial delta of the out-of-the-money calls was 0.36. As the call option got into the money and time expired, the delta of each call would approach 1.0. As the put option went out of the money and time expired, the delta of each put would approach zero.

ii. No change. The delta of both the put and call would approach zero as they both went out of the money and time was expiring.
iii. Index declines to 635. The initial delta of the out-of-the-money put was -0.44. As the put option got into the money and time expired, the delta of each put would approach 1.0. As the call option went out of the money and time expired, the delta of each call would approach zero.

16(c). Pricing
Compared with the historical volatility of the market (12 percent), the calls (Part ii) are priced relatively cheaper (11 percent) and the puts (Part i) are priced relatively more expensive (14 percent). The suggestion is that the collar (Part iii) is relatively expensive.

16(d). Wasting Asset
The options used to replicate the collar are “wasting assets” because they are worthless after the expiration date. Therefore, after 30 days, the strategy would have to be assessed and reestablished if desired. The profit or loss on the trade would depend on the market price as outlined in Part A and would be equal to the option’s intrinsic value. Collars do not hedge away risk; they change the risk profile of the portfolio.

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