AN INTRODUCTION TO DERIVATIVE MARKETS AND SECURITIES

1. Since call values are positively related to stock prices while put values are negatively related, any action that causes a decline in stock price (e.g., a dividend) will have a differential impact on calls and puts. Specifically, an impending dividend will boost put values and depress call values.

Another way to consider the situation is to represent the difference between the theoretical price of a call option (C) and the theoretical price of a put option (P) as C P. This is the same as a portfolio that is long a call option and short a put option. For a firm that pays dividends, we expect that the price of its stock will decline by the amount of the dividend on the last day before the stock goes ex dividend. A decline in stock price makes a call less valuable and a put more valuable, so C P will decrease.

This portfolio has the same payoff as being long a forward contract with a contract price equal to the strike price. Since there is no guarantee that the strike price is the forward price, this forward contract will typically have a non zero value (i.e. the call and put will have different prices). A dividend will decrease the up front premium for a long position in a forward contract because the expected stock price at expiration decreases. Consequently, C P is decreased by dividends.

2. It is generally true that futures contracts are traded on exchanges whereas forward contracts are done directly with a financial institution. Consequently, there is a liquid market for most exchange traded futures whereas there is no guarantee of closing out a forward position quickly or cheaply. The liquidity of futures comes at a price, though. Because the futures contracts are exchange traded, they are standardized with set delivery dates and contract sizes.

If having a delivery date or contract size that is not easily accommodated by exchange traded contracts is important to a future/forward end user then the forward may be more appealing. If liquidity is an important factor then the user may prefer the futures contract.

Another consideration is the mark to market property of futures. If a firm is hedging an exposure that is not marked to market, it may prefer to not have any intervening cash flows, hence it will prefer forwards.

3. For forwards, calls and puts, what the long position gains, the short position loses, and vice versa. However, while payoffs to forward positions are symmetric, payoffs to call and put positions are asymmetric. That is to say, long and short forwards can gain as much as they can lose, whereas long calls and puts have a gain potential dramatically greater than their loss potential. Conversely, short calls and puts have gains limited to the option premium but have unlimited liability.

For example, if the price of wheat declines by 10%, the losses to a long position in a futures contract on wheat would be the same as the gains if the price were to increase by 10%. For an at the money call option, there would be an asymmetric change in value. A long position in an at the money call option on wheat would decline in value less for a 10% fall in wheat prices than it would increase from a 10% rise in wheat prices.

Position Loss Potential Gain Potential Symmetry
Long Forward K unlimited symmetric
Short Forward unlimited K symmetric
Long Call Call premium unlimited asymmetric
Short Call unlimited Call premium asymmetric
Long Put Put premium K asymmetric
Short Put K Put Premium asymmetric

4. CFA Examination III (1993)

4(a). Derivatives can be used in an attempt to bridge the 90 day time gap in the following three
ways:

(1) The foundation could buy (long) calls on an equity index such as the S&P 500 Index and on Treasury bonds, notes, or bills. This strategy would require the foundation to make an immediate cash outlay for the “premiums” on the calls. If the foundation were to buy calls on the entire $45 million, the cost of these calls could be substantial, particularly if their strike prices were close to current stock and bond prices (i.e., the calls were close to being “in the money”).

(2) The foundation could write or sell (short) puts on an equity index and on Treasury bonds, notes, or bills. By writing puts, the foundation would receive an immediate cash inflow equal to the “premiums” on the puts (less brokerage commissions). If stock and bond prices rise as the committee expects, the puts would expire worthless, and the foundation would keep the premiums, thus hedging part or all of the market increase. If the prices fall, however, the foundation loses the difference between the strike price and the current market price, less the value of the premiums.

(3). The foundation could buy (long) equity and fixed income futures. This is probably the most practical way for the foundation to hedge its expected gift. Futures are available on the S&P 500 Index and on Treasury bonds, notes, and bills. No cash outlay would be required. Instead, the foundation could use some of its current portfolio as a good faith deposit or “margin” to take the long positions. The market value of the futures contracts will, in general, mirror changes in the underlying market values of the S&P 500 Index and Treasuries. Although no immediate cash outlay is required, any gains (losses) in the value of the contracts will be added (subtracted) from the margin deposit daily. Hence, if markets advance as the committee expects, the balances in the foundation’s futures account should reflect the market increase.

4(b). There are both positive and negative factors to be considered in hedging the 90-day gap before the expected receipt of the Franklin gift.

Positive factors

(1). The foundation could establish its position in stock and bond markets using derivatives today, and benefit in any subsequent increases in market values in the S&P Index and Treasury instruments in the 90¬ day period. In effect, the foundation would have a synthetic position in those markets beginning today.

(2). The cost of establishing the synthetic position is relatively low, depending on the derivative strategy used. If calls are used, the cost is limited to the premiums paid. If futures are used, the losses on the futures contracts would be similar to the amounts that would be lost if the foundation invested the gift today. Writing the puts is the riskiest strategy because there is an open-ended loss if the market declines, but here again the losses would be similar if the foundation invested today and stock and bond markets declined.

(3). Derivative markets (for the types of contracts under consideration here) are liquid.

Negative factors

(1). The Franklin gift could be delayed or not received at all. This would create a situation in which the foundation would have to unwind its position and could experience losses, depending on market movements in the underlying assets.

(2). The committee might be wrong in its expectation that stock and bond prices will rise in the 90 day period. If prices decline on stocks and bonds, the foundation would lose part or all of the premium on the calls and have losses on the futures contracts and the puts written. The risk of loss of capital is a serious concern. (Given that the current investment is primarily bonds and cash, the foundation may not be knowledgeable enough to forecast stock prices over the next 90 days.)

(3). Because there is a limited choice of option and futures derivative contract compared to the universe that the committee might wish to invest in, there could be a mismatch between the specific equities and bonds the foundation wishes to invest in and the contracts available in size for $45 million. Unless the 90-day period exactly matches the 90-day period before expiration dates on the contracts, there may be a timing mismatch.

(4). The cost of the derivatives is potentially high. For example, if the market in general shares the committee’s optimistic outlook, the premiums paid for calls would be expensive and the premiums received on puts would be lean. The opportunity cost on all derivative strategies discussed would be large if the committee is wrong on the outlook for one or both markets.

(5). There may exist regulatory restrictions on the use of derivatives by endowment funds.
Evaluation

The negative factors appear to outweigh the positive factors if the outlook for the market is neutral; therefore, the committee’s decision on using derivatives to bridge the gap for 90 days will have to be related to the strength of its conviction that stock and bond prices will rise in that period. The certainty of receiving the gift in 90 days is also a factor. The committee should certainly beware that there is a cost to establish the derivative positions, especially if its expectations do not work out. The committee might want to consider a partial hedge of the $45 million.

5. CFA Examination II (1991)

Because Chen is considering adding either short index futures or long index options (a form of protective put) to an existing well diversified equity portfolio, he evidently intends to create a hedged position for the existing portfolio. Both the short futures and the long options positions will reduce the risk of the resulting combined portfolios, but in different ways.

Assuming that the short futures contract is perfectly negatively correlated with the existing equity portfolio, and that the size of the futures position is sufficient to hedge the risk of the entire equity portfolio, any movement up or down in the level of stock market prices will result in offsetting gains and losses in the combined portfolio’s two segments (the equity portfolio itself and the short futures position). Thus, Chen is effectively removing the portfolio from exposure to market movements by eliminating all systematic (market) risk (unsystematic (specific) risk has already been minimized because the equity portfolio is a well diversified one). Once the equity portfolio has been perfectly hedged, no risk remains, and Chen can expect to receive the risk-¬free rate of return on the combined portfolio. If the hedge is less than perfect, some risk and some potential for return beyond the risk free rate are present, but only in proportion to the completeness of the hedge.

If, on the other hand, Chen hedges the portfolio by purchasing stock index puts, he will be placing a floor price on the equity portfolio. If the market declines and the index value drops below the strike price of the puts, the value of the puts increases, offsetting the loss in the equity portfolio. Conversely, if the stock market rises, the value of the put options will decline, and they may expire worthless; however, the potential return to the combined portfolio is unlimited and reduced only by the cost of the puts. As with the short futures, if the long options hedge is less than perfect, down¬side risk remains in the combined portfolio in proportion to the amount not covered by the puts.

In summary, either short futures or long options (puts) can be used to reduce or eliminate risk in the equity portfolio. Use of the options (put) strategy, however, permits unlimited potential returns to be realized (less the cost of the options) while use of the short futures strategy effectively guarantees the risk free rate but reduces or eliminates potential returns above that level. Neither strategy dominates the other; each offers a different risk/return profile and involves different costs. Arbitrage ensures that, on a risk adjusted basis, neither approach is superior.

6. CFA Examination II (1997)

Three prominent pricing inconsistencies are apparent in the table for the Furniture City call options:

1. The June call option at a strike price of $110 is undervalued. A call option that is in the money should be worth at least as much as its intrinsic value. The intrinsic value of a call option is the maximum of either zero or the difference between the security price and the exercise price (S – E). The June $110 option, therefore, should be worth at least $119.50 – $110.00, or $9.50. The current price of $8 7/8 implies that the option is undervalued.

2. The August call option at a strike price of $120 is undervalued. Call options having the same strike price but with longer maturities are more valuable than those with shorter maturities because the stock has more time in which to rise above the strike price; that is, the time value increases with maturity. The August $120 option of $3 is below the July $120 option of $3 3/4; therefore, the August $120 option is undervalued. Alternatively, the July $120 option could be said to be overvalued.

3. The September call option at a strike price of $130 is overvalued. Call options having the same maturity but with higher strike prices that are more out of the money are worth less because a larger and less likely move in the stock price will be needed for the option to pay off. The September $130 option is priced higher than the September $120 option; therefore, the September $130 option is overvalued. Alternatively, the September $120 option could be said to be undervalued.

(Note: Candidates receive full credit for identifying other, less prominent, pricing inconsistencies.)

7. The important distinction is whether the option is a covered or uncovered position. If the option is added to a portfolio that already contains the underlying asset (or something highly correlated), then the option will frequently be a covered position and, consequently, lower overall risk. For example, selling a call without owning the underlying asset leaves the seller open to unlimited liability and (probably) increases portfolio price fluctuation (risk). But if the seller of the call owns the underlying asset, then selling the call neutralizes the portfolio from price changes above the strike price and (probably) decreases risk.

Because options are bets that an asset’s price will be above or below some level (the strike price), they represent a way of leveraging one’s subjective view on the asset’s future price. Options always cost less than the underlying asset and consequently can change in price more (on a percentage basis) than the underlying asset does. This provides the same effect as borrowing money to buy the asset or selling the asset short and investing the proceeds in bonds. (In fact, this is how option pricing theory values options, by replicating the price of an option using the underlying asset and bonds.)

8. Call options differ from forward contracts in that calls have unlimited upside potential and limited downside potential, whereas the gains and losses from a forward contract are both unlimited. Therefore, since call options do not have the downside potential of forwards, they represent only the “good half” (the upside potential) of the forward contract. The “bad half” of the long forward position is the unlimited downside potential that is equivalent to being short a put. This is consistent with put call parity where being long a call and short a put yields the same payoff as a forward contract.

9. Since options have nonlinear (kinked) payoffs, broad market movements may have different relative effects on the value of a portfolio with options depending on whether the market moves up or down. For example, a portfolio that is put protected may not move down much if the market declines 10% but may move up nearly 10% if the market rises 10%. Consequently, the returns are asymmetric or skewed. This makes standard deviation a less informative statistic because it reveals information only about the degree of variation and not the “direction” of the variation. Since investors usually care about downside risk (standard deviation), investors probably will not care as much if all of the variation is in the upside return. The standard deviation statistic could be modified to only measure the variation in negative returns (the so called semi variance) so that it was a measure of downside risk only.

10. A synthetic off market forward contract with a forward price of $25 could be created using put call parity. Buying a call struck at $25 and selling a put struck at $25 assures the investor of buying the stock at the expiration date for $25. This portfolio then has the same requirements as the off market forward at $25. C P is one half of the put cal1 parity relationship, and we know it has to equal S PV(K). Since S = $32 and the risk free rate can be calculated as (35 32)/32 = 9.375%, we can calculate that the off market forward is worth S P($25) = $32 $25 / (1 + .09375) = $9.14.

CHAPTER 21

Answers to Problems

l(a).
(i). A long position in a forward with a contract price of $50.

Expiration Date Sophia Long Forward Initial Long
Stock Price (S) (K=$50) Payoff=S 50 Forward Premium Net Profit
25 ($25.00) $0.00 ($25.00)
30 ($20.00) $0.00 ($20.00)
35 ($15.00) $0.00 ($15.00)
40 ($10.00) $0.00 ($10.00)
45 ($5.00) $0.00 ($5.00)
50 $0.00 $0.00 $0.00
55 $5.00 $0.00 $5.00
60 $10.00 $0.00 $10.00
65 $15.00 $0.00 $15.00
70 $20.00 $0.00 $20 00
75 $25.00 $0.00 $25.00

(ii). A long position in a call option with a exercise price of $50 and a front end
premium expense of $5.20.

Expiration Date Sophia Long Call (K=$50) Initial Long
Stock Price (S) Payoff = max (0,S 50) Call Premium Net Profit
25 $0.00 ($5.20) ($5.20)
30 $0.00 ($5.20) ($5.20)
35 $0 00 ($5.20) ($5.20)
40 $0.00 ($5.20) ($5.20)
45 $0 00 ($5.20) ($5.20)
50 $0.00 ($5 20) ($5 20)
55 $5.00 ($5 20) ($0.20)
60 $10.00 ($5.20) $4.80
65 $15.00 ($5.20) $9.80
70 $20.00 ($5.20) $14.80
75 $25.00 ($5.20) $19.80

(iii). A short position in a call option with an exercise price of $50 and a front end
premium receipt of $5.20.

Expiration Date Sophia Short Call (K=$50) Initial Short
Stock Price (S) Payoff = max (0,S 50) Call Premium Net Profit
25 $0.00 $5.20 $5.20
30 $0.00 $5.20 $5.20
35 $0.00 $5.20 $5.20
40 $0.00 $5.20 $5.20
45 $0.00 $5.20 $5.20
50 $0.00 $5.20 $5.20
55 ($5.00) $5.20 $0.20
60 ($10.00) $5.20 ($4.80)
65 ($15.00) $5.20 ($9.80)
70 ($20.00) $5.20 ($14.80)
75 ($25.00) $5.20 ($19.80)

l(b). (i). A long position in a forward with a contract price of $50.

Long Forward
$25.00
$20.00
$15.00
$10.00
$5.00
$0.00
($5.00) 25 50 75
($10.00)
($15.00)
($20.00)
($25.00)

(ii.) A long position in a call option with an exercise price of $50 and a front¬-end premium
expense of $5.20:
Long Call
$25.00
$20.00
$15.00
$10.00
$5.00
$0.00
($5.00) 25 50 55 75
($10.00)
($15.00)
($20.00)
($25.00)
(iii.) A short position in a call option with an exercise price of $50 and a front end premium receipt of $5.20
Short Call
$25.00
$20.00
$15.00
$10.00
$5.00
$0.00
($5.00) 25 50 55 75
($10.00)
($15.00)
($20.00)
($25.00)

THE BREAKEVEN POINT FOR THE CALL OPTIONS IS $55.20.

l(c). The long position in a forward with a contract price of $50: The purchaser believes that the price of Sophia Enterprises stock will be above $50.

The long position in a call option with an exercise price of $50 and a front-¬end premium expense of $5.20: The purchaser believes the price will be above $55.20.

The short position in a call option with an exercise price of $50 and a front-end premium receipt of $5.20: The seller believes the price of Sophia Enterprises stock will be below $55.20.

2(a).
(i). A short position in a forward with a contract price of $50.

Expiration Date Sophia Short Forward Initial Short
Stock Price (S) (K=$50) Payoff= S 50 Forward Premium Net Profit
25 $25.00 $0.00 $25 00
30 $20.00 $0.00 $20.00
35 $15.00 $0.00 $15.00
40 $10.00 $0.00 $10.00
45 $5.00 $0.00 $5.00
50 $0.00 $0.00 $0.00
55 ($5.00) $0.00 ($5.00)
60 ($10.00) $0.00 ($10.00)
65 ($15.00) $0.00 ($15.00)
70 ($20.00) $0.00 ($20.00)
75 ($25.00) $0.00 ($25.00)

(ii). A long position in a put option with a exercise price of $50 and a front-end premium
expense of $3.23.

Expiration Date Sophia Long Put (K=$50) Initial Long
Stock Price (S) Payoff= max (0,50 S) Put Premium Net Profit
25 $25.00 ($3.23) $21.77
30 $20.00 ($3.23) $16.77
35 $15.00 ($3.23) $11.77
40 $10.00 ($3.23) $6.77
45 $5.00 ($3.23) $1.77
50 $0.00 ($3.23) ($3.23)
55 $0.00 ($3.23) ($3.23)
60 $0.00 ($3.23) ($3.23)
65 $0.00 ($3.23) ($3.23)
70 $0.00 ($3.23) ($3.23)
75 $0.00 ($3.23) ($3.23)

(iii.) A short position in a put option with an exercise price of $50 and a front end
premium receipt of $3.23.

Expiration Date Sophia Short Put (K=$50) Initial Short
Stock Price (S) Payoff= max (0,50 S) Put Premium Net Profit
25 ($25.00) $3.23 ($21.77)
30 ($20.00) $3.23 ($16.77)
35 ($15.00) $3.23 ($11.77)
40 ($10.00) $3.23 ($6.77)
45 ($5.00) $3.23 ($1.77)
50 $0.00 $3.23 $3.23
55 $0.00 $3.23 $3.23
60 $0.00 $3.23 $3.23
65 $0.00 $3.23 $3.23
70 $0.00 $3.23 $3.23
75 $0.00 $3.23 $3.23

2(b). (i). A short position in a forward with a contract price of $50:

Short Forward
$25.00
$20.00
$15.00
$10.00
$5.00
$0.00
($5.00) 25 50 75
($10.00)
($15.00)
($20.00)
($25.00)
(ii). A long position in put option with an exercise price of $50 and front-end premium
expenses of $3.23:

Long Put
$25.00
$20.00
$15.00
$10.00
$5.00
$0.00
($5.00) 25 50 75
($10.00)
($15.00)
($20.00)
($25.00)

(iii). A short position in a put option with an exercise price of $50 and a front end
premium receipt of $3.23:

Short Put
$25.00
$20.00
$15.00
$10.00
$5.00
$0.00
($5.00) 25 50 75
($10.00)
($15.00)
($20.00)
($25.00)

THE BREAKEVEN POINT FOR BOTH PUT OPTIONS IS $46.77.

2(c). A short position in a forward with a contract price of $50: The seller believes the price of Sophia Enterprises will be below $50.

A long position inch put option with an exercise price of $50 and front end premium
expense of $3.23: The buyer of the put believes the price will be below $46.77.

A short position in a put option with an exercise price of $50 and a front end premium receipt of $3.23: The seller of the put believes the price will be above $46.77.

3(a). (i). A short position in a forward option with a exercise price of $50.

Expiration Date Sophia Short Forward (K=$50) Initial Short
Stock Price (S) Payoff =max (0, S-50) Forward Premium Net Profit
25 $25.00 $0.00 $50.00
30 $20.00 $0.00 $50.00
35 $15.00 $0.00 $50.00
40 $10.00 $0.00 $50.00
45 $5.00 $0.00 $50.00
50 $0.00 $0.00 $50.00
55 ($5.00) $0.00 $50.00
60 ($10.00) $0.00 $50.00
65 ($15.00) $0.00 $50.00
70 ($20.00) $0.00 $50.00
75 ($25.00) $0.00 $50.00

(ii). A long position in a put option with a exercise price of $50 and a front-end
premium expense of $3.23.

Expiration Date Sophia Long Put (K=$50) Initial Long Put
Stock Price (S) Payoff = max (0,50 S) Put Premium Net Profit
25 $25 00 ($3 23) $46.77
30 $20.00 ($3.23) $46.77
35 $15.00 ($3.23) $46.77
40 $10.00 ($3 23) $46.77
45 $5.00 ($3 23) $46.77
50 $0.00 ($3.23) $46.77
55 $0.00 ($3 23) $51.77
60 $0.00 ($3.23) $56.77
65 $0.00 ($3 23) $61.77
70 $0.00 ($3.23) $66.77
75 $0.00 ($3.23) $71.77

(iii). A short position in a call option with an exercise price of $50 and a front-¬end premium receipt of $5.20.

Expiration Date Sophia Short Call (K=$50) Initial Short
Stock Price (S) Payoff = -max (0,S-50) Call Premium Net Profit
25 $0.00 $5.20 $30.20
30 $0.00 $5.20 $35.20
35 $0.00 $5.20 $40.20
40 $0.00 $5.20 $15.20
45 $0.00 $5.20 $50.20
50 $0.00 $5.20 $55.20
55 ($5.00) $5.20 $55.20
60 ($10.00) $5.20 $55.20
65 ($15.00) $5.20 $55.20
70 ($20.00) $5.20 $55.20
75 ($25.00) $5.20 $55.20
3(b). (i). A short position in a forward with a contract price of $50:

Short Forward
$80.00
$70.00
$60.00
$50.00
$40.00
$30.00
$20.00
$10.00
$0.00
25 50 75

(ii). A long position in a put option with an exercise price of $50 and a front-end premium expense of $3.23:

Long Put
$80.00
$70.00
$60.00
$50.00
$40.00
$30.00
$20.00
$10.00
$0.00
25 50 75

(iii). A short position in a call option with an exercise price of $50 and a front end premium expense of $5.20:

Short Call
$80.00
$70.00
$60.00
$50.00
$40.00
$30.00
$20.00
$10.00
$0.00
25 50 75

3(c). F0,T = Call – Put + PV(Strike)
$50.00 = 5.20 – 3.23 + PV($50)
$48.03 = PV ($50)
PV = 1.041
S = F/(PV Factor)
$50 = F / 1.041
$50 x 1.041 = F
$52.05 = F

The zero value contract price, $52.05, differs from the $50 contract price because the put and the call prices are not the same. If they were, the combination of the two would yield a zero value forward price.

4(a). With $13,700 to spend, one could:
(1) Purchase 100 shares of Breener Inc. stock (@ $137 per share); or
(2) Purchase 1370 call options with exercise price of $140
Potential payoff is unlimited in both cases, however the leverage that options provide will translate into a higher percentage gain than purely purchasing stock. However, leverage works both ways.

4(b). (1) Stock price increases to $155
a. Stock return = ($155 – $137)/$137 = 13.14%
b. Option return: Exercise option @ $140, sell stock at $155
Sell Stock@ $155 x 1370 = $212,350
Cost of stock @ $140 (191,800)
Option purchase @ $10 ( 13,700)
Profit $ 6,850
Rate of return = $6850/$13,700 = 50%

(2) Stock price decreases to $135
a. Stock return = ($135 – $137)/$137 = -1.5%
b. Option return: Option would not be exercised, lose entire option purchase price
(-100%)

4(c). Breakeven on this call option is $150. In other words, the writer of the call option will receive the premium of $10, that is, the maximum amount the seller will receive. If the seller does not currently own the stock, his/her loss is potentially unlimited.

5(a). Givens:
Current Price of XYZ = $42
Put ($40) = $1.45
Call ($40) = $3.90
RFR = 8% (annual); 4% (semiannual)

(i). Buy one call option

Expiration Date Long Call (K=$40) Initial Long
XYZ Stock Price (S) Payoff = max (0,S 40) Call Premium Net Profit
20 $0.00 ($3.90) ($3.90)
25 $0.00 ($3.90) ($3.90)
30 $0.00 ($3.90) ($3.90)
35 $0.00 ($3.90) ($3.90)
40 $0.00 ($3.90) ($3.90)
45 $5.00 ($3.90) $1.10
50 $10.00 ($3 90) $6.10
55 $15.00 ($3.90) $11.10
60 $20.00 ($3.90) $16.10

Long Call
$20.00

$15.00

$10.00

$5.00

0.00
20 40 60
($5.00)

(ii). Short one call option

Expiration Date Short Call (K=$40) Initial Short
XYZ Stock Price (S) Payoff =-max (0,S 40) Call Premium Net Profit
20 $0.00 $3.90 $3.90
25 $0.00 $3.90 $3.90
30 $0.00 $3.90 $3.90
35 $0.00 $3.90 $3.90
40 $0.00 $3.90 $3.90
45 ($5.00) $3.90 ($1.10)
50 ($10.00) $3.90 ($6.10)
55 ($15.00) $3.90 ($11.10)
60 ($20.00) $3.90 ($16.10)

Short Call
$5.00

$0.00
20 40 60
($5.00)

($10.00)

($15.00)

($20.00)

Both call positions will break even at a stock price of $43.90.

5(b). (i). Buy one put option

Expiration Date Long Put (K=$40) Initial Long
XYZ Stock Price (S) Payoff =max (0,40-S) Put Premium Net Profit
20 $20.00 $1.45 $18.55
25 $15.00 $1.45 $13.55
30 $10.00 $1.45 $8.55
35 $5.00 $1.45 $3.55
40 $0.00 $1.45 ($1.45)
45 $0.00 $1.45 ($1.45)
50 $0.00 $1.45 ($1.45)
55 $0.00 $1.45 ($1.45)
60 $0.00 $1.45 ($1.45)

Long Put
$20.00

$15.00

$10.00

$5.00

0.00
20 40 60
($5.00)

(ii). Short one put option

Expiration Date Long Put (K=$40) Initial Long
XYZ Stock Price (S) Payoff =-max (0,40-S) Put Premium Net Profit
20 ($20.00) $1.45 ($18.55)
25 ($15.00) $1.45 ($13.55)
30 ($10.00) $1.45 ($8.55)
35 ($5.00) $1.45 ($3.55)
40 $0.00 $1.45 $1.45
45 $0.00 $1.45 $1.45
50 $0.00 $1.45 $1.45
55 $0.00 $1.45 $1.45
60 $0.00 $1.45 $1.45

Short Put
$5.00

$0.00
20 40 60
($5.00)

($10.00)

($15.00)

($20.00)

Both put positions will break even at a stock price of $38.55.

5(c). Does Call – Put = S – PV(K)?

$3.90 – $1.45  $42 – 40/(1.04)

$2.45  $3.53

NO! Put-call parity does not hold for this European-style contract.

6. CFA Examination III (1987)

6(a). The manager wishes to be protected from any decline in the government bonds’ price (any rise in rates) while maintaining a participation in a price advance, should it occur. The manager also wishes to hold the existing bond position over the next six months. The option strategy best suited for this goal is a protected put strategy.

The manager should purchase the put options in size sufficient to protect the $1.0 million portfolio. The manager should purchase 10 put contracts. In this strategy, the bond portfolio will be protected against any price decline below 98, (the put strike price – the put premium) yet will participate with any price advance on the bond less 2.00 (the put premium).
6(b). The manager is now willing to sell the existing bonds in order to create a portfolio structure that achieves the goal. Two strategic options structures equivalent to the structure in Part A above can be designed as follows:

Alternative 1

• Sell the government 8% bonds
• Invest the proceeds in the T bills and buy the appropriate amount of 8% bond futures
• Buy the put options to protect against price declines while participating in price advances.

Buying T bills and futures is equivalent to holding the bonds. The manager could purchase $1.0 million in T bills and 10 futures contracts.

The manager would complete the option strategy alternative 1 by buying 10 put options.

Alternative 2

• Sell the government 8% bonds
• Invest the proceeds in the T bills
• Buy the appropriate amount of call options

A T-bill plus call option position is equivalent in structure to a bond plus put option position. The bill plus call options provides protection against bond price declines since the investor can only lose the premium on the call option. Participation in any bond price advance is achieved since the call option premium will increase.

Specifically, the manager could purchase $1.0 million in T bills and 10 call options.

6(c). Given the put call parity pricing relationship, the put options and call options appear misvalued versus each other.

If the call is correctly priced at 4.00, the put should be priced at:

put = 4.00 – bond price + present value of strike
= 4.00 – 100 + 100/1.03
= 4.00 – 100 + 97.09
= 1.09

If the put is correctly priced at 2.00, the call should be priced at:

call = 2.00 + 100 – 100/1.03
= 2.00 + 100 – 97.09
= 4.91

The put is overpriced versus the call. Therefore:

(l) in the put buying strategies the put premium appears very fully priced.
(2) in the future buying put buying strategy, similar comments hold. Also the future
appears overvalued.
(3) in the buy bills/buy calls strategy, the call option price appears attractive.

In addition, the price of the future at 101 appears high; however, a fair price for the future would be:

price future = price bond + (bill income – bond income) price
= 100+(-1) = 99

The option strategy involving buying T-bills and the call options is recommended.

7(a). To solve this problem, express put call parity in the following form:

C(K) – P(K) – S + PV(K) = 0

Now we can equate put-call parity for two different options:

C(40) – P(40) – S + PV(40) = C(50) – P(50) – S + PV(50)

Note that the S on each side cancels.

Then the only unknown is P(50). Solving for P(50) yields:

P(50) = C(50) + PV(50) – C(40) + P(40) – PV(40)
P(50) = 2.47 + 50/1.03 – 8.73 + .59 – 40/1.03
P(50) = $4.03

The same can be done to find the value of C(45):

C(40) – P(40) – S + PV(40) = C(45) – P(45) – S P PV(45)
C(45) = C(40) – P(40) + PV(40) +P(45) – PV(45)
C(45) = 8.73 – .59 +40/1.03 + 1.93 – 45/1.03
C(45) = $5.21

7(b). If the future price of Commodity Z is $48, then the spot price (S) should be
S = PV($48) = 48/1.03 = $46.60.

Then solving for the theoretical price differential between calls and puts
(K=$40) gives:
C(40) – P(40) = S – PV(40)
C(40) – P(40) = $46.60 – 4/1.03
C(40) – P(40) = $7.77 which is less than the $8.14 difference observed in the market.
By selling the “overpriced” portfolio C(40) – P(40) = $8.14 and buying the “underpriced” portfolio S – PV(40) = $7.77, we can obtain an arbitrage profit of $8.14 – $7.77 = $0.37 since we know by put call parity that these two portfolios must have the same payoff at expiration.

This total portfolio position is:

• short the call
• long the put
• long the stock
• borrow PV(40)

8. CFA Examination III (2000)

8(a). Delsing should sell stock index futures contracts and buy bond futures contracts. This strategy is justified because buying the bond futures and selling the stock index futures
provide the same exposure as buying the bonds and selling the stocks. This strategy assumes high correlations between the movements of the bond futures and bond portfolio as well as the stock index and the stock portfolio.

8(b). The correct number of contracts in each case is:
i. 5 x $200,000,000 x .01 = $100,000, and $100,000/97.85 = 1022 contracts
ii. $200,000,000/($1,378 x 250) = 581 contracts.

8(c)i. The advantages of using financial futures for asset allocation are:
• execution speed in terms of lower transaction time
• execution efficiency in terms of lower market impact and brokerage fees
• less disruption of external manager’s performance
• less reallocation of funds among managers
• no direct cost to establishing the hedge
• high liquidity of index futures.

The disadvantage of using financial futures for asset allocation is:
• exposure of portfolio to basis risk or tracking error.

ii. The advantages of using index put options for asset allocation are:
• Less disruption of external manager’s performance
• Less reallocation of funds among managers

The disadvantages of using index put options for asset allocation are
• The cost of index put options, which must be borne regardless of the subsequent stock index movement, i.e., significant up-front cost/cash needed to satisfy premium
• Need for frequent rebalancing as delta changes with price of underlying asset
• Basis risk or tracking error.

8(d). The stock return is: $28/$1,378 = 2% (or 2.03%)
the bond return is: -Dmod x (basis point change/100) = -5 x (10/100) =-0.5%

where Dmod = modified duration.

i. For a 50/50 allocation, the capital gain return is:
(0.5 x 2%) + (0.5 x –0.5%) = 0.75% (or 0.765%)

ii. For a 75/25 allocation, the capital gain is;
(0.75 x 2%) + (0.25 x –0.5%) = 1.375% (or 1.3975%)

9. CFA Examination III (1990)

9(a). Alternative 1 (Buy Puts)

• Buy S&P 500 put options with market exposure equal to equity holdings to protect these holdings.
• Buy Government bond put options with market exposure equal to the bond holdings to protect these holdings.

This could be done by (1) buying $150 million of S&P 500 put options. Since each option is equivalent to $35,000 market exposure, this could be done by buying:

$150,000,00/($35,000/option) = 4,286 S&P 500 put options

and (2) buying $150 million of Government bond put options. Since each option is equivalent to $ 100,000 market exposure, this could be done by buying:

$150,000,000/($100,000/option) =1,500 Government bond put options

By buying 4,286 S&P 500 put options and 1,500 Government bond put options, the portfolio is protected and upside participation is achieved.

Alternative 2 (Sell Futures/Buy Calls)

• Sell S&P stock index futures equal to the equity exposure in the portfolio and buy S&P 500 call options equal to the exposure.
• Sell Government bond futures equal to the bond exposure in the portfolio and buy bond call options equal to the exposure.

This could be done by:

(1) Selling $150 million of S&P futures. Since each future is equivalent to $ 175,000 of equity exposure, this could be done by selling:

$150,000,000/($175,000/future) = 857 S&P futures

Buying $150 million of S& P 500 call options or 4,286 call options (see above calculation).

(2) Selling $ 150 million of Government bond futures contracts. Since each bond future is equivalent to $100,000 of bond exposure, this could be done by selling:

$150,000,000/($100,000/future) = 1,500 bond futures

Buying $150 million of Government bond call options or 1,500 call options (see above calculation).

By selling 857 S&P futures and 1,500 bond futures, and buying 4,286 S&P call options and 1,500 bond options, the portfolio is protected from loss and upside participation is achieved.

9(b). Given the put-call parity relationship, the put options appear misvalued compared to the call options.

Given the S&P 500 call price, the put should be priced at:

put = 8.00 – index price + present value of (strike + income)
= 8.00 – 350 + (.01) x (350)
= 8.00 – 350 + 1.015
= 6.28

Given the bond call price, the bond put option should be priced at:

put = 2.50 – bond price + present value of (strike + income)
= 2.50 – 100 + (.02) x (100)
= 2.50 – 100 + 1.015
= 2.99

For both the S&P 500 options and the Government bond options, the put options appear overvalued compared to the prices of the calls.

The prices of the futures also appear high. A fair price for the S&P 500 future would be:

S& P 500 future = index price + (bill income – dividend income)
= 350 + [(.015 – .01) x (350)]
= 351.75

A fair price for the bond future would be:

Bond future price = bond price + (bill income – bond income)
= 100 + [(.015 – .02) x (100)]
= 99.5
From this analysis the futures are somewhat overvalued and the put options are relatively overpriced compared to the call options.

Alternative 1 involves buying relatively expensive assets (the put options).
Alternative 2 involves selling expensive assets (the futures contracts) and buying relatively inexpensive assets (the call options).

Alternative 2 where protection is gained by selling futures and buying call options is recommended.

10(a). A synthetic put can be created by shorting the stock, buying a call, and lending PV(x) – S.

+ + =

Since the synthetic put and an actual put have the same payoff at maturity, they must have the same price today or there would be an arbitrage opportunity. The arbitrage would be to sell the more expensive “put” and buy the less expensive “put” netting the difference in price.

10(b). Buying a call and selling a put gives the same payoff as a long forward:

+ =

11(a). Sum of T-bill, call & put

Payoff
60 T-bill
Call

0 60 Stock Price
Put

-60

11(b). Using put-call parity the “no arbitrage price” is
S= PV(K)+C-P
S = 0.6*97 + 3.18 – 3.38
S = $58.00

11(c). Since put-call parity indicates the “no arbitrage” price of the stock is $58 and the stock selling at $60, the arbitrage would be to sell the over valued portfolio (the stock) and use the proceeds ($60) to buy the undervalued portfolio (.6 t bills, long 1 call, short 1 put). This set of trades yields $2 in arbitrage profits. Since by put call parity we know that the two portfolios will have exactly offsetting terminal payoffs, the trade is riskless.

12(a). Expiration Date Long Put (K=$55) Initial Long
Stock Price(S) Payoff = max(0,55-S) Put Premium Net Profit
35 $20.00 ($1.32) $53.68
40 $15.00 ($1.32) $53.68
45 $10.00 ($1.32) $53.68
50 $5.00 ($1.32) $53.68
55 $0.00 ($1.32) $53.68
60 $0.00 ($1.32) $58.68
65 $0.00 ($1.32) $63.68
70 $0.00 ($1.32) $68.68
75 $0.00 ($1.32) $73.68

Long Put
12(b). $80.00

$60.00

$40.00

$20.00

$0.00
35 50 75
Using put call parity, the position could have been replicated by selling the portfolio, buying the call option at the ask price, and investing the balance in T bills yielding 7%.

12(c). Expiration Date Short Call (K=$55) Initial Short
Stock Price(S) Payoff = max(0,S-55) Call Premium Net Profit
35 $0.00 $2.55 $37.55
40 $0.00 $2.55 $42.55
45 $0.00 $2.55 $47.55
50 $0.00 $2.55 $52.55
55 $0.00 $2.55 $57.55
60 ($5.00 ) $2.55 $57.55
65 ($10.00) $2.55 $57.55
70 ($15.00) $2.55 $57.55
75 ($20.00) $2.55 $57.55
Short Call
12(d). $80.00

$60.00

$40.00

$20.00

$0.00
35 50 75

Using put call parity, the position could have replicated by selling the portfolio, selling the put option at the bid price, and then investing the balance in T bills yielding 7%.

13(a). Since the forward price is the same as the strike price, going long a call and short a put is the same as a zero-value forward contract. Hence, the call price must equal the put price, so C =3.22 for the “no arbitrage” price. Rounding to the nearest eighth yields:
bid ask
Call $3.125 $3.250

13(b). Using put-call parity: S = C – P+ PI
from part (a): C-P=O,
so S = 45/(1.065).75
S = 42.92

13(c). To make an arbitrage profit you want to sell the overvalued portfolio (short 0.45 T-bills, short call, long put) and use the proceeds to buy the undervalued portfolio (the stock).
The arbitrage profit would be:
Profit = 45/(1.065).75 + 3.125 – 3.25 – 41
Profit = $1.80

13(d). The present value of the dividend would have to be equal to the difference in theoretical and actual prices:
PV(div) = 42.92 – 41.00
div = (1.065).75*1.92
div = $2.01

14. CFA Examination III (2000)

14(a)i. The transactions needed to construct the synthetic T-Bill would be to long the stock, long the put and short the call.

14(a)ii. Assuming the T-Bill yield was quoted on a bond equivalent basis, the synthetic Treasury bill’s annualized yield can be calculated on the same basis:

$77.50 + $4.00 – $7.75 = $73.75 initial investment and $75 ending value.
[$75.00 – $73.75)/473.75] x 4 = 6.78%

14(b)i. The strategy would be to short 75 actual T-bills and to long 100 synthetic T-bills.

14(b)ii. Assuming the actual T-bill was quoted on a bond equivalent basis, the actual T-bill
gives a 1.25% quarterly return.

Immediately, the short actual T-bill position pays:

$750/1.0125 = $740,741

At the time of creation, the long position in the synthetic T-bill would be:

Long stock -$775,000
Long put -$ 50,000
Short call $ 77,500
-$737,500

Therefore the net cash flow is:

$740,741 – $737,500 = $3,241

14(c). The approach to calculating net cash flow gives the same result whether the calculation is done for three months or six months. At the three-month expiration, the value of the long synthetic position is:

E = P + S – C

where E = exercise price, P = put price, C = call price

At expiration E = P + S – C
= $0 + $80 – $5
= $75 per share or $7,500 per contract

Total cash flow of the long synthetic position = 100 x $7,500
= $750,000

Total cash flow of the short Treasury bill position = 75 actual Treasury bills x $10,000
= 4750,000

Net cash flow = $750,000 – $750,000 = $0

$0 cash flow at three months would be worth $0 at six months. Alternatively, if the stock price at expiration is $80:

Long position = $80
Short call position = $75 – $80 = $ 5
Long put position = $ 0
Short Treasury bill position = -$75
Net position $ 0

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