Topics 39-45 Flashcards

Question

Underlying assets in options

Answer 1

Exchange-traded options trade on four primary assets: individual stocks, foreign currency, stock indices, and futures. * **Stock options**. Stock options are typically exchange-traded, American-style options. Each option contract is normally for 100 shares of stock. * **Currency options**. Investors holding currency options receive the right to buy or sell an amount of foreign currency based on a domestic currency amount. For calls, a currency option is going to pay off only if the actual exchange rate is above a specified exercise rate. For puts, a currency option is going to pay off only if the actual exchange rate is below a specified exercise rate. The majority of currency options are traded on the over-the-counter market, while the remainder are exchange traded. * **Index options**. Options on stock indices are typically European-style options and are cash settled. Index options can be found on both the over-the-counter markets and the exchange-traded markets. The payoff on an index call is the amount (if any) by which the index level at expiration exceeds the index level specified in the option (the strike price), multiplied by the contract multiplier (typically 100). * **Futures options**. American-styie, exchange-traded options are most often utilized for futures contracts. Typically, the futures option expiration date is set to a date shortly before the expiration date of the futures contract. The market value of the underlying asset for futures options is the value of the underlying futures contract. The payoff for call options is calculated as the futures price less the strike price, while the payoff for put options is calculated as the strike price less the futures price.

Answer 2

Nonstandard option products include flexible exchange (FLEX) options, exchange-traded fund (ETF) options, weekly options, binary options, credit event binary options (CEBOs), and deep out-of-the-money (DOOM) options. * **FLEX options**. FLEX options are exchange-traded options on equity indices and equities that allow some alteration of the options contract specifications. The nonstandard terms include alteration of the strike price, different expiration dates, or European-style (rather than the standard American-style). FLEX options were developed in order for the exchanges to better compete with the nonstandard options that trade over the counter. The minimum size for FLEX trades is typically 100 contracts. * **ETF options**. While similar to index options, ETF options are typically American-style options and utilize delivery of shares rather than cash at settlement. * **Weekly options**. Weeklys are short-term options that are created on a Thursday and have an expiration date on the Friday of the next week. * **Binary options.** Binary options generate discontinuous payoff profiles because they pay only one price ($100) at expiration if the asset value is above the strike price. The term binary means the option payoff has one of two states: the option pays $100 at expiration if the option is above the strike price or the option pays nothing if the price is below the strike price. Hence, a payoff discontinuity results from the fact that the payoff is only one value - it does not increase continuously with the price of the underlying asset as in the case of a traditional option. * **CEBOs**. A CEBO is a specific form of credit default swap. The payoff in a CEBO is triggered if the reference entity suffers a qualifying credit event (e.g., bankruptcy, missed debt payment, or debt restructuring) prior to the option’s expiration date (which always occurs in December). Option payoff, if any, occurs on the expiration date. CEBOs are European options that are cash settled. * **DOOM options**. These put options are structured to only be in the money in the event of a large downward price movement in the underlying asset. Due to their structure, the strike price of these options is quite low. In terms of protection, DOOM options are similar to credit default swaps. Note that this option type is always structured as a put option.

Answer 3

In general, options are not adjusted for cash dividends. This will have option pricing consequences that will need to be incorporated into a valuation model. Options are adjusted for *stock splits*. For example, if a stock has a 2-for-1 stock split, then the strike price will be reduced by one-half and the number of shares underlying the option will double. In general, if a stock experiences a *b-for-a* stock split, the strike price becomes (*a/b*) of its previous value and the number of shares underlying the option is increased by multiples of (*b/a*). Stock dividends are dealt with in the same manner. For example, if a stock pays a 25% stock dividend, this is treated in the same manner as a 5-for-4 stock split.

Answer 4

The number of options a trader can have on one stock is *limited by the exchange*. This is called a position limit. Additionally, short calls and long puts are considered to be part of the same position. The exercise limit equals the position limit and specifies the maximum number of option contracts that can be exercised by an individual over any five consecutive business days.

Answer 5

* **Commissions** Option investors must consider the commission costs associated with their trading activity. Commission costs often vary based on trade size and broker type (discount vs. full service). Brokers typically structure commission rates as a fixed amount plus a percentage of the trade amount. Commission costs fail to account for the cost embedded in the bid-offer spread. The cost associated with this spread for options can calculated by multiplying the spread by 50%. For example, if the bid price is $12 and the offer price is $12.20, the associated cost for both the option buyer and option seller would be $0.10 per contract [(= $12.20 - $12.00) x 50%]. This cost is also present in stock transactions. * **Margin Requirements** Options with maturities nine months or fewer cannot be purchased on margin. This is because the leverage would become too high. For options with longer maturities, investors can borrow a maximum of 25% of the option value. Investors who engage in writing options must have a margin account due to the high potential losses and potential default. The required margin for option writers is dependent on the amount and position of option contracts written. **Naked options** (or *uncovered options*) refers to options in which the writer does not also own a position in the underlying asset. The size o f the initial and maintenance margin for naked option writing is equal to the option premium plus a percentage of the underlying share price. Writing *covered calls* (selling a call option on a stock that is owned by the seller of the option) is far less risky than naked call writing. * **The Options Clearing Corporation** Similar to a clearinghouse for futures, the **Options Clearing Corporation** (OCC) guarantees that buyers and sellers in the exchange-traded options market will honor their obligations and records all option positions. Exchange-traded options have no default risk because of the OCC, while over-the-counter options possess default risk. * **Other Option-Like Securities** Exchange-traded options are not issued by the company and delivery of shares associated with the exercise of exchange-traded options involves shares that are already outstanding. *Warrants* are often issued by a company to make a bond issue more attractive and will typically trade separately from the bond at some point. Warrants are like call options except that, upon exercise, the company receives the strike price and may issue new shares to deliver. The same distinction applies to *employee stock options*, which are issued as an incentive to company employees and provide a benefit if the stock price rises above the exercise price. When an employee exercises incentive stock options, any shares issued by the company will increase the number of shares outstanding. *Convertible bonds* contain a provision that gives the bondholder the option of exchanging the bond for a specified number of shares of the company’s common stock. At exercise, the newly issued shares increase the number of shares outstanding and debt is retired based on the amount of bonds exchanged for the shares. There is a potential for dilution of the firm’s common shares from newly issued shares with warrants, employee stock options, and convertible bonds that does not exist for exchange-traded options.

Answer 6

The following six factors will impact the value of an option: 1. S_o= current stock price. 2. X = strike price of the option. 3. T = time to expiration of the option. 4. r = short-term risk-free interest rate over T. 5. D = present value of the dividend of the underlying stock. 6. G = expected volatility of stock prices over T. *The Time to Expiration* For American-style options, increasing time to expiration will increase the option value. With more time, the likelihood of being in-the-money increases. A general statement cannot be made for European-style options. !!! Time to expiration (T) is not necessarily an increasing function for European calls/puts on _dividend_-_paying_ stocks. Time to expiration is an increasing function of (i) American calls/puts (on both dividend- or non-dividend-paying stocks) and (ii) European calls/puts on **non**-dividend-paying stocks. *The Risk-Free Rate Over the Life of the Option* As the risk-free rate increases, the value of the call (put) will increase (decrease). The intuition behind this property involves arbitrage arguments that require the use of synthetic securities. *Dividends* The option owner does not have access to the cash flows of the underlying stock, and the stock price decreases when a dividend is paid. Thus, as the dividend increases, the value of the call (put) will decrease (increase).

Answer 7

The derivation of put-call parity is based on the payoffs of two portfolio combinations, a fiduciary call and a protective put. A *fiduciary call* is a combination of a pure-discount (i.e., zero coupon), riskless bond that pays X at maturity and a call with exercise price X. The payoff for a fiduciary call at expiration is X when the call is out of the money, and X + (S — X) = S when the call is in the money. A *protective put* is a share of stock together with a put option on the stock. The expiration date payoff for a protective put is (X — S) + S = X when the put is in the money, and S when the put is out of the money. **c + Xe^-rT = S + p**

Answer 8

Put-call parity only holds for European options. For American options, we have an inequality. This inequality places upper and lower bounds on the difference between the American call and put options. ## Footnote **S₀- X \< C - P \< S₀- Xe^-rT**

Answer 9

c \>= S₀ - D - Xe^-rT p \>= D + Xe^-rT - S₀ Put-call parity is adjusted for dividends in the following manner: p + S₀ = c + D + Xe^-rT S₀- X - D \<= C - P \< =S₀- Xe^-rT

Answer 10

Holding an asset and a put on the asset is a strategy known as a protective put. Value at expiration: V_T = S_T + max(0,X – S_T) Profit: Π = V_T – S₀ – p₀ Maximum profit = ∞ Maximum loss = S₀ + p₀ – X Breakeven: S_T\* = S₀ + p₀ You can use a protective put to limit the downside risk at the cost of the put premium

Answer 11

An option strategy involving the holding of an asset and sale of a call in the asset. Value at expiration: V_T = S_T – max(0,S_T – X) Profit: Π = V_T – S₀ + c₀ Maximum profit = X – S₀ + c₀ Maximum loss = S₀ – c₀ Breakeven: S_T\* = S₀ – c₀ Finally, we should note that anecdotal evidence suggests that writers of call options make small amounts of money, but make it often. The reason for this phenomenon is generally thought to be that buyers of calls tend to be overly optimistic, but that argument is fallacious. The real reason is that the expected profits come from rare but large payoffs. Following this line of reasoning, however, it would appear that sellers of calls can consistently take advantage of buyers of calls. That cannot possibly be the case. What happens is that buyers of calls make money less often than sellers, but when they do make money, the leverage inherent in call options amplifies their returns. Therefore, when call writers lose money, they tend to lose big, but most call writers own the underlying or are long other calls to offset the risk.

Answer 12

A **bull spread** is designed to make money when the market goes up. In this strategy we combine a long position in a call with one exercise price and a short position in a call with a higher exercise price. To summarize the bull spread, we have: Value at expiration: V_T = max(0,S_T – X₁) – max(0,S_T – X₂) Profit: Π = V_T – c₁ + c₂ Maximum profit = X₂ – X₁ – c₁ + c₂ Maximum loss = c₁ – c₂ Breakeven: S_T\* = X₁ + c₁ – c₂ Bull spreads are used by investors who think the underlying price is going up.

Answer 13

If one uses the opposite strategy, selling a call with the lower exercise price and buying a call with the higher exercise price, the opposite results occur. The graph is completely reversed: The gain is on the downside and the loss is on the upside. This strategy is called a **bear spread**. The more intuitive way of executing a bear spread, however, is to use puts. Specifically, we would buy the put with the higher exercise price and sell the put with the lower exercise price. To summarize the bear spread, we have Value at expiration: V_T = max(0,X₂ – S_T) – max(0,X₁ – S_T) Profit: Π = V_T – p₂ + p₁ Maximum profit = X₂ – X₁ – p₂ + p₁ Maximum loss = p₂ – p₁ Breakeven: S_T\* = X₂ – p₂ + p₁ The bear spread with calls involves selling the call with the lower exercise price and buying the one with the higher exercise price. Because the call with the lower exercise price will be more expensive, there will be a cash inflow at initiation of the position and hence a profit if the calls expire worthless.

Answer 14

In both the bull and bear spread, we used options with two different exercise prices. There is no limit to how many different options one can use in a strategy. As an example, the butterfly spread combines a bull and bear spread. Consider three different exercise prices, X₁, X₂, and X₃. Suppose we first construct a bull spread, buying the call with exercise price of X₁ and selling the call with exercise price of X₂. Recall that we could construct a bear spread using calls instead of puts. In that case, we would buy the call with the higher exercise price and sell the call with the lower exercise price. This bear spread is identical to the sale of a bull spread. In summary, for the butterfly spread Value at expiration: V_T = max(0,S_T – X₁) – 2max(0,S_T – X₂) + max(0,S_T – X₃) Profit: Π = V_T – c₁ + 2c₂ – c₃ Maximum profit = X₂ – X₁ – c₁ + 2c₂ – c₃ Maximum loss = c₁ – 2c₂ + c₃ Breakeven: S_T\* = X₁ + c₁ – 2c₂ + c₃ and S_T\* = 2X₂ – X₁ – c₁ + 2c₂ – c₃ Butterfly spread is a strategy based on the expectation of low volatility in the underlying. Of course, for a butterfly spread to be an appropriate strategy, the user must believe that the underlying will be less volatile than the market expects. If the investor buys into the strategy and the market is more volatile than expected, the strategy is likely to result in a loss. If the investor expects the market to be more volatile than he believes the market expects, the appropriate strategy could be to sell the butterfly spread. Doing so would involve selling the calls with exercise prices of X₁ and X₃ and buying two calls with exercise prices of X₂.

Answer 15

A **calendar spread** is created by transacting in two options that have the same strike price but different expirations. The investor profits only if the stock remains in a narrow range, but losses are limited. In this case, the losses are not symmetrical as they are in the butterfly spread. A calendar spread based on calls is created in similar fashion. Calendar spreads are categorized differently depending on the relationship between the strike price and the current stock price. The strategy is referred to as a **neutral calendar spread** if the strike price is close to the current stock price. A **bullish calendar spread** has a strike price above the current stock price, and a **bearish calendar spread** has a strike price below the current stock price. A **reverse calendar spread** produces a payoff that is opposite of the graph shown in Figure 6. Instead of selling a short-dated option and buying a long-dated option, the investor of a reverse calendar spread will buy a short-dated option and sell a long-dated option. The investor will profit when the stock is well above or below the strike price and will suffer a loss if the stock is near the strike price.

Answer 16

A **diagonal spread** is similar to a calendar spread except that instead of using options with the same strike price and different expirations, the options in a diagonal spread can have different strike prices in addition to different expirations.

Answer 17

A **box spread** can also be used to exploit an arbitrage opportunity but it requires that neither the binomial nor Black–Scholes–Merton model holds, it needs no estimate of the volatility, and all of the transactions can be executed within the options market, making implementation of the strategy simpler, faster, and with lower transaction costs. In basic terms, a box spread is a combination of a bull spread and a bear spread. Suppose we buy the call with exercise price X₁ and sell the call with exercise price X₂. This set of transactions is a bull spread. Then we buy the put with exercise price X₂ and sell the put with exercise price X₁. This is a bear spread. Intuitively, it should sound like a combination of a bull spread and a bear spread would leave the investor with a fairly neutral position, and indeed, that is the case. (X₂−X₁)/(1+r)^T=c₁−c₂+p₂−p₁ If some combination of the options was such that the net premium is more than the present value of the payoff, then the box spread would be overpriced. So to summarize the box spread, we say that Value at expiration: V_T = X₂ – X₁ Profit: Π = X₂ – X₁ – (c₁ – c₂ + p₂ – p₁) Maximum profit = (same as profit) Maximum loss = (no loss is possible, given fair option prices) Breakeven: no breakeven; the transaction always earns the risk-free rate, given fair option prices

Answer 18

Suppose the investor buys both a call and a put with the same exercise price on the same underlying with the same expiration. This strategy enables the investor to profit from upside or downside moves. Its cost, however, can be quite heavy. In fact, a straddle is a wager on a large movement in the underlying. Only when the investor believes the market will be more volatile than everyone else believes would a straddle be advised. In summary, for a straddle: Value at expiration: V_T = max(0,S_T – X) + max(0,X – S_T) Profit: Π = V_T – (c₀ + p₀) Maximum profit = ∞ Maximum loss = c₀ + p₀ Breakeven: S_T\* = X ± (c₀ + p₀) As we have noted, a straddle would tend to be used by an investor who is expecting the market to be volatile but does not have strong feelings one way or the other on the direction. An investor who leans one way or the other might consider adding a call or a put to the straddle. Adding a call to a straddle is a strategy called a **strap**, and adding a put to a straddle is called a **strip**. It is even more difficult to make a gain from these strategies than it is for a straddle, but if the hoped-for move does occur, the gains are leveraged. Another variation of the straddle is a **strangle**, in which the put and call have different exercise prices. This strategy creates a graph similar to a straddle but with a flat section instead of a point on the bottom.

Answer 19

A **strangle** (or bottom vertical combination) is similar to a straddle except that the options purchased are slightly out-of-the-money, so it is cheaper to implement than the straddle. The payoff is similar to the straddle except for a flat section between the strike prices, as shown in Figure 8. Because it is cheaper, the stock will have to move more relative to the straddle before the strangle pays off. Strangles are also symmetric around the strikes. Answer: profit = max(0, S_T — X_H) + max(0, X_L—S_T) — C₀ — P₀ profit = max(0,40 —$50) + max(0,45 — 40) —1.50 —2 = $1.50

Answer 20

A **strip** involves purchasing two puts and one call with the same strike price and expiration. Figure below illustrates a strip. Notice the asymmetry of the payoff. A strip is betting on volatility but is more bearish since it pays off more on the downside. A **strap** involves purchasing two calls and one put with the same strike price and expiration. A strap is betting on volatility but is more bullish since it pays off more on the upside.

Answer 21

In effect, the holder of the asset gains protection below a certain level, the exercise price of the put, and pays for it by giving up gains above a certain level, the exercise price of the call. This strategy is called a **collar**. When the premiums offset, it is sometimes called a **zero-cost collar**. In summary, for the collar: Value at expiration: V_T = S_T + max(0,X₁ – S_T) – max(0,S_T – X₂) Profit: Π = V_T – S₀ Maximum profit = X₂ – S₀ Maximum loss = S₀ – X₁ Breakeven: S_T\* = S₀ Collars are virtually the same as bull spreads.

Answer 22

An **interest rate cap** is an agreement in which one party agrees to pay the other at regular intervals over a certain period of time when the benchmark interest rate (e.g., LIBOR) exceeds the strike rate specified in the contract. This strike rate is called the cap rate. For example, the seller of a cap might agree to pay the buyer at the end of any quarter over the next two years if LIBOR is greater than a cap rate of 6%. The buyer of a cap has a position similar to that of a buyer of a call on LIBOR, both of whom benefit when interest rates rise. Because an interest rate cap is a multi-period agreement, a cap is actually a portfolio of call options on LIBOR called **caplets**. The cap buyer pays a premium to the seller and exercises the cap if the market rate of interest rises above the cap strike. A long cap is *equivalent* to a portfolio of long put options on fixed-income security prices.

Answer 23

An **interest rate floo**r is an agreement in which one party agrees to pay the other at regular intervals over a certain time period when the benchmark interest rate (e.g., LIBOR) falls below the strike rate specified in the contract. This strike rate is called the **floor rate**. For example, the seller of a floor might agree to pay the buyer at the end of any quarter over the next two years if LIBOR is less than a floor rate of 4%. The buyer of a floor benefits from an interest rate decrease and, therefore, has a position that is similar to that of a buyer of a put on LIBOR, who benefits when interest rates fall and the price of the instrument rises. Once again, because a floor is a multi-period agreement, a floor is actually a portfolio of put options on LIBOR called **floorlets**. A long floor is *equivalent* to a portfolio of long call options on fixed-income security prices.

Answer 24

An **interest rate collar** is a simultaneous position in a floor and a cap on the same benchmark rate over the same period with the same settlement dates. There are two types of collars: * The first type of collar is to purchase a cap and sell a floor. For example, an investor with a LIBOR-based liability could purchase a cap on LIBOR at 8% and simultaneously sell a floor on LIBOR at 4% over the next year. The investor has now hedged the liability so that the borrowing costs will stay within the “collar” of 4% to 8%. If the cap and floor rates are set so that the premium paid from buying the cap is exactly offset by the premium received from selling the floor, the collar is called a “zero-cost” collar. * The second type of collar is to purchase a floor and sell a cap. For example, an investor with a LIBOR-based asset could purchase a floor on LIBOR at 3% and simultaneously sell a cap at 7% over the next year. The investor has now hedged the asset so the returns will stay within the collar of 3% to 7%. The investor can create a zero-cost collar by choosing the cap and floor rates so that the premium paid on the floor offsets the premium received on the cap.

Answer 25

A **package** is defined as some combination of standard European options, forwards, cash, and the underlying asset. Bull, bear, and calendar spreads, as well as straddles and strangles, are examples of packages. Packages usually consist o f selling one instrument with certain characteristics and buying another with somewhat different characteristics. Because packages often consist of a long position and a short position, they can be constructed so that the initial cost to the investor is zero.

Answer 26

Recall that standard exchange-traded American options can be exercised at any time prior to expiration. If some of the available expiration periods are restricted, or changes are made to other standard features, standard options become what we refer to as **nonstandard options**. Nonstandard options are common in the over-the-counter (OTC) market. There are three common features that transform standard American options into nonstandard options: * The most common transformation can be made to restrict early exercise to certain dates (e.g., a three month call option may only be exercised on the last day of each month.) This type of transformation results in a **Bermudan option**. * Early exercise can be limited to a certain portion of the life of the option (e.g., there is a “lock out” period that does not allow a 6-month call option to be exercised in the first three months of the call’s life). * The option’s strike price may change (e.g., the strike price of a 3-year call option with a strike price of 40 at initiation may rise to 44 in year 2 and 48 in year 3).

Answer 27

A **gap option** has two strike prices, X₁and X₂. (X₂ is sometimes referred to as the trigger price.) If these two strike prices are equal, the gap option payoff will be the same as an ordinary option. If the two strike prices differ and the payoff for a gap option is non-zero, there will be a gap in the payoff graph that is either increased or decreased by the difference between the strike prices. Gap options can be valued with a slight modification to the Black-Scholes-Merton option pricing model. * For a *gap call option*, if X₂is greater than X₁and the stock price at maturity, S_T, is greater than the trigger price, X₂, then the payoff for the call option will be equal to S_T — X₁. If the stock price is less than or equal to X₂, the payoff will be zero. Note that a negative payoff can occur if the stock price is greater than X₂ and X₂ is less than X₁. In this case, the payoff will be reduced by X₂ — X₁. * For a *gap put option*, if X₂ is less than X₁ and the stock price at maturity, S_T, is less than the trigger price, X₂ then the payoff for the put option will be equal to X₁ — S_T . If the stock price is greater than or equal to X₂, the payoff will be zero. A negative payoff can occur if the stock price is less than X₂ and X₂ is greater than X₁ Like with a gap call option, if this is the case, the payoff will be reduced by X₂ — X₁.

Answer 28

**Forward start options** are options that begin their existence at some time in the future. For example, today an investor may purchase a 3-month call option that will not come into existence until six months from today. Employee incentive plans commonly incorporate forward start options in which at-the-money options will be created after some period of employment has passed. Note that when the underlying asset is a nondividend paying stock, the value of a forward start option will be identical to the value o f a European at-the-money option with the same time to expiration as the forward start option.

Answer 29

**Compound options** are options on options. There are four key types of compound options: * A *call on a call* gives the investor the right to buy a call option at a set price for a set period of time. * A *call on a put* gives the investor the right to buy a put option at a set price for a set period of time. * A *put on a call* gives the investor the right to sell a call option at a set price for a set period of time. * A *put on a put* gives the investor the right to sell a put option at a set price for a set period of time. Compound options have two levels of the underlying that determine their value - the value of the underlying option, which in turn is determined by the value of the underlying asset. ! On the initial exercise date, the call price may be low (i.e., the asset price is low) so the owner of the compound put-on-a-call will exercise and sell the call. However, he/she is short a call option with (theoretically) **unlimited****loss** if the asset price reverses such that the long option holder can exercise for a gain on the second date. All of the other three compound options have limited downside: the call-on-a-call and call-on-a-put, if exercised, only leave the buyer with an option. And the put-on-a-put can “only” lose the strike price if the asset goes to zero.

Answer 30

This interesting option allows the owner, after a certain period of time has elapsed, to *choose whether the option is a call or a put*. The option with the greater value after the requisite time has elapsed will determine whether the owner will choose the option to be a put or a call.

Answer 31

**Barrier options** are options whose payoffs (and existence) depend on whether the underlying’s asset price reaches a certain barrier level over the life of the option. These options are usually less expensive than standard options, and essentially come in either *knock-out* or *knock-in* flavors. Specific types o f barrier options are: * *Down-and-out call (put)*. A standard call (put) option that ceases to exist if the underlying asset price hits the barrier level, which is set below the current stock value. * *Down-and-in call (put)*. A standard call (put) option that only comes into existence if the underlying asset price hits the barrier level, which is set below the current stock value. * *Up-and-out call (put).* A standard call (put) option that ceases to exist if the underlying asset price hits a barrier level, which is set above the current stock value. * *Up-and-in call (put).* A standard call (put) option that only comes into existence if the underlying asset price hits the above-current stock-price barrier level. Barrier options have characteristics that can be very different from those of standard options. For example, vega, the sensitivity of an option’s price to changes in volatility, is always positive for a standard option but may be negative for a barrier option. Increased volatility on a down-and-out option and an up-and-out option does not increase value because the closer the underlying gets to the barrier price, the greater the chance the option will expire. Note that the value of a down-and-out call combined with the value of a down-and-in call is equal to the value of a standard call option. In other words, by knowing the value of two of these three options you can calculate the value of the remaining option (e.g., down-and-out call = standard call — down-and-in call). Similarly, the value of a standard put option is equal to the value of an up-and-out put plus the value of an up-and-in put.

Answer 32

**Binary options** generate discontinuous payoff profiles because they pay only one price at expiration if the asset value is above the strike price. The term binary means that the option payoff has one of two states: the option pays a set dollar amount at expiration if the option is above the strike price, or the option pays nothing if the price is below the strike price. Hence, a payoff discontinuity results from the fact that the payoff is only one value — it does not increase continuously with the price of the underlying asset as in the case of a traditional option. In the case of a **cash-or-nothing call**, a fixed amount, Q, is paid if the asset ends up above the strike price. Since the Black-Scholes-Merton formula denotes N(d₂) as the probability of the asset price being above the strike price, the value of a cash-or-nothing call is equal to Qe^-rTN(d₂). An **asset-or-nothing call** pays the value of the stock when the contract is initiated if the stock price ends up above the strike price at expiration. The corresponding value for this option is S₀e^-qTN(d₁), where *q* is the continuous dividend yield.

Answer 33

Lookback options are options whose payoffs depend on the maximum or minimum price of the underlying asset during the life of the option. A **floating lookback call** pays the difference between the expiration price and the minimum price of the stock over the horizon of the option. This essentially allows the owner to purchase the security at its lowest price over the option’s life. On the other hand, a **floating lookback put** pays the difference between the expiration and maximum price of the stock over the time period of the option. This translates into allowing the owner of the option to sell the security at its highest price over the life of the option. *! strike price is dynamic, S_T is fixed (actual price at expiration)* Lookback options can also be fixed when an exercise price is specified. A **fixed lookback call** has a payoff function that is identical to a European call option. However, for this exotic option, the final stock price (or expiration price) in the European call option payoff is replaced by the maximum price during the option’s life. Similarly, a **fixed lookback put** has a payoff like a European put option but replaces the final stock price with the minimum price during the option’s life. *! strike price is fixed, but S_T is dynamic*

Answer 34

A **shout option** allows the owner to pick a date when he “shouts” to the option seller, which then translates into an intrinsic value of the option at the time of the shout. At option expiration, the owner receives the maximum o f the shout intrinsic value or the option expiration intrinsic value. In other words, for a shout call option, even if the price of the stock falls after the shout, the investor has locked in the difference between the price of the stock and the shout price. If the stock continues to rise, the shout option will have a payoff consistent with a standard call option. Note that most shout options allow for one “shout” during the option’s life.

Answer 35

**Asian options** have payoff profiles based on the average price of the security over the life of the option. *Average price* calls and puts pay off the difference between the average stock price and the strike price. Note that the average price will be much less volatile than the actual price. This means that the price for an Asian average price option will be lower than the price of a comparable standard option. *Average strike* calls and average strike puts pay off the difference between the stock expiration price and average price, which essentially represents the strike price in a typical intrinsic value calculation. If the average price or strike price for an Asian option is based on a geometric average, then using an option pricing model is not a problem because a geometric average is lognormal. However, most Asian options base their average calculations on arithmetic averages, which complicates the pricing process. In this case, a lognormal distribution of prices is assumed, which provides an adequate approximation.

Answer 36

A common use of an option to exchange one asset for another, often called an **exchange option**, is to exchange one currency with another. For example, consider a U.S. investor who holds an option to purchase euros with yen at a specified exchange rate. In this particular case, the option will be exercised if euros are more valuable to the U.S. investor than yen. Other applications, such as tender offers to exchange one stock for another, also arise in certain situations.

Answer 37

Basket options are simply options to purchase or sell baskets of securities. These baskets may be defined specifically for the individual investor and may be composed of specific stocks, indices, or currencies. Any exotic options that involve several different assets are more generally referred to as **rainbow options**.

Answer 38

Hedging positions in barrier and other exotic options are not so straightforward. This type of hedging requires the replication of a portfolio that is exactly opposite to the option position. When the replication portfolio requires frequent adjustments to the holdings in the underlying assets, the hedging procedure is referred to as dynamic options replication. **Dynamic options replication** requires frequent trading, which makes it costly to implement. As an alternative, a **static options replication** approach may be used to hedge positions in exotic options. In this case, a short portfolio of actively traded options that approximates the option position to be hedged is constructed. This short replication options portfolio is created once, which drastically reduces the transaction costs associated with dynamic rebalancing.

Answer 39

**F_0,T = E(S_T) e^(r-α)T** where: * r represents the risk-free rate of return * α represents the discount rate for the S_T cash flow at time T * E(S_T) is the expected spot price at time T Thus, the forward price today is a biased estimate of the expected commodity spot price at time T. The bias is a function o f the risk premium on the commodity, r — α . This equation is used to calculate the net present value (NPV) of commodities with available forward prices.

Answer 40

A **cash-and-carry** arbitrage consists of buying the commodity, storing/holding the commodity, and selling the commodity at the futures price when the contract expires. The steps in a cash-and-carry arbitrage are as follows: At the initiation of the contract: * Borrow money for the term of the contract at market interest rates. * Buy the underlying commodity at the spot price. * Sell a futures contract at the current futures price. At contract expiration: * Deliver the commodity and receive the futures contract price. * Repay the loan plus interest. If the futures contract is *overpriced*, this 5-step transaction will generate a riskless profit. The futures contract is overpriced if the actual market price is greater than the no-arbitrage price. If the futures price is *too low* (which presents a profitable arbitrage opportunity), the opposite o f each step should be executed to earn a riskless profit. This is **reverse cash-and-carry arbitrage**. The steps in reverse cash-and-carry arbitrage are as follows. At the initiation of the contract: * Sell commodity short. * Lend short sale proceeds at market interest rates. * Buy futures contract at market price. At contract expiration: * Collect loan proceeds. * Take delivery of the commodity for the futures price and cover the short sale commitment. *It may help to remember “_buy low, sell high._ ” If the futures price is “too high, ” sell the future and buy the spot. If the futures price is “too low, ” buy the future and sell the spot.*

Answer 41

A **lease rate** is the amount of interest a lender of a commodity requires. The lease rate is defined as the amount of return the investor requires to buy and then lend a commodity. From the borrower’s perspective, the lease rate represents the cost of borrowing the commodity. The lease rate and risk-free rate are important inputs to determine the commodity forward price. The lease rate in the pricing of a commodity forward is very similar to the dividend payment in a financial forward. The commodity forward price for time T with an active lease market is expressed as: **F_0,T= S₀e^{(r - δ_l)T}** where: S₀ = commodity current spot price r — δ_l = risk-free rate less the lease rate The lease rate, δ_l, is income earned only if the commodity is loaned out.

Answer 42

An upward-sloping forward curve indicates that forward prices more distant in time are higher than current forward prices. The market is described as being in contango with an upward-sloping forward curve. A **contango** commodity market occurs when the lease rate is less than the risk-free rate. Based on the commodity forward formula, F_0,T = S₀e^(r-δ_l)T, if r \>δ_l the _forward price must be greater than the spot price_. The market is described as being in **backwardation** with a downward-sloping forward curve. A backwardation commodity market occurs when the lease rate is greater than the risk-free rate. Based on the commodity forward formula, F_0,T = S₀e^(r-δ_l)T, if r \<δ the _forward price must be less than the spot price_.

Answer 43

When holding a commodity requires storage costs, *the forward price must be greater than the spot price* to compensate for the physical storage costs (i.e., costs associated with constructing and maintaining a storage facility) and financial storage costs (i.e., interest). If *storage costs are paid continuously and are proportional to the value of the commodity*, the noarbitrage forward price becomes: **F_0,T = S₀e^{(r + λ)T}** where: λ = continuous annual storage cost proportional to the value of the commodity Notice the *approximation* used in the previous example: F_0,T = S₀e^rT ~ S₀ x (1+r)^T Using either approach will produce similar results.

Answer 44

If the owners of the commodity need the commodity for their business, holding physical inventory of the commodity creates value. For example, assume a manufacturer requires a specific commodity as a raw material. To reduce the risk of running out of inventory and slowing down production, excess inventory is held by the manufacturer. This reduces the risk of idle machines and workers. In the event that the excess inventory is not needed, it can always be sold. Holding an excess amount of a commodity for a non-monetary return is referred to as **convenience yield**.

Answer 45

Because gold can earn a return by being loaned out, strategies for holding synthetic gold offer a higher return than holding just the physical gold without lending it out. When a positive lease rate is present, the synthetic gold is preferred to physically holding the gold because the lease rate represents the cost o f holding the gold without lending it.

Answer 46

Corn is an example of a commodity with seasonal production and a constant demand. Corn is produced every fall, but it is consumed throughout the year. In order to meet consumption needs, corn must be stored. Thus, interest and storage costs need to be considered. The price of the corn will fall as it is being harvested and then rise to reflect the cost of storage over the next 12 months until it is harvested again. Thus, the forward curve is increasing until harvest time, and it drops sharply and slopes upward again after harvest time is over.

Answer 47

As previously mentioned, electricity is not a storable commodity. Once it is produced, it must be used or it will likely go to waste. In addition, demand for electricity is not constant and will vary with time of day, day of the week, and season. Given the non-storability characteristic of electricity, its price is set by demand and supply at a given point in time. Since arbitrage opportunities do not exist with electricity (i.e., the inability to buy electricity during one season and sell it during another season) futures prices on electricity will vary much more during the trading day than financial futures.

Answer 48

Natural gas is an example of a commodity with constant production but seasonal demand. Natural gas is expensive to store, and demand in the United States peaks during high periods of use in the winter months. In addition, the price of natural gas is different for various regions due to high international transportation costs. Storage is at its peak in the fall just prior to the peak demand. Therefore, the forward curve rises steadily in the fall.

Answer 49

The physical characteristics of oil make it is easier to transport than natural gas. Therefore, the price of oil is comparable worldwide. In addition, demand is high in one hemisphere when it is low in the other. Lower transportation costs and more constant worldwide demand causes the long-run forward price to be more stable. In the short-run, supply and demand shocks cause more volatile prices because supply is fixed. For example, the Organization of Petroleum Exporting Countries (OPEC) may decrease supply to increase prices by causing a shortage in the short run. Supply and demand adjust to price changes in the long run.

Answer 50

A **commodity spread** results from a commodity that is an input in the production process of other commodities. For example, soybeans are used in the production of soybean meal and soybean oil. A trader creates a **crush spread** by holding a long (short) position in soybeans and a short (long) position in soybean meal and soybean oil. Similarly, oil can be refined to produce different types of petroleum products such as heating oil, kerosene, or gasoline. This process is known as “cracking,” and thus the difference in prices of crude oil, heating oil, and gasoline is known as a **crack spread**. For example, seven gallons of crude oil may be used to produce four gallons of gasoline and three gallons of heating oil. Commodity traders refer to the crack spread as 7-4-3, reflecting the seven gallons of crude oil, four gallons of gasoline, and three gallons of heating oil. Thus, an oil refiner could lock in the price of the crude oil input and the finished good outputs by an appropriate crack spread reflecting the refining process. However, this is not a perfect hedge because there are other outputs that can be produced such as jet fuel and kerosene.

Answer 51

An oil producer may enter into a contract to supply a fixed amount of barrels o f oil per month at a fixed price. The oil producer could set up a **strip hedge** by buying futures contracts that match the maturity and quantity for every month of the obligation. To help reduce transaction costs, the oil producer might instead utilize a **stack hedge**. To form a stack hedge, the oil producer would enter into a one-month futures contract equaling the total value of the year’s promised deliveries. As transactions costs are less for short-term (e.g., one-month) contracts, the total cost of implementing this strategy is less than for a comparable strip hedge. At the end of the first month, the producer rolls into the next one-month contract, and so forth, each month setting the total amount of the contract equal to the remaining promised deliveries. This strategy of continually rolling into the next near-term contract is referred to as **stack and roll**. A stack hedge has the advantage when near-term contracts are more readily available due to heavier volume and more liquidity. Another advantage o f near-term contracts is that distant futures on commodities often have wider bid-ask spreads and therefore larger transaction costs. In addition, an oil producer may prefer a stack hedge in order to speculate on the shape of the forward curve. For example, assume the forward curve looks unusually steep. The oil producer would then enter into a stacked hedge with a large near-term contract. If the forward curve later flattens, the oil producer locks in all the oil at a relatively cheap near-term price compared to the more expensive futures using the strip strategy.

Answer 52

In some cases, a futures contract with an underlying instrument that is exactly the same as the position to be hedged will not exist. For example, there are no contracts for jet fuel futures in the United States. Therefore, hedging jet fuel requires a **cross hedge**. Three factors are relevant when making a cross hedge decision: * The liquidity of the futures contract (since delivery may not be an option). * The correlation between the underlying for the futures contract and the asset(s) being hedged. * The maturity of the futures contract. A cross hedge is also applied when firms use **weather derivatives**. Weather risk is a business risk that is faced by agricultural firms as well as many firms involved with providing recreational services. It refers to any financial losses, explicit and implicit, that a firm faces from changes in the weather. The use of weather derivatives by other investors is growing, but one of the biggest problems is basis risk. That is, it is difficult to accurately match up the exposure of other assets to the weather with that specified by the contracts.

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