Chapter 6 Fundamentals of options hedging.
at the money
in the money
synthetic futures hedge
The objective of this chapter is to introduce one of the most powerful price risk management tools--options.
Options have existed for thousands of years. They have been known throughout history as "rights to buy" or "privileges." In the early 1900s privileges incurred a nasty reputation as highly speculative instruments with a high degree of default. The U.S. Congress banned privileges in agricultural and natural resource commodities through the 1936 Commodity Exchange Act. Options continued to be popular in real estate and in the securities business but never evolved in agricultural commodities after 1936 due to the ban. However, during the early 1970s several traders found a catch in the 1936 law that exempted "international" commodities. An active off-exchange market for options on rubber, cocoa, coffee, and some metals was created. It crashed in the late 1970s amid similar circumstances in the 1920s and 1930s-defaulted contracts. Meanwhile, options on equity stocks flourished and the Chicago Board Options Exchange became a major player in exchange-traded stock options when it opened for business in 1973. Real estate options and options on the actuals, especially on metals, continued to be popular. However, because of the hullabaloo over the failed return of options in the 1970s, the Commodity Futures Trading Commission (CFTC) banned all options on commodities, even the ones used successfully by physical traders. It took several court cases, which were decided by the U.S. Supreme Court, to restore option trading on the actuals. The CFTC opened pilot programs on options on futures in 1980 and by 1982 most exchanges had some option contracts available for trade. By the mid-1980s most futures contracts had options contracts. All major agricultural futures contracts now have options contracts and additionally there are numerous option contracts being traded off-exchange on most major agricultural commodities.
Futures contracts derive their value from the underlying cash commodity. An option on a futures contract derives its value from the underlying futures contract, which in turn derives its value from the underlying cash, thus the name double derivative. Double derivatives are also known as two-step derivatives. Options on the physicals such as equity stocks, real estate, and actual commodities are, however, only one-step derivatives because their value is directly derived from the underlying commodity. Only options on futures contracts are double derivatives.
These double derivatives have emerged as one of the foremost tools in managing the risk of price change in agriculture. Options are much more versatile than futures contracts and can be structured to provide price risk management through a wide spectrum of price levels.
The buyer of an option is buying the right, but not the obligation, to buy (call) or sell (put) something at some point in the future. The seller of the option has an obligation to perform if the buyer exercises the option. If the buyer decides to let the option expire then the seller's obligation is dissolved. The seller will demand compensation for taking on the responsibility to perform. That compensation is called the premium. The price that the commodity will exchange at should the option be exercised is called the strike price.
Option sellers are also called writers, underwriters, grantors, and the ubiquitous short. Option buyers are simply buyers or long. Options on futures are just like futures in that they are strong contracts and can be retraded. An initial buyer of an option can do three things: exercise, let expire, or retrade by selling. An initial seller of an option can actively do only one thing--retrade by buying. If the initial seller does not retrade the option, he will have to passively wait for the buyer to either exercise or let the option expire.
The Insurance Concept
The basic premise of insurance is that the owner of a policy has purchased the right but not the obligation to use the benefits of the coverage. The underwriter of the policy has agreed to be obligated to provide the coverage in the event the owner wants the coverage. The buyer pays a premium to the underwriter who agrees to provide a certain level of coverage. With a typical homeowner policy for fire coverage the seller would be the insurance company while the buyer would be the homeowner, the owner of the policy. The buyer pays the insurance company a premium for a certain level of coverage (strike price). If no fire occurs, the buyer would let the option expire and the insurance company would keep the premium. The buyer would then buy another policy for the next year. If a fire occurs, the buyer would probably exercise the right to the coverage (but is not required to). The insurance company would then be obligated to perform according to the contract terms.
Because options are so similar to insurance, the terms and concepts are almost interchangeable. Options on futures contracts can correctly be called price insurance. The buyer buys a certain level of price protection (strike price) and pays a premium for that protection. If the price protection is needed, the buyer exercises the option and if the protection is not needed, the option is allowed to expire. Futures contracts cannot provide this insurance-like protection. Furthermore, options and insurance are similar in that insurance companies are regulated by state insurance departments and options are regulated by exchanges and the CFTC.
Premium and Strike Prices
Options on futures are traded or offered on exchanges at various strike prices at fixed intervals above and below the futures price with correspondingly different premiums. Each day the exchanges always offer three strike prices above the current futures price, one near the current price and three below. As a minimum there will always be seven strike prices available for each delivery month. As the market price of the underlying futures contract changes, additional strike prices will be added. Toward the expiration date, literally dozens of strike prices will be available for trading. Figure 6-1 illustrates the concept with a December corn futures contract.
The July 1 relationships show three strike prices above the December corn futures price of $2.40 per bushel and three below. On July 2 the December corn futures price had moved up to $2.50 per bushel. A new $2.80 strike price had to be added to get three above the market and three below. Also, the $2.10 strike price is still active, providing four strike prices below the market. As the December corn futures price changes, the number of potential strike prices will increase.
In Figure 6-1 the strike price that is the same as the underlying futures price ($2.40 per bushel on July 1 and $2.50 per bushel on July 2) is called at the money. For puts (calls), the strike prices that are above (below) the at the money are said to be in the money. Out-of-the-money strike prices for puts (calls) are below (above) the at-the-money strike price. Some traders refer to in-the-money option strike prices as on the money and out of the money as off the money.
An at-the-money strike price is composed of only time value. On July 1, an at-the-money strike price on a call option on December corn futures is trading at a price of 15 cents per bushel. The buyer of the call option pays a premium of 15 cents per bushel to the writer for the right, but not the obligation, to have a long position in December corn at the strike price of $2.40 per bushel when the underlying December corn futures is trading at $2.40 per bushel. If buyers exercise the option immediately, they would have a long position in December corn at $2.40 per bushel for which they paid 15 cents per bushel. They could bypass the option and simply buy a December corn futures at $2.40 per bushel and not have to pay the 15 cents premium. Clearly the option buyer in this case has to have some value for the 15 cents and that value is time. The buyer is speculating that over time the December corn futures price will increase and having the right to buy it at $2.40 per bushel will be worth more than 15 cents per bushel.
The length of time to expiration of the option is directly related to the time value--the longer the time to expiration, the higher the premium. Obviously an option buyer will want the premium as low as possible, thus it is the seller of the option that plays the major role in premium values. If time is short to expiration, the probability that the price will move such that the buyer will exercise the option is lower than if the time frame is longer. Given a sufficiently long enough time horizon, prices can move very dramatically. The probability that a large meteor will hit the earth tomorrow is very small; however, if the time horizon is long enough, the probability will be one--it will happen with certainty. Time value for options is really the probability of a price move. Longer time periods have higher probabilities that prices will move, and vice versa.
Option premiums also have intrinsic value. Intrinsic value can be positive or negative, but only positive values are counted. All in-the-money options have positive intrinsic value. All out-of-the-money options have negative intrinsic value. In Figure 6-1 on July 1, the $2.20 strike price for a call has a premium value of 27 cents per bushel. The underlying futures contract is trading at $2.40 per bushel. If the buyer of the call exercised the call option, he would have a long position in December corn futures at the strike price of $2.20 per bushel. He could then sell the futures contract at the market price of $2.40 and have a net positive difference of 20 cents per bushel. The seller of the $2.20 per bushel strike price when the market is $2.40 per bushel will demand at least the 20-cent difference between the strike price and the market price for a premium. The difference between the strike price and the underlying market price is labeled intrinsic value. The premium for the $2.20 strike price is 27 cents. This 27 cents represents 20 cents of positive intrinsic value and the remainder, 7 cents, is time value.
On the other hand, the $2.70 strike price on July 1 for the call option is trading for 3 cents premium. If the buyer exercises the option, they will have a long position in the futures at $2.70 per bushel and have to sell at the current market price of $2.40 per bushel for a 30-cent loss. This 30-cent loss is a negative intrinsic value and consequently not counted. This out-of-the-money option will have value only if the market price moves so it only has time value (3 cents).
Positive intrinsic value is merely a bookkeeping concept. Sellers demand as a minimum the built-in profit (intrinsic value) of an in-the-money option. All that matters in the option premium is the time value. Every seller and buyer will have a different set of values for what they think will occur over time with the price. Since time value is really just the probability of a price move it becomes a function of how variable prices are during a given period and the length of the time periods. Therefore, a good rule of thumb for assessing time value of options is to look at the variability of prices and the length of time to maturity. Other things being equal, the more variable the prices, the higher the premium; and the longer the time period, the higher the premium, and vice versa. Option premiums are very important and over the last several years pricing models have been developed to address the trade-off between price variability and time.
Options Pricing: The Premium
The premium that a writer receives for an option and thus the amount that the buyer pays is very simply the price at which the options contract trades. Like every other good or service in an economy, the price at which the option trades is determined by supply and demand--the interaction of the interests of buyers (and their willingness to take on the risk associated with writing either a put or a call relative to the compensation they will receive to accept the additional risk) and sellers (and the cost of shifting the risk of adverse price movements relative to the benefits of risk avoidance). However, there is a third category of market participants, arbitragers, that play a role in options markets just like they do in every other market.
Before we consider the role of arbitragers, think about the major players in the markets--the buyers and the sellers. The buyers observe a price (the premium) in the market. For any given trading day, the strike prices are fixed, the length to maturity is fixed, market interest rates are generally steady (they will not usually change substantially in one day), and even though the price of the underlying futures contract is changing in the futures market, that price usually does not change within a range as large as it might over a week, over a month, or over six months. But, as all of these variables change, so does the premium--the trading price of the contract. All of these factors affect the supply of and demand for options contracts. Any one buyer or any one seller has relatively little market power--power sufficient to change price--especially if the buyer is seeking to hedge this year's crop of corn or calves or this month's wheat needs for milling.
Consider a single farmer selling his grain cartload of soybeans at the local grain elevator. That farmer has little power by himself to negotiate with the elevator manager to change his posted daily soybean price. The elevator manager is likely to believe that the next few farmers are already unloading their soybeans and taking the posted price even while our negotiator is laying out his story. The buyer (the elevator manager) himself has little power because he must retrade the soybeans, and market forces up the channel from him limit his power to pay more. He can, of course, pay the individual seller more but that will only cut into his profit. Similarly, the farmer could accept a price lower than the posted price, but that won't help him sell more.
In a very similar fashion, any one options buyer and any one options writer face a market price determined by market forces just like those faced by participants in the local grain market (Figure 6-2). If the premium increases, options sellers want to write more contracts (increase quantity supplied), if all other factors (underlying futures price, time to maturity, interest rates, and price volatility) are held constant in the short run. Conversely, options buyers would buy less as the price increases. Quickly, equilibrium--the trading price or premium--will be reached, or very few trades will take place. What would happen if price (premium) were too high to clear this market? Buyers would attempt to buy more than sellers would be willing to sell. The mechanism is the same for options contracts as it is for futures contracts or for the actual commodity. But how can a buyer or a seller really determine whether a given price, market-clearing or not, will meet his needs, especially if that buyer or seller finds himself in the position of being a price taker?
The Risk and Mitigation Profile process used in previous chapters is the best approach for a hedger to determine whether any particular options contract, along with its associated strike price and premium, meets his needs. Because the seller is assuming additional risk rather than offsetting it through a hedge, the RMP process is not the correct approach, but a similar assessment of the risk/return trade-off is necessary.
Arbitragers are aware of and participate in every market, from corn to cotton to sugar futures to soybean options to toys at garage sales and flea markets. Arbitrages keep markets honest and liquid. An arbitrager is always seeking an opportunity to exploit market imperfections. Suppose Elevator A was offering to sell corn at $3.25 per bushel and Elevator B (over 100 miles away from Elevator A with at least 25 other country elevators between) was offering to buy at $3.40 per bushel. Arbitrager C, who had stopped by Elevator A to check today's prices and have a cup of coffee, knows that regardless of the long-run volatibility of corn and diesel prices or of interest rates and options premiums he can load his truck at $3.25 per bushel and deliver to Elevator B and return home to Elevator A for another load for $0.08 per bushel. Before the morning is over, he can net $0.07 per bushel. He won't be able to do this too often before Elevator A raise its price or Elevator B lowers its price, or both.
[FIGURE 6-2 OMITTED]
Options traders arbitrage in a fashion that is more mathematically sophisticated than the one used by our grain hauler, but the process is directly analogous. The primary tool that they use to determine if there are arbitrage opportunities (no matter how brief or how minute) is called the Black-Scholes Options Pricing Model (BSOPM). Although it has been modified somewhat over the years and subjected to a great deal of criticism, research, and refinement attempts, the BSOPM used in the industry today is remarkably the same as the model originally developed by Fischer Black and Myron Scholes in their seminal article published by the Journal of Political Economy in 1973. (1)
Before we investigate the BSOPM, it is helpful and reasonably easy to determine the maximum and minimum price of an option. Assume the premium to be zero for a moment. What is the maximum price a rational buyer would be willing to pay? Since the call buyer receives the right to buy a futures contract, the maximum price she would be willing to pay is the price of the futures contract. Conversely, the minimum she could ever expect to pay is the intrinsic value, which is futures price minus strike price but always a non-negative amount. For a put buyer, what are the maximum and minimum prices he would be willing to pay or expect to pay? If futures prices fell to zero, the maximum value of his option contract is the strike price, so the maximum price of an option would be its strike price. Its minimum price, like the minimum price of a call, would be its intrinsic value. Of course, the actual price of an option, either a call or a put, will fall somewhere between the theoretical maximum and minimum. The exact amount depends on many factors, factors that are included in the BSOPM. Figure 6-3 describes the BSOPM.
Figure 6-3 The Black-Scholes Options Pricing Model CP = P N([d.sub.1]) - S [e.sup.-rt] N([d.sub.2]) where: [d.sub.1] = In (PIS) + [(r + [[sigma].sup.2]/2).sup.t] / [sigma] [square root of f] and [d.sub.2] = [d.sub.1] - [sigma] [square root of t] CP = call premium P = current security price S = option strike price e = base of natural logarithms r = riskless interest rate t = time until option expiration [sigma] = standard deviation of returns of the underlying security N(*) = cumulative standard normal distribution functions evaluated at * In = natural logarithm Express r, t, and [sigma] on the same temporal basis. Use days, weeks, months, exclusively; do not mix time periods.
The mathematical equation looks like it is difficult to compute and to interpret, but modern spreadsheets, especially those that can be programmed onto handheld computers, reduce the mathematical complexity into a very accessible, readily available, and very useful tool. For example, check out www.fis-group.com/finmodls.htm for a variety of BSOPM applications.
The BSOPM was developed for European options on stocks--corporate equity securities, not commodity futures. This influence is clearly seen in Figure 6-4. But, the model can be adapted easily to options on futures contracts, and is in practice. Pit traders do not use their spreadsheets to recompute the mathematical model for small changes in every variable, just like equities traders do not recompute the discounted present value model of future cash flows for every stock market trade. But, they fully and intimately know how the model reacts to a change in any of the variables. Their success as pit traders, as arbitragers, depends on it.
The BSOPM was built and remains dependent upon several assumptions:
1. All relevant markets (for options, futures, and underlying actuals) are efficient.
2. Interest rates are constant and known.
3. Returns are lognormally distributed random variables.
4. No commissions are paid.
5. The option can only be exercised on expiration.
The last assumption forces the model to specifically address European options rather than the retradeable, strong form American options used in the agricultural commodities markets. Current BSOPM forms adjust for this relaxation of the original assumption; exercise before expiration date shortens it. Commissions are not important to a pit trader and are simply an additional charge to be entered into a hedger's RMP. Interest rates are constant (or nearly so) on any given day.
The BSOPM yields a theoretical premium that may or may not be exactly the same as the market premium (the premium being quoted currently when buyers and sellers enter the market). Market imperfections or differences between reality and assumption cause disparity between theoretical and market premiums. Especially relevant are:
1. The true underlying distribution of prices is not exactly lognormally distributed.
2. Borrowing and lending interest rates are different.
3. Volatility expectations differ among market participants.
4. The market is less liquid than it usually is or is theoretically expected to be.
Regardless of relatively minor theoretical/market inconsistencies, all traders use the BSOPM or one of its close variants. Consequently, the BSOPM remains the cornerstone of modern options premium determinations in the marketplace.
Astute arbitragers assess differences between theoretical and market premiums, as well as relationships between call and put premiums on the same maturity and relationships among premiums across maturities. But more important than the assessment phase, astute arbitragers exploit those differences and inconsistencies. Exploiting temporary differences forces markets relentlessly toward a fundamental equilibrium.
Options Pricing: The Greek Values
The BSOPM is based upon several major variables. From Figure 6-3, we see P (the price of the underlying futures contract), S (the strike price), r (the interest rate), t (length of time to option maturity), and [sigma] (a measure of the volatility of futures contract prices). If all other things are held constant, as the futures price P increases, CP (the premium) would be expected to increase. That relationship between premium and futures price is referred to as delta ([DELTA]), as shown in Figure 6-4. Delta is computed as the rate of change of the premium relative to the futures price, [DELTA]CP/[DELTA]P or more correctly, [partial derivative]CP/[partial derivative]P in a continuous mathematical model. Gamma ([GAMMA]) measures the rapidity of the change in premium relative to the futures price, thus it is the second derivative, [GAMMA] = [[partial derivative].sup.2]CP/[partial derivative][P.sup.2]. Theta ([THETA]) measures how quickly the time value of an option decays (see Figure 6-5), and is computed as [THETA] = [partial derivative]CP/[partial derivative]t. The responsiveness of premium to volatility in futures prices is captured by lambda ([LAMBDA]), thus [LAMBDA] = [partial derivative]CP/[partial derivative][sigma]. And finally, rho ([rho]) reflects the responsiveness of premium to the interest rate; thus [rho] = [partial derivative]P/[partial derivative]r.
[FIGURE 6-5 OMITTED]
Buying Puts and Calls
Buyers of options have at least seven different strike price/premium combinations to select from and usually many more depending upon how long the option contract has been trading. Speculative buying of options is attractive to investors because the most that can be lost due to trading is the amount of the premium. Option buyers have the right but not the obligation to exercise the option so rational traders will only exercise those options that have a gain, not a loss. Option buying truncates the loss of trading at the premium level. However, gains are unlimited. Buying of options is attractive to risk averse traders and those that must know in advance what the loss factor will be for any speculative investment.
Table 6-1 reveals the limited loss factor and the unlimited gains associated with buying a call on December corn. The example in Table 6-1 shows two price increases and two price decreases. Notice that no matter what price December corn drops to, the most the buyer of the call can lose is the 15-cent premium. Similarly, as price increases the buyer gets the full amount of the price increase, less the premium. Speculators who buy call options have thus limited their total loss to the amount of the premium in bear markets but will participate in all of the bull market price increase, less the premium. Figure 6-6 shows the truncation of losses and unlimited gains potential. Notice that the most the buyer can lose is the 15-cent premium, no matter how low the price drops. Likewise, the solid sloping line up and to the right shows that once the price moves above the strike price of $2.40 per bushel, gains will occur for the buyer. The buyer will breakeven when prices move enough to cover the 15-cent premium ($2.55 per bushel). The short dashed line sloping down and to the left represents the loss potential of holding a long futures position at $2.40 per bushel. Losses are potentially unlimited with a futures position, but holding an option on having a long futures position (call) limits the loss. Buyers of call options are speculating that the price of the underlying futures contract will increase. If it does increase they can exercise the option and capture the price movement. If prices remain the same or decrease, they can simply let the option expire.
[FIGURE 6-6 OMITTED]
A speculator who thinks prices will decrease would buy a put option. Table 6-2 has the results of both a price decrease and increase with a put option. Just like with a call option, losses are truncated at the premium, but gains remain unlimited. Figure 6-7 reveals the gains associated with a price movement down from $2.40 per bushel but no matter how high prices increase, the loss is fixed at the premium level of $0.15 per bushel. The dashed line represents the potential loss if a short futures was held versus buying a put option. Buyers of put options are speculating that the price of the underlying futures contract will decrease.
Buying either a put or call limits the maximum loss to the amount of the premium and provides for unlimited gains. This is possible because an option buyer has the right but not the obligation to receive a futures contract. If a trader had directly purchased or sold a futures contract rather than an option on the futures contract, their maximum loss and gain are unlimited. Buying options allows traders to have their cake and eat it too-for the price of the premium.
Writing Puts and Calls
To write an option is to accept the responsibility of performing in case the buyer decides to exercise the option. Writers of options have the obligation to perform whereas buyers have the right but not the obligation to perform. The buyer pays a premium for the right and the writer gets the premium for taking on the obligation. The most income that a writer will get is the premium and the maximum loss is unlimited. Option writing truncates the gains of trading at the premium level. Why would any rational trader accept limited gains and unlimited losses? The concise answer is that writers have a better chance of earning limited gains versus facing an unlimited loss. Buyers of options must guess correctly which direction that price will move before they can earn any positive income. But if prices don't move at all or move in the other direction, the option will be allowed to expire. Enter the writer. Writers will collect their premiums if prices don't move and/or move in the opposite direction of what buyers want. Thus, writers generally have a higher probability of earning a limited income--the old "a bird in the hand is worth more than two in the bush" idea. Consequently there is no shortage of writers of options.
[FIGURE 6-7 OMITTED]
Writers of options on futures must decide if they want to write the option covered or naked. A covered writer will have the underlying futures contract that the option is written against. A naked writer does not have the underlying futures and is promising to provide one in case the buyer exercises the option. Writing an option covered or naked will depend upon what the writer's attitude is toward price movements.
Table 6-3 shows the effects of writing a put naked with both a price decrease and a price increase. The writer gains the maximum possible (the premium) when the price of December corn futures increases because the put option has become worthless to the buyer; thus, he allows it to expire. However, when the price decreases the buyer exercises the option and the writer has to perform. Since the option was written naked, there is no futures position to provide to the buyer. The clearing company of the exchange where the futures position was placed will create a short futures position at the strike price and give it to the option buyer. One half of a futures position cannot be created, so the writer gets assigned the opposite side (long). Writers of naked puts are bullish because they lose money when the price falls and make the maximum possible when prices rise.
Table 6-4 provides the results of writing puts covered. Notice that the effects are just the opposite of writing puts naked. Covered put writers are bearish because they lose money when the price increases and earn the maximum possible when prices decrease. The covered writer must pass along the futures contract to the buyer of the option when it is exercised complete with all of the paper profits the futures position has accrued. That, of course, is the reason the buyer exercises the option--it has profits. However, the writer still earns the maximum possible--the premium. A put option writer will cover the option if he or she is bearish and leave the option uncovered (naked) if he or she is bullish. This concept is easily visualized by looking at Figures 6-8 and 6-9. The covered writer earns the premium regardless of how low prices go but loses money as prices increase (Figure 6-8). When the option is naked, just the opposite occurs--the premium is earned regardless of how high prices rise, but losses occur as prices dip below the strike price as illustrated in Figure 6-9.
[FIGURE 6-8 OMITTED]
A call writer who is bullish will write the call covered, as shown in Table 6-5. The writer earns the maximum possible when prices increase and loses when prices decrease. The writer must pass along all of the paper profits in the futures position to the buyer, but earns the premium. When the market goes down, the writer must offset the losing futures position.
If a call writer is bearish, he or she will simply leave the option naked. If prices fall, as shown in Table 6-6, the buyer will let the option expire and the naked writer earns the premium. However, if prices increase, the buyer will exercise the option and receive a long futures position from the futures clearing corporation and the writer will be assigned the opposite (short) side and therefore incur a loss when the futures position is offset. A call option writer will cover the option if he is bullish and leave the option uncovered (naked) if he is bearish. Figure 6-11 displays the constant premium income regardless of how low prices dip and the losses that occur as prices increase when the option is naked. When the option is covered as shown in Figure 6-10, premium income is earned as prices increase and losses occur when prices decrease below the strike price.
[FIGURE 6-9 OMITTED]
[FIGURE 6-10 OMITTED]
[FIGURE 6-11 OMITTED]
A Note on Writing versus Buying Options
If an option writer is bullish, he can write a call covered or write a put naked. Go back to Figures 6-9 and 6-10 and notice that they are identical in terms of the pattern of returns--writing a call option covered has the same return profile as writing a put option naked. They are mechanically different, but alike financially. Likewise, if they are bearish, they can write a call naked or write a put covered. If the writer guesses correctly, he will earn the maximum possible--the premium. If he guesses wrong, he will be subject to unlimited losses. However, since options on futures are retradable, the wise option writer will trade out of the option with only limited losses. When the term covered is used, most rational individuals will assume that some amount of risk has been managed or covered. Yet one of the paradoxes of option writing is the meaninglessness of the term covered as it pertains to risk management. The previous examples clearly show that when an option is written covered, regardless of whether or not it is a put or call, the writer can have unlimited losses if prices move in the direction opposite of what the writer expected. Thus, writing options covered or naked is simply done to reverse the direction of price expectation and not done for any risk management reason.
Therein lies the problem with using option writing as a price risk management tool--losses are potentially unlimited, but gains are strictly limited to the premium regardless of whether or not the option was written covered or naked. The concept of counterbalance requires that the loss in one market or financial instrument be compensated by gains in another, and vice versa. Yet that is impossible when written options are used because gains are truncated while losses are unlimited. Consequently, the use of written options as a stand-alone risk management tool is impossible. Only when written options are combined into a more complex strategy can they have some value in risk management. Sadly, written options are widely touted as a hedging strategy when in fact they have very limited value as a hedge.
Buying options is a valid risk management tool because losses are limited and gains are unlimited. The concept of counterbalance holds with option buying because they behave just like insurance--losses are covered, but gains are not limited. The insurance policy is used when necessary, otherwise it is not. Written options, however, assume the role of the insurance provider--losses can be unlimited and gains limited to premiums. An insurance policy holder (the option buyer) has shifted the risk of financial losses from an event (managed the risk) to the insurance company (the option writer). The insurance company has absorbed the risk, not managed the risk, which is exactly what option writing does. Insurance companies contract with other insurance companies (re-insurers) to partially manage their risk and option writers can develop sophisticated strategies to offset or pass some of their risk to others. But, the initial act of writing the option, selling the auto insurance policy, or writing the fire insurance policy is a risk acceptance strategy, not a hedge or a risk management/minimization plan.
Because gains are limited with written options, only the buying of options will be used in simple hedging. Since the buying of options involves the right but not the obligation to have a futures position, hedging with options creates the right but not the obligation to hedge with a futures contract. With a simple futures hedge, when prices moved in favor of the cash position, the futures position lost an approximately equal amount. However, with an option hedge the holder can always let the option expire and not take a futures position. It is this simple characteristic that makes options very popular as hedges.
Put Option Hedging
A corn producer is concerned that the price of corn will decrease during the growing season and be at a very low level when the corn is harvested in the fall. A proper futures hedge would be to sell a corn futures contract during the growing season. The short futures position would gain in value if the corn price decreased. The corn producer has managed the risk of price change. The corn producer could also hedge with options. Instead of selling a futures contract, an option hedge for the corn producer would involve buying a put option on corn futures. A put is the right but not the obligation to sell a corn futures contract. Thus, the corn producer has bought the right but not the obligation to hedge. If corn prices go down, the producer will exercise the option and have a futures hedge. On the other hand, if prices go up and the hedge is not needed, the producer can let the option expire. Table 6-7 shows the effects of both a price decrease and a price increase.
The corn producer in the example would never receive anything less than $2.35 per bushel regardless of how low prices went. Yet when the cash price increased, the option can be allowed to expire and the producer will only lose the premium. This put hedge sets a floor below which the net effect to the producer will be fixed at the floor, but the upside is unlimited and reduced only by the amount of the premium. The producer simply takes the strike price and subtracts the premium amount to set the floor. Since numerous combinations of strike prices and premiums are available at any one time, the producer can adjust the floor to the level that he wants.
Call Option Hedging
A large feed company has forward sold pre-mixed feed for delivery in two weeks. The major ingredient is corn valued at $2.50 per bushel even though the company will not buy the corn until a day before delivery. The company has the risk that between the forward pricing of the corn at $2.50 per bushel and the actual purchase of the corn, that the price will increase and thus squeeze and/or eliminate the profit margin. Table 6-8 shows the effects of using a call option to hedge the risk.
The company's ceiling price was $2.65 per bushel. The company would never pay any more than that amount for corn regardless of how high prices increase. However, when prices fell, the company actually paid less than $2.50 per bushel for the corn. Call option hedging sets a ceiling on prices. In this particular case, if the company has any market power, they could have valued the corn in the ration at the ceiling price of $2.65 per bushel rather than $2.50 per bushel. The company simply would take the current price of $2.50 per bushel and add the call option premium of 15 cents per bushel and value the corn at $2.65 per bushel. The cost of the option is thus passed on to the buyer of the feed. Depending upon how price sensitive the buyer is will determine whether or not the seller could include the price of the option in the valuing of the feed.
Floors and Ceiling with Option Hedging
Using the two previous examples in Tables 6-7 and 6-8, Figures 6-12 and 6-13 illustrate the concept of price floors and ceiling with option hedging. Figure 6-12 shows that with a put option hedge, the lowest net hedged price (price floor) the corn producer will receive is $2.35 per bushel regardless of how low market prices get. Likewise in Figure 6-13 the highest price the feed company will pay for corn (price ceiling) is $2.65 per bushel no matter how high prices get. The figures show the real attraction of hedging with options--price protection, but also the ability to have a gain when prices move favorably. With Figure 6-12 the corn producer clearly got a price floor protection but was also able to keep all of the market price increase less the cost of option premium. The same goes for the feed company with call options as revealed in Figure 6-13. The feed company received the protection of a price ceiling but got the advantage of lower market prices when market prices decreased.
Hedging with futures certainly provides a price floor for a corn producer, but at the same time it furnishes a price ceiling. In other words, the producer is protected against price movements both favorable and unfavorable. Option hedging provides either a floor or ceiling for unfavorable price movements, and lets the hedger take advantage of favorable movements. Of course this attribute of options has a cost--the premium.
[FIGURE 6-12 OMITTED]
Hedging to a Certain Price/Cost
Similar to hedging price floors and ceilings is the use of options as a hedging tool to achieve a certain price or cost coverage. Since options have several strike prices to pick from when structuring a hedge, it is important to establish criteria for determining which strike price to hedge. Put option hedgers can subtract the premium from the strike price and derive the price floor, or also commonly called the target price. Table 6-9 shows hypothetical strike prices, premiums, and target prices. The formula for a target price is
target price = put strike price - premium
[FIGURE 6-13 OMITTED]
Call option hedgers are concerned about a price ceiling and thus if the premium is added to the strike price a target cost can be calculated as shown in Table 6-10. The formula for target costs is:
target cost = call strike price - premium
Both Table 6-9 and 6-10 use the same product, feeder cattle, and the same set of strike prices and premiums (reversed for puts and calls). Feeder cattle can be a final product for a rancher and thus need downside price protection and they can also be an input for cattle feeders and thus need upside price protection. Feeder cattle options thus provide a clear concept of both target prices and costs.
Target Price Hedging
Consider a rancher who is trying to decide whether to sell his calf crop as stockers or to carry them to heavier weights and sell them later as feeder cattle. The rancher can now use the information in Table 6-9 as a management tool in setting his criteria for making the decision to carry the animals to heavier weights. The rancher calculates that if he transforms the stocker cattle into feeder cattle by carrying them longer on grass and supplemental feed, his estimated cost of production is $95 per hundredweight. By using the information in Table 6-9, he has very valuable information in advance of making the decision. All of the information about the target prices in Table 6-9 is available in advance. The rancher, for example, could hedge by buying a one-step out-of-the-money put option at a strike price of $96 for a premium of 45 cents per hundredweight for a calculated target price of $95.55 per hundredweight as shown in Table 6-11. If the rancher's estimated cost of production is correct and he hedges with the $96 put option, his gross profit will be 55 cents per hundredweight. If his estimated final feeder weight is 700 pounds, his gross profit per head will be $3.85. This value is, of course, a minimum. If feeder cattle prices increase during the time the rancher held the animals, the option hedge could be allowed to expire and the rancher would gain from the cash price increase as revealed in Table 6-11. The target price hedge becomes the price floor. The rancher could also hedge with a $100 strike price, in which he would have a target price of $96.90 or a gross profit of $1.90 per hundredweight or $13.30 per head. The $100 strike price hedge involves a larger insurance premium than the $96 strike price and thus a larger negative cash flow, but it guarantees a larger profit potential--a classic management decision--higher guaranteed profits, but at a cost.
Target Cost Hedging
A cattle feeder needs feeder cattle as an input. Table 6-10 shows the array of possible costs of feeder cattle relative to the initial strike price used as a hedge. A cattle feeder, for example, might have the potential to forward sell the finished fed cattle for a certain price and then would estimate the feed and other costs involved, set a target profit margin, and determine that they could pay no more than $98 per hundredweight for the feeder cattle. Should the cattle feeder proceed? The target cost array shows several possible call option hedges that would satisfy the feeders' goals. If the cattle feeder hedges by buying the $97 strike price call, the target cost of feeders will be $97.50 per hundredweight as shown in Table 6-12. The cattle feeder achieves the profit margin she wanted with a feeder cattle cost of $98 per hundredweight and the hedge with the call option beat the price by fifty cents per hundredweight. Assuming a 700-pound feeder, the cattle feeder will add an additional $3.50 per head to the estimated profit margin. The hedge also allowed for the cattle feeder to capture a lower feeder cattle cost when prices move lower by letting the option expire as Table 6-12 reveals. The cattle feeder could also hedge with a $94 strike price at a substantial premium of $3.10 per hundredweight for a target cost of feeder cattle at $97.10 per hundredweight. The cattle feeder would gain a reduced cost of 40 cents per hundredweight for an additional cost of $2.60 per hundredweight. Given that the target price of feeders is achieved with the higher strike price and lower premium, the cattle feeder might not opt for the lower target cost with its corresponding higher negative cash flow. However, that is a management decision and can be made prior to the production decision using the concept of target cost hedging.
Calculating target prices and costs can be done in advance of many business, financial, and production decisions and thus can be very powerful as a management tool. Consider the simple process of placing cattle on feed. If a cattle feeder calculates that they can pay no more than $85 per hundredweight for the feeder cattle and assuming the prices in Table 6-10 are the only hedging choices, a cattle feeder has a potent bit of information. No possible hedge exists with call options that would guarantee a profit. The cattle feeder would have to lower their profit expectations and or find cost savings elsewhere, or simply not feed cattle.
Using the target price and costs arrays also leads to another potential type of option hedging--partial coverage (also known as worst-case or catastrophe hedging). Partial coverage hedging is the idea that only a certain level of protection is desired. This could be because of the cost of full coverage or the hedger may want to speculate that prices will move in their favor and thus they want only a worst-case type of coverage.
Partial Coverage Hedging
In the example in Table 6-11, a rancher is looking at carrying his stockers to feeder cattle weights and hedges with a put option and achieves a target price of $95.55 per hundredweight. In that example the rancher calculated that he needed to receive at least $95 per hundredweight. Assume instead that the rancher's total cost of production was $97 per hundredweight and that the rancher's fixed costs were estimated to be $93 per hundredweight. The rancher is generally bullish on feeder cattle prices, but wants to make sure he can make his land mortgage payments. He wants protection for only the fixed cash portion of the costs. Using the information in Table 6-9, the rancher could hedge with the $94 three-step out-of-the-money put option with a target price of $93.85 per hundredweight. This hedge would allow the rancher to cover his fixed costs and have 85 cents per hundredweight extra as a contribution to variable costs as a minimum. The rancher has partial coverage of his total costs with the possibility of more if his bullish expectations pan out, as shown in Table 6-13. In the example in Table 6-13, the rancher is protected at a price of $93.85 per hundredweight that doesn't cover all of his costs, but does cover his fixed costs, but in the event prices increase, he can take advantage of that as well.
The astute reader can clearly see how partial coverage hedging with options relates to car or home insurance. Full replacement insurance costs more than partial coverage, and likewise hedgers with options can use the concept to provide financial protection at various levels. How much property and casualty insurance coverage a business needs to carry is related to many factors--how much debt the business carries, owners' attitude toward risk, forecasts of risky events occurring, cash flow, and premium costs. Likewise, how much price insurance a business should carry is related to many of the same factors. Option hedging provides an almost limitless array of possible partial coverage opportunities for agribusinesses.
Option Multiple Hedging
To properly use option as a hedging tool requires knowing that options are double derivatives. Options on futures derive their value from the underlying futures contract that in turn derives its value from the underlying cash market. The relationship between the cash market price and the futures market price, as explained in Chapter 4, is called basis. Knowledge of basis and basis movements is critical to effective price risk management using futures contracts. The same is true for options. The relationship between the value of the option (the premium) and the underlying futures price is called delta (expressed as [DELTA]). The formula for delta is
delta = change in the option premium/change in futures price
Deltas range from zero to one. A delta of zero means that the option premium did not change when there was a change in the price of the underlying futures contract. Deep out-of-the-money options will have deltas close to or at zero. A delta of one implies that for every dollar change in the underlying futures price, the option premium changed a dollar as well. Deep in-the-money options have deltas that approach one or are one. As an option moves from at the money to in the money, intrinsic value is picked up and deltas start increasing and vice versa.
Deltas impact the concept of counterbalance. The idea of proper hedging stems from having the potential loss in the cash market value offset with a similar gain with a financial instrument used as a hedge. If the hedge involves an option with a delta of 0.5, then for every dollar the futures price changed, the option premium only changed by 50 cents, thus a major mismatch in counterbalance. Table 6-14 shows the impact of the change in the value of the option with a price decrease and increase. In the example, the futures hedge fully protected the cash position with either a price increase or decrease (assuming a zero basis and zero basis change). The option hedge only protected half of the cash price movement when prices moved down because the option premium changed by 50 percent.
To provide for dollar equivalency the option hedge in the example in Table 6-14 would need to be twice as large as the futures hedge. The process of achieving dollar equivalency between futures hedges and option hedges is called the multiple. The formula is:
multiple = 1/delta
With a delta of 0.5 in Table 6-14, the multiple would be 2 (1/0.5). The multiple is simply the inverse of delta. In the example in Table 6-14, two puts would need to be purchased instead of just one.
A proper multiple option hedge will mimic the effects of a futures hedge when the price moves against the cash position as it did when it decreased in the example in Table 6-14. In the example in Table 6-14, a multiple option hedge would be two put options purchased, for a total gain on the options of hedge of 10 cents (5 cents x 2). However, as the example in Table 6-14 also shows, when prices move in favor of the cash position, the option hedge will likewise mimic the futures hedge and the loss of 5 cents would double to 10 cents.
The central idea behind deltas and multiples is to understand how the value of the option as a hedge changes as the futures price changes. Hedgers that desire to have dollar equivalency throughout the hedging period will need to scale the hedge. As the futures price changes during the hedge period, so will the delta, and thus the multiple. If the hedge is not scaled (adjusted for size) then the hedge will not achieve dollar equivalency. The example in Table 6-15 displays a hedge that started out with deep out-of-the-money options with a delta of 0.1, but as the futures prices moved, and moved the options to at the money, the delta rose to 0.5. The hedge needs to be scaled down by offsetting. The end result of the scaling process in Table 6-15 produced a net hedged price of $2.82 per bushel which is close to the $2.80 per bushel strike price that was initially purchased as the level of desired protection.
Table 6-16 shows an option hedge that was scaled up to reflect a decreasing delta as the market price moved up and thus against a put position. The level of protection was initially set at $2.90 with the beginning strike price and the final net hedged price of $3.07 per bushel is considerably better since the cash price increased.
The process of scaling option hedges attempts to achieve delta neutral hedges. A true delta neutral hedge would achieve perfect dollar equivalency and thus a perfect hedge. A perfect hedge will mitigate all price risk, but will not allow any profit potential. Most option hedgers, therefore, are aware of the importance of delta as a measure of how option premiums change as the underlying futures price changes, but generally shun trying to achieve delta neutral hedges.
A Final Word on Deltas
Deltas of necessity have to be calculated after the fact, thus they are always historic and static. Markets are dynamic and therefore deltas will constantly be changing. Option hedges, consequently, are in constant need of being scaled. Scaling increases the cost of hedging by having to pay additional brokerage fees. Furthermore, as options hedges are scaled they more closely pattern a futures hedge. If the end result of an options hedge is similar to a futures hedge, then a futures hedge would be superior since option scaling involves higher brokerage fees. A wise options hedger is aware of the importance of deltas as well as their limitations.
Synthetic Futures Hedging
Option writing when combined with option buying can, if constructed properly, mimic a futures hedge. If a short futures hedge is desired, the combination of buying a put and simultaneously selling a call on the same futures contract will exactly reproduce the effects of a futures hedge as shown in Table 6-17. The opposite is true for a long futures hedge--buy a call and sell a put. This combination will exactly mimic the long futures hedge. In the example in Table 6-17 the futures hedge netted a price of $2.50 per bushel regardless of whether or not the price increased or decreased. Likewise, the synthetic set of options produced the exact same results.
A synthetic futures hedge is favored by some brokers because the client will have to pay two commissions for the options versus just one for a futures hedge. A synthetic futures hedge that has the options positions at the same strike price and roughly the same premium should be avoided. This type of hedge simply parrots a futures hedge and costs more to place. They should be avoided.
Obviously, limitless combinations of strike prices and premiums exist for the synthetic hedge. One of the most popular synthetic futures hedge involves buying an out-of-the-money put (call) and selling an in-the-money call (put) such that the hedge initially earns a net cash flow for the hedger. Table 6-18 illustrates this concept. This strategy is recommended by some brokers as a way for the hedger to earn an initial positive cash flow and thus it appears that the hedge actually makes a profit right off the bat. However, as the example shows, the effects are the same if prices go up, down, or do nothing.
To be sure, some times playing the game of putting on a synthetic hedge with differing premium values might earn the hedger a small net gain, but other combinations will cause a small net loss. As a rule of thumb, a hedger should merely hedge with a futures instead of a synthetic hedge with options. Unfortunately, the synthetic hedge is alive and well in the brokerage community and a prudent hedger should find another broker if the broker continues to recommend synthetic hedges.
Options Hedges Versus Futures Hedges
It might appear from the previous discussions and examples that options are clearly superior forms of hedging versus futures contracts; nevertheless, that is not the case. Options are a super way to hedge sometimes, and an inferior way to hedge at other times. Academics call options "second best" inasmuch as an option hedge is less effective than a futures hedge when prices move against the cash position. If prices move in favor of the cash position, an option hedge is more costly than not being hedged at all. The option hedge will always cost the premium and accordingly will always be "second best."
This idea of second best sounds correct and in fact is correct with one glaring fault-it can only be judged after the fact. Consider the example in Table 6-19. In the situation where the price moved against the cash position (down), the futures hedge was superior to the options hedge by the amount of the premium. When prices moved in favor of the cash position, the options hedge was superior to the futures position, but inferior to not being hedged at all--by the amount of the premium. Ergo the idea that options are always second best. But this is clearly unfair to both the hedger and options. No one knows in advance what prices will do. To say that an option hedge is second best to not being hedged at all is true, but in reality a worthless comparison. Options are used as a risk management tool and being unhedged is absorbing risk, not managing risk.
Instead of viewing options as second best, they should be considered a tool to be used in conjunction with futures contracts. The general rules for using one tool over the other are as follows: If prices are forecasted to move against the cash position, hedge with futures contracts. If prices are forecasted to move in favor of the cash position, hedge with options.
1. A March corn call option has a strike price of $2.50 per bushel. For the call to be in the money, what would the underlying March corn futures have to be?
2. What is the relationship between the strike price and the premium?
3. Explain why option writing has very limited risk management potential?
4. Why is option hedging similar to price insurance but futures hedging is not?
5. What determines the value of an option?
Table 6-1 Effects of Buying a Call Option July 1, buy 1 call option on December corn $2.40 Strike Price, 15 cents premium option is at the money Effect of Price Increase 1) Price increase to $2.70/bushel EXERCISE OPTION RECEIVE long December corn futures position @ $2.40/bushel; sell December corn futures @ $2.70/bushel $0.30/bushel gain less 15 cents per bushel premium = $0.15/bushel net gain 2) Price increase to $2.90/bushel EXERCISE OPTION RECEIVE long December corn futures position @ $2.40/bushel; sell December corn futures @ $2.90/bushel $0.50/bushel gain less 15 cents per bushel premium = $0.35/bushel net gain Effect of Price Decrease 1) Price decreases to $2.00/bushel Let option expire Net loss = $0.15 (the premium) 2) Price decrease to 1.80/bushel Let option expire Net loss = $0.15 (the premium) Table 6-2 Effects of Buying a Put Option July 1, buy 1 put option on December corn $2.40 Strike price, 15 cents premium option is at the money Effect of Price Increase 1) Price decreases to $2.70/bushel Let option expire Net loss = $0.15 (the premium) 2) Price decrease to $2.90/bushel Let option expire Net loss = $0.15 (the premium) Effect of Price Decrease 1) Price increase to $2.00/bushel EXERCISE OPTION, receive a short December corn futures position @ $2.40/bushel; buy December corn futures @ $2.00/bushel $0.40/bushel gain less 15 cents per bushel premium = $0.25/bushel net gain 2) Price increase to $2.90/bushel EXERCISE OPTION, receive a short December corn futures position @ $2.40/bushel; buy December corn futures @ $1.80/bushel $0.60/bushel gain less 15 cents per bushel premium = $0.45/bushel net gain Table 6-3 Effects of Writing a Put Naked Price increases to $2.80 Seller gains the premium 5 (Buyer lets the option expire) cents/bushel Write a Put December Corn Strike price @ $2.40/bushel (at the money) Premium 5 cents/bushel Price decreases to $2.00/bushel (Buyer exercises the option) Seller is assigned a long (buy) position in December corn futures at $2.40/bushel; seller offsets the buy position by selling a December corn futures @ $2.00/bushel (Loss of $0.40/bushel, gains the 5 cent/bushel Seller loses 35 cents/bushel premium for a net loss of $0.35/bushel) Table 6-4 Effects of Writing a Put Covered Price increases to $2.80/bushel Writer loses 35 cents/bushel (Buyer lets the option expire) Writer offsets futures position by buying December corn futures @ $4.80/bushel; loses $0.40/bushel but gains the premium of 5 cents for a net loss of $0.35/bushel Writes a put December corn Strike price @ $2.40/bushel (at the money) Premium 5 cents/bushel simultaneously sells a December corn futures @ $2.40/bushel to cover option; posts margin Price decreases to $2.00/bushel (Buyer exercises the option) Writer passes the sell position in December corn futures to buyer Writer gains the premium Seller loses 35 cents/bushel Table 6-5 Effects of Writing a Call Covered Price increases to $2.80/bushel Seller gains the premium (5 cents/bushel) (Buyer exercises the option) Writer passes the buy position in December corn futures to buyer Writer gains the premium Writes a call on December corn Strike price @ $2.40/bushel (at the money) Premium 5 cents/bushel simultaneously buys a December corn futures @ $2.40/bushel to cover option; posts margin Price decreases to $2.00/bushel (Buyer lets the Option expire) Writer offsets the long futures position by selling a December corn futures @ $2.00/bushel for a loss of $0.40/bushel Seller loses 35 cents/bushel but gains the premium of 5 cents for a net loss of $0.35/bushel Table 6-6 Effects of Writing a Call Naked Price increases to $2.80/bushel (Buyer exercises the option) Seller is assigned a short (sell) position in December corn futures @ $2.40/bushel Seller offsets the sell position by buying a December corn futures @ $2.80/bushel Loss of $0.40/bushel, gains the 5 cent premium for a net loss of $0.35/bushel Write a call on December corn Strike price @ $2.40/bushel (at the money) Premium 5 cents/bushel Price decreases to $2.00/bushel (Buyer lets the option expire) Seller gains the premium (5 cents/bushel) Table 6-7 Effects of a Put Option Hedge Cash Option July 1, corn crop growing current Buys a put option on December cash corn price @ $2.50/bushel corn futures at a strike price of $2.60/bushel and premium of $0.15/bushel (at-the-money option) Price Decrease Oct 1, harvests and sells corn Exercises the option; receives a locally @ $2.00/bushel sell position in December corn futures @ $2.60/bushel; buys December corn futures @ $2.10/ bushel; gains $0.50/bushel Less $0.15 premium/ $0.35 net gain Net hedged option price = $2.00 + $0.35 = $2.35/bushel. Price Increase Oct 1, harvests and sells corn Lets option expire locally @ $3.00/bushel Loss of premium = $0.15/bushel Net hedged option price = $3.00 - $0.15 = $2.85/bushel Table 6-8 Effects of a Call Option Hedge Cash Option Jan 2, forward sells pre-mixed Buy a call option on March corn feed for delivery in 2 weeks futures at a strike price of with corn valued @ $2.50/bushel $2.60/bushel Premium of $0.15/ bushel (at-the-money option) Price Decrease Jan 15, buys corn @ $2.30/bushel Lets the option expire Loss of premium of $0.15/bushel Net hedged option price = $2.30 + $0.15 = $2.45/bushel Price Increase Jan 15, buys corn @ $2.90/bushel Exercises the option; receives a buy position in March corn futures @ $2.60/bushel; sell March corn futures @ $3.00/bushel Gains $0.40/bushel Less $0.15/bushel premium/ $0.25/bushel gain Net hedged option price = $2.90 - $0.25 = $2.65/bushel Table 6-9 Target Prices for Feeder Cattle Put Options Put Strike Price Premium Target Price /cwt. /cwt. /cwt. $100.00 $3.10 $96.90 $99.00 $2.15 $96.85 $98.00 $1.35 $96.65 $97.00 $0.50 $96.50 $96.00 $0.45 $95.55 $95.00 $0.25 $94.75 $94.00 $0.15 $93.85 Table 6-10 Target Prices for Feeder Cattle Call Options Put Strike Price Premium Target Price /cwt. /cwt. /cwt. $100.00 $0.15 $100.15 $99.00 $0.25 $99.25 $98.00 $0.45 $98.45 $97.00 $0.50 $97.50 $96.00 $1.35 $97.35 $95.00 $2.15 $97.15 $94.00 $3.10 $97.10 Table 6-11 Put Target Price Hedge Cash Financial Decides to carry stockers to Buys put @ $96/cwt. heavier weights (feeder Pays premium of $0.45/cwt. cattle cash price $97/cwt.) Target Price = $96 - $0.45 = $95.55/cwt. Cash Price Decrease Net hedged price = $80 + $15.45 = $95.55 Cash Price Increase Sell feeder cattle for $105/cwt. Let option expire Lose premium of $0.45/cwt. Net hedged price = $105 - $0.45 = $104.55/cwt. Table 6-12 Call Target Cost Hedge Cash Financial Current feeder cattle Buys a call @ $97/cwt. price @ $98/cwt. Pays premium of $0.50 Target cost = $97 + $0.50 = $97.50 Cash Price Increase Buys feeder cattle @ $101/cwt. Exercise option Receives long feeder cattle futures @ $97/cwt. Sells @ $101/cwt. Gross margin of $4/cwt. less premium of $0.50/cwt. Net margin = $3.50/cwt. Net hedged price = $101 - $3.50 = $97.50/cwt. Cash Price Decrease Buys feeder cattle @ $93/cwt. Lets option expire Lose premium of $0.50/cwt. Net hedged price = $93 - (-$0.50) = $93.50/cwt. Table 6-13 Partial Coverage Hedge Cash Financial Decides to carry stockers to Buys put @ $94/cwt. heavier weights (feeder Pays premium of $0.15/cwt. cattle cash price $97/cwt.) Needs $97/cwt. to cover all Target price = $94 - costs and $93/cwt. to cover $0.15 = $93.85/cwt. fixed costs. Cash Price Decrease Sells feeder cattle for $80/cwt. Exercise option Receives short feeder cattle futures @ $94/cwt. Buys back @ $80/cwt. Gross margin of $14/cwt. less premium of $0.15/cwt. Net margin = $13.85/cwt. Net hedged price = $80 + $13.85 = $93.85/cwt. Cash Price Increase Sells feeder cattle for $105/cwt. Lets option expire Lose premium of $0.15/cwt. Net hedged price = $105 - $0.15 = $104.85/cwt. Table 6-14 Impact of Delta on Effectiveness on Option Hedging Cash Futures Options Buys @ $3.00 Sells @ $3.00 Buys put strike price @ $3.00 Pays premium $0.05, Delta 0.5 Price decrease Futures price change = $0.10 Option premium increases ($0.10 x 0.5 (delta)) = $0.05 Sells @ $2.90/ Buys @ $2.90/ Sells Put for 10 cents -$0.10 +$10.00 (5 cents + 5 cents)/+0.05 Price increase Futures price change = $0.10 Option premium decreases ($0.10 x 0.5 (delta)) = $0.05 Sells @ $3.10/ Buys @ $3.10/ Put premium = 0/ +$0.10 -$0.10 -$0.05 Table 6-15 Scaling an Option Hedge Down Cash Financial Buy corn (5,000 bushels) Buy put options (10) @ $2.80 @ $3/barrel strike price (out of the money) Premium = $0.02, delta = 0.1 Multiple = 10 Corn price goes to $2.80/bushel Delta = 0.5 Multiple = 2 Scale down Hedge Sell 8 put options @ $2.80 strike price (now at the money) Premium $0.05 $0.05 - $0.02 = $0.03 gain $0.03 x 40,000 (8 x 5,000) = $1,200 Sell corn @ $2.75/bushel/ Sell 2 puts @ $2.80 strike price -$0.25/bushel x 5,000 = -$1,250 (now at the money) Premium of 6 cents $0.06 - $0.02 = $0.04 gain $0.04 x 10,000 (2 x 5,000) = $400 $1,200 + $400 = $1,600 Total Gain Net hedged price = $1,600 - $1,250 = $350/5,000 = $0.07/bushel Table 6-16 Scaling an Option Hedge Up Cash Financial Buy corn @ $3/bushel Buy put options (4) @ $2.90 strike price (out of the money) Premium = $0.04, delta = 0.25 Multiple = 4 Corn price goes to $3.20/bushel Buy put options (6) Premium = $0.02, delta = 0.1 Multiple = 10 Sell corn @ $3.25/bushel Sell 10 put options -$0.25/bushel x 5,000 = -$1,250 Premium = 0.01 Buy @ 4 cents - 1 cents = - 3 cents x 20,000 = -$600 Buy @ 2 cents - 1 cents = - 1 cents x 30,000 = -$300 -$900 Net hedging gain = $1,250 - $900 = $350/5,000 = $0.07/bushel. Net hedged price = $3 + $0.07 = $3.07/bushel Table 6-17 Synthetic Futures Hedge Cash Futures Options Buy corn @ $2.50 Sell December Buy a put Sell a call corn @ $2.60 Strike price Strike price $2.60 $2.60 Premium $0.10 Premium $0.10 (at the money) (at the money) Price Increase Sell @ $3.00 Buy @ $3.10 Let put expire Buyer of call +$0.50 -$0.50 Lose $0.10 exercises option; receives sell position in December corn futures @ $2.50. Buy back @ $3.10/ -$0.50 Receive $0.10 -$0.10 Premium - $0.40 Net loss = $0.50 Net hedged futures price = Net hedged option price = $3.00 - $0.50 = $2.50 $3.00 - $0.50 = $2.50 Price Decrease Sell @ $2.00 Buy @ $2.10 Exercise put, Buyer lets -$0.50 +$0.50 receive expire sell position Earn premium December corn @ $0.10 @ $2.60. Buy back @ $2.10/ +$0.50 less Premium ($0.10) $0.40 Net gain on put plus $0.10 on call = $0.50 Net hedged futures price = Net hedged option price = $2.00 + $0.50 = $2.50 $2.00 + $0.50 = $2.50 Table 6-18 Synthetic Futures Hedge with Different Premiums Cash Buy corn @ Sell a call $2.50/bushel Buy a put Strike price $2.30 Strike price $2.30 Premium $0.23 Premium $0.03 Net Premium Gain of $0.20 (0.23 - $0.03) Price Decrease Sell @ Buyer lets expire Exercise the put, Gains $0.23 $2.00/bushel receive sell -0.50 futures position @ $2.30 Buys back @ +2.00/ +0.30 Less $0.03 Premium/ +0.27 +0.23 Net hedged price = Net gain of $0.50 $2.00 + $0.50 = $2.50 ($0.27 + $0.23) Price Increase Sell @ Let put option expire Buyer exercises option Lose $0.03 Seller assigned a sell $3.00 / bushel futures position @ $2.30 +0.50 Buys back @ $300/ -0.70 +$0.23/ -$10.47 Net hedged price = Net loss $0.50 $3.00 - $0.50 = $2.50 No Price Change Sell Let put option expire Buyer exercises option Lose - $0.03 Seller assigned sell futures position @ $2.30 $2.50/ $0.00 Buys back @ $2.50/ -$0.20 +$0.23/ +$0.03 Net = 0 ($0.03 x $0.03) Net hedged price = $2.50 + $0.00 = $2.50/bushel Table 6-19 Comparing Futures Hedges with Options Hedges Cash Futures Options Buy @ $2.50 Sell @ $2.60 Buy a put Strike price @ $2.60 Premium $0.05 (at the money) Price Decrease Exercise option; receive Sell position @ $2.60 Sell @ $2.00/ Buy @ $2.10/ Buy back @ -$0.50 +$0.50 $2.40/ +$0.50 Loss $0.05 premium/ +$0.45 Net hedged futures price = $2.00 + $0.50 = $2.50 Net hedged options price = $2.00 + $0.45 = $2.45 Price Increase Let option expire Lose premium = - $0.05 Sell @ $3.00/ Buy @ $3.10/ +$0.50 -$0.50 Net hedged futures price = $3.00 - $0.50 = $2.50 Net hedged options price = $3.00 - $0.50 = $2.95 Figure 6-1 Option strike prices and premium values for December corn futures contract Premiums Strike Price Puts Calls July 1 2.70 33 3 2.60 27 6 December 2.50 21 9 Corn futures $2.40 2.40 15 15 Price 2.30 9 21 2.20 6 27 2.10 3 33 July 2 2.80 33 3 2.70 27 6 December 2.60 21 9 Corn futures $2.50 2.50 15 15 Price 2.40 9 21 2.30 6 27 2.20 3 33 2.10 1 42 Figure 6-4 Options premium factors Factor Measure Computation Values Premium change [DELTA] [DELTA] = [DELTA]CP/ 0 < [DELTA] < 1 relative to [DELTA]P = [partial [DELTA] [right futures derivative]CP/ arrow] 1 Deep price change [partial derivative] in the money P [DELTA] [right arrow] 0 Deep out of the money [DELTA] [right arrow] 1 At the money Rate of change [GAMMA] [GAMMA] = [[partial 0 < [GAMMA] < of [DELTA] derivative].sup.2] [infinity] CP/[partial [DELTA] [right derivative][P.sup.2] arrow] 0 Deep in or out of the money Time [THETA] [THETA] = [partial 0 < [THETA] < option derivative]P/ value, decays as [partial option approaches derivative]t expiration date Volatility [LAMBDA] [LAMBDA] = [partial 0 < [LAMBDA] < derivative]P/ option value, [partial decays as option derivative][sigma] approaches expiration date Interest rate [rho] [LAMBDA] = [partial 0 < [rho] < derivative]P/ [infinity] [partial derivative]r