22.7Hedging with Options

Options are used in two ways to hedge risk. In the first, a covered option strategy,

one option is issued or bought per unit of the asset or liability generating the risk expo-

sure. The resulting one-to-one hedge ratio places either a floor on losses or a cap on

gains. With the alternative, delta hedging, one first computes the option’s delta where

delta () is the number of units of the underlying asset in the option’s tracking port-

folio (see Chapter 7). Then, options in the quantity 1are issued or bought per unit

of the asset or liability generating the risk exposure. Delta hedging can, at least theo-

retically, eliminate all risk. Such hedging typically has a greater than one-to-one hedge

ratio because is generally between 0 and 1.

Why Option Hedging Is Desirable

For a variety of reasons, the payoff from an option hedge is sometimes preferred to the

payoff from hedged positions with futures or forwards. For example, a covered option

hedge can be used when managers want to partake in some upside risk. Portfolio insur-

ance (see Chapter 8), which offers this desirable payoff, can be created by acquiring

put options to partly offset the risk from holding assets.

Alternatively, options may be appropriate when the risk being hedged has some

option-like component. For example, many companies purchase swap options when

they enter into swaps that are designed to offset the interest rate risk of callable bonds

issued by the firm. Once the bond is called by the issuing firm, the interest rate swap

that formerly hedged the bond now hedges nothing. The interest rate swap now creates

rather than mitigates interest rate risk. In such a case, the previously purchased swap

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Markets and Corporate			Companies, 2002
Strategy, Second Edition

800Part VIRisk Management

option can be exercised when the bond is called to eliminate the risk from the interest

rate swap.¹¹

Metallgesellschaft represents another case where option-based hedging may have

been useful. If heating oil prices declined substantially, Metallgesellschaft may have

had to deal with a set of irate customers who demanded renegotiation of their contracts

to purchase heating oil at exorbitant prices. Faced with this option-like risk, Metallge-

sellschaft might have found a more suitable hedge by issuing call options on oil instead

of selling oil futures.

As Chapter 21 noted, options may be particularly useful in cases where firms would

like to hedge to minimize the probability of financial distress but do not want to elim-

inate all of the upside associated with favorable outcomes. For instance, in Example

21.3, National Petroleum wanted to eliminate the possibility of financial distress but

also wished to have sufficient cash flow in the event of an oil price increase to inter-

nally fund new exploration. Oil options are particularly useful in such cases because

they place a floor on the company’s oil revenues while allowing them to generate higher

revenues, and hence fund more exploration activity, as oil prices rise above the strike

price of the options.

Covered Option Hedging: Caps and Floors

The use of a single option in combination with a single position in the underlying asset

(or liability) implicitly creates what is known as a capor a floor,first discussed in

Chapter 2. Afloor, illustrated in panel Aof Exhibit 22.8, can be thought of as a call

option in combination with a risk-free security. It eliminates some downside risk—

namely, all values below the floor value—so one generally pays more to acquire a

EXHIBIT22.8Floors and Caps

	Panel A: Floor		Panel B: Cap
Position		Position
value		value

Cap value

Floor value

Future	Future
value of	value of
hedged	hedged
variable	variable

¹¹In many cases, such option exercise is suboptimal. Chapter 8 indicated that there are correct rules

for when to exercise an option early and when to defer exercise. The time at which the investor should

optimally exercise a swap option often depends on interest rate risk alone, while the exercise of a call

provision of a bond often occurs because the credit health of the company has improved. Hence, the call

of a bond by the issuing firm does not mean that the firm should exercise its swap option if it is trying

to maximize the option’s value.

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Markets and Corporate			Companies, 2002
Strategy, Second Edition

Chapter 22

The Practice of Hedging

801

floorlike position than a comparable payoff without a floor. Acap, illustrated in panel

B of Exhibit 22.8, can be thought of as a short position in a put option plus a risk-free

security. It eliminates some upside volatility—eliminating all outcomes above the cap

value—and thus costs less (or one is paid more) than the comparable payoff without

the cap.

Put-Call Parity.The put-call parity relation discussed in Chapter 8 is useful for

understanding the construction of caps and floors. Afloor is created by buying an option

on some underlying value or exposure of a firm. For example, if the value of the firm

decreases when the price of oil declines, buying a put on oil prices creates insurance

against the loss in the value of the firm that results from oil prices declining too much.

If the value of the firm decreases when the price of oil increases (for example, when

oil is a major input, not an output, in the production process), buying a call creates

insurance against the loss in firm value that results from oil prices rising too much.

Consider now a firm whose future value at date T,V, can be represented as a con-

T

stant value aplus the product of a coefficient band the price of oil S; that is

T

VabS	(22.1)
TT

Then bis positive when there is a positive relation between oil prices and firm value;

bis negative when there is a negative relation between oil prices and firm value. In

Chapter 8’s discussion of put-call parity, we learned that the future difference between

the date Tvalue of a call and a put is

c pS K

TTT

or

Sc pK	(22.2)
TTT

We now show how to create floor values or cap values that are related to the option

strike price of K.

Creating Floors.Substituting equation (22.2) into equation (22.1) tells us that the

firm’s value at a given future date Tis

VabKbc bp	(22.3)
TTT

The expression abKcan be thought of as a risk-free bond. With a positive value

for b,acquiring bputs at a cost of bpconverts the firm’s date Tvalue from that shown

0

in equation (22.3) to

*^T

VabK bp(1 r)bc

T0fT

where rdenotes the risk-free interest rate per period.

f

This payoff is like a call option plus a risk-free bond, where the risk-free bond has

the payoff

abK bp(1 r)^T

0f

The floor value of abK bp(1 r)^T

is generated when Sis small in this case.

0fT

If bis negative, buying bcalls, a positive number of calls, makes the firm value

*^T

VabKbc(1 r) bp

T0fT

Since bis negative, this is like having a risk-free bond plus a positive number of puts.

In this case, the floor value of abKbc(1 r)^T

takes effect when Sis large.

0fT

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Markets and Corporate			Companies, 2002
Strategy, Second Edition

802Part VIRisk Management

Example 22.11 provides a numerical illustration of how to create a floor by acquir-

ing options.

Example 22.11:Using Options to Create Floors on Losses

Assume that Metallgesellschaft has an obligation to deliver 1.25 million barrels of oil one

year from now at a fixed price of $25 per barrel.If oil prices rise to $45 a barrel, Metallge-

sellschaft loses $20 per barrel on this promise.How can Metallgesellschaft insure itself

against oil prices exceeding $30 per barrel, yet profit if oil prices decline, as its analysts are

forecasting?

Answer:Acquiring a call option to buy 1.25 million barrels of oil one year from now at

$30 per barrel will cap oil prices for Metallgesellschaft at $30 per barrel, and put a floor on

Metallgesellschaft’s losses should oil prices rise.

Creating Caps.Caps are constructed by shorting options. Starting with the firm

value shown in equation (22.3), shorting bcalls when bis positive generates a new

firm value of

*^T

VabKbc(1 r) bp

T0fT

This has the payoff of a risk-free bond and a short position in puts. This new value is

capped at abKbc(1 r)^Tbecause the short put position can never have a pos-

0f

itive value and the remaining terms on the right-hand side of the equation, the cap

value, are certain.

Similarly, with bnegative, shorting bputs, a positive number of puts, creates a

firm value of

*^T

VabK bp(1 r)bc

T0fT

This has the payoff of a risk-free bond and a short position in calls. The new firm

value is now capped at abK bpr)^T

(1 because the short call position can never

0f

have a positive value and the remaining terms on the right-hand side of the equation,

the cap value, are certain.

Currency Caps and Floors.Multinational companies often use currency option con-

tracts to hedge transaction exposure. Options on spot currency are traded on organized

exchanges, such as the Philadelphia Stock Exchange, while options on currency futures

trade on the Chicago Mercantile Exchange. Options also trade over the counter through

large commercial banks and other financial institutions.

Options on foreign currencies provide corporate foreign exchange managers with

a unique hedging alternative to the forward or the futures contract. The purchase of

options can create a floor. Selling options creates a cap. Options to buy a portfolio of

currencies, known as basket options, are also popular because—with a diversified, and

thus less volatile, basket of currencies underlying the option—they are less expensive

than buying a portfolio of single currency options.

Example 22.12 illustrates how to use a single currency option to create a floor on

foreign currency exposure.

Example 22.12:Using Currency Options to Create Floors on Losses

Return to the hypothetical case of Disney from Example 22.4, which needs to hedge the FFr

1 billion expected loss spread out over the next year.As the earlier example pointed out,

this is similar to a FFr 1 billion loss six months from now.How can Disney use currency

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Markets and Corporate			Companies, 2002
Strategy, Second Edition

Chapter 22

The Practice of Hedging

803

options to ensure that a six-month drop in the value of the dollar below FFr 5.15 per US$

will not make the dollar loss even larger?

Answer:Acquiring call options to buy 1 billion French francs with strike prices of

US$0.1942 per FFr (which is FFr 5.15 per US$) creates a floor on the dollar loss.The option

allows the firm to lock in the cost of purchasing French francs for up to six months at a spec-

ified price (the strike price).

Contrast the option in Example 22.12 with a forward contract. If FFr 1 billion was

purchased with a six-month forward contract at FFr 5.15 per US$ and the value of the

dollar increases from FFr 5 to FFr 6, the firm would be bound by the terms of the for-

ward contract and would not benefit from a dramatic rise in the dollar vis-à-vis the

franc.

Assume that a six-month call option to buy FFr 1 billion at the forward price of

US$0.1942 per FFr costs US$10,000 (or equivalently FFr 50,000 at FFr 5 per US$).

Such a call option insures against a drop in the dollar below the forward rate of FFr

5.15 per US$. The option costs US$10,000, which contrasts with the forward contract,

which costs nothing, because it enables a company like Disney to earn additional dol-

lar profits if the U.S. dollar appreciates against the French franc. This $10,000 cost

must be weighed against the benefit of being able to partake in an increase in the value

of the dollar (versus the franc).

It is interesting to note that the reference to calls or puts with currency options is

discretionary: the right to buyBritish pounds in exchange for a prespecified number of

dollars (a call) can also be viewed as the right to selldollars in exchange for a pre-

specified number of British pounds (a put). While both views of the option are correct,

this dual view raises the question of which risk-free interest rate to use for valuation:

the domestic interest rate or the foreign rate? With foreign exchange options, the inter-

est rate differentialbetween the two countries ultimately determines option values. In

addition, it is this differential that determines whether American currency options

should be exercised prior to their maturity date.¹²

Delta Hedging with Options

The caps and floors created above leave risk on one side—upside or downside risk.

However, as Chapter 8 indicated, options are tracked by a dynamic portfolio of the

underlying security and riskless bonds. The tracking portfolio’s investment in the under-

lying security, the option’s delta, referred to here as the spot delta, can be used in a

dynamic trading strategy to eliminate all risk exposure from the underlying asset or

liability.

Spot Delta versus Forward Delta.In the context of hedging currency or commod-

ity risk associated with some future obligation, it is sometimes useful to think of

options as a dynamic portfolio of forward contracts in the underlying security and risk-

less bonds. The forward deltarepresents the number of forward contracts that track

the option. In the case of a non-dividend-paying stock, the forward delta and the spot

delta of the option are the same. This means that if the option’s stock-bond tracking

¹²The

interested reader is referred to Margrabe (1978), who values exchange options. Subsequent

researchers have pointed out that European options to buy currency can be viewed as a special case of

exchange options.

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Markets and Corporate			Companies, 2002
Strategy, Second Edition

804Part VIRisk Management

portfolio contains two-thirds of a share of stock, the option’s forward contract-bond

tracking portfolio contains a forward contract to acquire two-thirds of a share of

stock.¹³However, if the stock pays dividends or if the underlying asset is a commodity

with a convenience yield or a currency with an interest rate that exceeds the interest

rate on the currency with which the strike price is paid, then the forward delta gen-

erally exceeds the spot delta.

Delta Hedging with the Forward Delta.We now illustrate how to apply forward

deltas to perfectly hedge risk. The use of forward deltas allows us to skirt the issue of

how convenience yields affect delta hedging. Consider once again a firm with a future

value of abS, where Sis the uncertain future spot price of a barrel of oil at date

TT

T. If (delta) represents thenumber of forwardbarrels of oil that track one option,

and is the number of risk-free dollars implicit in the option’s tracking portfolio, then

shorting boptions creates a firm with a riskless future value of

*^TT

Va(1r)(option cost)bS (b) S (1r)b

TfTTf

a(1r)^TT

(option cost) (1r)b

ff

The firm also may use options to alter the risk exposure from a commodity like oil

without completely eliminating the exposure. For the case above, shorting fewer than

boptions reduces but does not eliminate risk.

Example 22.13 assumes that Metallgesellschaft’s bis 1.25 million, which is the

number of barrels of oil generated by Metallgesellschaft’s delivery agreement. It enters

into option agreements to completely eliminate its exposure to oil price risk.

Example 22.13:Using an Option’s Delta to Perfectly Hedge Oil Price Risk

Assume that Metallgesellschaft has an obligation to deliver 1.25 million barrels of oil one

year from now at a fixed price of $25 per barrel.European options to buy oil in one year at

a price of $30 per barrel have a forward delta (according to the Black-Scholes formula of

Chapter 8) of .25.How many of these options should Metallgesellschaft buy to eliminate oil

price risk generated by the delivery agreement?

Answer:Acquiring call options to buy 5 million barrels of oil one year from now at $30

a barrel eliminates oil price risk.Each option has the same sensitivity to oil price changes

as one-fourth of a forward contract to deliver a barrel of oil.Thus, the firm needs four times

the number of options relative to forward contracts to perfectly hedge this risk.

Delta Hedges Are Self-Financing.The option hedging strategy in Example 22.13 is

a dynamic strategy that hedges only instantaneous changes in oil prices. As oil prices

change, the forward delta changes, implying that the number of options required for

the hedge needs to change. While an increase in the delta implies that cash is needed

to acquire additional options as the delta rises, the additional cash is balanced by the

profit on the present value of the promise to deliver 1.25 million barrels of oil at $25

a barrel. The reverse is true as well.

We summarize the results of this subsection as follows:

Result 22.6

If a firm’s exposure to a risk factor is eliminated by acquiring bforward contracts, then thefirm also can eliminate that risk exposure by acquiring boptions, where represents theoption’s forward delta.

¹³The

risk-free bond position in the two tracking portfolios always differs.

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Markets and Corporate			Companies, 2002
Strategy, Second Edition

Chapter 22

The Practice of Hedging

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