by Mark Rubinstein and Hayne E. Leland

Repiicating Options uuitiiPositions in Stocii aod Casii

In most situations of practical relevance, the price behavior of a call option is verysimilar to a combined position involving the underlying stock and borrowing. Thecall price and the stock price will change in the same direction. The effect on the callprice of a one dollar change in the stock price, however, will depend on the currentprice of the stock; the number of shares of stock in the replicating portfolio mustequal the slope of the call price curve at that price.

When the call is deep out of the money —i.e., when the stock price is muchlower than the striking price—a one dollar change in the stock price has little effecton the call price. When the stock price is equal to the striking price, a one dollarchange in the stock price produces roughly a half-dollar change in the call price. Ifthe stock price rises until the call is deep in the money a one dollar move in the stockprice results in nearly a one dollar move in the call price.

Because the call price behaves this way we must revise the replicating portfolio asthe stock price changes—selling stock as the share price falls and buying stock as theshare price rises. Since we are never fully invested when the stock price rises, norfully disinvested when the stock price falls, this process will deplete our initialinvestment. By the call's expiration date, the accumulated depletion will, in princi-ple, exactly equal the initial value of the call.

This concept permits one to replicate, not only calls, but many other optionpositions. Using replicating portfolios, institutions can create for themselves protec-tive puts and covered calls on stocks for which there is no options market.

THE VOLUME of trading in exchange-traded puts and calls {in terms of share

equivalents) now rivals share volume on theNew York Stock Exchange.' Yet, for most inves-tors, options remain an arcane or complex sub-ject. One thing is obvious, however: An optionprovides a comparable alternative to a direct in-

1. Footnotes appear at end of article.

Hayne Leland and Mark Rubinstein are Professors of Fi-nance at the Graduate School of Business, University ofCalifornia at Berkeley, and cofounders, abng with JohnO'Brien, of Leland O'Brien Rubinstein, Associates, an in-vestment counselling firm specializing in risk managementthrough dynamic investment strategies.

John Cox has not been listed as an author of this article,although the authors feel his contribution merited more thanequal coauthorship. The authors also thank John O'Brien forhis many useful comments.

vestment in its underlying stock. To decide be-tween the two alternatives, we would like toknow

—how to value an option,—how to measure the expected return and risk

of an option position,—how the margin requirements and transac-

tion costs of option positions compare withcommon stock,

—who should be buying and who should beselling options, and

—how to create an option position if optionson a stock or on a portfolio do not exist.

Is it quixotic to hope that there is a single,simple principle that can provide satisfactory an-swers to all these questions? We think not. Thekey insight to modem option pricing theory is

FINANCIAL ANALYSTS JOURNAL / JULY-AUGUST 1981 D 6 3

that, in most situations of practical relevance, theprice behavior of an option is very similar to a portfolioof the underlying stock and cash that is revised in aparticular way over time. ̂ That is, there exists areplicating portfolio strategy, involving stock andcash only, that creates returns identical to thoseof an option.

Why the Principle Makes SenseSuppose we want to replicate a purchased calloption using only stock and cash. To succeed, ourreplicating strategy must satisfy three condi-tions:

(1) for small changes in the stock price, theinitial out-of-pocket investment must givethe same absolute, dollar return as a call;

(2) to equalize the rate of return as well, theinitial out-of-pocket investment mustequal the value of the call; and

(3) thereafter, since a call requires no furtherinvestment, the replicating strategy mustbe self-financing.

If it satisfies these three conditions, we have rea-son to believe our strategy will resemble a pur-chased call at any time prior to expiration as wellas at expiration.

A brief study of exchange-traded options andtheir underlying stocks shows that their respec-tive daily price movements tend to parallel eachother. In particular, (a) call prices and stock priceschange in the same direction, but (b) a one dollarchange in the stock price causes a change of less thanone dollar in the call price. To satisfy Condition 1,the value of our stock-cash portfolio must, at aminimum, share these properties. This will beeasy to achieve if we have a long position in lessthan one share of stock.

Further observation of call prices shows thatthey have additional important properties ourreplicating portfolio will have to match. Moreprecisely, how a call option responds to a onedollar change in the stock depends largely on therelationship of its striking price to the stock price,(c) When the stock price is much lower than the strik-ing price (deep out of the money), a one dollar changein the stock price has little effect on the call price. If thestock price rises and becomes equal to the striking price(at the money), a one dollar change in the stock priceproduces about a half-dollar change in the call price. Ifthe stock price rises further so the call becomes deep inthe money, then a one dollar move in the stock priceresults in almost a one dollar change in the call price.

Because the call price behaves this way, wewill have to revise our replicating portfolio as the

stock's price changes. We will hold almost noshares when the stock price is low, and we willbuy more shares as the stock price rises. In par-ticular, when the call is at the money, we willhold about half a share. As the stock price risesfurther and the call becomes deep in the money,we will gradually buy in until we hold almost oneshare. Conversely, whenever the stock pricefalls, we will reduce the number of shares held.

Exhibit A compares the value of a call withvarious positions in stock and cash. Both thestriking price and current stock price are $30, sothe call is currently at the money. The straightline from the origin with a slope of one showshow the value of an unlevered position in oneshare of stock depends on the stock price; in thisvery simple case, these values are identical. Incontrast, the value of a fixed, fully levered posi-tion in the stock is represented by the straightline with a slope of one cutting the horizontal axisat 30.

The curve describing the cal! price as the stockprice changes is positively sloped throughout(corresponding to property a). Although theslope of this curve is always lower than the unle-vered stock line {property b), it increases con-tinually as the stock price rises (property c).-̂Indeed, the slope of the call price curve at the currentstock price is equal to the number of shares in thereplicating stock-cash portfolio. At very low stockprices, the slope is almost zero; at a stock price of30 the slope is one-half; and at very high stockprices the slope is almost one.

Equalizing dollar return is not enough. Wemust also equalize the rate of return (Condition2). Observation of call prices shows that (d) a oneper cent change in the stock price causes a more thanone per cent change in the call price. Our stock-cashportfolio will share this property if the stock posi-tion is financed partly through borrowing.

For example, suppose the initial at-the-moneycall value equals three dollars and the stock priceis $30. In this case, we will buy one-half share byinvesting three dollars and borrowing $12. Thecurrent value of our portfolio is thus three dollars— $15 worth of stock minus the $12 owed onborrowing. If the stock price then goes up by onedollar, the value of the portfolio will increase byonly 50 cents. However, this represents a 16%per cent increase for our portfolio, comparedwith a 3V3 per cent increase for the stock.

Exhibit A shows that the amount of borrowingin the replicating portfolio ($12) can be read offthe vertical axis by extending the dashed line

FINANCIAL ANALYSTS JOURNAL / JULY-AUGUST 1981 D 6 4

Exhibit A: Call Option vs. Positions in Stock and Cash

Value ofPosition

At-the-MoneyCall Price

Borrowingin replicating

portfolio

CurrentStockPrice

At the Money

Out of the Money

Slope of Ihisline is numberof shares inrepliciting portfolio

StockPrice

In Ihe Money

tangent to the call price line at the current stockprice ($30). The distance between the corres-ponding call price ($3) and the amount borrowed($12) equals the dollar value of stock in the repli-cating portfolio ($15).

Finally, to satisfy Condition 3, our strategymust be self-financing from this point on. Toaccomplish this, we borrow more to buy moreshares as the stock price rises and, as the stockprice falls, we sell some of our shares and use the

proceeds to retire a portion of our loan. Exhibit Ashows what is happening. As the stock pricerises, the dashed tangent line pivots coun-terclockwise, taking on increasing slope and anintercept farther from zero along the vertical axis.As the stock price falls, the tangent line pivotsclockwise, with decreasing slope and an inter-cept closer to zero.

By the expiration date, if the call ends up in themoney, we will find ourselves owning one share

FINANCIAL ANALYSTS JOURNAL / JULY-AUGUST 1981 D 6 5

of stock and owing from our borrowing an of interest, the stock's volatility or cash divi-amount exactly equal to the striking price. If the dends on the stock. For example, while an unan-call finishes out of the money, we will find our- ticipated increase in volatility wUl increase theselves fully disinvested with our borrowing fully value of a caU, it is certainly conceivable that suchrepaid. This is, of course, equivalent to the posi- a change could occur without affecting the pricetion of the caU buyer at expiration. In either case, of the underlying stock. Consequently, the valuesmce, prior to expiration, we were never fully of the call would change but the value of ourinvested when the stock price rose nor fully dis- stock-cash portfolio would not, no matter howinvested when the stock price fell, we will have we revised it.

depleted out initial out-of-pocket investment. It Although it may be impossible to find aIS a fact of modern option pricing theory that, strategy that will allow a portfolio of stock andsubject to certain conditions, in either case this cash to duplicate exactly the returns of a call"shortfall" will always be the same and exactly under all possible conditions, we nonethelessequal to the initial value of the caU. strongly believe that the concept of an option

being equivalent to a carefully adjusted positionSome Additional Factors in the underlying stock and cash is close enoughIn conclusion, we can replicate a purchased to being true in most situations of practical inter-

call position by a strategy of buying shares plus est to make it an invaluable tool for understand-borrowing, where we buy (sell) shares and increase ing options.(decrease) our borroiving as the stock price rises (falls).Of course, the current level of the stock price and Translating Option Positions intoits relation to the striking price wUl affect how Stock-Cash Equivalentsmuch stock we should be holding and how much If we can replicate a call, we can replicate anywe should be borrowing. In addition, the exact other type of option position as well. Exhibit Bcomposition of our replicating portfolio wUl also which translates the language of options into thedepend on other factors. For example, if the call more familiar language of stock and cash, showsIS sufficiently in the money, we should be hold- what we would need to do. For each option posi-rng more shares, the closer the option is ioexpira- tion in the center of the table, the correspondingtion or the lower the stock volatility, because stock-cash portfolio is given at the top of itsprofitable exercise is then more likely. Since we column and the appropriate adjustment strategyare borrowing against purchased shares, it is given at the ends of its row. For example, theshould be obvious that we wUl also need to take exhibit shows that buying a put is equivalent to aaccount of interest rates and cash dividends. The short position in the stock combined with lend-appendix shows how a call option can be repli- ing, which will be revised by lending more andcated exactly by a properly adjusted stock-cash shorting more stock when the stock price fallsportfolio. and by lending less and buying back stock to

The accuracy of the replicating strategy de- reduce the short position when the stock pricepends on four considerations. First, since the rises.strategy may involve frequent trading, it is nee- Some option positions are likened to "insur-essary that transaction costs be relatively insig- ance." For example, an at-the-money put pur-nificant. Second, it must be possible to borrow chased against a long share of stock protects thewhatever is required to form the replicating investor against loss if the stock price falls. Ourstock-cash portfolio (or, in the case of other op- analysis implies that this insurance effect arisestion positions, it must be possible to short the because the protective put is equivalent to a longstock). Third, the possibility of gap openings or stock-lending portfolio that is systematicallyjump movements in the stock price means that a shifted (1) away from stock and into cash as thecall can provide something that a levered stock stock price falls, providing a floor on losses, andposition cannot. To take an extreme case, sup- (2) into stock and away from cash as the stockpose a catastrophic event suddenly causes the price rises, permitting future gains or losses to bestock price to collapse to zero. This may happen realized.too fast for us to adjust our stock-cash position. To generalize, any hedged option positionAcall, on the other hand, will pay off our borrow- whose replicating portfolio involves shiftinging even in such a catastrophe. Fourth, there may away from a long or short stock position towardbe significant uncertainty surrounding future rates no stock as the stock price falls implicitly involves

FINANCIAL ANALYSTS JOURNAL / JULY-AUGUST 1981 D 6 6

the purchase of insurance. Conversely, anyhedged option position whose replicatingportfolio involves shifting further into a long orshort stock position as the stock price falls im-plicitly involves the sale of insurance. Therefore,the far left and far right upper (lower) optionpositions in Exhibit B amount to buying (selling)insurance. For example, buying protective putsis similar to the purchase of insurance, whilev/htin^ covered calls (buy one share and write onecall) is similar to the sale of insurance.

Option positions have also been compared toforward contracts, which promise delivery of anunderlying asset on a given date (delivery date)in the future at a currently agreed price (forwardprice). Like a call, a forward contract is equiva-lent to a levered position in the underlying asset.In contrast to a call, however, a forward contractrequires unconditional delivery, rather thanexercise at the option of the buyer. A forwardcontract could bte replicated by borrowing to fi-nance the entire holdings of the underlying assetand leaving this stock-cash position unrevisedthrough the delivery date."This is represented inExhibit A by the straight line cutting the horizon-tal axis at 30.

Is there an option position that also has thesetwo properties? If so, we can replicate forwardcontracts with options. Exhibit B shows that thereplicating portfolios for both long calls and shortputs involve long stock and borrowing.Moreover, the revision strategies for these twopositions move in opposite directions. There-fore, we might suspect that the proper combina-tion of the two positions would neutralize therequired revisions. As the stock price changes wewould find ourselves simply transferring stockbetween the two replicating positions with nonet purchases or sales required. Indeed, as itturns out, we can replicate a purchased forwardcontract exactly by buying one call and shortingone put with a common expiration date equal tothe delivery date and a common striking priceequal to the forward price.^

We are now prepared to answer the questionsposed at the beginning of the article. In eachcase, we simply need to examine the composi-tion of the stock-cash portfolios that replicateoption positions.

How to Value an OptionIf we can exactly duplicate an option with a

stock-cash position, we can also accomplish thereverse. If it turns out that the current market

price of an option differs from that of the replicat-ing portfolio, then we will have found an arbi-trage opportunity, since both the option and itsreplicating portfolio (which is self-financing) aresure to have identical payoffs at expiration.^Thus the value of an option is equal to the valueof its replicating stock-cash portfolio.

From this perspective, the problem of valuingan option is the same as the problem of determin-ing the composition of its current replicatingportfolio. The appendix provides an examplewhere the current replicating portfolio consistsof 5/7 shares of stock at $50 per share financedpartially with $22.50 of trorrowing. This impliesthat the current value of the call must be $13.20a$50x5/7] - $22.50).

The Black-Schoies option pricing model pro-vides another way of determining the composi-tion of the replicating portfolio. The Black-Scholes formula for a call takes the followingform:

call value = (stock price x delta)- borrowing.''

The "delta" is the standard terminology used inthe options market for the number of shares inthe replicating portfolio. If the market price of thecall exceeds this value, the call is overpriced; if itis less, the call is underpriced.

How to Measure the Expected Returnand Risk of an Option PositionIf we know how to measure the risk and return

of stock-cash portfolios, we can easily measurethese variables for option positions as well. Theleverage of any stock-<ash portfolio will be:

^ stock price x number of shares(stock price x number of shares)

- borrowing

where the leverage a will exceed one if we areborrowing and be less than one if we are lending.From this formula we can derive the expectedreturn, volatility and beta of any stock-cashportfolio:

a X stock expectedexpected return = return +

(1-a) X interest rate ,

volatility = a x stock volatility ,

beta = a X stock beta .Measuring the return and risk of an option

position (on a given underlying stock) entails

FINANCIAL ANALYSTS JOURNAL / JULY-AUGUST 1981 D 6 7

three steps. First, we translate each option intoits current replicating stock-cash portfolio; sec-ond, we aggregate across all the options to findtheir replicating net position as a group in stockand cash; third, we calculate the leverageparameter a for this netted position and applythe formulas for return and risk."

For example, since a properly priced pur-chased call is replicated by a margined long posi-tion in the stock, a will exceed one and the callwill be both more risky and have greater ex-pected return (provided the stock's expected re-turn is greater than the interest rate) than thestock itself. Similarly, a covered call will haveboth lower risk and lower expected return thanthe stock, since a will be less than one.^

Moreover, observe that the leverage measure aof the option position's replicating portfolio willtypically change continually through the future.Therefore, even if the expected return and risk ofthe underlying stock remain unchanged, the ex-pected return and risk of the option position willtypically change over time.

How the Margin Requirements andTransaction Costs of Option PositionsCompare with Common StockMargin requirements on common stock in-

volve (1) limits on borrowing against long posi-tions, (2) limits on the use of proceeds of shortsales and (3) collateral to guarantee performanceof short positions. An examination of theirequivalent stock-cash positions shows how op-tions can be used to relax each of these require-ments.

For example, buying a cal] will often prove away to relax the first requirement. Currently aninvestor can borrow only up to 50 per cent toinitiate purchase of stock. In contrast, one of ourprevious examples showed that one at-the-money call selling at three dollars was equivalentto the purchase of one-half share for $15, $12 ofwhich was borrowed. In other words, the callimplicitly allows borrowing 80 per cent (12/15) ofthe price of the stock.'" Moreover, the call mayimplicitly permit borrowing at more favorablerates than otherwise obtainable. Indeed, this islikely to be the case for retail investors if it is theinterest rates available to professionals that de-termine option prices. In this event, the lowerborrowing rates available to professionals will bepassed along to the public through lower callprices.

Buying puts may relax the second and third

margin requirements. Remember that the repli-cating portfolio of a put consists of selling stockshort and lending. Unlike many professionals,most retail investors cannot earn interest on theproceeds of short sales. Again, however, if theseprofessionals determine option prices, then theinterest they can earn on short sale proceeds willbe passed along to the public in the form of lowerput prices.'^ In addition, the lending containedin the replicating portfolios for many puts will beless than the collateral required to guarantee per-formance of short stock positions.

With respect to transaction costs, options andstock can be compared (1) dollar for dollar ofinvestment, (2) option contract vs. round lot ofstock or (3) option vs. replicating stock-cashportfolio. Under the first approach, optionscome out unfavorably; under the second, op-tions look very good for holding periods not ex-ceeding the life of the option. However, if wewant to compare positions of similar expectedreturns and risks, then neither of these ap-proaches is correct. The third approach, whichwill generally have implications intermediate be-tween the first two, is what we want. An analysisof the commissions usually charged for ex-change-traded options shows that options tendto dominate stock for short holding periods, butthat the advantage shifts to stocks for longer termpositions that exceed the life of the option. ̂ ^

Who Should be Buying andWho Should be Selling OptionsThe most frequently given reason for trading

options is that they offer new desired patterns ofreturns. Yet, as we have seen, much of whatoptions offer can be replicated by properly ad-justing a stock-cash position over time. Theremust be other considerations that incline inves-tors toward options.

As we have just seen, options may offer inves-tors more favorable implicit borrowing or lending op-portunities, margin requirements, transaction costs ortax exposure. Also, as we mentioned earlier,changes in stock volatility or dividends may verywell leave the current stock price unchangedwhile affecting the option price." Thus optionscan offer opportunities either to take advantage ofinformation about stock volatility or dividends or tohedge against their impact.

The question remains, however, whether theinvestor who decides to hold an option positionshould be buying or selling. The correspondencebetween options and replicating stock-cash

FINANCIAL ANALYSTS JOURNAL / JULY-AUGUST 1981 D 6 8

Exhibit B: Stock-Cash Portfolios to Replicate Option Positions

S Buy stock•f^ financed byoj borrowing

^ Sell stock^ and lend•̂ proceeds

Long Stock(no more than one share)+ +

Lending . Borrowing

long stock(one share)

long one put'

long stock(one share)

+short one callt

longonecall

shortoneput

Short Stock(no more than one share)+ +

Lending . Borrowing

longoneput

shortonecall

short stock(one share)

+long one call

short stock(one share)

+short one put

Sell stock >and lend ^proceeds o

o'Buy stock '̂

financed by ^borrowing "̂

NOTE; In all cases, any dividends received will be used to increase lending or reduce borrowing. Restitution fordividends paid while stock is held short will be financed by reduced lending or more borrowing.

•Protective put. tCovered call.

portfolios can help answer this question. Forexample, an average investor might want simplyto buy and hold stock, with no borrowing orlending. But if he were more risk-averse thanaverage, he might not want to assume the riskinherent in holding a typical stock. He can re-duce his risk by investing only part of his moneyin the stock and lending the remainder. If thestock price subsequer\tly rose, his risk wouldtend to increase as the relative dollar value of thestock-cash position shifted toward the stock. Atthe same time, as his position became more valu-able, he might become willing to accept morerisk. Indeed, if he were average in this respect,he would find that the increase in the stock priceautomatically injected just the desired amount ofrisk into the portfolio. He would then be contentto buy and hold, and would have no need foroptions.

However, suppose that, as the stock price roseand he became wealthier, the investor's willing-ness to accept more risk were less than the aver-age investor's. Then he would want to shift fromstock to cash gradually as his position becamemore valuable, and into stock from cash as hisposition became less valuable. He could, ofcourse, do this by continually revising hisstock-cash position. On the other hand, he couldlet a fixed covered call position achieve the sameresult automatically. Which strategy he preferswill typically depend on the comparative trans-action costs.

in brief, covered call writers should typically

be investors whose risk aversion does not decreaseas rapidly as the average investor's as the value oftheir portfolios increases. Conversely, protectiveput buyers should typically be investors whoserisk aversion decreases more rapidly than the aver-age investor's as the value of their portfolios in-creases.*" Similar reasoning applies to the otheroption positions in Exhibit B.

A completely separate reason for a preferencebetween buying or selling options rests on cer-tain technical theories of stock price behavior. Ifinvestors believe in trends, they may want to buyprotective puts or buy uncovered calls. If theybelieve in reversals, they should prefer writingcovered calls or writing puts.

What to Do When the CorrespondingOption Doesn't ExistIf options on a particular stock or on a portfolio

do not exist, we can create them by using theappropriate strategy for the underlying asset andcash. For example, we can effectively create anat-the-money protective put option on ourequity portfolio. We would begin by placing partof our capita] in the equity portfolio and part incash and then, without changing the composi-tion of the equity portfolio, shift between theportfolio and cash as the equity portfolio valuechanges and as the "expiration date" ap-proaches. Such an investment strategy would betantamount to insuring the equity portfolioagainst losses by paying a fixed premium to aninsurance company.'^

FnsIANCIAL ANALYSTS JOURNAL / JULY-AUGUST 1981 D 6 9

Even in the unlikely event that exchange-trad-ed put options of synchronous maturity existedon all the stocks in a portfolio, a combination ofput options could not match the above strategyin terms of cost. If we purchase put optionsagainst each stock in the portfolio, we would beinsuring each individual stock, rather than theportfolio as a whole, against loss; even if ourportfolio rose in value, as long as the price of atleast one stock fell the insurance would pay off.The insurance provided by a stock-cash strategywould pay off if and only if the portfolio fell invalue. As such, the implicit premium of the latterstrategy will be less because only the portfolio asa whole, not every individual stock, is insured.

Total portfolio option replicating strategies.

since they involve shifts between equities andcash, are similar to rebalancing strategies thatseek to maintain the risk level by keeping thesame relative amounts invested in equities andcash. In contrast, the total portfolio protectiveput option strategy systematically increases therisk of the overall position as the portfolio be-comes more valuable and decreases exposure torisk as the portfolio falls in value.

For many financial institutions, the replicatingstrategies for protective puts and covered callswill be feasible because they do not require bor-rowing or short selling. However, replication ofmany other option positions will not be possiblebecause they do require borrowing or shortselling. •

Appendix

A Simple Example of Replication of OptionReturns with a Portfolio of Stock and Cash

Suppose the current price of an underlying stockis $50 and that, over the next period, it will eithermove up to $70 or down to $35. If it moves up to$70 during the first period, then it will move to$100 or $50 during the second period. If it movesdown to $35 during the first period, then it willmove to $50 or $25 during the second period.The tree diagram in Exhibit AA illustrates thisbehavior.

Exhibit AA

Suppose that we can borrow money at an 11 percent rate of interest in each period. How can wereplicate the returns of an at-the-money pur-chased call that expires at the end of the secondperiod? First, let us see what we would do withjust one period remaining when the stock price isat $70. At expiration, the call will be worth $50($100 - $50) if the stock price goes up or zero if thestock price goes down (since we would then beindifferent to exercising it).

What mixture of stock and cash would pro-duce these same returns? Suppose we let A stand

50B=$22.50)

FINANCIAL ANALYSTS JOURNAL / JULY-AUGUST 1981 D 70

for the number of shares we would need to buyand B for the number of dollars we would need toborrow. Then our problem is to find values of Aand B such that:

(lOOxA) - (l .llxB) - 100 - 50 = 50 ,

(50xA) - ( l . l l x B ) = 0 .

The first equation insures that our stock-cashportfolio has the same return as the option if thestock goes up and the second assures us thereturns will also be equal if the stock goes down.It is easy to see that setting A equal to one and Bapproximately equal to $45 will solve both theseequations and therefore give us a stock-cash po-sition with the same returns as a call.

On the other hand, suppose we have oneperiod remaining and the stock price is at $35,then our problem is to find values of A and B suchthat:

(50xA)-( l . l lxB) = 0 ,

(35XA)-(l . l lxB) - 0 .

In this case the call has no chance of finishingin the money, so it makes sense that our solu-tions are A equals zero and B equals zero.

Finally, let us go back to the first period whenthe stock price was $50. Now we must find aportfolio of stock and cash that will (1) provide uswith just enough money to buy one share ofstock, financed by $45 of borrowing, if the stockprice goes up over the first period to $70, or (2)provide us with just enough money to buy zeroshares of stock, financed with zero borrowing, ifthe stock price goes down over the first period to$35. That is, in the first period we must choose Aand B so that:

(70xA) - (l .llxB) = (70x1) - 45 = 25 ,

(35xA)-( l . l lxB) = 0 .

In this case, A equals 5/7 and B equals approxi-mately $22.50.

To summarize, to replicate the call we willneed to start by buying 5/7 shares financed par-tially by borrowing $22.50. This implies we willhave to put up $13.20 ([50x5/7] - 22V2) of ourown money. If we do this, we will have justenough money to take the appropriate positionduring the second period, whether the stockprice goes up or down. If the stock price goes up,we will find ourselves subsequently buying in,financing the additional 2/7 (l-^h) shares by ad-ditional borrowing. If the stock price goes down,we will find ourselves subsequently completely

selling out and thereby raising just enoughmoney to repay our borrowing. In either case, onthe expiration date we will find that the value ofour replicating portfolio is exactly equal to thevalue of the call. •

Footnotes1. Acall option is a contract giving its owner the right

to buy a fixed number of shares of a specifiedcommon stock at a fixed price at any time on orbefore a fixed date. The act of making this transac-tion is referred to asexercising the option. The fixedprice is termed the striking price and the given date,the expiration date. The individual who issues a callis termed the writer and the individual who pur-chases the call is termed the buyer. A put option isidentical except it conveys the right to sell thestock.

2. The seminal articles developing the theory areFischer Black and Myron Scholes, "The Pricing ofOptions and Corporate Liabilities," Joumal of Polit-ical Economy, May-June 1973 and Robert C. Mer-ton, "Theory of Rational Option Pricing," Belljournal of Economics and Management Science, Spring1973. A considerably simplified development ofthis theory appears in John C. Cox, Stephen A.Ross and Mark Rubinstein, "Option Pricing: ASimplified Approach," Joumal of Financial Econom-ics, September 1979.

3. To simplify the figure, the interest rate on borrow-ing is assumed to be zero. If this were not the case,the call price curve would have the same zerovertical intercept, but be shifted somewhat to theleft. The exact position and shape of the curve isalso influenced by the time remaining to expira-tion and the stock volatility.

4. This conclusion presumes there are no carryingcosts or cash payouts involved in holding the un-derlying asset. However, the conclusion of thenext paragraph holds even with possibly uncer-tain carrying costs or cash payouts, as long as thereplicating options are not protected against thesecosts or payoffs and cannot be exercised prior toexpiration.

5. Readers may want to refer to Eugene Moriarty,Susan Phillips and Paula Tosini, "A Comparisonof Options and Futures in the Management ofPortfolio Risk," Financial Analysts Journal,January/February 1981. -—Ed.

6. indeed, many floor traders of options exchangesacting on their own account follow trading strate-gies based on this observation. They will almostsimultaneously buy or sell an option and take anopposing position in the stock (or a related deep-in-the-money option that behaves like the stock).Over time, they will adjust the composition of thisportfolio to keep its value insensitive to stock pricemovements, hoping to profit from mispricing ofthe option.

7. According to the original Black-Scholes formula,the current value of a call C depends only on itsunderlying stock price S, striking price K, time toexpiration t, the interest rate r - l , and the stock

FINANCIAL ANALYSTS JOURNAL / JULY-AUGUST 19S1 D 71

volatility (T. The composition of the replicatingportfolio consists of N{x) shares of stock, whereN(x) is the area under a standard normal distribu-tion function to the left of x and

, = log(S/Kr-') ^ , ,

The amount borrowed is Kr"*N(x-o-^t )_whereNix-a-JT) is the area to the left of x-tr^/t . Thevalue or a call is then

C = SN(x) - Kr- 'N(x-o-^) .

Although this formula applies only to stocks thatdo not pay dividends prior to the expiration date,it can he modified to include the effects of divi-dends.This technique will only work exactly over thevery short run, since the composition of the repli-cating portfolio changes as the expiration date ap-proaches. However, as shown in John C. Cox andMark Rubinstein, Options Markets (Prentice-Hall,1981), these short-run measures of return and riskwill usually be adequate approximations of thelonger run exact measures. This makes sense in-tuitively, since if the chances are roughly equalthat the stock price may rise or fall, then thechances are roughly equal that the short-run re-turn and risk measures will increase or decrease inthe future,Note that the conclusions of this paragraph willhold exactly in the long run as well. For example,since a covered call ahvays involves a replicating

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portfolio consisting of long no more than oneshare of stock and lending, it must always have\esBexpected return and risk than the stock by itself,although for some stock price outcomes its realizedreturn may be greater.A full analysis of margin requirements shouldconsider maintenance margins on stock and thatthe implicit leverage obtained through a call willchange as the stock price changes.Since the buyer of a put is implicitly lending, thenthe higher the interest rate he receives, the less hewill need to lend to come up with the striking priceon the expiration date if the put finishes in themoney. Therefore, the lower the current value ofthe put. Since calls involve implicit borrowing,similar reasoning shows their values will behigher, the higher the interest rate, other thingsequal.

A complete analysis would also consider any dif-ferences in the bid-ask spread typically sacrificedor gained by various classes of investors in optionsand stock.This will also be true of interest rates. But, in thiscase, interest rate futures would typically besuperior to currently listed options for hedging ortaking advantage of special information.

14. This correspondence was first discussed by HayneLeland, "Who Should Buy Portfolio Insurance?"Joumal of Finance, May 1980.Remember that the analogy to insurance breaksdown under a sudden catastrophic loss that doesnot leave sufficient time to adjust the replicatingportfolio.

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FINANCIAL ANALYSTS JOURNAL / JULY-AUGUST 1981 D 72