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American CappedCall Options onDividend-Paying AssetsMark BroadieColumbia University
two types of caps are analyzed constant capson-firstuals op-ingon-n be
s is artance.
mposium,o-thank theo anony-
the editor,is paper.e School
-191and caps with a constant growth rate. For cstant caps, it is optimal to exercise at the time at which the underlying assets price eqor exceeds the minimum of the cap and thetimal exercise boundary for the corresponduncapped option. For caps that grow at a cstant rate, the optimal exercise strategy caspecified by three endogenous parameters.
The valuation of American capped call optionproblem of theoretical as well as practical impoIndeed, in the past few years several securitibeen issued by financial institutions which incap features combined with standard Americoptions. One example is the Mexican Index-LEuro Security, or MILES. The MILES is an Ameri
This paper was presented at the 1993 Derivative Securities Sy1993 Western Finance AssociationMeetings, 1994 American Finance Assciation Meetings, Baruch College, and Universit de Genve. We participants of the seminars, Phelim Boyle, Bjorn Flesaker, and twmous referees for their comments. We are especially grateful toChi-fu Huang, for detailed suggestions which have improved thAddress correspondence to Mark Broadie, 415 Uris Hall, Graduatof Business, Columbia University, New York, NY 10027.
The Review of Financial Studies Spring 1995 Vol. 8, No. 1, pp. 161 1995 The Review of Financial Studies 0893-9454/95/$l.50McGill University and CIRANO
This art ic le addresses the problem of vaAmerican cal l opt ions with caps on d iv idpaying assets. Since early exercise is allothe valuation problem requires the detertion of optimal exercise policies. Options
the Bolsa has both a the option.payments.the security
ll (and put)roduced byer, 1991.ise featureaturity) and triggered ify. Flesakerex options.the optionhey are notall options
notes, and89) for a
ritten un-r unlimiteded liabilityissuer, or tosecurities inrstanding ofciples is im-for hedgings. Because beneficialunderstand
ly by stan-ation as tord binomialthe numbery is greatlyoptimal ex-
ll (1989)apped op-capped op-an cappedcall option on the dollar value of the Mexican stock index,Mexicana de Valores. The option is nonstandard because itcap and an exercise period that is less than the full life ofThe underlying asset is an index which involves dividend The decision to exercise the option is under the control of holder.
Other examples of options with caps are the capped caoptions on the S&P 100 and S&P 500 indices that were intthe Chicago Board of Options Exchange (CBOE) in NovembThese capped index options combine a European exerc(the holder does not have the right to exercise prior to man automatic exercise feature. The automatic exercise isthe index value exceeds the cap at the close of the da(1992) discusses the design and valuation of capped indThese options differ from the MILES because the holder of does not have any discretion In the exercise policy; i.e., tAmerican options. Other examples of European capped cinclude the range forward contract, collar loans, indexedindex currency option notes [see Boyle and Turnbull (19description of these contracts].
The motivation for introducing capped options is clear. Wcapped call options are inherently risky because of theiliability. By way of contrast, capped call options have limitand are therefore attractive instruments to market for an hold short for an investor. Wide acceptance of these new the marketplace, however, depends on a thorough undetheir features and properties. Understanding valuation prinportant not only for pricing these new securities, but also the risks associated with taking positions in the securitieearly exercise (i.e., before the options maturity) may bewith American capped call options, it is also essential to the structure of optimal exercise policies.
While these types of securities can be valued numericaldard techniques, numerical results alone offer little explanwhy certain exercise strategies are optimal. When standavaluation procedures are applied to capped call options, of iterations required to obtain a desired level of accuracincreased compared to uncapped options. In this article, ercise policies and valuation formulas are derived.
Options with caps were studied by Boyle and TurnbuTheir article provides valuation formulas for European ctions as well as insights about early exercise for American tions. The possibility of optimal early exercise with Americ162
American Capped Call Options on Dividend-Paying Assets
call options was recognized by Boyle and Turnbull. They point out the cap,
in Cox Reiner
ature andered if theautomaticd indexding day.ces them
aluationives anlas-ncappedature ishave the incentive
arisesAmericane writtenand Carr,
ise early. previousrcise re-changede policyets pricexer-For low-simplifiesfollows anal divi-
rcised delayedgrowinggle jump. potentialsuers whounction isthat when the underlying asset price substantially exceedsimmediate exercise dominates a waiting policy.
Barrier options are related contracts which were studiedand Rubinstein (1985) and more recently in Rubinstein and(1991). Barrier options combine a European-style exercise fean automatic exercise feature. The automatic exercise is triggprice of the underlying asset reaches the barrier (cap). The exercise feature is slightly different than for CBOEs cappeoptions, where exercise can only occur at the close of a traThe combination of exercise features of barrier options -plabetween European options and American options.The presence of a cap on a call option complicates the v
procedure. The floor on an option (i.e., the strike price) gincentive to exercise as late as possible. This is the source of the csical result that early exercise is suboptimal for American ucall options on non-dividend paying stocks. The American feworthless in this case, and American and European options same value. When the underlying asset pays dividends anto exercise early is introduced. The optimal exercise boundary from the conflict between these two incentives. For standard options without caps, the optimal exercise boundary can bas the solution to an integral equation [see, e.g., Kim (1990) Jarrow, and Myneni (1992)].
The introduction of a cap adds a further incentive to exercOn the surface, it seems that the cap could interact with theincentives to completely alter the form of the optimal exegion. However, we show that the optimal exercise region is in a straightforward way. We show that the optimal exercisis to exercise at the first time at which the underlying assequals or exceeds theminimum of the cap and of the optimal ecise boundary for the corresponding uncapped call option. and non-dividend paying assets the optimal exercise policy to exercising at the cap. When the underlying asset price geometric Brownian motion process with constant proportiodends, an explicit valuation formula is given.
The valuation formula is then generalized to options with delayedexercise periods, that is, American options that cannot be exebefore a prespecified future date. The MILES contract has aexercise period, which can be viewed as a time-varying or cap. In this case, the growing cap is a step function with a sinGrowing caps may be preferred by investors since the upsideincreases over time. These caps may also be attractive to iscan accept a larger potential liability as time passes. A step f163
The Review of Financial Studies / v 8 n 1 1995
an extreme form of a growing cap. We proceed to analyze caps thatgrow at a constant rate. For these caps we derive optimal exercisestrategies and show that they can be completely characterized in termsof only three endogenous parameters.
The article is organized as follows. American capped call optionswith constant caps are analyzed in the next section. Results are given
e analysisand to thelue of ant grow at4 provides Americanonclusionsn in Sec-
dividend-ock or anic
ds a constant
t any dateed optionfor finite and infinite maturity options. Section 2 extends thto capped call options that have delayed exercise periods MILES contract, that is, a capped call option on the dollar vaindex with a delayed exercise period. The case of caps thaa constant rate is considered in detail in Section 3. Section numerical results comparing European capped calls to theircounterparts. A comparison of hedge ratios is also given. Cand remarks on possible extensions of the model are givetion 5. Proofs of results and some of the more lengthy forcollected in the Appendix.
. Valuation of American Capped Call Options
We consider a class of derivative securities written on a paying underlying asset which may be interpreted as a stindex. The price of the underlying asset, St, satisfies the stochastdifferential equation
where is a Brownian motion process athe coefficients , d, and s are constants. Equation (1) implies the stock price follows a geometric Brownian motion (lognormprocess with a constant dividend rate of d. We also assume that funcan be invested in a riskless money market account bearingpositive rate of interest denoted r.
Let represent the value of an American capped call otime t. The option has a strike price of K, a cap of L, and a maturity ofT. Throughout the article, we assume that L K > 0. Exercise maytake place, at the discretion of the owner of the security, aduring the life of the option [0, T]. The payoff of the cappwhen exercised at time t is where the operator x yden