Feedback effects of dynamic hedging strategies in the presence of transaction costs

Andergassen, Rainer; Agliardi, Elettra Working Paper Feedback effects of dynamic hedging strategies in the presence of transaction costs Quaderni - Working Paper DSE, No. 445 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Andergassen, Rainer; Agliardi, Elettra (2002) : Feedback effects of dynamic hedging strategies in the presence of transaction costs, Quaderni - Working Paper DSE, No. 445, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/4851 This Version is available at: https://hdl.handle.net/10419/159286 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/3.0/ Feedbacke¤ectsof dynamichedgingstrategies inthepresence oftransactioncosts ElettraAgliardi*,RainerAndergassen** DepartmentofEconomics,UniversityofBologna PiazzaScaravilli 2,40126 Bologna *[email protected] **[email protected] Abstract Westudythedestabilisinge¤ectofdynamichedgingstrategieson theprice oftheunderlyinginthepresence ofsunkcostsoftransaction. Once sunkcostsoftransactionaretakeninto account,continuousportfolio rehedgingisnolongeranoptimalstrategy.Usinganon-optimising(local intime)strategyforportfoliorebalancing,explicitdynamicsfortheprice oftheunderlying arederived,focusingin particularonthe excess volatility and feedbacke¤ectsoftheseportfolioinsurance strategies.Further,we showhowtheselatterdepend ontheheterogeneityoftheinsured payo¤s. Finally,conditionsarederived underwhichitmaystill bereasonable, fromapracticalviewpoint,toimplementBlack-Scholes strategies. J.E.L.Classi…cation numbers:G10,G11,G12. Keywords:dynamichedging,volatility,Black-Scholesmodel, transactioncosts. 1 1Introduction Standardoption pricingliteraturereliesonthehypothesisthat thedynamics oftheunderlying assetareindependentofthehedgingstrategy.Dynamicdelta hedgingstrategiesrequiretosell theunderlying assetifitsprice decreases, whiletheyrequiretobuyifitsprice increases.Thehypothesisofindependency betweenstrategiesand price dynamicsoftheunderlying assetcorrespondsto theassumptionthat themarketfortheunderlying assetisperfectlyliquid. Positivefeedbacke¤ectsfromdynamicdeltahedgingstrategieshavebeen studiedrecentlyassumingthat theassetmarketfortheunderlying assetis only…nitelyliquid(see,forexample,Freyand Stremme(1997),Schoenbucher and Wilmott (2000)and their references).Ithasbeenshownthatinthiscase portfolioinsurance activityhasadestabilisinge¤ectonthedynamicsoftheprice oftheunderlying asset.In particular,thesestrategiesincreasethevolatilityof theprice oftheunderlying asset.Theabove-mentioned papersassumethat programtraderscan buyand sell assetswithoutincurringtransactioncosts. But,asamatterof fact,transactioncostsarenon-negligibleinassetmarkets. Ifweintroduce transactioncosts,thenitisnolongeroptimalto adjust the portfoliocontinuously.Therearetwomainapproachesintheliteraturetaking the e¤ectsoftransactioncostsinto account:the…rstconsidersdiscreteadjustmentsoftheportfolio,wherethetimestepofportfoliorebalancingisexogenouslygiven,whilethesecond considerstradersascontinuouslymonitoringthe price oftheunderlying asset,althoughadjustingtheirportfolio onlyifthegain fromadjustmentisgreaterthanthe costofadjustment.Thislatterapproach can besubdividedintotwofurtherapproaches:the…rstiscalledlocal intime, whilethesecond iscalledglobal intime.Theformerisanon-optimising approach,whilethelatterisanoptimising one(see,forexample,Wilmott (2000) forareview). Inthispaperwewill followalocal intimeapproachand assumethat transactioncostsaresunkcosts1. Supposethatanagent’sportfolioisgiven byalongpositiononacall option and ashortpositiononsomeunitsoftheunderlying asset,suchthatariskless portfolioisobtained.IfwedenotebyV(S;t)theoptionvalue,whereSisthe valueoftheunderlying asset,thentheperfectBlack-Scholeshedgeisgiven by ¢=@V @S>0(1) Considernowachangeintheprice oftheunderlying asset.Giventhischange, intheabsence oftransactioncosts, itwould beoptimaltorehedgetheposition inordertomaintainariskless portfolio.Butonce weassumethat therearesunk costsofadjustment, itisnolongeroptimaltorehedgeimmediatelytheportfolio astheprice oftheassetchanges.Thus,wehavetode…neanewstrategyfor portfoliorehedging. 1Themodelcould be extendedtriviallyinordertoconsideralsoproportionaltransaction costs. 2 FollowingWhalleyand Wilmott (1993)and Henrotte(1993)wede…nea con…dence levelforthedeviationoftherisky assetpositionfromtheperfect hedge.Supposeanagentiholds¡Giunitsoftheunderlyingrisky asset.The riskoftheimperfectlyhedged position,measured bythevariance overatimestep±t, isgiven by¾2S2³Gi¡@Vi @S´2±t,toleading order.Ifsuch positionis perfectlyhedged, i.e., Gi=@Vi @S,thentheportfolioriskiseliminatedcompletely, otherwisethepositionis still risky. Wehavenowtodeterminetherehedgingstrategy.De…ne´i(S;¿;K)= @Vi @S¡Giasthehedge-unbalance levelofagenti,where¿=T¡tdenotes timetomaturityand Kisthestrikeprice.Whalleyand Wilmott (1993)and Henrotte(1993)de…neatolerance level~ Hi 0suchthat ¾S¯¯¯¯Gi¡@Vi @S¯¯¯¯6~ Hi 0(2) whereinactionisoptimalaslong as(2)is satis…ed,whilethepositionshould berebalancedonce (2)isviolated.Theparameter~ Hi 0givesameasureofthe maximumexpectedriskintheportfolio. Inwhatfollows,wewill assumethat~ Hi 0ispartlydeterministicand partly stochastic:theformercapturesthein‡uence oftransactioncosts,whilethe lattercaptures stochastic contingencies.Giventheseassumptions,we can de…ne anadjustmenthazardfunctionforeach programtrader.FollowingCaballero and Engel(1993,1999)wewill studytheaggregatedynamicsoftheadjustment hazardratesand theresultingprice dynamicsoftherisky asset. Thepaperisorganisedasfollows.In Section2theadjustmenthazardfunctionisformallyintroducedand themodel ispresented.Section3containsthe mainresults.Finally,Section4concludes. 2Thebasicmodel Supposethat therearetwotypesoftradersoperatinginamarket,wherethere isarisky assetand ariskless one(apurediscountbond):programtraders and reference traders.Programtradersuseadynamichedgingstrategy,while reference tradersaresmall price takers,whichincludemarketmakersand market timers,providingliquidityformarket transactions.Weassumethat thereisa continuumofreference traders,suchthat the e¤ectsoftransactioncostson theaggregatedemand functionofreference tradersarenegligible.Inother words,we canassumeasmoothaggregatedemand functionwithrespect tothe price oftherisky asset.Furthermore,wesupposethatreference tradershave perfectinformationabout thefundamentalsoftherisky asset.Theaggregate demand functionofreference tradersisdenoted byD(t;Ft;St),whereSisthe proposed price oftherisky asset,whileFcan beinterpretedin di¤erentways. Forexample,Freyand Stremme(1997)assumeFtobetheaggregateincome ofreference traders,whilePlatenand Schweizer(1994)assumeFtobean unspeci…edliquiditydemand,and othersassumeFtobethefundamentalvalue 3 ofthe…rm.Wefollowthislatterapproach,and makethefollowing assumptions about thedemand ofreference traders: Assumption1. a)D(t;Ft;St)isasmoothfunction b)@D @S6¡d<0and@D @F>0 Assumption1.b). indicatesthatastheprice oftheassetincreases,demand decreases,whileasthefundamentalvalueoftheassetincreases,demand increasesaswell. Letusnormalize thetotalsupplyoftherisky asset to one.Thus, inthe absence ofprogramtraders,equilibriumisguaranteed bythefollowingmarket clearingcondition D(t;Ft;S¤ t)=1(3) From(3)itfollowsthat the equilibriumprice oftheassetisS¤ t='(t;Ft). Thus,theprice oftheassetfollowsitsfundamental.value.Wewill call S¤ tthe normalprice oftheasset. Wearelookingforadi¤usion process fortheassetprice oftheform dSt=¹S(St;t)dt+¾S(St;t)dWt(4) whereWtdenotesaWienerprocess. Once wehavespeci…edthedemand functionofthereference tradersand thestochasticprocess forthefundamentalvalue we can determinethedynamicsoftheassetprice S¤ t. Wewill assumethat thedynamicsofthefundamentalvalueoftherisky asset followadi¤usion process oftheform dFt=¹F(Ft;t)dt+¾F(Ft;t)dWt:(5) Usingthe equilibriumcondition(3),thedynamicsofthefundamentalvalue(5) and thefact thatwearelookingforadi¤usion process fortherisky assetofthe type(4),wehavethat, inequilibrium,thefollowingcondition hastobesatis…ed 0=(DS¾S(S¤ t;t)+DF¾F(Ft;t)) dWt+ +(DS¹S(S¤ t;t)+DF¹F(Ft;t)+ +1 2DSS (¾S(S¤ t;t))2+1 2DFF (¾F(Ft;t))2+ +DSF¾S(S¤ t;t)¾F(Ft;t)+Dt)dt (6) Inordertosatisfycondition(6)weneedthestochasticaswell asthedeterministicterminequation(6)equaltozero;therefore,weobtainthefollowing momentsfortherisky assetprice dynamics: ¾S(S¤ t;t)=¡¾F(Ft;t)DF DS 4 ¹S(S¤ t;t)=¡1 DS"Dt+DF¹F(Ft;t)+1 2DSS µ¾F(Ft;t)DF DS¶2 +(7) +1 2DF F (¾F(Ft;t))2¡DSF(¾F(Ft;t))2DF DS¸ Thus,theprice dynamicsoftherisky assetS¤ tfollowadi¤usion process (4), where¾S(S¤ t;t)and ¹S(S¤ t;t)aregiven byexpressions(7). InthenextSectionwearegoingtostudytheaggregatedemand ofprogram traders.Then,weplugtheaggregatedemand ofprogramtradersintoexpression (3)and studyitsimplicationsfortheprice dynamicsoftherisky asset. 2.1Aggregate demand ofprogramtraders Wearguedin Section1thatinthe caseofnotransactioncosts,continuousadjustmentoftheportfolioisoptimal. Butonce weintroduce transactioncosts, and in particularsunkcostsofadjustment,continuousadjustmentoftheportfolioisnolongeroptimal. Letusde…neacon…dence level~ H0asinexpression(2),and supposeitisa function~ H0(c;H0),wherec>0isthedeterministic component,whileH0isthe stochastic component.Thedeterministic componentccapturesthein‡uence of thesize ofthetransactioncostsonthe con…dence level. Anincreaseinthe transactioncostsincreasesc.In particular,cmaydepend onthesize ofthebidaskspread.Thestochastic componentH0capturesthein‡uence ofstochastic contingenciesonthe con…dence level. Wede…neaprobabilityofadjustingtheportfoliointhefollowingway Pr³~ H0(c;H0)6¯¯´t+dt¯¯¯¯¯j´tj<~ H0(c;H0)´=h(´;c)dt where´=´(S;¾;¿;K)=¢(S;¾;¿;K)¡G,¢(S;¾;¿;K)=@V(S;¾;¿;K) @S,¾is theinputvolatility, i.e.thevolatilityusedforthe computationofthehedging strategy(Freyand Stremme,1997),¿isthetimetomaturity,whileKisthe strikeprice. Weassumethat thedistributionofH0isthesameforeach programtrader, and thus,theprobabilityofportfoliorehedgingdependsjustonthehedging unbalance level´and onthelevelofthedeterministic componentc.Wemake thefollowing assumptionsabout theadjustmenthazardfunctionh(´;c): Assumption2. a)h(¢)isasmoothfunction b)h´(´;c)>0,andhc(´;c)<0 c)limc!0h(´;c)=1andlimc!1 h(´;c)=0. 5 Assumption2.b)statesthat theprobabilityofadjustmentincreasesasthe hedgingunbalance level increasesand/orasthesize ofthetransactioncosts decreases.Assumption2.c)impliesthat, ifthesize ofthetransactioncostsis vanishingsmall, thentheprobabilityofportfolio adjustmentconvergestowards one,thatis,wehave continuousadjustmentordynamicdeltahedging,while, ifthetransactioncostsarein…nitelylarge,thentheprobabilityofportfolio adjustmentbecomesvanishingsmall. Thehedgingunbalance level´hasacommonelementforeach program trader,whichistheprice oftheunderlyingS,whiletherearealsoidiosyncratic components suchasthestrikeprice Kand timetomaturity¿.Thus,astheprice oftheunderlying assetchanges,thehedgingunbalance levelforeach program traderi,´i,changesaswell, whilethewayitchangesdependsonthedistribution ofstrikepricesand ofthetimetomaturity. Byaggregatingprogramtradersovertheirhedgingunbalance levelswehave that theaveragedemand oftherisky assetoveratimeintervaldtbecomes g(S;¾;c;t)=Z< ´h(´;c)f(´)d´ wheref(´)isthedistributionofthehedgingunbalance levelovertheprogram traders.Thislatterdistributionisnot time-invariantsince,asthetimegoeson ¿changes,and thus,accordingtothedistributionof¿,thedistributionof´ changes.Wearegoingto assumeacontinuousand randomin‡uxand out‡ux ofprogramtradersfromtheassetmarket,and heterogeneityinthedistribution ofstrikes.Thus,wehavethat theaveragedemand overasmall timeperioddt ofprogramtraderisgiven by ª(S;¾;c)=Z <2 +£R ´(S;¾;¿;K)h(´(S;¾;c;¿;K)) f(´)À(dK©d¿©d´)(8) whereÀhasasmooth densityfunctionwithrespect to a Lebesgue-measure. Thus,expression(8)representsthedemand fortheunderlying assetofthe programtradersoverasmall timeperioddt.Weareinterestedtosee howthis demand changesastheprice oftheunderlyingchanges. Considerthe changeintheprice oftherisky assetofsize dS.Usingexpression(8)wehave: ª(S+dS;¾;c)=R <2 +£R [¢(S+dS;¾;¿;K)¡¢(S;¾;¿;K)+´]£ £h(¢(S+dS;¾;¿;K)¡¢(S;¾;¿;K)+´;c)£ £f(´)À(dK©d¿©d´) TakingTaylorexpansionof¢(S+dS; :)and h(¢(S+dS; :)¡¢(S; :)+´)around Sand ´respectively,wehave ¢(S+dS; :)¡¢(S; :)=¢SdS+1 2¢SS (dS)2 6 and h(¢(S+dS; :)¡¢(S; :)+´;c)=h(´;c)+h´(´;c)³¢SdS+1 2¢SS (dS)2´+ +1 2h´´ (´;c)³¢SdS+1 2¢SS (dS)2´2 Now we cancalculatedª(S;¾;c)= ª(S+dS;¾;c)¡ª(S;¾;c)asfollows dª(S;¾;c)=R <2 +£Rnh¢SdS+1 2¢SS (dS)2i[h(´;c)+h´(´;c) (¢SdS+ +1 2¢SS (dS)2´+1 2h´´ (´;c)³¢SdS+1 2¢SS (dS)2´2¸+ +´h´(´;c)³¢SdS+1 2¢SS (dS)2´+ +1 2´h´´ (´;c)³¢SdS+1 2¢SS (dS)2´2¾f(´)À(dK©d¿©d´) Since (dS)µ=0forµ>2weobtain,after rearrangingterms dª(S;¾;c)=dSR <2 +£R [h(´;c)+´h´(´;c)]¢Sf(´)À(dK©d¿©d´)+ +1 2(dS)2R <2 +£R f[h(´;c)+´h´(´;c)]¢SS+ +[2h´(´;c)+´h´´ (´;c)](¢S)2of(´)À(dK©d¿©d´) Thislatterexpressioncan berewrittenasfollows dª(S;¾;c)=H1(S;¾;c)dS+1 2H2(S;¾) (dS)2(9) where H1(S;¾;c)=Z <2 +£R ~ h(´;c)¢S(S;¾;K;¿)f(´)À(dK©d¿©d´)(10) H2(S;¾;c)=Z <2 +£Rn~ h(´;c)¢SS +~ h´(´;c) (¢S)2of(´)À(dK©d¿©d´) and where~ h(´;c)=@ @´´h(´;c)=h(´;c)+´h´(´;c)and ~ h´=@ @´~ h(´;c)= 2h´(´;c)+´h´´ (´;c). Letusintroduce thefollowing assumption: Assumption3.There existfunctionsÀ1andÀ2suchthatÀ(dK©d¿©d´)= À1(d´)À2(k;¿)dKd¿whereÀ1andÀ2have asmooth densityfunctionwithrespect toaLebesguemeasure;À2hasacompactsupportinR+£[0;1): 7 WithAssumption3,we canrewrite expression(10)asfollows H1(S;¾;c)=~ H(c)~ ¡(S;¾)(11) where ~ H(c)=Z< ~ h(´;c)f(´)À1(d´)(12) ~ ¡(S;¾)=Z<2 + ¢S(S;¾;K;¿)À2(k;¿)dKd¿(13) ~ H(c)indicatesthestationaryaveragesize oftheadjustment,givenachange inthehedgeunbalance level. ¢S(S;¾;K;¿)isknownintheoption pricingliteratureastheparametergamma,and itindicates, intheabsence oftransaction costs,howoftenapositionmustberehedgedinordertomaintainadelta-neutral position.Thus,~ ¡(S;¾)isthestationaryaveragevalueofthegamma,which indicates, intheabsence oftransactioncosts,howofteninthestationarystate, apositionmustbeadjustedonaverageinordertokeep delta-neutralpositions. H1(S;¾;c)indicatesthestationaryaverageadjustment,givenachangein theprice oftherisky asset.H1(S;¾;c)depends:a)ontheaveragesize ofadjustment,and thusonthepropertiesoftheadjustmenthazardfunctionand onthestationarystatedistributionofunbalance levels,and b)onthefrequencyofadjustment,which dependsonthestationaryaveragesensibilityof thedeltawithrespect totheprice oftheunderlying.Notice thatAssumption2impliesthat@ @cH1(S;¾;c)<0and limc!0H1(S;¾;c)=~ ¡(S;¾),while limc!1 H1(S;¾;c)=0.Inotherwords,H1(S;¾;c)isadecreasingfunctionof c. Furthermore, ifthesize ofthetransactioncostsisvanishingsmall, then H1(S;¾;c)convergestowardsthestationaryaveragevalueofthegamma and so wearebacktothe caseofdynamichedgingstrategies,whileifthetransaction costsareverylarge,then noportfolio adjustmentoccursatall. Finally,by Assumption2wehaveH1(S;¾;c)>0. 3Positivefeedbacke¤ectsfromhedging Aswepointedoutbefore,reference tradershaveperfectinformationabout the fundamentalvalueoftherisky asset.Thus,areductioninthefundamental valueleadsto a decreaseintheprice oftherisky asset.Giventhisdecrease, someprogramtraderswill sell therisky assetsinorderto adjust theirportfolio. Thislatterleadstoafurtherprice reductionoftherisky asset,which now will belowerthanitsnormal level, i.e.St<S¤ t.Thus,theactionofprogram tradersleadstopotentialgainsforliquidityproviders,suchasmarketmakers and market timers(see Grossman,1988).Theselattercould buytheassets since theiractualprice isnowlowerthantheirnormalprice.Inthisway, liquidity 8