Static replication of barrier options: some general results

Transcription

1 Arcle 7/6/ 3: pm Page ac replcaon of barrer opons: some general resuls Lef B. G. Andersen Managng Drecor, Banc of Amerca ecures, 9 Wes 57h ree, New York, NY 9, UA Jesper Andreasen Drecor, Nordea Markes, randgade 3, Copenhagen, 9 C, Denmark Davd Elezer c/o L.B.G. Andersen, Managng Drecor, Banc of Amerca ecures, 9 Wes 57h ree, New York, NY 9, UA hs paper presens a number of new heorecal resuls for he replcaon of barrer opons hrough a sac porfolo of European pu and call opons. Our resuls are vald for opons wh compleely general knock-ou/knock-n ses, and allow for me- and sae-dependen volaly as well as dsconnuous asse dynamcs. We llusrae he heory wh numercal examples and dscuss praccal mplemenaon. Inroducon he classcal approach o he hedgng of dervaves nvolves mananng an everchangng poson n he underlyng asses. he consrucon of such dynamc hedges s a key argumen n he semnal paper by Black and choles (973), and s a sandard echnque for praccal hedgng of dervave producs. A leral nerpreaon of dynamc hedgng sraeges, however, requres connuous radng, whch would generae enormous ransacon coss f mplemened n pracce. Insead, mos real-lfe radng sraeges nvolve me-dscree rebalancng, exposng he hedger o some rsk, parcularly f he gamma of he opon hedged s hgh. For some dervaves, urns ou ha s possble o consruc a hedge ha does no nvolve connuous rebalancng. uch sac hedges normally nvolve seng up a porfolo of smple, European opons (ypcally pus and calls) ha s guaraneed o mach he payoff of he nsrumen o be hedged. I s far o say ha less s known abou sac hedgng han dynamc hedgng, alhough recen papers have made some progress. Derman, Ergener and Kan (995) descrbe a numercal algorhm for sngle barrer opons n he conex of a bnomal ree represenng he evoluon of a sock wh me- and level-dependen volaly. Bowe and Carr (994), Carr and Chou (997) and Carr, Ells and Gupa (998) examne n deal he sac replcaon of barrer opons n he Black choles hs paper was compleed whle all hree auhors were employed by General Re Fnancal Producs.

2 Arcle 7/6/ 3: pm Page Lef Andersen, Jesper Andreasen, and Davd Elezer (973) model. For marngale sock processes, Brown, Hobson and Rogers (998) demonsrae how o se up model-free over- and underhedges for ceran smple classes of sngle-barrer opons. he approach n hs paper exends he resuls n prevous leraure n a number of ways. Frs and foremos, we derve exac, explc expressons for he composon of sacally replcang porfolos n asse models ha allow for boh jumps and me- and sae-dependen dffuson volaly. econd, we are able o derve sac hedgng porfolos no only for smple, connuously monored barrer opons bu we also allow for almos arbrarly complcaed knock-ou regons and ermnal payoffs (and can easly handle curved, dscree, paral, and double-barrer opons). All of our heorecal resuls are derved under he assumpon (or approxmaon) ha European opons are raded n nelasc supply for all maures and srkes. hs s no rue n pracce and we herefore devoe a secon of he paper o reang some ssues ha arse n he praccal mplemenaon of he sac hedgng sraeges suggesed n he frs par of he paper. he res of he paper s organzed as follows: econ derves sac hedges for general barrer opons wren on an asse wh a volaly ha depends deermnscally on me and he asse self. In econ 3 we exend our resuls o he case of dsconnuous asse dynamcs. econ 4 nvesgaes some ssues relang o he praccal mplemenaon of sac hedgng sraeges and presens numercal resuls. Fnally, econ 5 conans he conclusons of he paper. he frs of wo appendces demonsraes how he resuls n he paper whch are derved usng probablsc echnques can alernavely be proven by he more radonal ools of dfferenal forms and crculaon heorems. he second appendx brefly consders he case of sochasc volaly and demonsraes ha our echnque does no lead o a sac hedge for hs case. Fnally, le us pon ou ha he sudy a hand s largely appled n naure. As our man focus s new formulas and he deas behnd hem, we have de-emphaszed echncales and se he paper n a relavely loose mahemacal frame. In parcular, we have pu lle emphass n he specfcaon of echncal regulary condons, whch we rus ha readers can supply hemselves. Deermnsc volaly In hs secon we derve sac hedgng porfolos for barrer opons wren on an underlyng sock (or foregn exchange rae) characerzed by a local volaly ha s only a funcon of me and sock prce level. uch asse prce dynamcs are dscussed n deal n Dupre (994). For ease of noaon we make he smplfyng assumpon ha all neres raes and dvdend yelds are zero. Nex we assume ha he underlyng sock (or foregn exchange rae) evolves accordng o () () = σ (, ( )) dw ( ) () Journal of Compuaonal Fnance

3 Arcle 7/6/ 3: pm Page 3 ac replcaon of barrer opons: some general resuls 3 where σ s a connuous, deermnsc funcon, and W s a Brownan moon under he rsk-neural measure. We assume ha σ s posve and suffcenly regular for () o have a unque, non-explosve, posve soluon. We furher assume ha we can rade European opons on he sock wh all maures and srkes. We wll le C(, K ) and P(, K ) denoe he me- prces of European call and pu opons, respecvely, wh maury and srke K. We le C(;, K ) and P(;, K ) denoe he same opons prces a me. We noe ha European opon prces are lnked o he rsk-neural margnal densy of he sock prce. pecfcally, f we le f (, ) denoe he me- margnal densy of () aken n, we have ha B [ ] = = f(, ) E δ( ( ) ) C (, ) P (, ) f(, ) E C (, B) P (, B) = [ ] = = ( ) B K K B = [ ] = + = f(, ) E C (, B) P (, B) ( ) B K K () where subscrps denoe paral dervaves, δ(.) s Drac s dela funcon, E(.) s he me- rsk-neural expecaons operaor, and A denoes he ndcaor funcon on he se A.. Connuous barrers Consder he funcon F = F(, ), defned as he soluon o F(, ) + σ(, ) F(, ) =, <, > B() F (, ) = R (), <, B () F (, ) = g ( ), (3) where g s a funcon of he sock prce only, and B s a connuous funcon of me on [, ]. We recognze (3) as he PDE formulaon of he problem of prcng a down-and-ou barrer opon wh me-dependen rebae R() and medependen connuously observed barrer level, B(). Here, and hroughou he paper, we assume ha R s a dfferenable funcon. Noe ha we le g defne he ermnal value of F(, ) for all values of, ncludng he knock-ou regon B(). o, f for example we consder a down-and-ou call opon, hen g() = ( K) + >B( ) + R() B( ) I should be sressed ha F() s he value of a barrer opon naed a me, e, f G(; s) s he me- value of a barrer opon orgnally ssued a me s, hen F()=G(; ). hs means ha F, unlke G, s no a marngale under he rsk-neural measure (as wll be evden shorly). If we know ha dd no Volume 5/Number 4, ummer

4 Arcle 7/6/ 3: pm Page 4 4 Lef Andersen, Jesper Andreasen, and Davd Elezer breach he barrer n [, ], hen F() = G(; ); hs relaon obvously only holds up o he frs me hs he barrer. Usng (3) and he fac ha F s connuous, bu no generally connuousdfferenable, a = B we ge from anaka s formula (Karazas and hreve, 99) and (3) ha ( ) = ( ) ( ) + df, () F, () () σ, () dw () R() d > B() < B() + σ( ) ( + ) ( ), () B () F B, () δ () B () d (4) where R =dr d s assumed o exs, and F (, B()+) s he lm of F (, B() +ε) for ε. Inegrang (4) n he me-dmenson yelds g ( ( )) F(, ( )) F, ( ) σ, ( ) ( ) dw ( ) akng expecaons and rearrangng yelds he relaon = [ ] [ () < B ()] F(, ( )) E g( ( )) R ( ) E d Noce ha we have here used he fac ha F s a deermnsc funcon around he barrer; were he sock volaly sochasc, hs would no hold. 3 he formula above relaes he barrer opon prce o he volaly and he (rsk-neural) margnal densy. Ineresngly, he frs passage-me denses and condonal probables are no drecly nvolved here. 4 he margnal densy can be synheszed usng opon posons by use of (). We ge F(, ( )) = g( ) P (, ) R ( ) P (, ) d + R () d + F (, B() + ) σ(, B() ) B() δ( () B ()d ) = ( ) ( ) () > B () () < B () σ ( ) ( ) ( + ) B, ( ) B ( ) f B, ( ) F B, ( ) d σ ( B, ( )) B ( ) F( B, ( ) + ) P( B, ( )) d (5) where we have arbrarly chosen o synhesze he densy from pu opons. We noe ha (5) expresses he value of a barrer opon as a lnear combnaon of pus, specfcally: B () Journal of Compuaonal Fnance

5 Arcle 7/6/ 3: pm Page 5 ac replcaon of barrer opons: some general resuls 5 long a connuum {g()} << of -maury buerfly pu spreads P (, ); shor a double connuum {R ()} < < of buerfly pu spreads P (, ) wh srkes n [, B()]; and shor a connuum {σ(, B()) B() F (, B()+)} < < of buerfly spreads wh srkes along he barrer. Consder now usng he pu porfolo suggesed by (5) as a hedge for a barrer opon G(; ) naed a me. pecfcally, f τ =nf{: ()=B()} s he frs me he sock ouches he barrer, we hold he pu porfolo up o τ and, f τ <, sell off he ousandng porfolo a he me he barrer s breached. As menoned earler, F()=G(; ) up o (and ncludng) he mnmum of τ and, whereby such a sraegy would clearly generae he correc cashflow a τ. For he pu porfolo o qualfy as a sac hedge, we need o verfy ha he porfolo does no generae any oher cashflows a mes < τ. Bu as all he pu posons wh maures less han only nvolve srkes a or below he barrer, clearly no such cashflows are generaed; whence, he pu porfolo n (5) qualfes as a sac hedge. Alhough (5) s a sac hedge, s no necessarly he mos convenen. In parcular, we noce ha he second erm n (5) can be smplfed o whch represens a poson of pu spreads along he barrer. hs poson does no generae cashflows before he opon maures or knocks ou, 5 and he hedge remans sac. We can smplfy he hedge even furher by relang he buerfly spreads o calendar spreads hrough he forward equaons of Dupre (994): = C + σ(, K) K C ; = P + σ(, K) K P We can now rewre (5) as smply B () R () P (, ) d = R () P (, B() ) d F(, ( )) = g( ) P (, ) F, B( ) + P, B( ) d ( ) R () P, B() d K As calendar pu spreads on he barrer do no produce cashflows as long as he barrer opon s alve, (6) represens a sac hedge, where he barrer opon s now replcaed by a European opon payng g a maury, mnus he (deermnsc) connuum {F (, B()+)} < < of calendar spreads along he barrer, and mnus a connuum {R ()} < < of pu spreads along he barrer. As menoned earler, he opons posons mus be unwound when he barrer s h. If he model s correc e, f he dela (F ) along he barrer of alve opons s compued correcly hen he unwnd gan equals he rebae. Volume 5/Number 4, ummer K ( ) ( ) (6)

6 Arcle 7/6/ 3: pm Page 6 6 Lef Andersen, Jesper Andreasen, and Davd Elezer As wren n (6), hedgng he European payoff payng g s accomplshed hrough buerfly spreads. Alernavely, we assemble he European payoff drecly from he hockey-sck buldng blocks of pus and calls. Followng Carr and Chou (997), hs can be accomplshed by wrng g ( ) = g( κ) + g ( κ)( κ) + g ( K)[ K ] dk+ g ( K)[ K] dk for some arbrary posve consan κ. eng κ = B() and negrang over he densy of yelds [ ] = E g( ( )) R( ) P (, B( )) K + g ( K) C(, K) dk B ( ) + κ + + κ + g ( B ( ) + ) CB (, ( )) C ( B, ( )) gb ( ( ) + ) K (7) (7) represens a sac hedge conssng of a connuum of calls wh srkes above he barrer, plus a fne number of calls and pu call spreads wh srkes a he barrer. Noce ha f g has knks or dsconnues, he dervaves of g n (7) mus, of course, be nerpreed n he sense of dsrbuons. I s worh nong ha (6) (7) only requre model-based compuaon of he dela along he barrer for nsance by a fne-dfference scheme (see, eg, Andersen and Broheron-Raclffe (998) for a dscusson of he mplemenaon of he dynamcs () n a fne-dfference scheme); all oher erms n he hedgng porfolo can be deduced from he marke prces of sandard European opons. he echnque oulned above s easy o apply o many ypes of barrer opons, ncludng n -syle barrer opons. omemes we can also rely on pary resuls; for nsance, a down-and-n opon can be wren as a European opon mnus a down-and-ou opon (wh no rebae), whereby he resuls derved above can be used drecly o sacally hedge a down-and-n opon. Applcaons o double-barrer opons are smple as well, and would merely nvolve ncludng n (6) an exra negral of call maury-spreads and an exra negral of call spreads along he second barrer. 6 We wll reurn o more general barrer shapes n a laer secon.. Dscreely monored barrers Consder now he case when he down-and-ou barrer of he prevous secon s only monored on a dscree se of daes: < < n < he PDE formulaon of he prcng problem s Journal of Compuaonal Fnance

7 Arcle 7/6/ 3: pm Page 7 ac replcaon of barrer opons: some general resuls 7 F(, ) + σ(, ) F(, ) =, (, ) (, ) { }, B( ) F (, ) = R ( ), B ( ) { } F (, ) = g ( ), (8) nce he opon prce s dsconnuous n he me dmenson across every barrer me for all B( ), Iô-expandng he funcon defned by (8) gves ( ) = ( ) ( ) df, ( ) F, ( ) σ, ( ) ( ) dw ( ) {} or () > B() n + δ( ) F( +, ( )) R( ) d = [ ] ( ) B ( ) (9) Inegrang, akng expecaons, and rearrangng yelds n F(, ( ) ) = E[ g( ( )) ] E F( +, ( )) R( ) ( ) B( ) = [[ ] ] he expecaons can be subsued wh negrals and European opon spreads o gve n B ( ) F(, ( ) ) = E[ g( ( )) ] F( +, ) R( ) C (, ) = [ ] () We have now arrved a an equaon ha explcly specfes a sac hedge for he barrer opon. As for he connuous barrer case we need a poson ha replcaes a European payoff g() (for nsance, he pu call porfolo (7)) and a number of buerfly spread posons along he barrer. 7 he number of spreads ha we need s agan dependen on he value of he barrer opon along s barrer. As n (6), only barrer opon values along he barrers depend drecly on he model for sock prce evoluon..3 A general resul he resuls n econs. and. have been proved by anaka s formula. As one would expec, s possble o prove he resuls by more radonal mehods. Appendx A shows how hs can be done hrough he use of dfferenal forms and crculaon heorems. he crculaon heorems se ou n Appendx A allow for a compac and compleely general represenaon of barrer opons wh almos arbrarly complcaed knock-ou regons. uch exensons can also be accomplshed usng he anaka formula. pecfcally, we can summarze he resuls of he prevous wo subsecons n he followng heorem (where we arbrarly have used European calls as he hedgng nsrumens). Volume 5/Number 4, ummer

8 Arcle 7/6/ 3: pm Page 8 8 Lef Andersen, Jesper Andreasen, and Davd Elezer HEOREM uppose ha he underlyng sock evolves accordng o () and consder an opon ha has he value g(()) a me and knocks ou on a se B Ω, Ω =[,] (,), wh a once-dfferenable rebae funcon, R, ha depends only on me. Assumng ha Ω\B s an open submanfold n Ω, a sac hedge for he opon value s defned by F(, ( ) ) = g( ) C (, ) F(, + ) F(, ) σ(, ) C(, ) d ( B) (,) ( B) (, ) where B and nb denoe, respecvely, he boundary and neror of B, R = dr d, and where we use he convenon F(+,.)=F(,.). Furher, we defne { } B = (, ) B ε> :(, + h) B, h < ε B = B \ B [ ] [ ] F ( +, ) F (, ) C (, ) R () C (, ) dd n B and f A Ω, we le { } A (, ) = (, ) (, ) A { } A(, ) = [, ](, ) A Alhough heorem looks complcaed, s really jus a smple exenson of he prevous resuls. In parcular, he barrer prce s spl no a conrbuon from he ermnal maury (frs erm), he non-vercal pars of he barrer (second erm), he vercal pars of he barrer (hrd erm), and he rebae (fourh erm). Noce ha he second erm nvolves boh F (, B+) and F (, B ); he former s requred for down-and-ou porons of he barrer, he laer for up-and-ou pars of he barrer. Also, he echncal requremen ha Ω\B s a submanfold s smply o ensure an alve regon ha s genunely wo-dmensonal, rulng ou arbrarly crnkly or even fracal barrers. As we have seen n he prevous wo secons, s ofen possble o smplfy he expressons above by eher compleng negrals 8 or applyng he forward equaon of Dupre (994). However, care mus be aken o ensure ha he resulng expressons represen sac hedges wh no cashflows beng generaed on he alve regon of he opon. Journal of Compuaonal Fnance

9 Arcle 7/6/ 3: pm Page 9 ac replcaon of barrer opons: some general resuls 9 We emphasze ha heorem holds for dffuson processes of he ype () bu does no generalze o he case of sochasc volaly. Appendx B shows ha, whle a prce represenaon smlar o ha n heorem s possble, he resulng expresson does no represen a sac hedge. For smple barrer opons on marngale sock processes, sochasc volaly models can be accommodaed by he sac overhedges developed n Brown, Hobson and Rogers (998). If s known ha volaly s resrced o a specfed band (as n Avellaneda, Levy and Paras, 995), s possble o combne he Hamlon Jakob Bellman equaon wh he approach aken n hs paper o develop an overhedgng sraegy. Deals of hs are avalable from he auhors on reques..4. Comparson wh exsng resuls he hedge suggesed by heorem generally nvolves akng on an nfne number of posons n European opons wh maures n [, ], all sruck a he barrer B. In conras, he sac hedges suggesed by, for nsance, Carr, Ells and Gupa (998) only nvolve akng posons n European opons ha maure a me. Alhough he hedge hey propose has so far only been proved o be possble for he farly lmed dynamcs of he underlyng (zero neres raes and dvdends and a local volaly sasfyng a ceran symmery condon), s worh demonsrang ha he wo sac hedges are, ndeed, conssen. Furher maeral on hs can be found n Chou and Georgev (998). Le us focus on he case of a down-and-ou call opon wh consan barrer, srke K > B, and no rebae, for whch he hedgng relaon (6) can be wren as For he case when volaly s consan and dvdends and raes are zero, he negrand s gven by BK F (, B ) P (, B ) ( x ) ln( ) + = φ / σ( ) F( ) = C(, K) + F (, B+ ) P(, B) d 3 ( ) BΦ( y+ σ ) Φ( y), x = y = ln( BK) + σ ( ) σ ln( B) σ σ () () can be proven by akng he cross-dervave of he barrer opon prcng expresson n Meron (973). In Fgure we gve an example of he profle {F (., B+)P(., B)}. Volume 5/Number 4, ummer

10 Arcle 7/6/ 3: pm Page Lef Andersen, Jesper Andreasen, and Davd Elezer FIGURE ac hedgng porfolo for down-and-ou call he fgure shows he profle {F (., B+)P(., B)} for he case of zero raes and dvdends, σ =.5, and a down-and-ou call opon wh =,B = 8, K =, and () =. One can show ha negrang expresson () yelds Meron s prcng formula: K B F( ) = Φ( a) KΦ( a σ ) Φ( b+ σ ) Φ( b), B K a = b = ln( K) + σ σ ln ( ( B K) ) σ σ Carr, Ells and Gupa (998) observe ha Meron s formula can be represened as F ( ) K C (, K ) B PB, = ( K ) () whch leads hese auhors o sugges he sac hedge: long one call wh maury and srke K; shor K B pus wh maury and srke B K. We pon ou agan ha he smpler represenaon () of (6) s possble only for very smple assumpons abou he sock process. Journal of Compuaonal Fnance

11 Arcle 7/6/ 3: pm Page ac replcaon of barrer opons: some general resuls 3 Dsconnuous asse dynamcs In hs secon we consder sac hedgng for he case where he process () s exended o allow he sock o jump. pecfcally, we wll assume ha he sock evolves accordng o () = + ( ) + ( ) ( ) λ () m() d σ, () dw() J() dn() (3) where N s a Posson process wh deermnsc nensy λ(), and {J()} s a sequence of ndependen posve random varables, each wh dsrbuon gven by he denses {ξ(,.)}. We assume ha W, N, and J are ndependen of each oher, and le m() = E[J() ] denoe he mean jump. Le us now consder he case of a connuous down-and-ou barrer opon F(, ()), equvalen o he dscusson n econ.. We wll need he defnon Iô anaka expanson of F yelds F() =F(, ()J()) F(, ()) df () = () > B () dm () + δ( () B ()) F(, B () + ) σ(, ()) B () d where M() s a (dsconnuous) marngale. Inegrang over me and akng expecaons yelds E g( ( )) F( ) F (, B() ) σ (, B() ) B() f(, B() ) d [ ] = + Usng (), we ge + R () d+ F() dn() () < B () F( ) = g( ) P (, ) σ (, B( ) ) B( ) F(, B( ) + ) P(, B( ) ) d [ ] [ ] ( ) < + E R () + λ() F() () B () d B () R () + λ() E F () = P (, ) d [ [ ]] (4) hs shows ha our sac replcaon resuls can be exended o he case of jumps. In hs case he sac replcang porfolo also ncludes an exra erm Volume 5/Number 4, ummer

12 Arcle 7/6/ 3: pm Page Lef Andersen, Jesper Andreasen, and Davd Elezer from below he barrer. o se up he sac hedge, we need a model o compue F a he barrer, as well as he quany E[ F ] below he barrer. Andersen and Andreasen (999a,b) show ha, under he model assumpons above, = C m( ) λ( ) KCK σ(, K) K C λ ( ) E [ C] = P m( ) λ( ) KPK σ(, K) K P λ ( ) E [ P] where λ () = λ()( + m()) and (5) E [ C](, K) = E [ P](, K) = he quanes E [ C] and E [ P] can be nerpreed as spreads on a connuum of European opons around a ceran srke. he fac ha hese spreads conan srkes ha le above he barrer means ha we canno generally use (5) o elmnae he erm σ(, B()) B() C n (4) whou nroducng cashflows on he alve regon of he barrer and hereby desroyng he sac hedge n (4). Neverheless, f ζ s known, (5) does provde us wh a way o compue he volaly funcon σ(, B()) n (3) from quoed opons prces; see Andersen and Andreasen (999a,b). We noe ha he hedgng expresson for dscree barrers () s unaffeced by jumps. hs ogeher wh (4) leads o he followng generalzaon of heorem. HEOREM uppose ha he underlyng sock evolves accordng o () and consder a barrer opon smlar o ha n heorem. A sac hedge for he opon s defned hrough F(, ( ) ) = g( ) C (, ) F(, + ) F(, ) σ(, ) C(, ) d ( B) (,) where he noaon s he same as n heorem. ( B) (, ) [ ] [ ] F ( +, ) F (, ) C (, ) R () + λ() E[ F() () = ] C(, ) d n B J ζ(, J) C(, K J) dj C(, K) + m () J ζ(, J) P(, K J) dj P(, K) + m () [ ] Journal of Compuaonal Fnance

13 Arcle 7/6/ 3: pm Page 3 ac replcaon of barrer opons: some general resuls 3 4 ome praccal consderaons he resuls so far have reled on he key assumpon ha pu and call opons exs n unlmed supply and a all srkes and maures. In pracce, hs assumpon s obvously no sasfed, makng he consrucon of perfec hedges mpossble. In hs secon we wll brefly deal wh hs ssue, and we also consder he problem (whch also affecs regular dynamc hedgng) ha ceran barrer opon conracs have delas a he barrer ha grow nfnely large as he opon approaches maury. 4. Fne number of European opons In praccal applcaons we only have a fne and ofen sparse se of acvely raded opons. hs means ha can be dffcul o pu ogeher a porfolo ha closely replcaes he barrer opon under consderaon. I s useful o consder he alernave of seng up sac over- or underhedges. As a frs example, consder he case of a down-and-ou call wh srke K, no rebae, and a dscreely monored, consan barrer B. he barrer observaon daes are { }. he hedgng equaon for hs opon conrac s F(, ( ) ) = C(, K) F( +, ) C (, ) I s clear ha, o overhedge he opon a me, we need o sell off a profle, p (.), ha sasfes B p () F(, ), p (), B > B for each barrer observaon dae. If for maury we can rade European call opons wh srkes K,,K m, we fnd ha he cheapes overhedge of he opon corresponds o he profle where he weghs {a j } are he soluon o he lnear programmng problem max aj C, K { aj } j ( j) + s.. aj ( Kj ) F(, ) B, j p ()= s a K j j ( j ) + Afer dscrezng n he sock prce dmenson he lnear problem can be solved numercally usng he smplex algorhm (see, eg, Press e al., 99). Volume 5/Number 4, ummer

14 Arcle 7/6/ 3: pm Page 4 4 Lef Andersen, Jesper Andreasen, and Davd Elezer Le us now urn o a slghly more complcaed example where he opon has a connuous down-and-ou barrer a a consan level B. Assume ha he rebae s consan over me bu allow for a general payoff g. We assume ha we can purchase enough -maury opons a varous srkes o allow for an overhedge of g. However, we can only ransac n B-srke European pu opons wh a fne number of maures =,,,, n, n =. Assumng deermnsc volaly of he underlyng sock, he hedgng equaon s, from (6), F(, ( ) )= E [ g( ) ] F ( u, B+ ) P ( ; u, B) du = E [ g ( )] F( B, + ) PB ( ;, ) + F( ub, + ) PuB ( ;, ) du = E [ g ( )] F( B, + ) PB ( ;, ) + F ( u, B+ ) P( ; u, B) du where he second equaon follows from negraon by pars and he fac ha P(;, B) = when () >B. For our process assumpon, European pu and call opon prces are ncreasng n maury, whereby we can now wre n : > V () F (, ()) V () (6) where V () = E [ g ( )] F( B, + ) PB (;, ) n + F (, B+ ) F (, B+ ) P( ;, B) : > [ ] (7a) V () = E [ g ( )] F( B, + ) PB (;, ) n + F (, B+ ) F (, B+ ) P( ;, B) : > [ ] (7b) =, F(, B+ ) F(, B+ ), F(, B+ ) > F(, B+ ), F(, B+ ) F(, B+ ) =, F(, B+ ) > F(, B+ ) o es he ghness of he above bounds on F, consder he specal case of a down-and-ou call opon wh srke K =, maury =, spo asse prce Journal of Compuaonal Fnance

15 Arcle 7/6/ 3: pm Page 5 ac replcaon of barrer opons: some general resuls 5 ABLE Under- and overhedgng of down-and-ou call Underhedge Md- Overhedge n V () sum V () he able shows under- and over-hedges for a one-year, connuously monored, down-and-ou call opon wh srke K = and barrer B = 9.he spo prce s () = and he rebae amoun s. Over- and under-hedge prces are compued from (7a,b) and are repored as a funcon of n, he number of maures a whch pus sruck a B can be purchased n he marke. he maures of he replcang pu porfolo are assumed o be equdsanly spaced on [, ]. () =, and barrer B = 9. We assume ha he sock volaly s consan a σ =.5. For hs case, a closed-form soluon exss for he opon (see econ.4), and all erms n (6) can be compued whou resolvng o numercal mehods. Noce also ha here F (, B+)P(;, B) =, E [g()] = C(;, K ), and F a he barrer whereby = and =. able shows he bounds n (6) as a funcon of he number of equally spaced pu maures, n. For reference we also repor he value of a hedge based on a smple md-sum approxmaon o he negral (ha s, we smply subsue P(;, B) for P(;, B) n (7a)). 4. Unbounded dela For many barrer opons, he ermnal payoff funcon g s dsconnuous a he barrer, resulng n an unbounded dela a he maury of he barrer opon. Common examples nclude connuously monored down-and-ou pus wh srke above he barrer, and connuously monored up-and-ou call opons wh srke below he barrer. he unbounded dela of such n-he-money barrer opons s no only a problem for radonal dynamc dela hedgng bu also affecs our sac hedges, whch nvolve opon spread posons of sze proporonal o he dela a he barrer. Volume 5/Number 4, ummer

16 Arcle 7/6/ 3: pm Page 6 6 Lef Andersen, Jesper Andreasen, and Davd Elezer Le us focus on he specfc example of an up-and-ou call opon wh a fla, connuous barrer B, no rebae, and a srke K < B. Our hedgng equaon s F( ) = C(, K) C(, B) + ( B K) C (, B) + F (, B ) C (, B) d Here we use calls n he replcang porfolo o avod cashflows before he opon expres or knocks ou. nce F (, B ) for, we would need o shor an nfne number of maury spreads n he hedge porfolo. o crcumven hs problem we noe ha, f we move he barrer slghly upwards by ε > whou changng he ermnal pay-off, 9 we no only ge an overhedge bu also are able o bound he dela a he barrer. In fac, he dela of hs opon a = B wll end o zero as we approach maury. he resulng overhedge s F( ) = C(, K) C(, B) + ( B K) CK (, B) + F (, ε+ B ) C (, B+ ε) d he choce of ε s a maer of compromse: he larger ε s, he more expensve he hedge becomes; he smaller ε s, he larger (n absolue magnude) he dela can become. A more scenfc approach o he problem of unbounded delas has been suggesed by Wysup (997), and chmock, hreve and Wysup (999). he auhors mpose consrans on he dela and show ha he cheapes super-replcang clam ha sasfes hs consran can be found as he soluon o a sochasc conrol problem. Ineresngly, Wysup (997) pons ou ha he smple sraegy of movng he barrer s a close approxmaon of he correc sraegy. He also gves an approxmae lnk beween he sze of he barrer shf (ε above) and he consran on dela. 5 Concluson K hs paper has dscussed he consrucon of sac hedges for generalzed barrer-ype clams on socks followng a jump dffuson process wh sae- and me-dependen volaly. he sac hedge akes he form of a lnear porfolo of European pus and calls ha exacly maches he cashflow from he opon o be hedged. Allowng for me-dependen rebaes, we have derved exac expressons for he composon of he hedgng porfolo, he form of whch depends boh on he opon o be hedged and on he sock process. Whle our heorecal resuls assume an unlmed supply of European opons and perfec knowledge of sock dynamcs, we have dscussed several praccal echnques for relaxng such dealzed assumpons. Fnally, we pon ou ha, alhough hs paper has focused on barrer opons, many oher opon ypes allow for a decomposon n erms of barrer opons whch agan allows our hedgng resuls o be appled. For nsance, lookback and rache opons can be synheszed by a ladder of connuously monored Journal of Compuaonal Fnance

17 Arcle 7/6/ 3: pm Page 7 ac replcaon of barrer opons: some general resuls 7 barrer opons (see, eg, Carr and Chou 997), and can hus be sacally hedged n our framework. mlarly, Bermudan opons can, afer he deermnaon of he early exercse froner, be reaed as a dscreely monored barrer opon (albe wh an asse-dependen rebae) and can be hedged by a sac poson of pus and calls ha maure a each exercse dae. Appendx A: Dervaon of hedgng equaon usng dfferenal forms Le f (, ) denoe he densy of ha sasfes (), and le F(, ) be he value of a knock-ou opon ha knocks ou on some se B Ω, Ω =[, ] (, ), where B s closed n Ω. Le B c = Ω\B denoe he complemen of B, and defne he open se Bˆ = B c \{(, ): =or = }. We assume ha Bˆ s a submanfold of Ω. Consder he dfferenal form ω = Q (, ) + P (, ) d, Q (, ) = f(, F ) (, ) P (, ) = F(, ) σ(, ) f (, ) F ( σ(, ) f (, )) LEMMA A Le he submanfold Bˆ be as defned above, and le here be gven a submanfold M Bˆ, wh boundary curve M lyng enrely n Bˆ. hen ω = M PROOF Gven he assumpons abou he opology of M, provng Lemma A s equvalen o showng ha ω s closed n M, e, ha for all (, ) M. Now Q = P Q + P = F + + f F f σ f F F ( σ f) = f F + F + f σ F ( σ f) On he submanfold M, F sasfes he backward equaon (), whereby he erm Volume 5/Number 4, ummer

18 Arcle 7/6/ 3: pm Page 8 8 Lef Andersen, Jesper Andreasen, and Davd Elezer mulplyng f s zero. By he Fokker Planck equaon, we also have, for >, f whence he erm on F s also zero. ( σ f) =, subjec o f(, ) = δ( ( )) As an applcaon of Lemma A, consder now he case of a down-and-ou barrer opon where B ={(, ): B(), [, ]}, for some connuous, posve funcon B(). e for wo parameers ε > and L, where everywhere L > B()+ε. Inegrang around he boundary of M, and leng L and ε, we ge from he lemma: F(, ( ) ) = B ( ) { [ ] } M = (, ): B() + ε, L, [ ε, ε] f( F, ) (, ) F ( B, () + ) σ ( B, () + ) ( B ()) f( B, ()) d + R () ( σ(, ) f (, )) where we have used ha = B() + d + f (, B( ) ) R( ) B ( )d (A) and assumed ha f (, ) des ou suffcenly fas when s ncreased o make he negral along = L vansh n he lm. In (A) we have nroduced he rebae R() = F(, B()). o complee he dervaon, negraon of he Fokker Planck equaon yelds Inserng hs no (A) and performng negraon by pars yelds he desred resul: F(, ( ) ) = E[ F (, )] F ( B, () + ) σ( B, () + ) B () f( B, ()) d E[ ] R () d [ σ(, ) (, ) ] = = B() L lm f (, ) F (, ) F, ( ) ε B+ ε = ( ) f f(, ) B f, B( ) B ( ) f(, ) B () () () B () = ( ) + Journal of Compuaonal Fnance

19 Arcle 7/6/ 3: pm Page 9 ac replcaon of barrer opons: some general resuls 9 Whle Lemma A s compleely general and can be appled o almos all ypes of barrer opons, s slghly nconvenen o work wh and requres some rearrangemens of he fnal resuls o yeld a sac hedge. Below, we have lsed a more convenen form of Lemma A expressed drecly n erms nvolvng pus, P(, K ): LEMMA A Le everyhng be as n Lemma A, and defne ω = F σ + (, ) (, ) P(, ) PK(, F ) (, ) d+ PK(, F ) (, ) = F(, ) σ (, ) P(, ) d+ PK(, ) df hen ω = M PROOF e Z(, ) = F(, ) f (, s)ds and noce ha dz (, ) = F (, ) fs (, ) dsd+ F (, ) f(, s) dsd + F (, ) f(, s) ds + F(, ) f(, ) (A) From Lemma, on M, ω = F (, ) σ(, ) f(, ) d ( σ (, ) f (, )) F (, ) d f(, ) F(, ) = F(, ) σ (, ) f(, ) d F(, ) f(, s) dsd f(, ) F(, ) = F (, ) σ (, ) f(, ) d dz(, ) + df(, ) f(, s) ds Here he frs equaon follows from he Fokker Planck equaon, and he second from (A). As dz s an exac dfferenal, he lemma follows by applcaon of (). Volume 5/Number 4, ummer

20 Arcle 7/6/ 3: pm Page Lef Andersen, Jesper Andreasen, and Davd Elezer As a smple example, consder applyng Lemma A o a down-and-ou opon wh a sngle sep-down dsconnuy a = *. pecfcally, we se B () = B(), * B(), < * where B and B are smooh funcons, wh B ( *)>B ( *+). Usng he same ype of negraon conour as n our prevous example, we now ge:. me + vercal pece:. Pece along B ()+, for (, *) (where df = R ()d): B( ) + PK(, ) F(, ) = F(, ) + F(, ( ) ) * (, ( ) + ) σ (, ( )) ( ) (, ( )) F B B B P B d * P, B ( ) R ( ) d + ( ) K 3. Horzonal pece from (, )=( *, B ( *)+) o (, )=( *+, B ( *)+): P, B ( ) F, B ( ) R( ) 4. me *+ vercal pece: K ( ) ( + ) B ( * + ) B ( * ) * * * * * P (, ) F ( +, ) = K * * P, B ( + ) R( ) P, B ( ) F, B ( ) K * * F( +, ) P (, ) B ( * ) ( ) 5. Pece along B ()+, for ( *,): ( ) ( ) ( + ) * * * * * * * K B ( * + ) * + (, ( ) + ) σ(, ( )) ( ) (, ( )) F B B B P B d + P (, B ( ) ) R ( ) d * K Journal of Compuaonal Fnance

21 Arcle 7/6/ 3: pm Page ac replcaon of barrer opons: some general resuls 6. me vercal pece: B( ) + 7. Horzonal pece a = L, L : PK(, ) F(, ) d = F(, ) F(, ) P (, ) F (, ) K Addng all peces, seng he sum o zero, and rearrangng yelds he desred sac hedge decomposon: F(, ( ) ) = g( ) P (, ) = F (, ) P( B, ( ) ) R ( ) g( ) P (, ) = F (, ) g( ) P (, ) F(, B( ) + ) σ (, B( ) ) B( ) P(, B( ) ) d ( ) ( + ) * * * P, B( ) R ( ) d F(, ) R( ) P (, ) K K B ( * + ) B ( * ) Fnally, we wsh o demonsrae ha s possble o formulae heorem n erms of crculaon negrals. Consder he followng: HEOREM A Le everyhng be as n Lemmas A and A. Le he conneced componens of he knock-ou se B be denoed as B. hen [ ] F(, ( ) ) = E F( ( ) ) B + (A3) where A+ for a se A ndcaes a conour nfnesmally close o A bu jus ousde he se A wherever A nω, and whch concdes wh A oherwse. he crculaon negral n (A3) should be performed counerclockwse. PROOF We defne ω and ω as n Lemmas A and A, bu, usng he rebae funcon R(), we exend her domans of defnon from Bˆ o all of Ω, he closure of Ω n. Ω s a compac space, whose boundary ncludes he pons a =. (We may alernavely oban hs ype of boundary by a sandard lmng proce- Volume 5/Number 4, ummer B( ) + ω

22 Arcle 7/6/ 3: pm Page Lef Andersen, Jesper Andreasen, and Davd Elezer dure, as demonsraed earler). he forms so defned conan sngulares (a B, and a = and = ); however, hese are all negrable sngulares as her componen funcons are producs of dervaves of pecewse-smooh funcons, bounded on compac subses. hese forms are closed everywhere n B c, and so we may choose a conour B c B c ha s nfnesmally close o B c. By Lemmas A and A we fnd ha Now, B c sasfes B c = Ω \B, so ha B c = Ω \ B n, and also, ha B c = Ω \ B +, so ha ω = ω ω = ; ω = ω ω = c B B + c Ω B Ω B + Noe ha he negrals on he rgh may run over much larger regons han hose on he lef, bu hese addonal negrals cancel ou. he exra peces are n Ω B (closure n ), and represen negrals along he lnes { =, [, ]} and { =, [, ]} or he oher pars of Ω. hese exra peces make use of he exenson of he forms ω and ω o Ω, because hey are no n B c, and so ω and ω on hese conours canno be obaned as a lm of values n B c. Fnally, we noe ha ω = ω +dz, wh Z defned n he proof of Lemma A. he funcon Z s well defned everywhere n Ω, and, as dz s an exac form on all of Ω, Usng he same echnque as used n he example afer Lemma A, s easy o verfy ha, negrang counerclockwse, ω = F(, ( ) ) + E F(, ( ) ) Ω ω Ω ω = = ω = c c B B Ω ω [ ] hus, we have he fnal resul. Appendx B: ochasc volaly Consder now he case where follows he process () () =σ() dw() where σ() s a sochasc process. As n econ., le F denoe he prce of a Journal of Compuaonal Fnance

23 Arcle 7/6/ 3: pm Page 3 ac replcaon of barrer opons: some general resuls 3 down-and-ou opon wh a connuous barrer. nce he volaly s allowed o be sochasc, F wll n general depend on oher varables han me and sock prce level, e, where x s a vecor of sae varables addonal o me and curren sock prce. o Iô anaka expanson of F yelds where M s a marngale. However, f he lm F x for B exss and s fne almos everywhere, hen connuy on { > B} and he fac ha F = B = R mply ha F x and hence we can gnore he erms n he sums. o, negrang over me and akng expecaons yelds [ ] [ ( u) < B( u) ] F () = E g ( ( )) R( u) E du E F( u, B( u) +, x( u) ) ( u) ( u) = B( u) B( u) C ; u, ( u) du (B) Alhough equaon (B) s a perfecly vald expresson for he prce of he barrer opon, does no consue a sac hedge. he reason s, of course, ha he erms E [F (u, B(u)+,x(u))σ (u) (u)=b(u)] are sochasc and move around as calendar me passes. As a consequence, any buerfly hedge se up o replcae he las negral n (B) would need rebalancng over me. We noe ha (B) may n some crcumsances lead o sac over- and underhedges, f one can fnd a robus way o bound E [F (u, B(u)+,x(u))σ (u) (u)=b(u)]. If he barrer s dscreely monored, as n econ., we fnd he expresson F () = E g ( ( )) df () = dm () + R () d + δ( () B ()) F ( B, () +, x ()) σ() B () d [ ] ( ) [ ] F () = F(, (), x ()) > B < B = + B Fx dx + Fx x dxdx j j j B ( ) E[ F ( +, x, ( ) )] R ( ) C ;, ( ) : ( ) ( ) Agan, hs expresson does no represen a sac hedge. (B) Volume 5/Number 4, ummer

24 Arcle 7/6/ 3: pm Page 4 4 Lef Andersen, Jesper Andreasen, and Davd Elezer. Noce ha, f raes and dvdend yelds are non-zero bu deermnsc, one can easly represen he evoluon of he underlyng as n () by smply modelng he forward sock prce. In hs case barrer levels mus be represened n erms of forward sock levels and ermnal paymens and rebaes n erms of her dscouned values. As our approach s vald for arbrary barrer shapes (no jus consan barrers), such ransformaons can easly be accommodaed n he framework of hs paper.. hs assumpon s made manly for convenence. In mos cases s possble o allow for rebae funcons wh knks and even dsconnues by nerpreng dervaves of R n erms of sep and dela funcons. 3. Appendx B akes a closer look a sochasc volaly models. 4. For a relaon beween F and passage mes n he Dupre forward PDE, see Chou and Georgev (998). 5. I s obvous from () ha a decomposon usng call spreads s possble, oo. However, hs would no consue a sac hedge as he call posons would generae random cashflows n he alve regon of he barrer opon. 6. For up-syle barrers, a sac hedge represenaon usng calendar spreads (such as (6)) mus be based on calls raher han pus o preven he hedge from generang cashflows before he barrer opons maures or knocks ou. uch consderaons are no necessary for he represenaon (5), whch can be based on eher pus or calls. 7. Obvously, we can use he same rck ha leads o (7) o rewre he poson n buerfly spreads o a more drec poson n pu and call opons. 8. pecfcally, as R s a funcon only of me, s clear ha we can wre he plane negral over he neror of B (las negral n he heorem) as a pah negral over he boundary of B. 9. ha s, we keep g() = ( K ) + < B.. By (), Lemma A can also easly be wren n erms of call opons. REFERENCE Andersen, L., and Andreasen, J. (999a). Jump dffuson processes: volaly smle fng and numercal mehods for prcng. Forhcomng n Revew of Dervaves Research. Andersen, L., and Andreasen, J. (999b). Jumpng smles. Rsk, November, Andersen, L., Andreasen, J., and Broheron-Raclffe, R. (998). he passpor opon. Journal of Compuaonal Fnance (3), Andersen, L., and Broheron-Raclffe, R. (998). he equy opon volaly smle: a fne dfference approach. Journal of Compuaonal Fnance (), Avellaneda, M., Levy, A., and Paras, A. (995). Prcng and hedgng dervave secures n markes wh unceran volaly. Appled Mahemacal Fnance, Black, F., and choles, M. (973). he prcng of opons and corporae lables. Journal of Polcal Economy 8, Bowe, J., and Carr, P. (994). ac smplcy. Rsk, Augus, Brown, H., Hobson, D., and Rogers, L. (998). Robus hedgng of barrer opons. Workng paper, Unversy of Bah. Journal of Compuaonal Fnance

25 Arcle 7/6/ 3: pm Page 5 ac replcaon of barrer opons: some general resuls 5 Carr, P., and Chou, A. (997). Hedgng complex barrer opons. Workng Paper, Morgan anley and MI Compuer cence. Carr, P., Ells, K., and Gupa, V. (998). ac hedgng of exoc opons. Journal of Fnance 53, Chou, A., and Georgev, G. (998). A unform approach o sac hedgng. Journal of Rsk (). Derman, E., Ergener, D., and Kan, I. (995). ac opons replcaon. Journal of Dervaves (4), Dupre, B. (994). Prcng wh a smle. Rsk, January, 8. Karazas, I., and hreve,. (99). Brownan Moon and ochasc Calculus. prnger. Meron, R. (973). heory of raonal opon prcng. Bell Journal of Economcs 4, Press, W., eukolsky,., Veerlng, W., and Flannery, B. (99). Numercal Recpes n C. Cambrdge Unversy Press. chmock, W., hreve,., and Wysup, U. (999). Valuaon of exoc opons under shorsellng consrans. Workng paper, Carnege Mellon Unversy. Wysup, U. (997). Barrer opons: a hedgng problem and a hedgng soluon. Workng paper, Carnege Mellon Unversy. Volume 5/Number 4, ummer

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