The price of liquidity in constant leverage strategies. Marcos Escobar, Andreas Kiechle, Luis Seco and Rudi Zagst
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1 RACSAM Rv. R. Acad. Cin. Sri A. Mat. VO , 2009, pp Matmática Aplicada / Applid Mathmatics Th pric of liquidity in constant lvrag stratgis Marcos Escobar, Andras Kichl, uis Sco and Rudi Zagst Abstract. In this papr w dvlop a formula for th iquidity Prmium of constant lvrag stratgis CS. Ths financial products ar path dpndnt options whr th undrlying typically is a hdg fund portfolio. W dscrib and xplain th functionality of CSs, showing a closd form xprssion for th pric of a CS on a hdg fund assuming a Gomtric Brownian Motion, discrt rbalancing for th hdg fund invstmnt as wll as stochastic intrst rats. Th risk of dfault bfor th nxt rbalancing dat lads to a liquidity prmium for th CS which incrass with th volatility of th undrlying hdg fund portfolio and th lvrag of th stratgy. An incrasing rbalancing priod first lads to a highr liquidity prmium, howvr, as th rbalancing priod is xtndd furthr th liquidity prmium bgins to shrink again. El prcio d liquidz n stratgias con apalancaminto constant Rsumn. a funcionalidad d las stratgias d apalancaminto constant CS s invstigada n st artículo. Estos productos financiros son opcions dpdndints dl camino, dond los típicos subyacnts son Hdg Funds. En particular s ncuntra una fórmula crrada para l prcio d liquidz d st drivado n l contxto d procsos brownianos gométricos con rajust discrto d la cartra y tasa d intrés stocástica. El risgo d bancarrota ants d un rajust conllva a un prcio d liquidz para l CS, l cual s proporcional a la volatilidad dl activo subyacnt y al apalancaminto d la stratgia. Un incrmnto n l priodo ntr rajusts implica un incrmnto inicial n l prcio, sin mbargo, l prcio disminuy para largos priodos d rajust. 1 Introduction Constant vrag Stratgis ar options whr th undrlying typically is a hdg fund portfolio. In this contxt lvrag is dfind as th ratio of dbt to quity. An invstmnt stratgy with constant lvrag works as follows. In addition to his own mony, which w will rfr to as th quity, th invstor taks a loan and invsts th sum of th quity and th loan in a portfolio. As th portfolio valu changs ovr tim, th lvrag of th portfolio changs as wll. Th aim of a constant lvrag stratgy is to kp th initial lvrag constant, which implis that th financing of th portfolio has to b adjustd. In th cas that th valu of th undrlying portfolio has incrasd mor than th intrst on th loan, th invstor has to tak an additional loan to uphold th lvrag. In th othr cas, th invstor pays back a fraction of th loan. In th nd, th invstor pays back th cumulativ loan including intrst and his payoff is th valu of th portfolio minus th loan and accrud intrst. A CS is a product whr th issur of th option slls a constant lvrag stratgy, typically on a hdg fund portfolio, to an invstor. Th undrlying portfolio is Prsntado por / Submittd by Aljandro Balbás. Rcibido / Rcivd: Fbruary 10, Acptado / Accptd: April 1, Palabras clav / Kywords: CPPI, hdg fund, liquidity prmium, constant lvrag stratgy. Mathmatics Subjct Classifications: 91B30, 47N30. c 2009 Ral Acadmia d Cincias, España. 373
2 M. Escobar, A. Kichl,. Sco and R. Zagst financd by a loan which is providd by th issur, and by a larg fraction of th pric of th CS i.. th quity which th issur chargs th invstor. Th rmaining fraction of th pric of th CS is calld th liquidity prmium and is rtaind by th issur. Th proprtis of continuous-tim CS stratgis hav bn studid in th litratur as particular cass of Constant proportion portfolio insuranc CPPI 1 stratgis. Som clarifying paprs on th topic of continuous CPPIs ar Bookstabr and angsam [8, 2000], Black and Prold [7, 1992] or Brtrand and Prignt [3, 2002]. Th litratur on CPPI also dals with th ffcts of jump procsss, stochastic volatility modls and xtrm valu approachs on th CS mthod, c.f. Brtrand and Prignt [4, 2002], Brtrand and Prignt [5, 2003]. Nvrthlss th issu of discrt-tim CS has bn barly covrd; in a rcnt working papr Baldr-Brandl-Mahayni [2, 2005] analyz a discrt-tim vrsion of a gnral CPPI stratgy which is usd for risk managmnt purposs. Thrfor risk masur statistics lik shortfall and xpctd shortfall givn dfault ar all computd undr th ral world masur. On important fatur of hdg funds is thir illiquidity s,.g., Inichn [10, 2002], which mans that buying or slling shars of a hdg fund is rstrictd to spcific dats. Th priod btwn two conscutiv dats can rang from on month to two yars. Accordingly, th issur can only chang his invstmnt in th undrlying hdg fund portfolio of th CS at ths spcific rbalancing dats. For th invstor, th distinction btwn carrying out a constant lvrag stratgy by himslf and buying a CS ariss in th cas whr th portfolio valu falls undr th loan btwn two conscutiv rbalancing dats. As for th constant lvrag stratgy, this cas corrsponds to a ngativ quity which th invstor ows to th issur of th loan. Rgarding th CS w will rfr to this cas as th dfault of th CS. Th hdg fund portfolio is liquidatd and th CS xpirs worthlss. Th payoff for th invstor is zro and th issur taks th rcipts to rdm th loan. Thus, th buyr of th CS is in possssion of a downsid protction on his invstmnt. Whil h could los mor than his original invstmnt by carrying out a constant lvrag stratgy on his own, this is not possibl if h buys th CS. Hr th option charactr of th CS coms into xistnc for th invstor. Th pric for this option or downsid protction is th liquidity prmium. Vic vrsa, th issur of th CS of cours bars th risk of losing part of his loan as h cannot liquidat th hdg fund portfolio immdiatly whn th portfolio valu drops blow th issud loan. Du to this liquidity risk, th issur will charg mor for th CS than th initial valu which h invsts into th portfolio. Th pric of th CS is composd by th mony invstd in th portfolio plus th liquidity prmium accounting for th option th invstor buys and which compnsats th issur for baring th risk of dfault. In contrast to approachs which dtrmin th liquidity prmium for stocks using th xpctd illiquidity s,.g., Acharya and ass [1, 2003], th liquidity prmium can b calculatd in advanc and is constant. Th thr factors thanflunc th probability of dfault of th CS and thus dtrmin th liquidity prmium ar th volatility of th undrlying portfolio, th lngth of th priod btwn two rbalancing dats, and th lvrag of th portfolio. Basd on th dscriptions abov, a CS can b sn as a string of call options with a maturity priod idntical to th timstp btwn two conscutiv rbalancing dats. Obviously, th strik pric corrsponds to th loan plus accrud intrst at a spcific rbalancing dat. As th lvrag is rst to a constant lvl ovr tim, th strik pric of th option maks up a particular fraction of th currnt portfolio valu. This mans that th rlation of portfolio valu and strik pric is constant at ach rbalancing dat, nglcting th cas of dfault. On th othr hand, th absolut valu of th strik pric, or th loan rspctivly, changs with th duration of th CS. In this papr w focus on a discrttim CS, and w pric this stratgy undr th risk nutral masur obtaining a closd form xprssion for th liquidity prmium. This papr procds as follows. In Sction 2 w dfin th CS mathmatically. In Sction 3 w calculat th payoff of th CS at maturity and driv th pric of a CS assuming a Gomtric Brownian Motion for th undrlying portfolio. To gt a bttr undrstanding of th snsitivity of th CS to markt paramtrs, w laborat its dynamic bhavior in Sction 4. Finally, w summariz our findings and draw a conclusion. 1 A CS could b sn as a CPPI with zro floor and multiplir qual to a function of th wantd lvrag 374
3 Th pric of liquidity in constant lvrag stratgis 2 Mathmatical Dfinition of th CS In this sction w dfin th constant lvrag stratgy and CS mathmatically and dmonstrat its functionality by giving a on-stp xampl from on rbalancing dat to th conscutiv rbalancing dat. All stochastic procsss ar dfind on a filtrd probability spac Ω, F, F t t [0,T], P which satisfis th standard hypothss. W assum that th undrlying S of th CS follows a Gomtric Brownian Motion in th risk-nutral world whr intrst rats ar stochastic 2 and th short rat is r t. Furthrmor, w assum that th marks complt 3, i.. that thr xists an quivalnt martingal masur s,.g., Zagst [11, 2002] and that th dynamics of th stock, S is: ds t = r t S t dt + σs t dw t, whr W s th standard Brownian Motion undr th risk-nutral masur s,.g. Hull [9, 2005]. t α t dnot th amount of S t hld from t to t + 1 so that th total mony invstd in th hdg fund portfolio in s α t S t. t Bt a b th valu of th loan in t, or rspctivly th mony borrowd from th risklss account, aftr rbalancing in t. Bt a grows to Bt+1 b = R t+1 t rsds Bt a bfor th rbalancing in t + 1 taks plac. Th quity V n s th diffrnc btwn th valu of th hdg fund portfolio and th loan, V t = α t S t Bt a. Not that th procss for V s slf-financing whras th procss for α t S s not. Th maturity of th CS and thus th constant lvrag stratgy is T. Rbalancing occurs in, i = 1, 2,..., N 1, with N := T/ t and s th timstp btwn two rbalancing dats. To illustrat th functionality of th CS, w giv an on-stp xampl from to +1. Aftr rbalancing in th portfolio valu of th undrlying hdg fund portfolio is α ti S ti. This amouns financd by th quity V ti and th loan Bt a i whr Bt a i is chosn in such a way that th ratio of Bt a i and V ti quals th targt lvrag, i.. th lvrag of th portfolio is = Bt a i /V ti. In +1 th valu of th portfolio bfor rbalancing is α ti, th nw loan bfor rbalancing is B b +1 = t rsds i Bt a i and th quity is V ti+1 = α ti B b +1. Dpnding on th prformanc of S w now hav to distinguish btwn two cass: Cas 1 α ti B b +1 In this cas th portfolio valu has droppd blow th loan. As th quity V ti+1 is zro or ngativ in +1 th CS dfaults and xpirs worthlss. Th ownr of th option will gt back nothing, i.. his payoff is zro. Th issur of th option loss th diffrnc of B b +1 α ti. Cas 2 α ti > B b +1 In this cas is vry likly that th lvrag has changd and thus th financing of th portfolio has to b adjustd. Th rbalancing of th portfolio occurs according to two conditions, th constant lvrag condition and th slf-financing condition for V ti+1. Th constant lvrag condition implis that must hold B a +1 = V ti+1 = α ti+1 B a Solving this for B a +1 lads to B a +1 = α ti+1, i.. α +1 = V ti Th calculations will show that th particular procss for th intrst rat has no influnc in th valu of th CS. 3 Th hdg fund is a drivativ in th markt not an undrlying, and th marktslf is arbitrag fr and complt. Th assts of th hdg fund, ar tradd continuously in th markt and th rbalancing of th hdg fund is discrt. 375
4 M. Escobar, A. Kichl,. Sco and R. Zagst Thus, th amount of mony invstd in th undrlying portfolio is a constant multipl of th quity. Hr w can s th similarity to th CPPI stratgy s.g. Black and Jons [6, 1987]. Du to th slf-financing condition for V t must also hold that By solving 3 for α ti+1 w obtain V ti+1 = α ti B b +1 = α ti+1 B a α ti+1 = V +1 + B a +1 = V +1 + V ti+1 last quality by 1 = V +1 = α B b +1 last quality by 3 = α t rsds i Bt a i 4 = α S ti 1+ rsds = α ti 1 = α ti + α ti 1 S by 2 R ti t rsds i t rsds i. Thus w gt 1 S t rsds i < 0 if S ti t < rsds i, Cas 1 and 1 S t rsds i > 0 if S ti t > rsds i, Cas 2 rsds is th bond prfor- whr /S ti dnots th prformanc of th hdg fund portfolio and manc in [, +1 ]. Hnc, α ti+1 < α ti in Cas 1 and α ti+1 > α ti in Cas 2. Using Equation 3 w thus gt Bt a i+1 = α ti+1 α ti + Bt b i+1 < Bt b i+1 in Cas 1 and Bt a i+1 = α ti+1 α ti + Bt b i+1 > Bt b i+1 in Cas 2. W can draw th following conclusion. In Cas 1 w gt α ti+1 < α ti and Bt a i+1 < Bt b i+1, which mans that th loan is dcrasd to match th initial lvrag of th portfolio. For Cas 2 w gt α ti+1 > α ti and Bt a i+1 > Bt b i+1, 376
5 Th pric of liquidity in constant lvrag stratgis which corrsponds to an incras of th loan to match th initial lvrag. By substituting th rsult for α ti+1 from Equation 4 into 2, w gt for B a +1 Bt a i+1 = α ti 1 = α ti 1 S ti S ti rsds t rsds i With this w can now stat th nw portfolio aftr rbalancing in +1 α ti+1 = V ti+1 + B a +1,. whr α ti+1 and Bt a i+1 ar givn by Equations 4 and 5. Bringing th two cass togthr, w can now giv th valu of th quity for th invstor in th CS aftr on stp from to +1. { α ti Bt b V ti+1 = i+1 if α ti > Bt b i+1 0 if α ti Bt b i+1 = max{α ti B b +1, 0}. Th valu of th loan for th issur aftr on stp from to +1 is Bt b t rs i+1 = min{ ds i Bt a i, α ti }. This quation for B b +1 clarly shows th risk of th bank to los part of its loan. Abov w argud that du to this risk th bank will charg a liquidity prmium. Hnc, th pric of th CS: COt 0 is COt 0 = V t0 + iquidity Prmium. 6 Th liquidity prmium ariss bcaus of th illiquidity of hdg funds. In th cas of continuous rturns and liquidity of th undrlying of th CS, th liquidity prmium disappars and th pric of th CS is V t0. In this cas thr is no rason for th issur to charg a prmium as th loan dosn t bar any dfault risk bcaus th hdg fund portfolio can b liquidatd as soon as th portfolio valu approachs th valu of th loan. As w can s from th prvious xampl, th Constant vrag stratgy is a pro-cyclical stratgy. In th cas that th undrlying portfolio has rturnd mor than th intrst on th loan from t to t + 1, th quity maks up a biggr fraction of th portfolio at th nxt rbalancing t + 1, or in othr words, th lvrag of th portfolio has dcrasd. Thrfor, an additional loan has to b takn in t + 1 to st up th portfolio with th initial lvrag. On th othr hand, in th cas th undrlying portfolio has rturnd lss than th intrst on th loan but has not dfaultd, th lvrag of th portfolio has incrasd and thus som of th loan will b paid back to rstor th initial lvrag. 5 3 iquidity Prmium and th pric of CS. With th stting from abov w can first calculat th pric of a CS and thn th valu of th liquidity prmium. This pric in t 0 = 0 is composd by th quity V 0 plus th liquidity prmium which compnsats th bank for th risk of losing part of its loan. W first driv th payoff of th CS at maturity and aftrwards calculat th pric of th CS as th discountd xpctd valu of th payoff at maturity. In ordr to do this, rcall that th CS can b sn as a string of singl call options, ach with a maturity of t. Thus, w hav to chck at ach rbalancing dat if th CS has dfaultd or not sinc th last tim 377
6 M. Escobar, A. Kichl,. Sco and R. Zagst w hav rbalancd. This mans that th payoff at maturity is conditional on th fact that th option has not dfaultd bfor. To rprsnt th conditional payoff w us th Indicator Function. This function only taks two valus, 1 for a crtain st of vnts, and 0 for th complmntary st of vnts: { 1 if x A 1 A x = 0 if x / A t us now calculat th payoff of th CS at maturity T. W transform this payoff and xprss in trms of a normally distributd random variabl. This transformation will facilitat th calculation of th xpctatd payoff in th following. Proposition 1 t Y ti := W ti W ti and z := ln maturity tim T = t N is givn by COT = N R tn V t0 t0 rs ds PROOF. S Appndix A σ2 σ2 2 t+σy 2 t /σ. Thn th payoff of a CS at 1 {Yti >z}. 7 With this rsult w can now calculat th pric of th CS in t 0 by calculating th discountd xpctation of th payoff drivd in th abov proposition s,.g., Zagst [11, 2002]. Proposition 2 Th pric of th CS spcifid by th sttings abov is whr d 1 and d 2 ar givn by COt 0 = N V t0 Nd 1 N Nd 2, 8 d 1 = PROOF. S Appndix A. ln 1+ + σ2 2 t σ, d 2 = d 1 σ t. t Not than particular th pric of th CS in 8 is indpndnt of th stochastic intrst rat r t. Proposition 3 Th pric of th CS, COt 0, can b writtn as [ ] N COt 0 = V t0 1 + Call, 1, 0, σ, t, whr Call S, X, r, σ, t dnots th Black-Schols call pric with undrlying pric S, xrcis pric X, risklss rturn r, volatility σ, and tim to maturity t. PROOF. S Appndix A. Th valu of th iquidity coss providd nxt as a rsult of Equation 6 and Proposition 3. Corollary 1 Th valu of th iquidity Prmium, It 0, can b writtn as: [ ] N It 0 = V t0 1 + Call, 1, 0, σ, t 1, whr Call S, X, r, σ, s as bfor. 378
7 Th pric of liquidity in constant lvrag stratgis Figur 1. Pric of th CPPI Option, T = 10 yars, = 3, V t0 = Dynamic Bhaviour of iquidity Prmium For a dpr undrstanding of th CS w now show how th pric of th CS which w hav dducd in th prvious sction dpnds on th volatility σ of th undrlying hdg fund portfolio, th priod btwn two conscutiv rbalancing dats t and th lvrag of th portfolio. First, rcall that th pric of th CS is composd by th initial invstmnt of th invstor plus th liquidity prmium du to th illiquidity of a hdg fund invstmnt. This liquidity prmium is strongly rlatd to th probability of dfault of th CS. For th invstor th option charactr of th CS coms into xistnc in th cas of dfault. Th thr factors thanflunc th probability of dfault of th CS and thus dtrmin th liquidity prmium ar σ, t, and. As w hav showd abov, th pric of th CS is indpndnt of th stochastic intrst rat r t. Figurs 1 and 2 show th pric of th CS in t 0 with a maturity of T = 10 yars dpnding on σ and t with a lvrag of = 3 Figur 1 and a lvrag of = 5 Figur 2. Th initial quity V t0 is on. W chos valus up to 15% for σ and up to 2 yars for t which ar typical for a hdg fund portfolio. As w can s in both figurs, th pric of th CS is on, which is th initial invstmnt of th invstor, for small valus of σ and t. This mans that th liquidity prmium is clos to zro. Th rason for this is than ths cass almost no dfaults occur. This bcoms clar if w rcall that dfault happns whn th undrlying portfolio loss mor than th valu of th quity from on rbalancing dat to th nxt on. Thus, dfauls unlikly if th portfolio xhibits a low standard dviation and is rbalancd frquntly. Figurs 1 and 2 show that th liquidity prmium incrass with inrasing valus for σ and t. As th volatility of th undrlying portfolio incrass and th distanc btwn two conscutiv rbalancing dats gts biggr th probability of dfauls rising as wll. In th cas of dfault, th buyr of th CS is bttr off than with a dircnvstmnn th undrlying portfolio and thrfor has to pay for th liquidity prmium of th CS. Figur 2 shows that th liquidity prmium can mak up for a larg fraction of th pric of th CS. For a lvrag = 5, σ = 15% and t = 2 yars for xampl th liquidity prmium accounts for 0.7 of th pric of th option which mans that th CS is almost twic as xpnsiv as a dirct invstmnt. Th rason for this high liquidity prmium is than this cas th probability of dfault for th CS during th maturity is 72%. Furthrmor, w can s that highr valus for th lvrag lad to a highr liquidity prmium. This is obvious as th quity rprsnts a smallr valu of a highly lvragd 379
8 M. Escobar, A. Kichl,. Sco and R. Zagst Figur 2. Pric of th CPPI Option, T = 10 yars, = 5, V t0 = 1.0 portfolio and thus smallr losss of th undrlying portfolio can rsuln dfault. In Figurs 1 and 2 th bhaviour of th pric of th CS is quit intuitiv and asy to undrstand. To giv a complt pictur, w xtnd th volatility for th undrlying portfolio to a maximum of 40% in Figur 3. At th first glanc th fact that th option pric is falling again with incrasing t for gratr valus of σ may b surprising and countr-intuitiv. To undrstand this bhaviour of th pric, w hav to hav a closr look at th occurrnc of dfaults. Figur 4 shows th probability of dfault for a CS with T = 10 yars and = 3 dpnding on σ and t. Illustrats that th probability of dfault firsncrass and thn dcrass with growing t. Rcall that with continuous rbalancing thr ar no dfaults. If on waits longr for th nxt rbalancing,.g. on or thr months, th probability of dfaulncrass. With a rbalancing priod of on yar th probability for this option to dfault paks at about 80%. Howvr, th surprising conclusion of this plot is thaf th rbalancing priod is incrasd furthr, th probability of dfauls dcrasing again.this bhaviour can b xplaind as follows. Imagin two not vry ralistic CSs with th sam maturity T = 10 yars, on rbalancing onc aftr fiv yars CO 1, th othr on with no rbalancing at all CO 2. t th othr paramtrs for this xampl b σ = 25% and = 3. Th probability of dfault for CO 1 is highr than for CO 2 simply bcaus thr ar two possibilitis for it to dfauln t = 5 yars and in t = 10 yars compard to only on in t = 10 yars for CO 2. Or in mathmatical trms, th probability of dfault for CO 1 aftr fiv yars is 41%. Th ovrall probability of dfault for CO 1 thus is 41%+100% 41%41% = 65%. Th probability for a dfault of CO 2 aftr tn yars is 51%. Hnc th probability of dfauls gratr if w conduct a rbalancing aftr fiv yars compard to th cas whr w do not rbalanc at all. Knowing that th pric of th CS is closly rlatd to th numbr of dfaults w can now undrstand why th pric of th CS bgins to dcras again for larg valus of n Figur 3. Th rason for this bhaviour of th pric is that th probability of dfauls bginning to dcras for larg t. Howvr, ths thortical xaminations may not b vry rlvann practic. ooking at th maximum of th probability of dfauln th prvious xampl w can rason that nobody would us a product that has such high probabilitis of dfault to invsn hdg funds. For a rasonabl rang of th probability of dfault.g. up to 5% or 10% th pric of th CS incrass with incrasing σ and t as is shown in Figur
9 Th pric of liquidity in constant lvrag stratgis Figur 3. Pric of th CPPI Option, T = 10 yars, = 3, V t0 = Summary and Conclusion In this papr w calculatd th valu of th liquidity prmium that appars as a rsult of discrt rbalancing of Constant vrag Stratgis. Th main assumptions ar a Gomtric Brownian Motion for th undrlying hdg fund portfolio and stochastic intrst rats. Du to th fact that hdg funds can b tradd only at spcific dats, th CS can dfault btwn two rbalancing dats and thus th pric of th CS must contain a liquidity prmium. Th thr paramtrs which influnc this prmium ar th volatility σ of th undrlying portfolio, th timstp t btwn two conscutiv rbalancing dats and th lvrag which dtrmins th financing of th undrlying portfolio. W found out that gratr valus for σ and th lad to a gratr liquidity prmium and thus a highr pric of th CS. Th rason for this is that with incrasing σ and th probability of dfault of th CS also incrass. Th influnc of t on th option pric dpnds on th spcific combination of all thr factors σ, t and. W discovrd that for highr but fixd valus of σ and th impact of t on th probability of dfault changs with incrasing t and lads to a hump in th function of th pric of th CS. Finally, it turnd out that th pric of th CS dos not dpnd on th stocastic intrst rat r t. Rfrncs [1] ACHARYA, V. AND ASSE, H. P., Asst Pricing and iquidity Risk, ondon Businss School, working papr. [2] BADER, SVEN, BRAND, MICHAE AND MAHAYNI, ANTJE, Effctivnss of CPPI Stratgis undr Discrt-Tim Trading, working papr. [3] BERTRAND, P. AND PRIGENT, J.-., Portfolio Insuranc Stratgis: OBPI vrsus CPPI, discussion papr, GREQAM and Univrsit Montpllir1. [4] BERTRAND, P. AND PRIGENT, J.-., Portfolio Insuranc: Th Extrm Valu Approach to th CPPI, Financ, 23,
10 M. Escobar, A. Kichl,. Sco and R. Zagst Figur 4. Probalility of dfault, T = 10 yars, = 3 [5] BERTRAND, P. AND PRIGENT, J.-., Portfolio Insuranc Stratgis: A Comparison of Standard Mthods Whn th Volatility of th Stoch is Stochastic, Intrnational Journal of Businss, 8, [6] BACK, F. AND JONES, R., Simplifying portfolio insuranc, Th Journal of Portfolio Managmnt, 14, 1, [7] BACK, F. AND PEROD, A. R., Thory of constant proportion portfolio insuranc, J. Econ. Dynamics Control, 16, [8] BOOKSTABER, R. AND ANGSAM, J. A., Portfolio Insuranc Trading Ruls Digst Summary, Journal of Futurs Markts, 20, 1, [9] HU, J. C., Options, Futurs and Othr Drivativs, Parson Prntic Hall. [10] INEICHEN, A. M., Absolut Rturns: Th Risk and Opportunitis of Hdg Fund Invsting, Wily Financ. [11] ZAGST, R., Intrst Rat Managmnt, Springr Financ. A Appndix PROOF OF PROPOSITION 1. nd two rsults: For th following calculation of th payoff of th CS at maturity T w first 1. t X ti := S ti /S ti, thn N s=1 X ts = S t 0 S tn, i.. S tn = S t0 tn X. 382
11 Th pric of liquidity in constant lvrag stratgis 2. Rcall that S ti = S 0 R σ2 0 rs ds 2 ti+σw. Thn, X ti = S S ti = R ti rsds+ σ2 2 t σy. 9 Rcall that W t N0, t, which implis that Y ti N0, t. Not than particular Y ti and Y tj ar indpndnt for i, j = 1,..., N and i j. Now w can giv th payoff of th CS at maturity T = t N. Th payoff in T is th xcss of th portfolio valu in T ovr th loan BT b on th condition that thr has bn no dfauln, i = 1,..., N. As th Y ti ar indpndnt, w can simply multiply th probabilitis of dfault to obtain th conditional payoff in T. Furthrmor, du to th constant lvrag proprty, ths probabilitis of dfault ar th sam for ach, i = 1,..., N. Th payoff of th CS at maturity T = t N on th condition that no dfault has occurd in, i = 1,..., N 1 is α tn S tn Bt b N. W now transform this payoff to xprss in trms of Y t which is normally distributd. COT = applying 2 aplying 4 applying 9 R tn + N α tn S tn tn rsds B a tn R tn = α tn S tn tn rsds αtn S tn R tn StN = α tn S tn tn rs ds S tn = α t0 N 1 α ti Sti > R rs ds B a ti 1 R ti rsdss ti S tn S ti R tn StN tn rsds S tn = α t0 N S tn N R tn StN tn rsds S tn = α t0 N S t0 N X 1 R tn tn rsds X tn 1 α ti Sti > R rs ds B a ti 1 α ti Sti > R rs ds B a ti 1 α ti Sti > R rs ds B a ti 1 R ti t rsds S ti i S ti N N 1 S ti 1+ S ti R > t rs ds i 1 R ti rsds Xti 1 R X ti < 1+ t rs ds i 383
12 M. Escobar, A. Kichl,. Sco and R. Zagst = α t0 N S t0 N X applying 9 again = α t0 N S t0 1 R ti rsds Xti R ti t rsds σ2 i 2 t+σy 1 R ti t rsds R i t rsds+ σ2 i 2 t σy = α t0 N S t0 = α t0 N R tn S t0 t0 rs ds 1 R X ti < 1+ t rs ds i 1 Y ti > ln 1+ + σ2 2 t σ =:z R ti t rsds σ2 i 2 t+σy R ti t rsds i σ2 2 t+σy 1 {Yti >z} 1 {Yti >z} Now w know from Equation 2 that α ti+1 = V ti+1. PROOF OF PROPOSITION 2. t z := ln 1+ + σ2 2 t /σ and f yti dnot th dnsity function of Y ti, i.. th normal distribution N0, t. Thn th pric of th CS today is th xpctd valu of th discountd payoff of th CS in T. E t0 [COT] = E[ R T t 0 rsds COT] applying 7 = α t0 N S t0 = α t0 N S t0 N = α t0 N S t0 = α t0 N S t0 z z z z σ2 2 t+σyi fy 1 fy 2 dy 1 dy N σ2 2 t+σyi 1 1 2π t 1 2 t y2 i 2σ tyi+σ2 t 2 dy i z σ t 1 N t 2π t y z 2 i 2 t dy i 1 y2 ti 2 t dyti 2π t z 1 N t z σ t = α t0 N S t0 1 N z N 1 N t t Substituting z lads to ln 1+ COt 0 = α t0 N S t0 N + σ2 2 t σ ln 1+ t N σ2 2 t N σ. t 384
13 Th pric of liquidity in constant lvrag stratgis PROOF OF PROPOSITION 3. Not that for th pric COt 0 of th CS w gt, using th put-call parity, CO 1 t 0 = N V t0 [ = N V t0 Put N 1+ N [ = N V t0 1 [ = V t0 1 + Call ln 1+ σ2 2 t σ t ln 1++ σ2 2 t σ t, 1, 0, σ, t + Call ] N N ] N, 1, 0, σ, t ] N., 1, 0, σ, t W thrfor gt a liquidity prmium pr rbalancing priod t of Call, 1, 0, σ, t. Marcos Escobar Dpartmnt for Mathmatics, Ryrson Univrsity, Munich scobar@ryrson.ca uis Sco Dirctor, Risklab Toronto, Sigma Analysis & Managmnt, Toronto Andras Kichl Munich Univrsity of Tchnology, Rudi Zagst Dirctor, HVB-Institut for Mathmatical Financ, Munich Univrsity of Tchnology, Munich, zagst@ma.tum.d 385
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