CAPITAL PROTECTION: MODELING THE CPPI PORTFOLIO.

Transcription

1 CAPIAL PROECION: MODELING HE CPPI PORFOLIO. ALESSANDRO CIPOLLINI Deparmen of Mahemaics Applicaions, Universiy of Milan-Bicocca P.zza Aeneo Nuovo 1, Milan, Ialy Fixed Income Relaive Value Research, Deusche Bank AG London Wincheser House, 4 Grea Wincheser Sree, London EC2N2B alessro.cipollini@db.com ABSRAC: he aim of he paper is o inroduce a rigorous mahemaical modellizaion of he Consan Proporion Porfolio Insurance CPPI, wih as principal aims: o capure he characerisic of he porfolio s disribuion, o analyze he properies of he reurn/risk profile o idenify a procedure hrough i is possible o produce reliable fuure arge reurns. Paricular sress has been made in order o produce closed formulas ha may lead o easy Mone-Carlo simulaion esimaes. Moreover as aemp o hedge he CPPI porfolio s risks, he approach followed relies on classical replicaion echniques: he closing ou effec he gap risk are faced by acing on he CPPI as on a self-financed hedging sraegy convering i in a synheic derivaive. In addiion o pu an exra proecion i has been sudied he case of Vanilla opions having as underlying he CPPI porfolio s value a mauriy as srike he proeced amoun. wo are he CPPI porfolios analyzed. As firs i has been considered he known case of a CPPI porfolio whose risky exposure is non-consrained whose floor consiss in he risk-less bond. As second, i has been esed he mahemaical model on he case of a CPPI whose porfolio s risky exposure is consrained whose floor is rebalanced a any ime of he financial horizon. KEYWORDS: mehods. Capial proecion, CPPI, Non-arbirage price, Mone-Carlo 1. Inroducion. he Consan Proporion Porfolio Insurance CPPI has been firs inroduced in he conex of capial proecion echniques, by F. Black R. Jones 1968 F. Perold 1986, as a paricular form of dynamic hedging. he invesor invess in a 1

2 porfolio for which he wans principal proecion eiher of he whole iniial amoun or of a leas a porion of i. he porfolio s manager shifs asse allocaion over he invesmen period beween a risk-free invesmen a collecion of risky asses, having as a scope he preservaion of a porion of money sufficien o acquire a mauriy he proecion requesed by he invesor. he posiion on he risk-less asse is equivalen o an ideal invesmen in a risk-less bond ha maures a he proeced amoun. he posiion on he risky marke is equivalen o a porfolio whose asse allocaion is based enirely on risky asses, such as socks, bonds, hedge funds credi producs. he amoun of money ha ideally has o be invesed in he risk-less bond is called floor depends sricly on he iming-o-mauriy of he invesmen. he difference beween he porfolio s invesmen he floor is called cushion. In order o produce ineresing performances, he cushion is invesed in he risky marke in a leveraged forma, whose leverage originally, see for insance F. Perold W. Sharpe 1995 F. Perold F. Black 1992, has been aken as consan during he whole life of he produc, hus moivaing he name consan proporion. he higher he leverage is, he higher poenial upside would be. However, high leverage increases he risk ha he value of he risky exposure declines oo fas for he porfolio s manager o readjus he asse allocaion. Indeed he porfolio, albei wih a small probabiliy, could fall below he value of he floor, hus incurring in a possible loss for he invesor. In such cases he porfolio s manager has o choose among wo alernaives: eiher he invess he floor in he risk-less bond ha provides a mauriy he graned capial or consumes small porions of he floor invesing again in he risky marke. In he firs case he produc ceases o exis merely becomes a passive risk-free invesmen, hus moivaing he name of closing-ou effec. In he second case, he he porfolio s manager, despie of is views, may incur in oher losses ha eiher compromise he possibiliy of having enough money o ge a a cerain poin he proecion, or erode compleely he floor hus moivaing a bankrupcy. his risk is commonly referred as gap risk, see for insance F. Sharpe o grasp he poenial of he CPPI porfolio, i is worhy o analyze is advanages disadvanages in comparison wih classical capial proecion mehods, such as saic porfolio hedging. Mainly saic porfolio hedging such as Opion Based Porfolio Insurance OBPI, following R. Booksaber J. Langsam 2000, iniially invess a porion of he porfolio in he risk-free zero coupon bond in order o ensure for cerain ha he ensured amoun is available a mauriy. herefore he asse allocaion is esablished a he beginning canno be changed during he life of he produc. Differenly he CPPI porfolio s asse allocaion may be adjused according o he performance of he risky marke, hus given more flexibiliy o he porfolio s manager see P. Berr J. Prigen Moreover i is easy o noe ha he saic porfolio hedging is equivalen o purchase a Vanilla Pu opion a he beginning of he financial horizon, having as underlying he porfolio s value. hus only he iniial erminal values of he risky asse deermine 2

3 he porfolio performance, regardless of he exac pah he risky asses followed. In conras he CPPI s risky exposure grows conracs in response o favorable poor performance. Because he amoun of he risky exposure is readjused based on he previous performance, he overall porfolio performance is pah-dependen. he CPPI configures as an ineresing alernaive o saic porfolio hedging, because of is flexibiliy ha resuls from he dynamic asse allocaion managemen. However he pah-dependency complicaes subsanially he produc s srucure, hus creaing difficulies regarding he mahemaical modellizaion ha is required by a cauious risk analysis of he invesmen. While he case of he saic porfolio hedging has been longly invesigaed, see for insance N.E. Karoui, M. Jeanblanc V. Lacose 2002, he issue of he CPPI s mahemaical modellizaion seems o be compleely open. he aim of he paper is o presen a rigorous mahemaical foundaion of he CPPI porfolio able o suppor an efficien risk managemen of he produc. In deail he problems ackled have been o capure he disribuion of he CPPI porfolio s value of is corresponding floor, o analyze he reurn/risk profile of he invesmen o produce efficien arge reurns of boh he porfolio he floor. he aemp is o develop closed formulas able o provide naurally simple Mone-Carlo esimaors of he value of he porfolio of is corresponding floor, a any ime of he invesmen period. In his fashion fuure arge reurns can be evaluaed as he corresponding Mone-Carlo esimaes. Moreover o manage he closing ou effec he gap risk, he approach followed consiss in classical replicaion echniques. he idea is ha he CPPI porfolio can be regarded as a classical self-financed hedging sraegy herefore can be convered in a synheic derivaive. In addiion o pu an exra proecion i has been considered he case of Vanilla opions having as underlying he value of he CPPI porfolio a mauriy as srike he proeced amoun. he firs par of he paper deals wih he classical case of a CPPI porfolio whose risky exposure is non-consrained in paricular he floor consiss simply in a risk-free bond mauring o he proecion s value. In he second par he idea is o es he abiliy of he mahemaical srucure in facing complex CPPI srucures. In deail i has been analysed he case of a Consan Proporion Porfolio Insurance wih Dynamic Adjusmen of he Floor CPPI-DAF, where in paricular he risky exposure is consrained he floor is rebalanced a any ime of he financial horizon, according o he performance of he risky marke. 2. he Consan Proporion Porfolio Insurance CPPI. Given 0, financial horizon, he reference marke consiss in nrisky asses wih prices S } 0, a in a risk-less asse S 0 } 0,, whose dynamics are specified by he equaions: ds = µs d + σs dw, 0, S 0 = s, a.s., 2.1 3

4 0 ds = rs 0 d, 0, S 0 0 = s 0, a.s., 2.2 where W is a Brownian moion defined on he probabiliy space Ω, F, P endowed wih he filraion F } 0,. Here µ R is he asses expeced reurn, σ R he volailiy r 0, 1 he yield ineres rae. Coherenly wih he hypohesis of absence of arbirage compleeness of he marke, boh diffusion consans are aken as bounded. he yield r has o be disinguished from he risk-free reference marke s spo ineres rae d 0, 1 o which corresponds he money marke accoun S 1 } 0, 1 ds = ds 1 d, 0, S = s 1, a.s.. In cases where he risk-less asse coincides wih he money marke accoun, he ideniy r = d is rue; in he oher cases eiher r > d or r < d. he Consan Proporion Porfolio Insurance CPPI is a self-financed porfolio in which he porfolio s asse allocaion a any ime is seleced in order o ensure he purchase of risk-free bond ha maures a he proeced amoun required by he invesor. Denoe by P he graned amoun a mauriy define he floor as he he presen value of a zero coupon bond wih mauriy referring noional of P : P = e d P, 0,. 2.4 Only a par of he available resources, called cushion, equal o he difference beween he curren porfolio s wealh process V } 0, he value of he floor, is invesed in he risky asses: C = V P,,. 2.5 Excep cases in which he marke s risk-free spo ineres rae d is a a very high level he cushion does no suffice o generae ineresing performances. As a consequence a common aiude is o leverage he invesmen. Se m 0, gearing consan he porfolio s exposure is defined as: E = mc = mv P, 0,. 2.6 he CPPI s porfolio is he self-financed rading sraegy V, α having as exposure E } 0, : where V = α ds S + V α ds0, 0,, 2.7 S 0 α = E = mv P, 0,

5 Subsiuing he risky risk-free asses dynamics ref , one obains he following Sochasic Differenial Equaion SDE: dv = rv + mv P σθ d + mv P σdw, 2.9a for θ = µ r/σ risk premium. Alernaively dv = rp d + V P mσθ + r d + mσv P dw, 0,. 2.9b Since i is desirable ha a mauriy he CPPI porfolio aains he graned amoun, he SDE has as erminal condiion V P, P a.s.. he porofolio s manager invess hrough a leverage effec herefore may be no able o re-adjus he porfolio s asse allocaion, hus incurring in he closingou-effec or in he gap risk. Boh risks can be included in he modellizaion, by imposing he following consrained SDE dv = mv P ds S + V mv P ds 0, 0, S 0 V P, a.s he CPPI s porolio equaion. Denoe by 0, he presen momen consider a CPPI porfolio, having as wealh process V,v }, such ha: dv,v = mv,v P ds,s S,s + V,v mv,v P ds 0,,s 0 S 0,,s0,,, 3.1 where V,v = v is he iniial invesed amoun. As explained before, he CPPI porfolio s wealh is compleely described by he SDE dv,v = rp d + V,v P mσθ + rd + mσv,v P dw,, = v, a.s., 3.2 V,v ha admis a unique soluion, since he risk premium is bounded. he wealh process is driven by he risky par wih B = 0 P a.s. explici form db = mσθ + rd + mσdw,,, 3.3 B = mσθ+r +mσw W = mσθ+r +mσw,,. 3.4 Indeed: dv,v = rp d + V,v P db,,

6 A cauious analysis of he represenaion 3.4 pinpoins ou he explici form of he wealh process, hrough which i is possible o sudy he CPPI porfolio s disribuion: heorem 3.1 Given he sochasic process B }, specified in 3.3, he CPPI s porfolio value V,v }, saisfies he following ideniy: V,v = P + r d + v p exp exp B m2 σ 2 } + 2 B m 2 σ 2 B 2 } P d. 3.6 Proof Consider he exponenial process ExpZ } 0,, defined as: For any, ExpB = exp B m2 σ 2 2 },,. 3.7 dexpb = ExpB db ExpB db db + m2 σ 2 ExpB d By consrucion: hus 3.8 becomes Se C,vp hus db db = m 2 σ 2,,, 3.9 dexpb = ExpB db + m 2 σ 2 ExpB d,, dv,v = V,v P = dc,vp dv,v, clearly 3.5 implies ha: + dp = dc,vp + dp d,, = C,vp db + rp d, 0,, 3.12 implies ha he cushion C,vp saisfies: dc,vp = C,vp db + r dp d, 0, C,vp = v p, a.s Muliply he process ExpB } 0, o boh sides of 3.13: dc,vp C,vp db ExpB = r dp ExpB d,, he Inegraion by Pars Formula jusifies ha for any, d C,vp ExpB = C,vp dexpb + ExpB dc,vp + dexpb dc,vp,

7 bu dexpb dc,vp so 3.15 becomes d C,vp ExpB = ExpB C,vp db db = m 2 σ 2 ExpB C,vp d, 3.16 = C,vp ExpB db + ExpB dc,vp,, As a consequence one may wrie: d C,vp ExpB = r dp ExpB d,,, 3.18 herefore C,vp ExpB = v p + r d ExpB P d,, he boundness of he diffusion coefficiens ensure he exisence uniciy of a soluion: V,v for any,. = P + r d + v p exp exp B m2 σ 2 } + 2 B m 2 σ 2 B 2 } P d, 3.20 Cases in which he risk-less invesmen is replaced by a posiion in he money marke accoun are easy o rea mahemaically: Corollary 3.2 ha: If he wo spo ineres raes r, d 0, 1 are equal, han i is rue V,v = P + v pe mµr+rm2 σ 2 /2+mσW W,, Noe ha as peculiariy, in he case of r = d he disribuion of he CPPI porfolio urns ou o be log-normal. A firs ineresing issue o ackle a his poin, is o verify if he CPPI porfolio is able a mauriy o ouperform a bond mauring a he graned amoun herefore can aain he proecion required. Assuming ha he porfolio s manager choices are inline wih he model, he graned amoun is aained a mauriy: heorem 3.3 If he following condiions are saisfied 1 i holds ha v p, 2 i is rue ha r d, han he CPPI porfolio s value V,v }, saisfies V,v 7 P P a.s..

8 Proof In boh cases of r > d r = d he proof is an easy consequence of respecively ; indeed V,v P P a.s. if only if V,v = P + v p exp B m2 σ 2 } + 2 B m 2 σ 2 + r d exp B 2 } P d P If he invesor ses up an iniial invesmen a mos equal o he proecion required, han he CPPI is able o provide he graned amoun a mauriy, independenly from he behavior of he marke. Differenly in cases in which he proecion requesed is higher han he iniial invesmen, he CPPI underperforms a bond mauring o he graned amoun. Anoher imporan issue of grea imporance for he developmen of a rigorous risk monioring is o rack he reurn/risk profile of he CPPI. Whenever he porfolio s manager choices are inline wih he model, formulas permi o compue explicily he reurn/risk profile of he invesmen: heorem 3.4 E V,v he CPPI s porfolio has expeced reurn variance given by: = P + v pe mµr+r + r dp + mµ r + r d emµr+r 1 e mµr+rd, 3.23 V arv,v = v p 2 e 2mµr+r e σ2 m p 2 r d 2 1 e mµr+rdσ 2 m 2 mµ r + r d σ 2 m 2 mµ r + r d 1 e 2mµr+2rdσ 2 m 2 2mµ r + 2r d σ 2 m 2 mµ r + r r + 1 e 2mµr+2rdσ 2 m 2 + 2mµ r + 2r d σ 2 m 2 mµ r + r d σ 2 m 2 emµr+rdσ 2 m 2 1 e mµr+rd mµ r + r d σ 2 m 2 mµ r + r d e2mµr+r 1 e mµr+rd 2 mµ r + r d }

9 Proof Firsly noe ha he Markov s propery implies ha: E V,v = E V V = v = E V V 0 = v, V ar = V ar V V = v = V ar V V 0 = v,,, 3.25 V,v hus i is sufficien o sudy he expeced value he variance in he case of = 0. Consider he exponenial process ExpB = e B σ2 m 2 /2. 0, Clearly ExpB } 0, is log-normal having as momens E ExpB = e σ2 m 2 /2 E e B = = e σ2 m 2 /2 e mµr+r e +σ2 m 2 /2 = = exp + mµ r + r } 0,, 3.27 E ExpB 2 = e σ2 m 2 E e 2B = = e σ2 m 2 e 2mµr+r e +2σ2 m 2 = = exp + 2 mµ r + r } e +σ2 m 2 0,, 3.28 so ha V ar ExpB = E = exp 3.29 E ExpB mµ r + r Look a he represenaion 3.6: V v = P p + e B m2 σ 2 /2 v p + r d ExpB 2 = } e +σ2 m 2 1 0, 0,. } e B m 2 σ 2 /2 P p d, 3.30 for any 0,. Fubini-onelli heorem ensures ha aking he expecaion 3.30 becomes E V v = P p + v pe mµr+r + r d E e B m 2 σ 2 /2 P p d herefore for any 0, E V v = P p + v pe mµr+r + r d e mµr+r P p d = 0 = P p + v pe mµr+r + pr d 1 e mµr+rd mµ r + r d.

10 o evaluae he second order momen, look a he equaion 3.6 C vp = v pe B m2 σ 2 /2 B + r d e B m 2 σ 2 /2 P p d. 0 For any, i is rue ha E C vp 2 = v p 2 E + 2v pr de e 2B σ2 m 2 + e B σ2 m 2 /2 0 e B B σ 2 m 2 /2 P p d r d 2 E e B B σ 2 m 2 /2 P p d, so ha: C vp 2 E = v p 2 E e 2B σ2 m v pr de e 2B B m 2 σ 2 2/2 P p + d 0 + r d 2 E e 2B B B ησ 2 m 2 2η/2 P p P η p ddη Since E e 2B σ2 m 2 = exp +2 mµr+r } e +σ2 m 2 0, moreover E e σmw +W W = E e σmw E e σmw W 0,, he firs inegral in 3.34 is equivalen o: E e 2B B σ 2 m 2 2/2 P p d = e mµr+r2 E e σmw +W W 0 0 e σ2 m 2 2/2 P p d = e mµr+rσ2 m 2 /22 e σ2 m 2 /2 e σ2 m 2 /2 P p d 0 p = mµ r + r d e2mµr+r 1 e mµr+rd, 0, he second inegral in 3.34 may be compued in similar fashion, so ha elemenary compuaions leads o he ideniy: E C vp 2 = v p 2 e 2mµr+r e σ2 m 2 2v ppr d + mµ r + r d e2mµr+r 1 e mµr+rd + p 2 r d 2 e 2mµr+r e σ2 m 2 1 e mµr+rdσ 2 m 2 mµ r + r d σ 2 m 2 mµ r + r d 10

11 1 e 2mµr+2rdσ 2 m 2 2mµ r + 2r d σ 2 m 2 mµ r + r d + 1 e 2mµr+2rdσ 2 m 2 + 2mµ r + 2r d σ 2 m 2 mµ r + r d σ 2 m 2 emµr+rdσ 2 m 2 1 e mµr+rd } mµ r + r d σ 2 m mµ r + r d Finally E C vp 2 = r d 2 p 2 mµ r + r d e2mµr+r 1 e mµr+rd 2 + 2v ppr d + mµ r + r d e2mµr+r 1 e mµr+rd + c 2 e 2mµr+r, 3.37 so ha he formula V arv v = V arc vp = E C vp 2 E C vp 2 0, implies In he case of r = d, he reurn/risk profile simplifies consisenly: Corollary 3.5 If he wo spo ineres raes r, d 0, 1 are equal, han he CPPI porfolio has expeced reurn variance, given by: for any 0,. Proof EV,v = P + v pe mµr+r, 3.38 V arv,v = v p 2 e 2mµr+r e σ2 m 2 1, 3.39 he proof is rivially obained seing r = d ino he formulas 3.23 I is ineresing a his poin o wonder how he leverage regime modifies he reurn/risk profile of he produc. As he inuiion suggess, an increase in he gearing consan ha deermines he leverage regime amplifies heavily he volailiy: Proposiion 3.6 he expeced value he variance of he CPPI porfolio s value, increase wih he leverage consan. In paricular i is rue ha for any,,v,v lim EV = + lim V arv = +, 3.40 m m lim m wih an order of oe σ2 m 2 m. EV,v V arv,v = 0, m

12 Proof I is immediae looking a he formulas o deduce ha as he leverage regime increases he expeced reurn he volailiy end o zero exponenially. In paricular: EV,v V arv,v e 2mµr+r e 2mµr+r e 2 σ2 m 2 eσ m 2 0, m, 3.42 ha is exacly In paricular equaion 3.41 implies ha a high leverage amplifies he volailiy, offseing he risk premium. herefore he gearing consan has o be seleced wih care in order o no incur in his undesirable effec. A mehod commonly used in pracice, is o choose a gearing consan value equal o hree, four or five according o he asses included in he porfolio. 4. he CPPI porfolio as an hedging ool. he CPPI porfolio is defined among he class of self-financed rading sraegies, wih wealh process asse allocaion V,v, α,v,p }, : dv,v = rv,v d + mv,v P σθd + mv,v P σdw, 0,, 4.1 α,v,p = mv,v P,,. 4.2 he fac ha he porfolio resembles a self-financed rading sraegy is paricularly useful whenever one would like o cover porfolio s risks, since i permis o look for elemenary hedges based upon saic/dynamic replicaion. In his fashion, he firs idea ha comes in mind is o invesigae if i possible o conver he CPPI porfolio in a synheic derivaive having as underlying he same risky asses. For insance he CPPI porfolio may be ransformed in he pay-off of a Pu opion wih a srike a leas equal o he he proecion required, so solving he problem of boh closingou-effec of gap risk. he parameer on which he porfolio s manager acs is he leverage regime, so ha i is reasonable o imagine ha he CPPI s wealh process is driven by: dv,v = rv,v d + m V,v P µ rd + m V,v P σdw, 0,, 4.3 wih m : Ω, R P a.s. bounded. Since he process m is bounded P a.s., he wealh process is sill well defined in paricular i saisfies he ideniy: V,v = P + v p exp B σ2 2 + v pr d exp } m 2 d B B σ } m 2 udu P d, 4.4

13 wih B = r + µ r m 2 d + σ m dw B 0 = 0,,. 4.5 Now, suppose ha η = gs,s is a coningen claim ha he porfolio s manager is aiming o have a mauriy. Can he CPPI porfolio be convered in a synheic derivaive wih pay-off specified by η = gs,s? heorem 4.1 Pick a map g : R n R sufficienly smooh. here exiss a unique self-financed gs,s hedging CPPI porfolio V,s, defined as V,s for v C 1,2 0, R n unique soluion of he PDE = v, S,s,,, 4.6 u, s + x u, ssr r 2 xxu, s sσ sσ } rsu, s = 0, u, s = gs,, s 0, R n, u C 1,2 0, R n. 4.7 In paricular he CPPI porfolio s gearing facor is given by: S,s m = xv, S,s v, S,s P,,. 4.8 Proof Consider V,s }, CPPI s wealh process specified in 4.1 dv,s = rv,s d + m V,s P σθd + m V,s P σdw,, V,s = v, a.s In order o have ha V is a self-financed gs,s hedging porfolio, i is enough o ensure ha a mauriy: V,s = gs,s, a.s Pick a map v C 1,2, R n se V,s v, S,s = V,s = gs,s P a.s., so ha: Secondly Io-formula implies ha: = v, S,s,. Firsly v, s = gs, s R n v, S dv, S,s,s = r 2 xxv, S,s S,s σ S,s σ } d + x v, S,s σdw,,

14 A comparison among implies ha v, S,s m = xv, S,s S,s v, S,s P,,, x v, S,s S,s r r 2 xx S,s 4.14 σ, S,s for any,. Leing, i follows 4.7. S,s σ, S,s } rv, S,s = 0, In a financial urmoil, he porfolio s manager acing on he leverage regime may conver he CPPI porfolio in a suiable synheic derivaive whose price is specified by Moreover he required dynamic gearing facor can be easily deermined, using 4.8. Anoher observaion ha reveals o be cenral in he analysis of possible porfolio s hedges is ha a any ime of he financial horizon he CPPI porfolio value may be regarded as a sard risky asse herefore as an underlying for any convenien coningen claim: heorem 4.2 Under he risk neural measure Q : F, 0, 1 he discouned CPPI porfolio s value V,v }, is a Maringale. M,v = e r V,v,,, 4.15 Proof Consider he risk neural measure Q : F, 0, 1 defined by he Radon- Nykodymn derivaive dq µ r F, dp = exp σ W 1 2 µ r I is clear ha he process B }, saisfies he SDE: σ 2 } db = mσθ + r d + mσdw B = 0,,, 4.17 ha under he measure Q becomes db = rd + mσdŵ B = 0,,, 4.18 for Ŵ = W + µ r,,, 4.19 σ 14

15 F, }, Brownian moion. han he CPPI s equaion becomes dv,v = V,v rd + mv,v P σdŵ V,v = v,, he inegraion by pars formula implies ha: d e r V,v = re r V,v + e r dv,v,,, 4.21 because he funcion e r has finie variaion. herefore: d e r V,v = me r V,v P dŵ,,, 4.22 so he process M,v = e r V,v = v + m e r V,v P dŵ,,, 4.23 is a QMaringale ino F,,. Indeed: E Q e 2r V,v P 2 d CE Q V,v C E Q sup, P 2 d V,v 2 + P 2} <, since V,v }, under Q is a soluion of an SDE P P for any,. Given any claim η = gv,v funcion of he erminal porfolio s price, here exiss a unique self-financed gv,v hedging sraegy: heorem 4.3 Pick a map g : R R sufficienly smooh. here exiss a unique hedging self-financed rading sraegy U,v, β,v defined as gv,v U,v = u, V,s β,v = x u, V,v,,, 4.25 where u C 1,2 0, R is he unique soluion of he PDE u, v + x u, vrv xxu, vm 2 v p 2 σ, s 2 ru, v = 0, u, v = gv, 4.26 for any s R n. Proof sraegy U,v Consider V,v, β,v du,v }, as an asse, pick a self-financed gv,v }, by seing: = β dv,v U,v + U,v,v = gv hedging β V,v rd,,, a.s

16 I is clear ha, given he dynamic of he asse V,v dv,v = rv,v d + mv,v he hedging porfolio s equaion may be rewrien as: du,v = ru,v +β mv,v P σθd + mσv,v P dw, 0,, 4.28 P µr d +β mv,v Pick u C 1,2, R se U,v For any,, Io-formula implies ha: u, V du, V,v,v = P σdw,, = u, V,v,. + x u, V,v rv,v + mµ rv,v P xxu, V,v m 2 v p 2 σ d + x u, V,v mv,v A comparison beween implies 4.25 wih in paricular: Moreover as, i is rue ha: u, v P σdw. β = x u, V,v,, x u, v rv + mµ rv p xxu, vm 2 v p 2 σ 2 = = ru, v + x u, vmµ rv p,, v 0, R, 4.32 wih he final condiion u, v = gv, v R, 4.33 ha is exacly he PDE he raionale in consrucing self-financed rading sraegies ha hedge he CPPI porfolio s erminal price, is ha here are coningen claims paricularly useful o conrol boh he closing-ou-effec he gap risk. As example consider he case of a Vanilla opions having as underlying he CPPI porfolio s value. For isance being long in an a-he-money Pu opion on he porfolio wih a srike a leas equal o he proecion required is a naural way o hedge gap risk. Similarly being long in an a-he-money Call opion on he porfolio is a naural way o inves in a CPPI s porfolio preserving he capabiliy o no purse forward he invesmen in he case of closed ou. he case of r, d 0, 1 idenical is paricular, since i is possible o obain a Black-Sholes like formula for he pricing of Vanilla opions having as underlying he CPPI porfolio: 16

17 Proposiion 4.4 If he wo spo ineres raes r, d 0, 1 are idenical, he price of a Vanilla Call/Pu opion on he whole CPPI porfolio s value a mauriy is compleely deermined by: lnv p/k P Call,v,, K = v pn e r K P r + σ2 m 2 /2 σm lnv p/k P N + r σ2 m 2 /2 σm lnk P P u,v,, K = e r K P N /v p r σ2 m 2 /2 σm lnk P v pn /v p r + σ2 m 2 /2 σm, 4.35 where Nx = R ez2 /2 / 2πdz x R K > P Proof Clearly if Q is he risk neural measure, he price of a Vanilla Call opion is given by: Call, v,, K = E Q e r V,v. K + = E Q e r V K + V = v, 4.36 for any 0,. he CPPI porfolio s value V,v }, is a Markov process so ha: Call, v,, K = Call0, v,, K, 0,, 4.37 i is sufficien o cover he case of he Vanilla Call opion s price a zero. Now consider he process: V v = P p + v pe mµr+rm2 σ 2 /2+mσW, 0, he risk neural measure Q is such ha he process: µ r Ŵ = W + d, 0, 4.39 σ is a Brownian moion herefore under Q V v = P p + v pe rm2 σ 2 /2+mσŴ, 0,, 4.40 since µ r W = Ŵ d, 0, σ 17

18 As a consequence: Bu: Call0,v,, K = e r E Q V p K + = Call0,v,, K = e r = R = v p = e r E Q v pe rσ2 m 2 /2 +σmŵ + K P p R v pe rσ2 m 2 /2 +σm + e z z 2 /2 K P dz = 2π + e v pe w e r w+σ 2 m 2 /2 2 /2 2 /2σ 2 m 2 K P 2σ2 m 2 π + e r K P + = v p e r K P lnkp /vpr + lnkp /vpr lnkp /vpr + e w ew+σ2 m 2 /2 2 /2 2 /2σ 2 m 2 2σ2 m 2 π dw e w+σ2 m 2 /2 2 /2 2 /2σ 2 m 2 2σ2 m 2 π e wσ2 m 2 /2 2 /2 2 /2σ 2 m 2 2σ2 m 2 π lnkp /vprσ 2 m 2 /2 σm + = v p lnkp /vpr+σ 2 m 2 /2 e r K P σm + e z/2 2π dz lnkp /vprσ 2 m 2 /2 σm herefore, if Nx = x ez2 /2 / 2πdz x R han: e z 2 /2 2π dz = dw dw = dw = e z 2 /2 2π dz lnv p/k P + r + σ 2 m 2 /2 Call0, v,, K =v pn σm lnv p/k e r P + r σ 2 m 2 /2 K P N σm 4.44 so for any 0, lnv p/k P + r + σ 2 m 2 /2 Call, v,, K = v pn σm lnv p/k e r P + r σ 2 m 2 /2 K P N σm ,

19 A compleely similar argumen shows he case of a Vanilla Pu opion. 5. CPPI Mone-Carlo simulaion echniques. In order o esimae fuure arge prices for he CPPI porfolio, i is necessary o inroduce a significan saisical esimaor for boh he porfolio s value he floor s value. Due o he complexiy of he CPPI srucure, i seems efficien in pracice o adop Mone-Carlo simulaion echniques. Consider dv,v = rv,v d +V,v P mµrd +mσv,v P dw,,, 5.1 wih iniial condiion V,v = v. Se 0 = < 1 <... < n = } pariion of he financial horizon,. he CPPI floor porfolio s rajecories may be approximaed by he following ieraive scheeme P n,, V n,,v }, : P n, i = P n, i1 + rp n, i1 i i1, i = 1,..., n, 5.2 V n,,v i = V n,,v i1 + rv n,,v i1 + mµ rv n,,v i1 P n, i1 i i1 + mσv n,,v i1 P n, i1 W i W i1 5.3 for any i = 1,..., n, wih iniial condiions P n, 0 = p V n,,v 0 = v. 5.4 Obviously o ge he value a a pariion poin i is sufficien o combine he ieraions in his way: ogeher wih V n,,v i = v + P n, i = p + i j=1 + mσv n,,v j1 for any i = 1,..., n. Now se: i j=1 P n, j1 + rp n, j1 j j1, i = 1,..., n, 5.5 V n,,v j1 + rv n,,v j1 + m µ r V n,,v j1 P n, j1 j j1 } P n, j1 W j W j1, 5.6 = supi : 1 i n such ha i },,

20 A he pariion inermediae poins, he floor he porfolio s rajecories are approximaed by V n,,v P n, = p + = V n,,v + mσ P n, + rv n,,v, S n,,s + rp n,,,, m µ r V n,,v V n,,v P n, P n, W W n ψ, 5.9 for any,. Now i is known see Bally alay 1996 for a formal proof ha in he space L p Ω, F,, P p 1, he following limis hold: lim P n, = P n lim V n,,v = V,v,,, 5.10 n wih a convergence order of o1/ n n. Obviously he convergence exends naurally o he case of P a.s. convergence. Now observe ha in order o obain deerminisic esimaes, i is necessary o simulae via Mone Carlo he se of rom variables W i W i1 : i = 1,..., n} or W W ψ n :, } ha share he disribuion eiher of N0, i i1 i = 1,..., n or of N0, ψ n,. As a consequence he ieraion scheme is esimaed by he define he Mone-Carlo esimaes p n,mc,j,, v n,mc,j,,v },, where: v n,mc,j,,v i p n, i = p n, i1 + rp n,mc,j, i1 i i1, i = 1,..., n, 5.11 = v n,mc,j,,v i1 + rv n,mc,j,,v i1 + m µ r v n,mc,j,,v i1 p n,mc,j, i1 i i1 + mσv n,mc,j,,v i1 for any i = 1,..., n, wih iniial condiions p n,mc,j, i1 i i1 z J, 5.12 p n,mc,j, 0 = p v n,mc,j,,v 0 = v, 5.13 In paricular he values a he pariion poins are approximaed by: p n, i = p + i j=1 p n, j1 + rp n, j1 j j1,

21 v n,mc,j,,v i = v + i j=1 v n,mc,j,,v j1 + + rv n,mc,j,,v j1 + mµ rv n,mc,j,,v j1 p n,mc,j, j1 j j1 + mσv n,mc,j,,v j p n,mc,j, j1 j j1 z J }, for any i = 1,..., n J 1. Similarly he values a he pariion inermediae poins are approximaed by: v n,mc,j,,v p n, = p + = v n,mc,j,,v p n, + rp n,,,, rv n,mc,j,,v + mσv n,mc,j,,v for any,. In disribuion i is rue he ideniy V n,mc,j,,v so ha i follows in L p Ω, F,, P p 1, : + mµ rv n,,v P n, p n,mc,j, ψ n z J, 5.17 = V n,,v, J 1, n 1,,, 5.18 lim V n,mc,j,,v = V,v,,, J n As a consequence, for a sufficienly big n 1 for any J 1, is ensured he convergence in L p Ω, F,, P p 1, hus also P a.s. of v n,mc,j,,v o he rue value of V,v,. 6. he Consan Proporion Porfolio Insurance wih a Dynamic Adjusemen of he Floor CPPI-DAF. In order o preven he gap risk he invesor in a CPPI porfolio usually fixes a priori he maximum he minimum proporion of he porfolio ha has o be invesed in he risky marke. he exposure of he CPPI porfolio is consrained o say ino cerain limis ha has o be respeced by he porfolio s manager wihou affecing he CPPI performances. Denoe by ρ 0, 1 he invesmen olerance percenage se: 0 E,vp ρv,v, 0,

22 Noe ha he cosrain ensures ha a any ime, he par invesed in he risky asse isn superior han ρ imes he value of he whole porfolio. Main problem o ackle is he new srucure obained ha heavily depends from a pah-dependen consrain. Consider he CPPI porfolio floor s value V,v = P + r d + v p exp B m2 σ 2 } + 2 B m 2 σ 2 exp B 2 } P d, 6.2 for any,, where B = 0 P a.s. P = e r P,,, 6.3 B = mµr+r +mσw W = mµr+r +mσw,,. 6.4 A consrain on he risky exposure on invesmen olerance ρ implies ha: dv,v = Equivalenly: while dv,v finally E ds,s ρv,v ds,s /S,s /S,s dv,v = rp d + V,v V,v + V,v + mv,v dv,v + V,v ds 0,,s0 /S 0,,s0, E = 0 E ds 0,,s0 /S 0,,s0, E 0, ρv,v ρv,v ds 0,,s0 /S 0,,s0, E = ρv,v, 6.5 = rv,v d when V,v = P, 6.6 P mµ r + rd+ P σdw, when V,v P, ρp,6.7 = V,v ρµ r + rd+ + ρv,v σdw, when V,v ρp =, 6.8 for any,. If he CPPI porfolio s value ouches he bond floor, han he porfolio s manager concenraes he asse allocaion in he risk-less asse. When he CPPI porfolio does no exceed he cosrain, he holder can implemen he sard CPPI s asse 22

23 allocaion sraegy. Finally as he porfolio s value ouches he limi of exposure, he porfolio s manager canno invess more han he limi iself. Whenever he porfolio s value reaches he consrain, a common aiude is o pu he pure excess E ρv,v /m = /m V,v /,, ino he floor. he resuling porfolio is said Consan Proporion Porfolio Insurance wih Dynamic Adjusmen of he Floor CPPI-DAF. One of he main difficuly in managing he srucure of a CPPI-DAF porfolio is ha he porfolio s value heavily depends from a pah-dependen consrain. In deail simple algebra leads o he following SDE for he wealh process: dv,v = rv,v d1 P } + rp d + V,v V,v P + V,v db 1 P dd 1 } ρp V,v + mρ, ρp V,v, 6.9 mρ for any,. Noe ha he diffusion coefficiens fail o be Lipschiz herefore he exisence uniciy of a soluion isn ensured a priori. I is possible o inroduce he CPPI-DAF in an alernaive manner. Noe a his proposal ha he CPPI-DAF porfolio is a paricular version of a CPPI porfolio hus anoher possibiliy seems o be o define i by replicaion echniques. Denoe by V d,,v },, P d, },, C d,vp },, E d,vp }, in he same fashion V,v }, P },, C vp },, E vp }, respecevely value, floor, cushion exposure of he CPPI-DAF porfolio of he CPPI porfolio. Se he exposure of he CPPI-DAF porfolio as or equivalenly E d,,vp E d,,vp = mine,vp, ρv,v },, 6.10 = E,vp he corresponding floor as P d, = P + 1 m E,vp E,vp ρv,v +,, , ρv,v, A Consan Proporion Porfolio Insurance wih Dynamic Adjusmen of he Floor CPPI-DAF is a CPPI porfolio having as exposure floor E d,,vp }, floor P d, },, i.e. dv d,,v herefore dv d,,v = E d,,vp ds,s S,s = rv d,,v +V d,,v E d,,vp 0 ds0,,s S 0,s0,,, µre d,,vp d +E d,,vp σdw,,

24 I is possible o describe compleely he CPPI-DAF in erms of a classical CPPI: heorem 6.1 Given he CPPI porfolio wih value V,v }, bond floor },, he CPPI porfolio wih Dynamic Adjusmen is defined by: P V d,,v = V,v µ r e r V,v +d σ 6.15 Proof P d, = P Look a he equaion: + V,v +,, m dv d,,v rv d,,v for any,. By consrucion one has: so dv d,,v dv d,,v rv d,,v E,vp d = E d,,vp µ rd + E d,,vp σdw, 6.17 d = E,vp µ rd + E,vp σdw ρv,v +µ rd E,vp ρv,v +σdw, 6.18 rv d,,v d = dv,v rv,v E,vp ρv,v he inegraion by pars formula implies ha: d e r V d,,v = e r dv,v e r E,vp e r E,vp E,vp ρv,v +µ rd +σdw,, rv,v ρv,v ρv,v d +µ rd +σdw,,, 6.20 since he funcion e r, has P a.s. finie variaion. As a consequence performing a simple inegraion: V d,,v = ve r + e r e r E,vp e r E,vp e r dv,v ρv,v +µ rd rv,v d ρv,v +σdw,,

25 herefore: V d,,v = V,v e r E,vp e r E,vp Sobsiuing he definiion of he exposure, one ges: V d,,v = V,v µ r ρv,v +µ rd ρv,v +σdw,, e r V,v +d σ e r V,v +dw,, In similar fashion he bond floor is given by: P d, = P + 1 m E,vp ρv,v + = P + m V,v +,, he sysem of equaions V d,,v P d, = V,v µ r σ = P + mρ m e r V,v V,v mρ e r V,v compleely characerizes he CPPI-DAF porfolio. 7. he CPPI-DAF reurn/risk profile. Consider he sysem of equaions 6.25 V d,,v P d, = V,v µ r σ = P + mρ m +d +dw 6.25,,, +,,. e r V,v V,v mρ e r V,v +d +dw,,, +,,. Using he CPPI porfolio reurn/risk profile i is possible o describe compleely he case of he CPPI-DAF: 25

26 heorem 7.1 Given he CPPI-DAF s porfolio, he wealh process saisfies he following wo equaions E V d,,v V ar V d,,v + 2 µ r 2 2 µ r + σcov V,v = E V,v µ r e r E V,v = V ar V,v + σ 2 2 e r V,v Cov e r V,v e r2η Cov e r Cov V,v +d Morever he dynamic floor saisfies: e 2r E V,v V,v V,v + d, d + Vη,v + d +dw σµ r e r V,v η + ddη } +dw. 7.2 E P d, = P + m E V,v +, 7.3 Proof Look a: V ar P d, = m 2V ar V,v V d,,v = V,v µ r e r V,v σ e r V,v +d +dw, 7.5 P d, II follows ha: = P + V,v +,,. 7.6 m E P d, = P + m E + V,v,,,

27 V ar P d, = m 2V ar V,v +,,. 7.8 Noe ha boh momens depend sricly on he excess E ρv,v /m, or equivalenly /mv,v /,. In order o prove he validiy of analogous formulas for he porfolio s value, look a for any, E V,v + E V,v = E V,v = v pe mµr+r + r dpemµr+r 1 e mµr+rd mµ r + r d 7.9 so ha for any, e r E E,vp ρv,v + d e r E E,vp = v pe r e mµr r dper d + mµ r + r d 7.10 e mµr 1 e mµr+rd d Fubini-onelli heorem allows o wrie: E V d,,v Define he process M = σ = E V,v µ r ρ e r E V,v σe e r V,v 7.11 e r E,vp = σ dw ρp = e r m V,v ρv,v d = P d <. + d ρp, +dw,,. +dw P,,. In order o ge an expression for he porfolio s firs momen, i is sufficien o show ha he process M }, is a Maringale. Indeed, once provided he Maringale propery, i follows ha: E V d,,v = E V,v µ r e r E V,v 27 + d, 7.13

28 since Now noice ha: E E,vp E e r V,v +dw = 0,, ρv,v 2 = E mc,vp ρc,vp ρp 2 = = 2 E C,vp 2 + 2ρP 7.15 he formulas developed in he heorem 3.4 ensures ha: so 2 e 2r E E,vp E C,vp + ρ 2 P 2. ρv,v 2 d <, 7.16 e 2r E V,v 2 d <, 7.17 ha implies he maringale s propery of he process M },. In similar fashion, in order o compue he porfolio s second order momen, look a: E V d,,v 2 = E V,v µ r 2 E + σ 2 2 E 2µ re 2σE V,v e r V,v e r V,v V,v + 2σ 2 µ re e r V,v +d 2 +dw 2 e r V,v e r V,v e r V,v +d +dw +d +dw,, herefore Io-formula ogeher wih some elemenary compuaions bring o: E V d,,v 2 = E V,v 2 + σ µ r 2 e r2η E e 2r E V,v V,v 28 + Vη,v 2 d η + ddη

29 2µ r Cov V,v e r E e r V,v Cov e r V,v for any,. Bu V ar V d,,v so V ar V d,,v + 2 µ r 2 2 µ r + σcov V,v V,v V,v +d = V ar V,v + σ 2 2 e r V,v Cov e r V,v d 2σ +dw + 2σ 2 µ r = E V d,,v e r2η Cov e r Cov V,v +d e r V,v +dw, d,,v 2 E V, e 2r E V,v V,v V,v 2 d + Vη,v + d +dw σµ r e r V,v η + ddη } +dw,,. Noice ha he porfolio s he floor s momens, depend sricly on he sochasic behaviour of he processes m ρv,v /m ρ + m ρ/mv,v / +, ha represen he excess he pure excess. he problem of deermining he firs order momens of he CPPI-DAF is cenral for evaluaing he reurn/risk profile of he porfolio. o ha purpose a useful echnique is o esablish simple accurae approximaions. Define he Mone Carlo ieraive scheme p n,, v n,mc,j,,v },, defined by he equaions v n,mc,j,,v i p n, i wih iniial condiions = p n, i1 + rp n, i1 i i1, i = 1,..., n, 7.21 = v n,mc,j,,v i1 + rv n,mc,j,,v i1 + mµ rv n,mc,j,,v i1 p n, i1 i i1 + mσv n,mc,j,,v i p n, i1 i i1 z J, i = 1,..., n, p n, 0 = p v n,mc,j,,v 0 = v,

30 for Z J } J 1 sequence of sard Normal rom variables i.i.d.. Consider: I = V,v +,, A Mone-Carlo esimae in L p Ω, F,, P p 1, hus P a.s. of he process I }, is given by he quaniy: since i n,mc,j = v n,mc,j,,v mpn, +,,, 7.25 lim n in,mc,j = V,v + = I,,, 7.26 wih a convergence rae of o1/n n, where v n,mc,j,,v p n, = v n,mc,j,,v 7.28 = p n, + rp n, ψ n,,, rv n,mc,j,,v + mσv n,mc,j,,v + mµ rv n,mc,,v p n, p n, ψ n z J,,. herefore he floor s firs order momen, is approximaed by he following Mone- Carlo esimae: E MC,M P d, = P + m 1 M M J=1 i n,mc,j,, Moreover he Mone-Carlo esimae for he floor s second order momen, is given by: E MC,M P d, 2 = 2 1 M hus is variance is approximaed by V ar MC,M P d, Now look a: H = = E MC,M P d, e r V,v M J=1 n,mc,j 2, i,, E MC,M P d, 2,, d,,

31 A Mone-Carlo esimae ino L p Ω, F,, P p 1, hus P a.s. of he process H }, is given by: ψ n h n,mc,j = for any,, since i=1 e ri1 v n,mc,j,,v i1 + e rψn v n,mc,j,,v + mpn, i1 i i1 mp n, + ψ n, 7.33 lim n hn,mc,j = e r V,v +d = H,,, 7.34 wih a convergence rae of o1/n n. In L p Ω, F,, P p 1, hus P a.s. he following approximaion of he firs order porfolio s momen holds: E V d,,v = E V,v µ r lim lim M n 1 M M J=1 h n,mc,j,, Consequenly a Mone-Carlo esimae of he porfolio s firs order momen is given by: E MC,M V d,,v Now look a: K = = E V,v 1 µ r M e r V,v M J=1 h n,mc,j,, dw,, In L p Ω, F,, P p 1, hus P a.s. process K }, is given by he quaniy: a Mone-Carlo esimae of he ψ n k n,mc,j = i=1 e ri1 v n,mc,j,,v i1 + e rψn v n,mc,j,,v + mpn, i1 i i1z J mp n, + ψ n z J, 7.38 since lim n kn,mc,j = e r V,v +dw = K,,,

32 wih a convergence rae of o1/n n. Consider, for any, E V d,,v 2 = E V,v µ r 2 E + σ 2 2 E 2µ re 2σE V,v e r V,v e r V,v V,v + 2σ 2 µ re e r V,v +d 2 +dw 2 e r V,v e r V,v e r V,v +d +dw +d +dw,, I is immediae o derive he following Mone-Carlo esimae of he porfolio s second order momen: E MC,M V d,,v 2 = E V,v µ r 2 1 M 1 + σ M 1 M J=1 1 M 2µ r M 1 2σ 1 M 1 MC,n,J 2 k M J=1 J=1 + 2σ 2 1 M µ r M v MC,n,J,,v h MC,n,J v MC,n,J,,v k MC,n,J + J=1 M J=1 MC,n,J 2+ h h MC,n,J,,v k MC,n,J,,. In order o deermine a Mone-Carlo esimaor for he porfolio s variance, i is sufficien o pu ogeher he porfolio s firs second order momen, wih he formula V ar MC,M V d,,v = E MC,M V d,,v 2 E MC,M V d,,v 2,. In he risk-neural world, he CPPI-DAF porfolio can be replicaed easily by a suiable sequence of Vanilla opions having as underlying he classical CPPI porfolio: 32

33 heorem 7.2 Given he risk-neural measure Q, i is rue ha: E Q e r V d,,v = v mρµr Call, v,, d,,, 7.42 E Q e r P d, = P + m Call, v,,,, Proof Consider: e r V d,,v = e r V,v µ r e r V,v +d σ e r V,v +dw,, Since he CPPI porfolio is a Maringale under he risk-neural measure Q i is immediae o see ha E e r V,v = E e r V V = v = v,. Moreover: E Q e 2r V,v herefore E Q er V,v E Q e r V d,,v = E Q e r V d d V = v µ r = v µ r d <,,, dw mρ = 0, = v = E Q e r V,v Call, v,, In similar fashion one proves he same formula for he dynamic floor. + d d,,. In absence of arbirages he CPPI-DAF porfolio is replicaed by he sard CPPI porfolio a suiable sequence of Call opions. In similar spiri he CPPI-DAF floor is replicaed by he sard CPPI floor by a suiable Call opion. Noe in paricular ha he srucure of he CPPI-DAF is posiively asymmeric: he upside gained in he Dynamic Floor is he upside of a Call opion on he CPPI porfolio while he downside is locked a he CPPI floor s level. 33

34 8. CPPI-DAF Mone-Carlo simulaion echniques. Due o he mahemaical complexiy of he CPPI-DAF, i seems hard o define a Mone-Carlo esimaion echnique in order o rack he porfolio he floor s value. he idea is o uilize he Mone-Carlo simulaion for he classical CPPI in he represenaion formula: V d,,v P d, = V,v µ r σ = P + mρ m e r V,v V,v mρ e r V,v +d +dw 8.1,,, +,,. Look a he Mone-Carlo esimae of he CPPI floor porfolio s rajecories: p n, = p n, + rp n,,,, 8.2 v n,mc,j,,v = v n,mc,j,,v for any,, where p n, v n,mc,j,,v i p n, i = p n, i1 + rv n,mc,j,,v + mσv n,mc,j,,v + mµ rv n,,v P n, p n,mc,j, ψ n z J, 8.3, v n,mc,j,,v }, is defined by + rp n,mc,j, i1 i i1, i = 1,..., n, 8.4 = v n,mc,j,,v i1 + rv n,mc,j,,v i1 + m µ r v n,mc,j,,v i1 p n,mc,j, i1 i i1 + mσv n,mc,j,,v i1 for any i = 1,..., n wih iniial condiions for any J 1. Consider he CPPI-DAF floor: he esimae p d,n,mc,j, P d, p n,mc,j, i1 i i1 z J, 8.5 p n,mc,j, 0 = p v n,mc,j,,v 0 = v, 8.6 = P = p n, + V,v +,,. 8.8 m + m v n,mc,j,,v 34 mpn, +,,, 8.9

35 sill saisfies for any J 1 lim P d,n,mc,j, = P d,,,, 8.10 n ino L p Ω, F,, P p 1, so also P a.s.. As a consequence for a sufficienly big n 1 J 1 i is rue ha p d,n,mc,j,, is sufficienly close o he real value of P d,,. Consider he CPPI-DAF porfolio s value: V d,,v = V,v µ r e r V,v +d σ e r V,v +dw,, Approximaing he firs inegral leads o: µ r e r V,v µ r for any,. In similar fashion: σ i=1 +d e rii1 v n,mc,j,,v i1 + µ re rψn v n,mc,j,,v σ e r V,v i=1 +dw e rii1 v n,mc,j,,v i1 + σe rψn v n,mc,j,,v for any,. he esimae v d,n,mc,j,,v = v n,mc,j,,v e rjj1 v n,mc,j,,v i1 v n,mc,j,,v mpn,mc,j, +i i1 i1 } + mp n,mc,j, ψ + n ψ n, 8.12 mpn,mc,j, + i } i1 i1 z J + mp n,mc,j, + ψ n Z J, 8.13 e rψn j=1 mpn,mc,j, + i1 µ r i i1 + σ i i1 z J mp n,mc,j, + µ r 35 } + σ ψ n z J, 8.14

36 sill saisfies lim V d,n,mc,j,,v = V d,,v,,, 8.15 n for any J 1 ino L p Ω, F,, P p 1, so also P a.s.. As a consequence for a sufficienly big n 1 J 1 he esimae v d,n,mc,j,,v, is sufficienly close o he rue value of V d,,v,. 10. References. F. Black R. Jones, Simplifying Porfolio Insurance, Journal of Porfolio Managemen, 14 Fall 1986, pp A. F. Perold, Consan Proporion Porfolio Insurance, working paper, Harvard Business School, A. F. Perold, W. Sharpe, Dynamic Sraegies for Asse Allocaion, Financial Analyss Journal, January-February 1995, pp N. E. Karoui, M. Jeanblanc, V. Lacose, Opimal porfolio managemen wih American capial guaranee, preprin, February A. F. Perold, F. Black, heory of Consan Proporion Porfolio Insurance, Journal of Economics Dynamics Conrol, 1992, pp R. Booksaber J. Langsam, Porfolio Insurance rading Rule, he Journal of Fuures Markes, Vol. 20, No. 1, 41-57, P. Berr J. Prigen, Porfolio Insurance Sraegies: OBPI versus CPPI working paper, Universiy of Cergy-Ponoise-HEMA, V.Bally D.alay, he law of he Euler scheme for sochasic differenial equaions I: convergence rae of he disribuion funcion, Probabiliy heory Relaed Fields, 104, 43-60,