Opimal ime o Sell in Real Esae Porfolio Managemen Fabrice Barhélémy and Jean-Luc Prigen hema, Universiy of Cergy-Ponoise, Cergy-Ponoise, France E-mails: fabricebarhelemy@u-cergyfr; jean-lucprigen@u-cergyfr Absrac his paper examines he properies of opimal imes o sell a diversified real esae porfolio he porfolio value is supposed o be he sum of he discouned free cash flows and he discouned erminal value (he discouned selling price According o Baroni e al (27b, we assume ha he erminal value corresponds o he real esae index he opimizaion problem corresponds o he maximizaion of a quasi-linear uiliy funcion We consider hree cases he firs one assumes ha he invesor knows he probabiliy disribuion of he real esae index However, a he iniial ime, he has o choose one deerminisic opimal ime o sell he second one considers an invesor who is perfecly informed abou he marke dynamics Whaever he random even ha generaes he pah, he knows he enire pah from he beginning hen, given he realizaion of he random variable, he pah is deerminisic for his invesor herefore, a he iniial ime, he can deermine he opimal ime o sell for each pah of he index Finally, he las case is devoed o he analysis of he ineremporal opimizaion, based on he American opion approach We compue he opimal soluion for each of hese hree cases and compare heir properies he comparison is also made wih he buy-and-hold sraegy Key Words Real esae porfolio, Opimal holding period, American opion JEL Classificaion C61, G11, R21 1
1 Inroducion he research concerning real esae holding period is raher limied and exclusively empirical For he US, he holding duraions are mainly deermined by ax laws as shown by Hendersho and Ling (1984, Gau and Wang (1994 or Fisher and Young (2 For small residenial invesmen, Brown and Geurs (25 examine empirically how long does an invesor own an aparmen building hey found ha he average holding period is around five years, hrough a sample of aparmen buildings of beween 5 and 2 unis over he period 197-199 in he ciy of San Diego For he UK marke, Rowley, Gibson and Ward (1996 emphasized he exisence of ex ane expecaions abou holding periods, for real real esae invesors or new propery developers Indeed, he holding period decision is relaed o depreciaion or obsolescence facors Brown (24 proves ha he risk peculiar o real esae invesmens may explain he behaviour of real esae invesors However, applying he CAPM for individuals o undersand heir porfolio managemen does no yield relevan resuls, as shown by Gelner and Miller (21 Colle, Lizieri and Ward (23 examine empirically he ex pos holding periods and show ha hese values are higher han hose claimed by invesors Knowing he holding period is imporan for invesmen in commercial real esae porfolios and moreover he analysis period has o be specified Using he daabase of properies provided by IPD in he UK over an 18-year period, heir empirical analysis shows ha he median holding period is abou seven years Addiionally, higher he reurn, lower he holding period However, his empirical sudy canno lead o conclusions abou he relaion beween asse volailiy and holding period his is due o he absence of proxy o measure his evenual relaionship In his paper, we search for heoreical opimal imes o sell in a real esae porfolio As a byproduc, our models provide analyical soluions whose properies illusrae mos of he previous empirical resuls For his purpose, we exend he model of Baroni e al (27a, b by considering wo oher maximizaion issues 1 Firs, we deermine he opimal holding period if he invesor is perfecly informed abou he growh rae dynamics his corresponds o he bes ideal case where he invesor, as a new period sars, would know exacly he price dynamics, and would be able o choose he bes ime o sell he asse his approach deermines he upper bound of he presen value of he porfolio as a funcion of holding period policy, in he sense ha i maximizes presen value using perfec forsigh his case hus serves as a sor of benchmark In Baroni e al (27b, he invesor does no know he dynamics, bu only knows is probabiliy disribuion he soluions are analyzed by using simulaions and quasi explici formulae Secondly, we sudy he opimal holding period for an ineremporal maximizaion, according o he American opion approach In ha case, a each ime during a given managemen period, he invesor compares he presen porfolio value (he discouned value of he selling price a his ime wih he maximal expeced value he could have if he would keep he asse He sells as soon as he presen value is higher han his expecaion his is a compleely differen issue han he random imes in a real esae porfolio conex examined by Bond e al (27 In heir model, he markeing period risk corresponds o ime-on-he-marke ha indicaes how long i akes o sell an asse once you pu i on he marke o sell i I is an exogenous random variable In our model, he imes o sell are endogenously deermined from he opimizaion problems 2
he srucure of he paper is laid ou as follows: Secion 2 presens resuls of Baroni e al (27b in a coninuous-ime framework Secion 3 provides he analysis of he perfecly informed invesor Secion 4 develops he American opion approach Comparisons of hese hree approaches are presened in Secion 5 In paricular, we emphasize he case for which he deerminisic opimal ime is no degeneraed (neiher equal o he iniial ime nor o he mauriy Moreover, in his secion, we also inroduce he buy-and hold sraegy Mos of he proofs are gahered in he Appendix 2 Opimal ime * o sell, chosen a ime In his secion, he ime of sale is pre-se, commied irrevocably a ime, based on he expeced dynamics of he porfolio value and is cash flow he real esae porfolio value is defined as he sum of he discouned free cash flows (FCF and he discouned erminal value (he selling price Denoe k as he weighed average cos of capial (WACC, which is used o discoun he differen free cash flows, and he erminal value We assume ha he free cash flow grows a a consan rae g 2 21 Coninuous-ime model As Baroni e al (27a, we suppose ha he price dynamics, which corresponds o he erminal value of a diversified porfolio (for insance a real esae index, follows a geomeric Brownian moion: dp = µ d + σdw, (1 P where W is a sandard Brownian moion We have: = exp 1 2 µ / σ + σ (2 2 P P W his equaion assumes ha he real esae reurn can be modelled as a simple diffusion process where parameers µ and σ are respecively equal o he rend and o he volailiy he expeced reurn of he asse a ime is given by: P E = exp( µ P hen he fuure real esae index value a ime, discouned a ime, can be expressed as: (3 exp 1 2 2 P P µ k σ σw = / +, (4 wih P E = exp( [ µ k] (5 P Denoe by FCF he iniial value of he free cash flow he coninuous-ime version of he sum of he discouned free cash flows FCF is equal o: which leads o s ks [ k g] s s (6 C = FCF e ds = FCF e ds, 3
FCF [ k g] C = 1 e k g Inroduce he real esae porfolio value process V, which is he sum of he discouned free cash flows and he fuure real esae index value a ime, discouned a ime : 22 Characerisics of V V = C + P (8 We deermine he porfolio value V for a given mauriy his assumpion on he ime horizon allows o ake accoun of selling consrains before a limi dae he higher, he less sringen his limi Addiionally, his hypohesis allows he sudy of buy-and-hold sraegies (see secion 5 he fuure porfolio value a mauriy, discouned a ime, is given by: FCF [ k g] 2 = + / + k g V 1 e P exp µ k 1 2σ σw he porfolio value V is he sum of a deerminisic componen and a Lognormal random variable 23 Deerminaion of We deermine he opimal soluion a ime, for a given mauriy and for an invesor maximising a quasi linear expeced uiliy Firs noe ha he sum of he discouned free cash flows ( C is always increasing due o he cash accumulaion over ime Second, we have o analyze he expecaion of he fuure real esae index value a ime, discouned a ime : if he price reurn µ is higher han he WACC k, hen, he opimal soluion is simply equal o he mauriy hus, in wha follows, we consider he case µ < k Consequenly, no selling he asse implies a [ k ] higher bu a smaller discouned expeced erminal value Pe µ Hence, he invesor has o balance beween more (discouned flows and less expeced discouned index value We also focus on he sub case g < µ, which corresponds o empirical daa 3 (7 he opimizaion problem is: Max E V [ ] (9, Since he expecaion of V is equal o: FCF [ k g] [ µ k] EV = 1 e + Pe, k g we deduce: EV [ k g] ( [ µ k ] = FCFe + P µ k e (1 (11 4
hen, he opimal holding period is deermined as follows ( µ Case 1: he iniial price P is smaller han e FCF k k µ hen, he opimal ime o sell corresponds o he mauriy Since he Price P Earning Raio (PER is oo small ( < ( k µ e FCF k µ mauriy, he sell is no relevan before ( Case 2: he iniial price P lies beween he wo values FCF e k µ FCF and hen, he opimal ime o sell is soluion of he following equaion: EV = From Equaion (11, we deduce 4 : 1 1 ln FCF = µ g P k µ k µ k µ (12 (13 In paricular, noe ha is a decreasing funcion of he iniial price P and of he difference beween he index reurn µ and he growh rae g of he free cash flows his laer propery was empirically observed by Brown and Geurs (25 I means ha invesors sell propery sooner when values rise faser han ren FCF Case 3: he iniial price P is higher han k µ hen, he opimal ime o sell corresponds o he iniial ime Since he PER P is sufficienly large ( > 1 FCF k µ, here is no reason o keep he asse P As an illusraion, he cumulaive value C of he FCF values, of he expecaion of he index value EP [ ] and he expecaion of he porfolio value EV [ ] are displayed in Figure 1 We consider wo ses of parameer values for a 2 year managemen period ( = 2 We noe ha he discouned expeced value V of he porfolio is concave he parameer values imply ha he opimal holding period,, is respecively equal o 913 years and 1611 years For hese wo examples, he opimal ime o sell is smaller han he mauriy In he second example, he discouned porfolio value varies up o 2% 5 Knowing he opimal ime o sell, which is deerminisic, he probabiliy disribuion of he discouned porfolio value V can be deermined he value V is equal o: FCF ( k g 2 V = 1 e P exp µ k 1 2σ σw ( k g + / + 1 ( k g FCF ( k g Denoe A= e he cumulaive discouned free cash flow value a Since, from (13, he opimal ime o sell saisfies: 5
hen, we deduce: and he cdf FV of V 1 FCF = ln, µ g P ( k µ k g ( µ g FCF FCF A = 1, ( k g P ( k µ is given by: if, v A FV ( v = 1 v A 2 N ln µ k 1 2σ if v A /, > σ P (14 where N denoes he cdf of he sandard Gaussian disribuion Fig 1 C, EP [ ] and EV [ ] as funcions of ime in [,2] Case 1: µ = 4 4%g, = 3%k, = 8 4%P, = 1, FCF = 1/ 22 Case 2: µ = 6%g, = 2%k, = 9 5%P, = 1, FCF = 1/ 15 he pdf of V are illusraed in Figure 2 for wo differen volailiies Higher he volailiy, larger he range of admissible discouned porfolio values Up o a ranslaion, hese are he pdfs of a Lognormal disribuion, wih a peak near he iniial 6
value of he index P = 1 Fig 2 Pdf of V * µ = 4 4%, σ = 5%, g = 3%, k = 8 4%, P = 1, FCF = 1/ 22 µ = 4 4%, σ = 1%, g = 3%, k = 8 4%, P = 1, FCF = 1/ 22 3 Opimal ime ** o sell for a perfecly informed invesor In his secion he invesor is assumed o have a perfec foresigh abou he enire fuure price pah Pahs are random (he invesor do no choose he realized pah bu, a ime, he knows he whole rajecory Knowing he pah, he can opimize on i, which leads o a deerminisic soluion condiionally o his informaion Bu, aking ino accoun he randomness of he pah jus before ime, he opimal ime o sell is a random variable his corresponds o an ideal case which is no realisic bu provides an upward benchmark Firs, we examine he disribuion of he opimal holding period Second, we indicae he disribuion of he opimal value V 31 Probabiliy disribuion of ** We search for he probabiliy disribuion of he random ime a which he pah of he process V reaches is maximum, where V is he presen value of he porfolio his disribuion is no explicily known bu can be simulaed he simulaed pdf of 7
he real esae porfolio value is illusraed in Figure 3 wo cases are examined: he firs one corresponds o a relaively small values of he drif µ and he iniial free cash flow FCF he second one corresponds o a higher value of he drif µ and o a higher value of he iniial free cash flow FCF Case 1 Early selling: µ = 3, σ = 3, g = 2, k = 84, P = 1, FCF = 1/ 22, = 2 Case 2 Lae selling: µ = 6, σ = 3, g = 2, k = 84, P = 1, FCF = 1/ 13, = 2 Fig 3 Simulaed pdf of (case 1 and case 2 he ime a which he discouned porfolio value reaches is maximum depends on he sign of is drif which is an increasing funcion of boh he drif µ and he iniial free cash flow FCF Hence, if he real esae price reurn and/or he iniial free cash flow FCF are no high enough, he opimal sraegy consiss on selling quickly For boh higher discouned cash flows and higher expeced discouned index value, he opimal sraegy is o sell laely his shape modificaion is modelled by he Brownian moion wih drif as shown in Appendix A1 32 Probabiliy disribuion of V Recall ha is random since he invesor does no choose he realized pah even he knows i (he jus observes he rial oucome wihou selecing i Now, we search for he probabiliy disribuion of he maximum value V We provide an explici formula by means of a mild approximaion his laer one is jusified as shown by Mone Carlo simulaions of he rue probabiliy disribuion (see Figures 4 and 5 Inroduce he funcion G defined by: 1 y 1 2my y G( m, y, = 1 Erfc m e Erfc m 2 2 2 +, 2 2 2 where he funcion Erfc is given by: 8
2 2 u Erfc( x = e du π x Denoe also FCF 2 v Av ( = + µ k 12 / σ, and Bv ( = ln v P hen, he approximaed cdf of V is given explicily by: for, v< P, PV [ v] = Av ( Bv ( (15 G,, for v> P, σ σ Figure 4 illusraes Relaion (15 while Figure 5 shows he qualiy of he approximaion 3 Fig 4 Approximaed Pdf of V ** 25 2 15 1 5 12 14 16 18 Fig 5 Simulaed Pdf of V ** From Figures 4 and 5, we deduce for example ha: he probabiliy ha he real esae porfolio value is higher han P is equal o 1 hus, whaever he pah, he invesor receive a leas P Indeed, if all he fuure discouned porfolio values are lower han he iniial price, he knows he 9
has o sell a ime and hen receives exacly P he median is abou 125 ( =+ 25%P he probabiliy o receive more han 15 ( =+ 5%P is abou 1% 33 Numerical illusraions he disribuion of is very sensiive o he volailiy parameer, hrough he erminal value his is no he case for he soluion o illusrae his feaure, we consider he parameer values: µ = 44, g = 3, k = 84, P = 1, FCF = 1/ 22, = 2, and hree volailiy levels σ = 5, σ = 5 and σ = 5 In Figure 6, each column corresponds o a differen volailiy level and provides simulaed pahs of V **, hen he Pdf of ** and finally he Pdf of V ** Fig 6 Simulaed pahs of V and Pdf of ** and V ** σ = 5 σ = 5 σ = 5 When he volailiy is small, he Pdf of ** is concenraed on he value of * as expeced When he volailiy is increasing, he range of ** becomes larger When σ = 5, he shape of Pdf is a mix of he Pdf of case 1 and 2 in Figure 3 his is due 1
o he Brownian moion (as illusraed in Figure 13 of Appendix A1 In erms of porfolio value, more volailiy implies higher benefis, as he invesor in his case is perfecly informed abou he pah Noe also ha when he discouned porfolio values are never higher han he iniial price P, he opimal ime o sell is = hen he opimal value is equal o is smalles value P 4 American opimal selling ime *** In his hird case, we allow ha he invesor may choose he opimal ime o sell, according o marke flucuaions and informaion from pas observaions In his case, he faces an American opion problem Recall ha he invesor preferences are modelled by means of linear uiliy (sandard assumpion in real esae lieraure A any ime before selling, he compares he presen value P wih he maximum of he fuure value he expecs given he available informaion a ime (mahemaically speaking he compues he maximum expeced value of his porfolio on all J -, measurable sopping imes τ I means ha he decides o sell a ime only if he porfolio value a his ime is higher han he maximal value ha he can expec o receive if he does no sell a his ime hus, he has o compare P wih supτ J EC τ C + Pτ J, where C, s denoes he FCF value a ime s All he proofs of his secion are deailed in Appendix B 41 he opimal soluion Inuiively, he opimal ime mus be he firs ime where he asse price P is sufficienly high A his price level, he fuure free cash flows (received in case of no sell will no be high enough o balance an expeced index value lower han he price P a ime (he expeced index value decreases wih ime as he discouned rend µ k is negaive he opimal ime corresponds exacly o he firs ime a which he asse price P FCF ( k g is higher han he deerminisic level k µ e (see Appendix B his resul generalizes he case 3 obained a ime =, in secion 23, where he invesor sells FCF direcly he asse if he price P is higher han k µ Since he reurn of he discouned ( free cash flows is equal o k g e, he price P has o be compared wih he value FCF ( k g k µ e Figure 7 shows he poins a which differen realized pahs of P cross he deerminisic level (black curve Five pahs are simulaed for four differen volailiy levels 11
Fig 7 Pahs of he index value P σ = 1 σ = 2 σ = 5 σ = 1 Using he previous characerizaion of we can deermine is probabiliy disribuion Noaions: g m µ 1 FCF = σ 12 / σ and y = ln σ P ( k µ FCF - For he case P 1 ( k µ, we have = FCF - For he case P k µ >, for any <, we have ( 1 1 y 1 y = 2 + +, 2 2 2 2 2 2my P Erfc m e Erfc m and (16 1 y 1 y = = 1 + 2 2 2 2 2 2 2my P Erfc m e Erfc m As illusraed in Figure 8 (for σ = 1 %, = 2, he probabiliy ha he opimal American ime o sell may be no negligible (abou 15% for his numerical case, which corresponds o he size of he jump of he cdf 12
Fig 8 Cdf of *** 1 8 6 4 2 5 1 15 2 25 42 he porfolio value V and is cdf he porfolio value V can be deermined by using he fac ha, if he opimal ime o sell is before he mauriy, hen he index value P is equal o he FCF k g k µ ( deerminisic level e Consequenly, we ge: FCF ( k g ( µ g V = 1+ e, if < ( k g ( k µ and (17 FCF ( k g 2 V = 1 e P exp µ k 1 2σ σw if ( k g + / +, = he previous deerminaion of he value V allows he explici (and exac compuaion of is cdf Denoing: ( ( 1 FCF k g FCF 1 v ( k g e 1 µ g k g zv ( = ln ; gv ( = ln FCF, σ P k g k µ v k g he cdf of V is defined by: k g FCF ( k g - If v 1 e, F V ( v = ; FCF ( k g FCF µ g ( k g k g 1 e v k g 1 k µ e - If ( < < +, 2 ym FV ( v N e N ( zv ( m zv ( m+ 2y = ; 13
FCF - If ( ( 1 µ g ( k g FCF g k g k e v k g 1 µ µ k µ - If FCF µ g ( 1 µ + +, k g + k < v F ( v = G( m, y, g( v G( m, y, V zv ( m 2 ym y m zv ( m y m + N e N N N ; 2 2 F ( v = 1 G( m, y, V zv ( m 2 ym y m zv ( m y m + N e N N N 2 2 (18 5 Comparison of he hree opimal sraegies and he buy-and hold one We examine boh he probabiliy disribuions of he opimal imes o sell, and and he corresponding discouned porfolio values V, V and V We inroduce also he comparison wih he buy-and hold porfolio V he sensiiviy analysis is done wih respec o he volailiy σ and o he mauriy Noe ha, since he porfolio value V dominaes he oher ones, is cdf is always below he cdf of he ohers ( firs order sochasic dominance he previous parameer values, µ = 44, g = 3, k = 84, P = 1, and FCF = 1/ 22, are considered 51 Cdf of and, according o he volailiy σ he mauriy is equal o 2 and hen we have = 913 Graphical resuls are presened in Figure 9 - For small volailiy level, he cdf of is raher concenraed near, whereas, for higher values, he shape of he cdf of looks similar o he cdf of - Addiionally, for he American case, he probabiliy ha he selling occurs a mauriy ( P [ = ] is almos equal o, for small volailiy levels However, for higher volailiy values, his probabiliy is abou 15% Indeed, his probabiliy converges o a limi which is FCF ln equal o 1 P ( k µ e - Moreover, higher he volailiy, higher he probabiliy of smaller American opimal imes, as usually observed empirically (see Colle e al, 23 14
Fig 9 Cdf of ** and *** wih respec o σ σ = 1 σ = 5 σ = 1 σ = 15 52 Cdf of V, V V andv wih respec o σ We assume also ha = 2 and hen = 913 Graphical resuls are presened in Figure 1 - For small volailiy level, he cdf of V is almos equal o he cdf of V For higher values, wo componens of he cdf of V can be clearly idenified he smalles values of V (approximaely values up o 95, correspond o an opimal ime o sell equal o he mauriy ( = 2 : he discouned porfolio values were oo small so ha i was no opimal o sell before mauriy - For any given hreshold L smaller han 15, he probabiliy ha he American porfolio value V is smaller han L, is always weaker han he corresponding value for V he American opion approach leads o smaller losses, when comparing he discouning porfolio value o he iniial price ( P = 1 More generally, his approach prevens from large flucuaions of he price dynamics When he index 15
price falls, he American crierion provides proecion agains downside risk When he index price rises significanly, his approach leads o an earlier sell, in order o insure he profi of his increase If afer he sell he index dynamics goes down, he invesor did righ avoiding hen lower profis or even losses Finally, if afer he sell he index dynamics goes up, his profi would have been higher if he had no sold - he shape and he values of he cdf of V (he buy-and hold approach, look like he cdf of V excep for very small volailiies Fig 1 Cdf of V, V V andv wih respec o σ σ = 1 σ = 5 σ = 1 σ = 15 53 Cdf of ** and *** wih respec o Graphical resuls are presened in Figure 11 ( σ = 5% - In he American opion approach, he probabiliy o sell a mauriy (given by P [ = ] is almos equal o 9% for small mauriy (see he jump of he cdf for = 2, whereas for long mauriy, i becomes very smaller (around 6%, for = 3 16
- For he perfecly informed case, he probabiliy o exercise a mauriy is null ( P [ = ] =, whaever he mauriy he shape of he cdf of looks similar o he cdf of for high mauriy Fig 11 Cdf of ** and *** wih respec o = 2 = 1 = 2 = 3 54 Cdf of V, V V andv according o he mauriy Graphical resuls are presened in Figure 12 ( σ = 5% - For small mauriy, he cdf of V is almos equal o he cdf of V he difference comes from he probabiliy o sell before mauriy, which modifies he probabiliy o receive high porfolio values - For longer mauriies, his difference is more imporan he cumulaive disribuion funcions diverge more and more, excep for he smalles porfolio values - Whaever he mauriy, he numerical upper bound of V is around 115 (his bound is sill presen for oher volailiy values, as seen in Figure 1 Hence, higher porfolio values may be obained wih he case When he a priori opimal ime o sell is higher han he mauriy ( >, he porfolio values V corresponding o he buy- 17
and-hold sraegy are exacly he same as he values V * of he sraegy When <, higher he mauriy, higher he difference beween he cdfs of V and V * Fig 12 Cdf of V, V, V andv wih respec o = 2 = 1 = 2 = 3 6 Conclusion his paper proves ha real esae porfolio value srongly depends on he opimal ime o sell hree kinds of such opimal imes are considered he firs one implicily assumes ha he invesor can only choose he opimal ime o sell a he iniial dae However, generally such a soluion is no ime consisen: he same compuaion of opimal ime o sell a a fuure dae leads o a differen soluion A second one corresponds o a perfecly informed invesor his is an upward benchmark bu no oo realisic Finally, a more raional approach is inroduced o ake accoun of ineremporal managemen and cumulaive informaion his is he American opion framework For each of hese models, he opimal imes o sell and porfolio values are explicily deermined (an approximaion is used for he perfecly informed case and also simulaed We compare he soluions, using various parameer values of he real esae markes, in paricular he volailiy index and he porfolio mauriy he American approach allows he reducion of he probabiliy o ge small porfolio values his propery is reinforced boh when he volailiy and he mauriy are increasing 18
Appendix Appendix A: he case ** A1 Commens abou he pdf f his pdf is no known since he process V o be maximized on he ime inerval [, ] has he following form: a b W V = c(1 e + Pe + σ, where a and b are non negaive consan However, for a 55process X defined as a geomeric Brownian moion wih drif by b+ σw X = Xe wih X > and σ >, he pdf of b is known Indeed, noe ha W σ X X e σ + = herefore, he maximizaion of X is equivalen o he maximizaion of he process Y defined by b W Y = e σ + Recall he probabiliy densiy funcion f 1 of he firs ime 1 before a which a ( m Brownian moion wih drif W reaches is maximum (see Borodin and Salminen, ( m 22 Le W be defined by: ( m W = m+ W, where W is a sandard Brownian moion hen, he probabiliy densiy of he random ( m ime 1 a which he pah of he process W reaches is maximum is given by: f ( 1 v = ( 2 mv 2 2 m ( v / 2 2 m v / e m m e m Erfc Erfc + πv 2 2 π ( v 2 2 (19 Figure 13 illusraes he inversion of he cdf curves, as observed in Figure 3 for he simulaed cdf of We consider wo cases: 1 m < (m=-1, =3 years Due o he negaive drif, he maximum is early achieved 2 m > (m=+1, =3 years Due o he posiive drif, he maximum is ardily achieved 19
3 25 2 15 1 5 12 1 8 6 4 2 2 4 6 8 1 12 14 2 4 6 8 1 12 14 Fig 13 Pdf approximaion of ** (negaive and posiive drif A2 Compuaion of he cdf f V he approximaed cdf of V is given by: P [ Sup V ] v = a 2 ( v c e + c P, exp µ k 1/ 2σ + σw, P where a= k g and c= FCF /( k g Denoe: P v α = ln, and β = ln 1 c c hen, he inequaliy a 2 ( v c e + c exp µ k 1/ 2σ + σw P is equivalen o: 2 exp a α + a + µ k 1 / 2σ + σw e β + + 1 Consider he following approximaion: β β+ x y β e 1+ e = e wih y ln 1+ e + x β 1+ e hen: P [ Sup V ] v β 2 β e P, exp α + a+ µ k 1/ 2σ + σw exp ln 1 e a + + β 1+ e Denoe: β 2 e β v Av ( = a+ µ k 12 / σ aand Bv ( ln1 e α ln β = + = 1+ e P Noe ha we have: ac 2 v Av ( = + µ k 12 / σ and Bv ( = ln v P herefore: Av ( Bv ( P[ Sup V ] b = P W, + σ σ (2 2
Recall (see Borodin and Salminen, 22, p 25 ha he maximum value of he ( m Brownian moion wih drif W before has he following cdf: ( m P[ Sup W ] y = 1 y 1 2my y G( m, y, = 1 Erfc m e Erfc + m 2 2 2 2 2 2 Consequenly, we deduce: Av ( Bv ( If v > P, P[ Sup V ] v = G,,, σ σ If v < P, P[ Sup V ] v = Denoe: P v α = ln, and γ = ln 1 c c Consider he following approximaion: γ γ+ x y γ e 1 e = e wih y ln 1 e + x γ 1 e hen he inequaliy a 2 ( v c e + c exp µ k 1/ 2σ + σw P is equivalen o: 2 exp a α a µ k 1 2σ σw 1 e γ + + + / + (21 (22 hus, we deduce he same equaliy as previously: Av ( Bv ( P[ Sup V ] v = P W, + σ σ Consequenly, we have: Av ( Bv ( If v > P, P[ Sup V ] v = G,,, σ σ If v < P, P[ Sup V ] v = Finally, he cdf of he maximum value V is (approximaely given by: for, v< P, PV [ v] = Av ( Bv (23 G,, for v> P, σ σ Using he definiions of coefficiens A, B, and c, he cdf of V is deduced 21
Appendix B: he American case *** B1 he American opion problem Denoe by V ( x, he following value funcion: V ( x, = sup E C τ C + Pτ P = x τ F, Noe ha we always have V ( x, x, since τ = J and, in ha case, V ( x, = x, As usual for American opions 6, wo regions have o be considered: he coninuiy region: he sopping region: + C = ( x, R [, ] V ( P, > P + S = ( x, R [, ] V ( P, = P he firs opimal sopping ime afer ime is given by hen: B2 Compuaion of he value funcion V = inf u [, ] V ( P, u = P u u { } = inf u [, ] P C o deermine, we have o calculae V ( x, 2 Noe ha ( exp 1 2σ σw herefore: Moreover, since we have: / + is a maringale wih respec o he filraion J Consequenly, we have o deermine: 2 2 E exp / 1 2σ τ + σwτ J = exp / 1 2σ + σw exp 1 2 2 P P µ k σ σw = / +, E Cτ C + Pτ P = x = E Cτ C + P exp ( µ k ( τ P = x ( a aτ sup E ce e P exp µ k ( τ P x + = τ J, In paricular, we have o search for he value τ for which he maximum ( aτ sup E ce + xexp µ k ( τ P = x, τ J, is achieved his problem is he dynamic version of he deerminaion of presened in Secion 2 22
Inroduce he funcion f x, defined by: ( ( a + θ fx, θ = ce + xexp[ bθ ] wihb= k µ (24 Case 1 he asse value x ( = is smaller han P ac e a e ( a b( b hen, he opimal ime τ corresponds o he mauriy and ( ( a a V x, = c e e + xexp b( Case 2: he asse value x lies beween he wo values ac e a e ( a b( b and ac a b e hen, he opimal ime τ is equal o ( + θ, where θ is he soluion of he following equaion: f x, ( θ = θ hen, from Equaion (25, we deduce: a 1 ace θ = ln a b bx herefore, knowing ha P = x a ime, he ime τ is deerminisic hen, he value funcion V ( x, is given by: from which, we ge: a aτ V ( x, = c e e + xexp b( τ, b ( a b a a a bac a ( a b V ( x, = ce + e x a b (25 (26 Case 3: he asse value x is higher han ac a b e hen, he opimal ime τ corresponds o he presen ime, and V ( x, = x Consequenly, from he hree previous cases, we deduce: a a b( ac a ( a b( ce ( e + xe, if x e e b b ( a b a b ac a ( a b ac ( ( ac V ( x, = ce + e x, if e e < x< e a b b b ac a x, if x e b a a a a b a (27 23
Figure 14 illusraes he funcion value V ( x, (for = / 2 and parameer values of ac a he basic numerical example Noe ha 65 5 b e 7 65 6 55 5 45 45 5 55 6 65 7 Fig 14 Value funcion V Finally, he American opimal ime is deermined by: = inf [, ] V ( P, = P FCF k g ( k µ e ( herefore, we can check ha V ( P, = P if and only if P hus, we have: FCF ( k g = inf [, ] P e ( k µ ( Using sandard resuls abou he firs ime m a which a Brownian moion wih drif ( m W reaches a given level y, we can derive he pdf and cdf of Indeed, he FCF ( k g condiion P e is equivalen o ( k µ y 2 FCF ( k g P exp µ k 1/ 2σ + σw e, ( k µ 2 FCF µ g 12 / σ + σw ln, P ( k µ µ g 1 FCF σ σ µ 12 / σ + W ln P ( k g Seing m µ 1 FCF = σ 12 / σ and y = ln σ P ( k µ, he cdf of he random y is given by: - For he case FCF P 1 ( k µ, we have: = FCF - For he case P 1 ( k µ >, we have: P y = 1 G( m, y, 24
hus, since ( m P = P y for any <, we deduce 1 y 1 2my y P = Erfc 2 m e Erfc m 2 2 + 2 + 2 2, and ( m ( m ( m P P y P y P y = = = + > = + > = 1 y 1 2my y = 1 Erfc m e Erfc + m 2 2 2 2 2 2 B4 Compuaion of he opimal value V A ime FCF ( k g, we have P = ( k µ e if < hus, he porfolio value V is given by: FCF ( k g ( µ g V = 1 e if +, <, ( k g ( k µ and (28 FCF ( k g 2 V = 1 e P exp µ k 1 2σ σw ( k g + / +, if = B5 Compuaion of he cdf FV We have: FV ( v = P V v P V v < + = i Compuaion of P V v < : Seing gv ( We have P V v < FCF ( k g ( µ g = P 1+ e v < ( k g ( k µ FCF 1 µ g k g gv ( = Log FCF, k g k µ v k g FCF µ g gv v + e k g k µ ( 1 ( k g 25
herefore, P V v P g( v < = < if, gv ( > = P g( v, if g( v (29 FCF µ g Bu P g( v =, if gv ( <, ha is if v > k g 1+ k µ ii Compuaion of P V v = P V v = = FCF ( k g 2 1 e P exp µ k 1 2σ σw + / + v P ( k g ( m sup Ws y s ( m W Recall ha, for a geomeric Brownian wih drif, he join disribuion of is maximum on [, ] and is value a ime is given by (see Borodin and Salminen (22, p 251, P 2 2 ( ( 1 2 ( 2 [ Sup W m y W m dz mz m z y y ] e / + / dz, = 2π Since 2 2 ( m ( m ( m 1 mz m / 2 ( z y + y / 2 P[ Sup W ] [ ] < y, W dz = P W dz e dz, 2π finally we ge: ( m ( m 1 2 2 mz m / 2 ( z y + y / 2 P [ =, W dz ] = P [ W dz ] e dz 2π k g FCF ( k g Consequenly, for v > 1 e, we have: P V v = FCF ( k g ( 1 v ( k g 1 e m ( k g = P =, W < ln + σ P σ (3 1 FCF Recall ha y = ln σ P ( k µ g and m µ = σ 12 / σ Denoe also FCF ( k g 1 v ( k g 1 e ( k g zv ( = ln + σ P σ 26
hen, wih P V v = z( v 2 2 ( m 1 mz m / 2 ( z y + y / 2 [ ] 2π = PW dz e dz, z( v ( ( m zv ( m P[ W dz] dz = P[ W z( v] = N m ( o deermine z 2 2 1 mz m / 2 ( z y + y / 2 e dz 2π, we have o compare y and ( according o he values of v : FCF ( k g µ g zv ( y v 1+ e ( k g k µ If zv ( y 1 z( v 2 2 mu m / 2 ( u y + y / 2 1 z 2 2 mu m / 2 (2 y u / 2 e du = e du 2π 2π ( 1 2 1 z v ym u m+ 2 y 2 2ym = e e du e N = 2π 2 ( zv ( ( m+ 2y zv, If zv ( > y 1 2π z( v e 2 2 mu m / 2 ( u y + y / 2 du 1 y 2 2 ( 2 2 2 (2 2 1 z v mu m / y u / mu m / 2 u / 2 = e du+ e du, y 2π 2π 2 ym y m z( v m y m = e N + N N 2π 2π 2π Noe also ha we always have: FCF ( k g FCF g ( k g FCF g 1 e 1 µ e 1 µ + k g k g k µ + k g k µ Consequenly, he cdf of V is defined by: k g FCF ( k g - If v 1 e, FV ( v = 27
FCF ( k g FCF µ g ( k g k g 1 e v k g 1 k µ e < < +, - If ( 2 ym FV ( v = N e N FCF - If ( ( 1 µ g ( k g FCF g k g k e v k g 1 µ µ k µ + +, ( zv ( m zv ( m+ 2y F ( v = G( m, y, g( v G( m, y, V zv ( m 2 ym y m zv ( m y m + N e N N N 2 2 k g + k < v - If FCF µ g ( 1 µ F ( v = 1 G( m, y, V zv ( m y m zv ( m y m 2 2 2 ym + N e N N N (31 Acknowledgmens We graefully acknowledge Shaun Bond (discussan and paricipans a he MI- Cambridge-Maasrich Conference, 27 Noes 1 hese opimizaion problems are specific o real esae invesmens and differ from sandard financial porfolio managemen problems (see Karazas and Shreve, 21, or Prigen, 27 Firs, he asse is no liquid (no divisible Second, he conrol variable is he ime o sell and no he usual financial porfolio weighs (see Oksendal 27, for a relaed problem abou opimal ime o inves in a projec wih an infinie horizon 2 his assumpion allows explici soluions for he probabiliy disribuions of he opimal imes o sell and of he opimal porfolio values he inroducion of sochasic raes would lead o only simulaed soluions 3 he wo oher cases µ < g < k and µ < k < g could be analyzed in he same way 4 his is he coninuous-ime version of he soluion of Baroni e al (27b 5 We can also examine how he soluion depends on he index value P For example, proporional ransacion coss imply a reducion of P For insance, for he case 2, a ax of 5% leads o an opimal ime o sell equal o 1739 years, insead of 1611 years when here is no ransacion cos Wih a 1% ax, he soluion becomes 1874 years his is in line wih he empirical resuls showing ha high ransacion coss imply longer holding periods (see for example Colle e al, 23 6 See Ellio and Kopp (1999, p 193 28
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