The recent financial crisis highlights the importance of market crashes and the subsequent market illiquidity

Transcription

1 MANAGEMENT SCIENCE Vol. 59, No. 3, March 2013, pp ISSN (prin) ISSN (online) hp://dx.doi.org/ /mnsc INFORMS Marke Crashes, Correlaed Illiquidiy, and Porfolio Choice Hong Liu Olin Business School, Washingon Universiy in S. Louis, S. Louis, Missouri 63130, Mark Loewensein Rober H. Smih School of Business, Universiy of Maryland, College Park, Maryland 20742, The recen financial crisis highlighs he imporance of marke crashes and he subsequen marke illiquidiy for opimal porfolio selecion. We propose a racable and flexible porfolio choice model where marke crashes can rigger swiching ino anoher regime wih a differen invesmen opporuniy se. We characerize he opimal rading sraegy in erms of coupled inegro-differenial equaions and develop a quie general ieraive numerical soluion procedure. We conduc an exensive analysis of he opimal rading sraegy. In conras o sandard porfolio choice models, changes in he invesmen opporuniy se in one regime can affec he opimal rading sraegy in anoher regime even in he absence of ransacion coss. In addiion, an increase in he expeced jump size can increase sock invesmen even when he expeced reurn remains he same and he volailiy increases. Moreover, we show ha misesimaing he correlaion beween marke crashes and marke illiquidiy can be cosly o invesors. Key words: marke crashes; porfolio choice; correlaed illiquidiy Hisory: Received Ocober 27, 2010; acceped March 15, 2012, by Wei Xiong, finance. Published online in Aricles in Advance Sepember 4, Inroducion The recen financial crisis highlighs several poenially imporan fundamenal elemens for opimal porfolio choice. Firs, even risks such as a marke crash may be significan; second, marke liquidiy may dry up afer a crash; hird, he probabiliy of anoher crash may increase afer a crash; and fourh, oher invesmen opporuniy se parameers (e.g., marke volailiy) may also change afer a crash. However, he opimal rading sraegy in he presence of marke crashes ha can rigger changes in he invesmen opporuniy se has no been sudied in he exising lieraure. In his paper, we develop a flexible porfolio choice model for a small invesor ha incorporaes correlaed marke crashes and changes in he invesmen opporuniy se. For example, boh liquidiy and volailiy may change afer a crash and crashes hemselves may be correlaed in our model. This model capures he essence of all he above-menioned imporan feaures bu sill remains racable. More specifically, we consider he opimal rading sraegy of a consan relaive risk averse (CRRA) invesor who derives uiliy from erminal wealh and can rade a riskless asse and a risky sock coninuously. Sock price crashes in a liquid regime can rigger swiching ino an illiquid regime where oher parameers such as crash inensiy, expeced reurn, and volailiy can also change. Similarly, large upward price jumps in he illiquid regime can rigger regime swiching ino he liquid regime. 1 Because of he possibiliy of price jumps, he coupled Hamilon Jacobi Bellman (HJB) equaions become inegro-differenial variaional equaions, which makes our problem much more difficul o solve, even numerically han ha of Jang e al. (2007), who do no consider even risks. Remarkably, we are able o develop an ieraive procedure ha solves for he value funcion as a sequence of soluions o ordinary differenial equaions, which significanly reduces compuaion inensiy. This ieraive procedure can be readily applied o many oher opimal porfolio choice problems and significanly simplifies compuaion, especially for hose involving coupled nonlinear HJB equaions. As in he pure diffusion case, he no-ransacion region is characerized by wo regime-dependen boundaries wihin which he invesor mainains he raio of he dollar amoun in he riskless asse o he dollar amoun in he risky 1 We ake hese changes afer a marke crash as exogenously given. There is a large lieraure on why liquidiy and oher parameers may change afer a crash (see, e.g., Geanakoplos 2003, Diamond and Rajan 2011). Our model can also be consisen wih a model where invesors learn from crashes and updae heir beliefs abou he invesmen opporuniy se afer a crash, alhough we do no explicily model his learning process o keep racabiliy. 715

2 Liu and Loewensein: Marke Crashes, Correlaed Illiquidiy, and Porfolio Choice 716 Managemen Science 59(3), pp , 2013 INFORMS asse whenever possible. In conras o he pure diffusion case, however, his raio can jump ouside hese boundaries, which requires an immediae discree ransacion back o he closes boundary. We characerize he value funcion and provide some analyical comparaive saics and an exensive numerical analysis o illusrae how various elemens of our model affec he opimal rading sraegy. In conras o sandard porfolio choice models, changes in he invesmen opporuniy se in one regime can affec he opimal rading sraegy in anoher regime even in he absence of ransacion coss. This is because he correlaion beween a marke crash (or an upward jump) and regime swiching makes he impac of sock invesmen in one regime dependen on he invesmen opporuniy se in he oher regime. In absence of his correlaion and ransacion coss, he porfolio choice is independen of changes in he invesmen opporuniy se of a differen regime. Thus, our model differs from hose wih invesmen opporuniy se changes ha are independen of he sock price risk (e.g., Meron 1971, Jang e al. 2007). 2 We illusrae quaniaive condiions under which an invesor should sell sock afer a crash even when he marke becomes less liquid. No surprisingly, his sale ypically occurs when he invesmen opporuniy se significanly worsens afer a marke crash (e.g., much higher volailiy or much greaer furher crash inensiy) and he worsened environmen may persis for a period of ime. Inuiively, his is because a significanly worsened invesmen opporuniy se and he expeced long duraion of he illiquid regime make he marginal benefi of selling he sock ouweigh he marginal cos of incurring he necessary ransacion cos. This finding is consisen wih fligh o qualiy afer a crash, bu in sharp conras o he conrarian syle predicion of he sandard porfolio selecion models wih independen and idenically disribued (i.i.d.) reurns (e.g., Meron 1971). We show ha even a small increase in he afer-crash volailiy (e.g., from 12% o 20%) may rigger a shif from sock o he risk-free asse. On he oher hand, i may also be opimal o buy more sock or no o rade a all upon a crash. In general, o deermine he opimal rading sraegy afer a crash, he invesor rades off he benefi of rebalancing due o he change in he invesmen opporuniy se and he cos of ransacion. Loosely speaking, he greaer he change in he invesmen opporuniy se and he greaer he 2 Alhough we presen he model wih wo regimes, our mehodology exends o more regimes in which all he parameers as well as jump disribuions can change across regimes. Therefore, our mehodology can be used o solve opimal porfolio problems wih ransacion coss where he sock price risk is correlaed wih a wide variey of changes in he invesmen opporuniy se. expeced duraion of he illiquid regime, he greaer he benefi of rebalancing. Depending on he relaive magniude of he benefi and cos, he invesor may choose o sell, o buy, or o wai ou he illiquid regime afer a crash. We show ha an increase in he expeced jump size may increase he opimal sockholding even if he expeced reurn remains he same and he reurn volailiy increases. Inuiively, increasing he expeced jump size (bu keeping he expeced reurn consan) may help an invesor by making reurns less negaively skewed and price jumps can help reduce rebalancing coss across regimes. To undersand he laer effec, suppose a large price drop riggers he illiquid regime and he opimal fracion of wealh ha should be invesed in sock decreases. Wih a large price drop, he fracion of wealh invesed in sock is already lower, so a ransacion may be no longer necessary. Therefore, his ransacion cos reducion effec may make he invesor hold more sock in he liquid regime. In addiion, we show ha misesimaing he correlaion beween marke crashes and marke illiquidiy can be cosly o invesors. For example, if an invesor underesimaes he correlaion beween marke crashes and marke illiquidiy and adops he corresponding opimal rading sraegy under he wrong esimaion, he cerainy equivalen wealh loss from his rading sraegy can be as high as 3.5% of he invesor s iniial wealh in some reasonable scenarios. Closely relaed works include he lieraure on porfolio selecion wih ransacion coss bu wihou even risks (e.g., Consaninides 1986, Davis and Norman 1990, Dumas and Luciano 1991, Shreve and Soner 1994, Liu and Loewensein 2002), and he lieraure on porfolio selecion wih even risks bu wihou ransacion coss (Liu e al. 2003). The closes works o ours are Liu e al. (2003), Jang e al. (2007), Framsad e al. (2001), and Øksendal and Sulem (2005). Liu e al. (2003) examine he opimal rading sraegy when he sock price follows a jump diffusion process wih sochasic volailiy. However, hey do no consider he join impac of he correlaed marke crashes and marke illiquidiy. Jang e al. (2007) use a regime swiching model o show ha ransacion coss can have a firs-order effec when an invesmen opporuniy se varies hrough ime. In conras o our model, hey do no consider he effec of marke crashes on rading sraegies. Framsad e al. (2001) sudy he opimal consumpion/invesmen problem wih an infinie horizon in a jump diffusion seing wih consan proporional ransacion coss. However, hey do no examine he effec of crashriggered invesmen opporuniy se changes, which are imporan feaures for undersanding he opimal rading sraegy in a financial crisis. In addiion, hey

3 Liu and Loewensein: Marke Crashes, Correlaed Illiquidiy, and Porfolio Choice Managemen Science 59(3), pp , 2013 INFORMS 717 do no offer a numerical procedure o solve for he opimal sraegy. Øksendal and Sulem (2005) provide heoreical resuls on some ypes of opimal conrol problems wih jump diffusions and offer some examples o illusrae he applicaion of heir heory. However, hey do no provide heoreical or numerical analysis on he porfolio choice problem in he presence of marke crashes and correlaed changes in he invesmen opporuniy se. The correlaion significanly complicaes he heoreical and numerical analysis and he heoreical mehods provided by Øksendal and Sulem (2005) no longer apply wihou significan changes, because he opimal rading sraegy in one regime can depend on he invesmen opporuniy se afer a crash even in he absence of ransacion coss. The res of his paper is organized as follows. In 2 we describe our porfolio choice model in a woregime framework. We provide characerizaion of he value funcion and he no-ransacion region. Secion 3 describes an ieraive procedure o compue he opimal rading sraegy. Secion 4 provides some analyical comparaive saics on he opima rading boundaries. We conduc an exensive numerical analysis, on he opimal rading sraegy in 5. We conclude in 6 and provide proofs in he appendix. 2. The Basic Model 2.1. The Asse Marke An invesor can rade wo asses in he financial marke: one risk free ( he bond ) and one risky ( he sock ). There are wo regimes wih differen liquidiy: regime 0 (liquid, lower ransacion coss) and regime 1 (illiquid), across which oher parameer values may also change. We use 0 1 as a sae variable o indicae he regime a ime. The ime ineres rae is r. The invesor can buy he sock a he ask price 1 + S and sell i a he bid price S where 0 and 0 < 1 represen he proporional ransacion cos raes in regime, and S denoes he sock price wihou ransacion coss. We assume ha he sock price S may jump. To capure he idea ha a downward jump may have a differen impac compared o an upward jump, we sor a sock price jump ino an up jump ( U ) and a down jump ( D ), occurring a he jump imes of independen Poisson processes N j wih inensiies j for j U D, respecively, and random jump sizes J D 1 1 0, J U 1 0. The sock price process hen evolves as where ds = S d + S dw + J U 1S dn U + J D 1S dn D (1) = U EJ U 1 + D EJ D 1 (2) represens he expeced reurn compensaion for he presence of jumps so ha he insananeous sock expeced reurn is wih > r, w is a onedimensional Brownian moion, is he sock reurn volailiy, and J U D are he ime realizaions of J U D. We assume for simpliciy ha he jump sizes are drawn from idenical independen disribuions a each ime. 3 Le J be he greaes lower bound ha saisfies ProbJ U D J = 1. To capure he idea ha liquidiy changes may be correlaed wih price jumps (e.g., downward jumps may be posiively correlaed wih swiching ino an illiquid regime), we decompose each of he jump processes ino wo independen componens. Specifically, le N U = N U 1 + N U 2 N D = N D 1 + N D 2 where if N1 U or N1 D jumps hen he curren regime swiches ino he oher regime, and N2 U and N2 D are independen of regime swiching. In addiion, N j i has an inensiy of j i wih j 1 + j 2 = j for i = 1 2, 0 1 and j U D. To model he possibiliy ha regimes can also change because of oher facors such as general macroeconomic condiions, we assume ha regime also swiches ino a differen regime a he jump imes of anoher independen Poisson jump process N R wih inensiy. Wih hese assumpions, we have ha he sae variable evolves (almos surely) as { dn1 D + dn R d = dn U 1 + dn R if = 0 if = 1 To undersand Equaion (3), suppose he curren regime is liquid (i.e., = 0). Equaion (3) hen indicaes ha if N1 D jumps hen we have a downward jump in he sock price and he regime swiches ino he illiquid regime ( = 1). On he oher hand, if N R jumps, hen he regime also shifs bu here is no jump in he sock price. Finally, if N2 D jumps hen here is a downward jump in he sock price bu he marke says in he liquid regime. Similarly, suppose he curren regime is illiquid (i.e., = 1). Equaion (3) implies ha if N1 U jumps hen we have an upward jump in he sock price and he regime swiches ino he liquid regime ( = 0). On he oher hand, if N R jumps, hen he regime also shifs bu he sock price does no jump. Finally, if N2 U jumps hen here is an upward jump in he sock price bu he marke says in he illiquid regime. Thus, we have a fairly parsimonious model ha ness many possible submodels and allows he invesmen opporuniy se including liquidiy o be 3 Alernaively, mainaining he independence of jump sizes hrough ime, we could le he jump disribuion vary wih he sae variable. (3)

4 Liu and Loewensein: Marke Crashes, Correlaed Illiquidiy, and Porfolio Choice 718 Managemen Science 59(3), pp , 2013 INFORMS correlaed wih sock prices in many ineresing ways. For example, a pure jump diffusion model wih consan proporional ransacion coss is obained by seing 1 D = U 1 = = 0. Our model also allows regime swiching and changes in he oher parameer values o be correlaed wih sock price jumps. For example, afer a downward price jump, he regime may swich, and he volailiy, expeced reurn, or furher crash inensiy may become higher. When + > 0, he above model gives rise o equaions governing he evoluion of he dollar amoun invesed in he bond, x, and he dollar amoun invesed in he sock, y : dx = r x d 1 + di + dd (4) dy = y d + y dw +J U 1y dn U +J D 1y dn D +di dd (5) where he processes D and I represen he cumulaive dollar amoun of sales and purchases of he sock, respecively. These processes are nondecreasing, righ coninuous adaped processes wih D0 = I0 = 0. Le x 0 and y 0 be he given iniial dollar amouns in he bond and he sock, respecively. We le x 0 y 0 denoe he se of admissible rading sraegies D I such ha (4) and (5) are saisfied given (3) and he invesor is always solven, i.e., 4 x + 1 max0 1y J 0 0 (6) which, as in Liu e al. (2003), resrics he raio x /y The Invesor s Problem The invesor s problem is o choose admissible rading sraegies D and I so as o maximize Eux + 1 y for an even ha occurs a he firs jump ime of a sandard, independen Poisson process wih inensiy. Thus, is exponenially disribued wih parameer, i.e., P d = e d This formulaion can capure beques, accidens, reiremen, and many oher evens ha happen on uncerain daes. 5 If is inerpreed o represen he invesor s uncerain lifeime (as in Meron 1971), he invesor s average lifeime is hen 1/ and he variance of he invesor s lifeime is accordingly 1/ 2. 4 Because he jump size is no bounded above, o mainain solvency, he invesor canno shor and so y 0. 5 We also used he mehod proposed by Liu and Loewensein (2002) o solve he case wih a deerminisic horizon. The soluion shows ha he exponenially disribued horizon case is a close approximaion o he case wih a long horizon (abou 20 years). Because his finding is similar o ha in Liu and Loewensein (2002), we do no repor i in he paper o save space. Assuming a CRRA preference, we can hen wrie he value funcion as vx y [ x + = sup E y 1 D I x y ] 0 = (7) As shown in he appendix, similar o Meron (1971), Liu and Loewensein (2002), and Jang e al. (2007), Equaion (7) can be rewrien as he following recursive form: vx y = sup E D I x y where [ 0 e ( D 1 vx y J D + D 2 vx y J D + U 1 vx y J U + U 2 vx y J U + vx y ) ] d (8) + x + y 1 = + + U + D (9) 2.3. Opimal Policies wih No Transacion Coss For he purpose of comparison, we firs consider he case wihou ransacion coss (i.e., = = 0, 0 1). Define he oal wealh W = x + y and le be he fracion of wealh invesed in he sock. The invesor s problem becomes [ vw = sup E W e d 0 = W 0 = W subjec o (3), he dynamic budge consrain dw = r + r W d + W dw + W J U 1 dn U ] (10) + J D 1 dn D (11) and he solvency consrain W 0. Wihou ransacion coss, he opimal rading sraegy is o inves a consan fracion of wealh in sock in regime and he value funcion in regime is of he form vw = M W 1 From he HJB parial differenial equaion, i is sraighforward o show ha for = 0 1, M solves and a M + h M + = 0 (12) = arg max ( a M + h M + ) (13)

5 Liu and Loewensein: Marke Crashes, Correlaed Illiquidiy, and Porfolio Choice Managemen Science 59(3), pp , 2013 INFORMS 719 where a = ( r + r ) and 1 + U 2 E1+J U D 2 E1 + J D 1 1 (14) h = U 1 E1 + J U D 1 E1 + J D (15) Equaions (12) and (13) yield four equaions for four unknowns M ( = 0 1) ha can be easily solved numerically. As in Meron (1971) and Liu e al. (2003), condiions on he parameers and he jump disribuion are required for he exisence of he opimal soluion. Assumpion 1. The soluion M o (12) and (13) is such ha M > 0 for = 0 1. The posiiviy of M rules ou he case where he invesor can achieve bliss levels of uiliy and ensures he exisence of an opimal porfolio. 6 We summarize he main resul for his no-ransacion-cos case in he following heorem. Theorem 1. Suppose ha 0 = 0 = 1 = 1 = 0. Then under Assumpion 1, for 0 < he opimal sock invesmen policy in regime is equal o as defined in (13) and he lifeime expeced uiliy is vw = M W 1 (16) where M solves (12) and (13) for = 0 1. In addiion o he sandard rade-off beween he excess reurn and variance, in deermining he opimal rading sraegy he invesor also akes ino accoun he impac of sock price jumps in choosing he opimal porfolio. More ineresingly, in conras o Jang e al. (2007), he opimal rading sraegy in one regime can depend on he invesmen opporuniy se in he oher regime even in he absence of ransacion coss. This cross-regime dependence comes from he dependence of h in Equaion (13) on, which is due o he key feaure of our model: he correlaion beween price jumps and regime swiching. Wihou his correlaion (e.g., h =, or h = 0), h would no depend on and he porfolio choice would be unaffeced by he invesmen opporuniy se in he oher regime. Inuiively, he impac of price jumps is regime dependen and wih he correlaion beween jumps and regime swiching, 6 I can be easily verified ha Assumpion 1 reduces o he wellknown Meron condiion in he absence of jumps and regime shifs. his impac becomes dependen on he invesmen opporuniy ses in boh regimes. Because of he crossregime dependence, he opimal porfolio consiss of an exra regime hedging componen compared o he exising lieraure (e.g., Meron 1971, Jang e al. 2007). Remark 1. If J = 0, hen he invesor never leverages (i.e., 1). In general, when J < 1, leverage is limied because solvency requires x + yj 0 or 1/1 J. This is why Equaion (13) is no wrien in erms of he firs-order condiions Opimal Policies wih Transacion Coss Suppose now ha + > 0 for = 0 1. As in Liu and Loewensein (2002), he value funcions are homogeneous of degree in x y. This implies ha for = 0 1, vx y = y 1 z where z x (17) y for some concave funcion In he presence of ransacion cos, he solvency region in each regime splis ino hree regions: buy region, sell region, and no-ransacion (NT) region. Because of he ime homogeneiy of he value funcion, hese regions can be idenified by wo criical numbers (insead of funcions of ime) z and z in regime. The buy region corresponds o z z, he sell region o z z, and he NT region o z < z < z. We illusrae hese hree regions in regime in Figure 1. As in he pure diffusion case, he invesor does no rade as long as he raio z remains inside he NT region. However, as soon as he raio z moves ou of he NT region, he invesor immediaely rades a minimum amoun o ge back o he NT region. Thus, he invesor only rades on a measure zero se of imes and follows a singular conrol sraegy in sock rading. 7 However, in conras o he pure diffusion cases previously sudied, he raio z can jump ou of he NT region, which is followed by an immediae lumpsum ransacion o he closes boundary of he NT region. Moreover, when he regime shifs, an invesor migh also need o make a lump-sum rade o he new boundary in he new regime. Following Jang e al. (2007), we have he following coupled HJB equaion: 8 maxlv v x v y 1 + v x + v y = 0 = 0 1 (18) 7 Alhough singular conrols are a racable way o solve for opimal porfolio sraegies wih ransacion cos, hey require rading a an infinie rae on a measure zero se of ime poins. 8 The HJB equaion follows from he fac ha e vx y + 0 e s ( f x s y s + x s + y s 1 ) ds is a maringale for he opimal policy. The expression of f x y follows direcly from (8).

6 Liu and Loewensein: Marke Crashes, Correlaed Illiquidiy, and Porfolio Choice 720 Managemen Science 59(3), pp , 2013 INFORMS Figure 1 x The Solvency Region in Regime Buy z(ι) No-ransacion z(ι) 3. An Ieraive Procedure o Find Opimal Trading Sraegy The fac ha he raio z can jump ou of he NT region (refleced by he presence of he erm gz in Equaion (22)) and he regime can shif complicaes he problem significanly. We now develop an ieraive echnique ha solves he invesor s problem using a sequence of closed-form expressions. Firs, we choose an iniial funcion v 0 x y ha is finie, concave, increasing, and homogeneous such ha v 0 x y vx y for = 0 1. For concreeness and ease of boundary condiions, we assume ha where 0 Sell Solvency boundary Lv = y 2 v yy + rxv x + yv y v + f x y + x + y1 (19) f x y = U 1 EvxyJ U 1 + U 2 EvxyJ U + D 1 Evx yj D + D 2 Evx yj D + vx y (20) Using (17), we can simplify (18) o ge he following inegro-differenial variaional equaions: maxl 1 z + z z z where z z z + z = 0 (21) L 1 = z 2 zz z + 2 z z z + 1 z + gz gz = U 1 Ez/J U J U 1 + U 2 Ez/J U J U 1 + D 1 Ez/J D J D 1 + D 2 Ez/J D J D 1 + z + y z = 2 /2 + 2 = 2 + r + (22) v 0 x y = M x + y1 (23) where M are he coefficiens for he no-ransacioncos case. Then o compue v i+1 x y, for i = we can solve he following recursive srucure: v i+1 x y = where sup E D I x y f i x y = U 1 vi x y J U [ e (f i x y 0 + x + y 1 + D 1 vi x y J D + D 2 vi x y J D ) ] d (24) 1 + U 2 vi x y J U + v i x y (25) Lemma 1 in he appendix guaranees he convergence of his ieraive procedure and he concaviy of he limi funcion ˆv ( lim i v i ). To faciliae he proof ha ˆv is indeed he value funcion, noe ha as before, for = 0 1, because of he homogeneiy of v i x y, here exiss a funcion i such ha v i x y = y 1 i ( x y ) Solving (24) reduces o finding funcions i z for = 0 1 such ha z 2 i zz + 2z i z + 1 i + g i 1 z = 0 where g i 1 z = U 1 Ei 1 z/j U J U 1 + U 2 Ei 1 z/j U J U 1 i = 1 n (26) + D 1 Ei 1 z/j D J D 1 + D 2 Ei 1 z/j D J D 1 + i 1 z + z + 1

7 Liu and Loewensein: Marke Crashes, Correlaed Illiquidiy, and Porfolio Choice Managemen Science 59(3), pp , 2013 INFORMS 721 and 1 and 2 are he same as in (21). We hen have he following resul: Theorem 2. As i, he funcions v i x y = y 1 i x/y converge o he value funcion vx y for = 0 1. Proof. See he appendix. Theorem 2 shows ha he ieraive procedure can indeed closely approximae he value funcion and he corresponding opimal rading sraegy. The basic inuiion for he convergence of his ieraive procedure is he monooniciy of he value funcion: v i x y v i 1 x y, as direcly implied by he opimizaion srucure (24) and he fac ha v i x y is bounded by he value funcion v 0 x y for he no-ransacion-cos case and he value funcion for invesing only in he risk-free asse. The ieraive procedure is formally similar o ha used o solve a discree ime infinie horizon dynamic programming problem, where a ime period in our model corresponds o he ime beween adjacen Poisson jumps. The proof of Theorem 2 implies ha his ieraive procedure can be readily applied o many oher opimal porfolio choice problems and can significanly simplify compuaion especially for hose problems ha involve coupled nonlinear HJB equaions Analyical Comparaive Saics The opimal rading sraegy in regime is no rading if z z z, selling sock o he boundary z if z < z, and buying sock o he boundary z if z > z. In conras o a diffusion model, i is possible ha z jumps ou of he NT region, which would be followed by an immediae ransacion back o he closes boundary. When he regime shifs, he opimal sraegy may or may no involve an immediae ransacion depending on how he boundaries change across regimes. I is helpful o discuss hree possible ypes of NT regions across regimes we will encouner in our numerical work laer. Case 1: Separaed. For example, z0 < z0 < z1 < z1. In his case invesors may sell some sock and buy more of he risk-free asse righ afer a sock price crash. This is consisen wih he so-called fligh-oqualiy phenomenon, bu in sharp conras wih he conrarian sraegy prediced by a model wih i.i.d reurns. This case occurs if he regime shifs from he liquid regime ( = 0) o he illiquid regime ( = 1) afer a downward price jump and he new raio z righ 9 For example, one could have a model wih n regimes in which he coefficiens and he jump disribuions vary across regimes. This procedure can hen solve an opimal porfolio problem wih ransacion coss wih correlaed sock price risk and invesmen opporuniy se. afer he crash says in he sell region of he illiquid regime. This case will ypically obain when he shif in he invesmen opporuniy se across regimes is large and he expeced ime spen in he new regime is long so ha he required ransacion cos is jusified. Case 2: Nesed. For example, z1 < z0 < z0 < z1. In his case, if he regime shifs from he liquid regime and he jump magniude a his ime is no oo large, hen he invesor will opimally no rebalance. However, if he regime shifs from he illiquid regime o he liquid regime, hen he invesor may buy or sell sock even wihou a price jump. In his case an invesor opimally reduces ransacion frequency unil marke condiions improve. Inuiively, his case will occur when he difference in he invesmen opporuniy se is relaively small, he ime spen in he illiquid regime is relaively shor, and he ransacion coss are relaively large. Case 3: Overlapping bu nonnesed. For example, z0 < z1 < z0 < z1. In conras o Case 2, in he absence of upward jumps, he invesor never sells he sock when he regime shifs from he illiquid regime o he liquid regime. This case lies beween Cases 1 and 2 and ypically occurs when he difference in he invesmen opporuniy se is moderae and he ransacion coss are relaively small. We now presen an upper bound on he lowes sell boundary of he wo regimes in erms of he Meron lines (i.e., he opimal porfolio in he no-ransacioncos case). Proposiion 1. Le z = 1/ 1, where is he opimal porfolio in he no-ransacion-cos case in regime. Then 1. for an i.i.d. reurns case, if h = 0, hen z z ; 2. for a general case, if h 0, hen minz z maxz z (27) Our nex resul provides lower bounds on he ransacion boundaries in he regime wih he highes uiliy. Proposiion 2. Suppose eiher h = 0 (i.i.d. case) or vx y vx y wih r = r. 10 Then he buy boundary saisfies ( ) 2 z r 1 (28) 10 Sufficien condiions for vx y vx y are given in Proposiion 3 in he appendix. The mehod of proof uses he ieraive consrucion of he value funcion and can be used o provide a variey of comparaive saics.

8 Liu and Loewensein: Marke Crashes, Correlaed Illiquidiy, and Porfolio Choice 722 Managemen Science 59(3), pp , 2013 INFORMS and he sell boundary saisfies ( ) 2 z 2 r 1 (29) Proposiions 1 and 2 give useful bounds on he ransacion boundaries. In he i.i.d. case or in he regime wih he highes uiliy, if > 0, hen he sell boundary is always below he Meron line, he opimal raio of bond o sock wih no ransacion coss. However, he sell boundary is no always a decreasing funcion of he ransacion cos rae. For example, if > 1 (i.e., wih leverage), hen he sell boundary can be above he Meron line for a large enough ransacion cos rae. This can be seen from he exreme case where = 1 and hus he sell boundary has o be on or above zero by (29). The bounds obained in Proposiions 1 and 2 can be also useful for validaing numerically compued boundaries. 5. Numerical Resuls To gain some undersanding of how he various elemens of our model are opimally raded off, we now presen a baseline case and perform comparaive saics o see how he opimal boundaries behave. To capure he idea ha large price jumps likely have greaer correlaions wih changes in he invesmen opporuniy se han small jumps, for our numerical analysis we consider a slighly more general seing where large jumps can have differen correlaions from small jumps. More specifically, we divide jumps ino large downward jumps, large upward jumps and moderae (upward or downward) jumps and allow large jumps o be correlaed wih regime swiching. In oher words, we assume where ds = S d + S dw + J U + J M 1S dn U + J D 1S dn D 1S dn M (30) = U EJ U 1 + D EJ D 1 + M EJ M 1 (31) and he moderae jump Poisson process N M is independen of all oher jumps. All he previous resuls are easily exended o his case. 11 For our numerical analysis, we use as our defaul parameers 0 = 05%, 1 = 25%, 0 = 1 = 0, 0 = 1 = 7%, r0 = r1 = 1%, = 5, and = 004. These parameers represen an equiy premium 11 The analyical resuls for his generalized case are presened in an earlier version of he paper ha is available from he auhors. of 6% in boh regimes and an expeced horizon of 25 years. The round-rip ransacion cos is 05% in he liquid regime and 25% in he illiquid regime. We also assume in he baseline case volailiies are he same in boh regimes, i.e., 0 = 1. In our baseline case we assume ha a jump arrives on average once every wo years and jump inensiies do no change across regimes, ha is, U + M + D = 05 wih U 0 = U 1 = U, M 0 = M 1 = M, and D 0 = D 1 = D. 12 For i U M D, log jump size logj i is assumed o be runcaed normal wih parameers J and J and suppor inerval a i b i, where a U = R > 0, b U =, a M = R < 0, b M = R, a D =, and b D = R. To deermine he remaining baseline parameers, we calibrae he model o mach he variance (0.0082), skewness ( 1.33), and excess kurosis (34.92) repored in Campbell e al. (1996, p. 21) for daily log reurns. This procedure leads o 0 = 1 = 01190, J = 00259, and J = As defaul parameer values, we se R = 003 and R = 003, which implies he average large up jump size is 70%, he average large down jump size is 78%, and he average moderae jump size is 00%. Using hese parameer values, we obain ha U = 01003, D = 02377, and M = These parameers indicae ha he probabiliy of a large down jump is greaer han he probabiliy of a large up jump, consisen wih he negaive skewness of he sock reurns. We assume regime 0 swiches o regime 1 if and only if a large down jump occurs. To accomplish his, we se 0 = 0, 1 U 0 = 0, U 2 0 = U, 1 D0 = D, and 2 D 0 = 0. These choices capure he idea ha worsened liquidiy condiions are usually accompanied by large downward jumps in he sock price. Thus, he raio x/y will jump up whenever here is a shif from he liquid o he illiquid regime. Our baseline assumpion is ha he expeced duraion of he illiquid regime is one year. To accomplish his, we se 1 = and 1 U 1 = 00633, which implies ha he correlaion beween a large upward jump and swiching ino he liquid regime is 20%. Our remaining parameers are se o be consisen wih our assumpions ha he jump inensiies, do no vary across regimes. The relaion ha 1 U 1 + U 2 1 = U = dicaes ha 2 U 1 = The moderae jump inensiy remains fixed a M = For he large down jump inensiies, we se 1 D1 = 0 so ha he regime does no shif back o he liquid regime coinciden wih a large down jump. Thus, D = 2 D 1 = The defaul parameer values are summarized in Table We also conduced analysis on differen baseline cases wih lower jump frequencies, which implies larger jump sizes on average. The qualiaive resuls are he same.

9 Liu and Loewensein: Marke Crashes, Correlaed Illiquidiy, and Porfolio Choice Managemen Science 59(3), pp , 2013 INFORMS 723 Table 1 Defaul Parameer Values Parameers J J r Values Parameers Values Parameers 1 U 1 0 D 1 1 D 2 0 U 2 0 U U 2 1 Values Parameers U 1 1 M D 1 0 D 2 1 D R R Values We now examine he opimal porfolio policies for our model. For ease of comparison wih he noransacion-cos case, we will presen he ransacion boundaries in erms of he fracion of wealh invesed in he risky asse: = y/x + y. Le = 1/ z + 1 and = 1/ z + 1. The opimal policy is equivalen o mainaining he fracion beween and Wihou Transacion Coss For our baseline case wih no ransacion coss i is opimal o hold 708% of wealh in he risky asse in boh regimes. In he absence of jumps (he i.i.d. Meron case), he opimal fracion of wealh invesed in sock is 847%. The reducion in he sock invesmen compared o he i.i.d. case is caused by he jumps ha resul in higher volailiy and negaive skewness. We now graphically illusrae he crossregime hedging effec in he absence of ransacion coss in boh regimes, as discussed in 2. In Figure 2, we plo he regime 0 opimal fracion of wealh 0 agains he volailiy in regime 1 for wo cases: 1 = 7% and 1 = 9%, in he absence of ransacion coss in boh regimes. Figure 2 shows ha in conras o sandard models (e.g., Meron 1971, Jang e al. 2007), he opimal rading sraegy in regime 0 depends on he invesmen opporuniy se in regime 1 even when boh regimes are perfecly liquid. For example, he opimal fracion of wealh invesed in sock in he liquid regime decreases from 708% o 696% when he volailiy in he illiquid regime increases from 119% o 20%. On he oher hand, if he expeced reurn in regime 1 is also higher (e.g., 009), hen he ne hedging demand may be posiive or negaive depending on wheher he expeced reurn increase or he volailiy increase dominaes Changes in he Volailiy in he Illiquid Regime Nex, we examine he effec of he pos-crash changes in marke volailiy and liquidiy. I is well documened ha boh volailiy and illiquidiy end o be greaer afer a crash. Accordingly, in Figure 3 we show how he opimal ransacion boundaries vary when volailiy rises and a marke becomes less liquid afer a crash. In he absence of ransacion coss, i is opimal o always keep 708% (in boh regimes) of he wealh in sock in our baseline case. Wih posiive ransacion coss, his policy is no longer opimal. Figure 3 implies ha i is opimal o keep he fracion of wealh invesed in sock beween 661% and 725% in he liquid regime and beween 635% and 875% in he illiquid regime. The ransacion boundaries of he wo regimes are hus nesed. In oher words, he invesor does no ransac when he regime shifs from he liquid regime o he illiquid regime unless he price drop is oo large in magniude. The rading frequency in Figure 3 Opimal Trading Boundaries as a Funcion of 1 Figure 2 Opimal Porfolio in Regime 0, 0, as a Funcion of 1 Wihou Transacion Coss (0) (0) * (0) (1) = 0.09 (1) = (1) Illiquid regime volailiy (1) Noe. This figure shows how he opimal fracions vary wih he volailiy in he illiquid regime 1 for parameers J = 00259, J = 00666, 0 = 01190, 0 = 007, r = 001, = 004, = 5, 0 = 0, 1 = 09367, 0 = 1 = 0, 0 = 1 = 0, R = 003, R = 003, U 0 = 1 D1 = 1 D0 = 0, 2 U0 = 2 U = 01003, U1 = 00370, 2 U1 = 00633, 1 M = 01620, and D0 = 1 D1 = 2 D = Illiquid regime volailiy (1) (1) Noe. This figure shows how he opimal rading boundaries vary wih he volailiy in he illiquid regime 1 for parameers J = 00259, J = 00666, 0 = 01190, 0 = 1 = 007, r = 001, = 004, = 5, 0 = 0, 1 = 09367, 0 = 1 = 0, 0 = 05%, 1 = 25%, R = 003, R = 003, U 0 = 1 D1 = 1 D0 = 0, 2 U0 = 2 U = 01003, U1 = 00370, 2 U1 = 00633, 1 M = 01620, and D0 = 1 D1 = 2 D =

10 Liu and Loewensein: Marke Crashes, Correlaed Illiquidiy, and Porfolio Choice 724 Managemen Science 59(3), pp , 2013 INFORMS he illiquid regime is lower, as implied by he wider no-ransacion region due o greaer ransacion coss. As he volailiy in he illiquid regime increases, all ransacion boundaries move downward, which implies ha he invesor decreases sock invesmen no only in he illiquid regime bu also in he liquid regime, because of he cross-regime hedging. As expeced, he ransacion boundaries in he illiquid regime are much more sensiive o he illiquid regime volailiy increase han hose in he liquid regime. Figure 3 shows ha even for modes increases in volailiy in he illiquid regime, he NT region in he illiquid regime can move well below ha in he liquid regime. This implies ha even small increases in he afer-crash volailiy can make invesors sell sock and buy more of he risk-free asse righ afer he price crash, behaving like fligh o qualiy, ha was commonly observed afer a marke crash. For example, afer a 5% price crash, if he price crash reflecs significanly worsened fundamenals (e.g., much greaer uncerainy of he sock payoff) and as a resul he afer-crash sock volailiy increases from 119% o 20%, hen he invesor will keep he fracion of wealh invesed in sock beween 560% and 637% in he liquid regime, and beween 235% and 437% in he illiquid regime. Therefore, upon he crash, he invesor will sell sock so ha he sock fracion becomes 437% afer rebalancing. In conras, sandard porfolio choice models wih i.i.d. reurns (e.g., Meron 1971) predic he opposie: Afer a price drop, he invesor should buy more o rebalance. Thus, he deerioraion of invesmen opporuniy se afer a crash may conribue o he fligh o qualiy behavior Changes in he Inensiy of a Large Downward Jump in he Illiquid Regime Our baseline case assumes ha only liquidiy changes afer a marke crash. However, afer a marke crash, he probabiliy of anoher crash migh also change. Figure 4 shows how he opimal rading boundaries vary as we vary he inensiy of a large downward jump in he illiquid regime (2 D 1). For higher values of he inensiy of anoher marke crash in he illiquid regime, he invesor is more likely o sell righ afer a marke crash in he liquid regime, all else equal. Recall ha varying he jump parameers does no affec he expeced reurn bu affecs variance, skewness, and kurosis. A large negaive jump accompanied by he ransiion ino he illiquid regime where large downward jumps occur more frequenly represens a significan deerioraion of he invesmen opporuniy se. Thus, he invesor migh opimally incur he ransacion cos o sell some sock upon a crash, similar o he effec of increased volailiy. For example, if he inensiy of furher crash increases o 20 afer a crash (i.e., on average one crash every six Figure Opimal Trading Boundaries as a Funcion of D Large down jump inensiy in regime 1 2 D (1) (0) (0) (1) (1) Noe. This figure shows how he opimal rading boundaries vary wih he inensiy of he downward jump D1 for parameers 2 J = 00259, J = 00666, 0 = 1 = 01190, 0 = 1 = 007, r = 001, = 004, = 5, 0 = 0, 1 = 09367, 0 = 1 = 0, 0 = 05%, 1 = 25%, R = 003, R = 003, U 0 = 1 D1 = 1 D0 = 0, 2 U0 = 2 U = 01003, U1 = 00370, 2 U1 = 00633, 1 M = 01620, and D0 = 1 D = monhs), hen he invesor will keep he fracion of wealh invesed in sock beween 568% and 645% in he liquid regime, and beween 326% and 551% in he illiquid regime. Therefore, upon a crash, he invesor may sell some sock o reach 551%. In addiion, Figure 4 suggess ha in anicipaion of his increase in he crash inensiy, he invesmen in he liquid regime is also reduced Changes in he Expeced Reurn in he Illiquid Regime Figure 5 shows how he ransacion boundaries vary as a funcion of he illiquid-regime expeced reurn 1. If he afer-crash expeced reurn goes up (as found in empirical sudies; e.g., Fama and French 1989, Ferson and Harvey 1991), he NT region can be higher in he illiquid regime han in he liquid regime, which implies ha he invesor would hold more sock in he illiquid regime. In his case he invesor will always buy more of he risky asse o ake advanage of he higher expeced reurn when he regime shifs from he liquid o he illiquid regime and liquidae he posiion when he marke becomes more liquid. For example, if he afer-crash expeced reurn increases o 9%, hen i is opimal for he invesor o keep he fracion of wealh in sock beween 681% and 754% in he liquid regime and beween 801% and 953% in he illiquid regime. Figures 3 5 sugges ha he effec of he increased expeced reurn on he opimal rading sraegy couneracs he effec of he increased volailiy and he increased crash inensiy. The fligh o qualiy behavior ensues when he volailiy and he crash inensiy effecs dominae. As he expeced reurn in he illiquid regime increases, he NT region shrinks. Inuiively, when

11 Liu and Loewensein: Marke Crashes, Correlaed Illiquidiy, and Porfolio Choice Managemen Science 59(3), pp , 2013 INFORMS 725 Figure 5 Opimal Trading Boundaries as a Funcion of 1 (1) Figure 6 Opimal Trading Boundaries as a Funcion of 1 (1) (1) (0) (0) (0) (0) (1) Expeced reurn in regime 1 (1) Noe. This figure shows how he opimal rading boundaries vary wih he expeced reurn in he illiquid regime 1 for parameers J = 00259, J = 00666, 0 = 1 = 01190, 0 = 007, r = 001, = 004, = 5, 0 = 0, 1 = 09367, 0 = 1 = 0, 0 = 05%, 1 = 25%, R = 003, R = 003, U 0 = 1 D1 = 1 D0 = 0, 2 U0 = 2 U = 01003, U1 = 00370, 2 U1 = 00633, 1 M = 01620, and D0 = 1 D1 = 2 D = an invesor deermines he no-ransacion region size, he rades off he ransacion cos paymen and he risk exposure variaion (i.e., he variaion in he fracion of wealh invesed in sock). As he expeced reurn increases, he sock price ends o go up faser and herefore he dollar amoun invesed in he sock ends o increase faser. As a resul, he probabiliy of hiing he lower boundary and incurring ransacion coss a he lower boundary becomes lower. In addiion, because he invesor sells a he upper boundary when he price and hus his wealh goes up and buys a he lower boundary when he price and hus his wealh goes down, he marginal uiliy cos of ransacion cos paymen a he upper boundary is lower han ha a he lower boundary. Therefore, o avoid oo much risk exposure variaion, he invesor can increase he lower boundary more han he upper boundary wihou significanly increasing he marginal uiliy cos of ransacion cos paymen. This causes he no-ransacion region o decrease wih he expeced reurn in he illiquid regime, as also shown in Liu and Loewensein (2002, Figure 6). When he expeced reurn in he illiquid regime is much higher han ha in he liquid regime, he probabiliy of hiing he lower boundary becomes much lower in he illiquid regime, and hus he no-ransacion region can become even smaller han ha in he liquid regime even hough he ransacion cos rae is much higher in he illiquid regime Changes in he Illiquidiy in he Illiquid Regime Figure 6 shows how he ransacion boundaries change as he ransacion coss vary in he illiquid Transacion cos in illiquid regime (1) Noe. This figure shows how he opimal rading boundaries vary wih he ransacions cos in he illiquid regime 1 for parameers J = 00259, J = 00666, 0 = 1 = 01190, 0 = 1 = 007, r = 001, = 004, = 5, 0 = 0, 1 = 09367, 0 = 1 = 0, 0 = 05%, R = 003, R = 003, U 1 0 = D 1 1 = D 2 0 = 0, U 2 0 = U = 01003, U 2 1 = 00370, U 1 1 = 00633, M = 01620, and D 1 0 = D 2 1 = D = regime. The illiquid regime NT region ness he liquid regime NT region. For large ransacion coss in he illiquid regime, he invesor significanly widens he NT region in he illiquid regime o reduce rading frequency. Thus, for large ransacion coss, i is opimal o ry o wai ou he illiquid regime. As ransacion coss in he illiquid regime increase, he invesor also opimally holds less sock in he liquid regime Changes in Mean of he Log Jump Size The nex se of resuls addresses he sensiiviy o he jump size disribuion. For his we reurn o our baseline model and mainain he assumpion ha he uncondiional log jump size logj is normally disribued wih mean J and volailiy J. As we vary J or J, we change he values of U, M, and D so ha as before large up jumps correspond o a greaer han 3% jump size, moderae jumps beween 3% and 3%, and large down jumps o less han 3%. We mainain all oher assumpions. Noe ha as J goes down, he jumps end o be more negaively skewed and, in addiion, he possibiliy of a large down jump goes up while he possibiliy of a large up jump decreases. Thus, as we decrease J, in our baseline model, i becomes more likely ha he liquid regime shifs o he illiquid regime. Figure 7 shows how he opimal ransacion boundaries vary agains J when he expeced reurn remains he same a 7% in boh he liquid and he illiquid regimes. This figure reveals, similar o he findings in Liu e al. (2003), ha he opimal rading boundaries are nonmonoonic wih some asymmery across posiive and negaive jumps. I also implies ha an increase in he expeced jump size may increase he

12 Liu and Loewensein: Marke Crashes, Correlaed Illiquidiy, and Porfolio Choice 726 Managemen Science 59(3), pp , 2013 INFORMS Figure Opimal Trading Boundaries as a Funcion of J Uncondiional expeced log jump size J (1) (0) (0) (1) Noe. This figure shows how he opimal rading boundaries vary wih J for parameers J = 00666, 0 = 1 = 01190, 0 = 1 = 007, r = 001, = 004, = 5, 0 = 0, 1 = 09367, 0 = 1 = 0, 0 = 05%, 1 = 25%, R = 003, R = 003, U 0 = 1 D1 = 1 D 2 0 = 0, and 1 = U 1 opimal sockholding even if he expeced reurns are held consan and he oal reurn volailiy increases. Inuiively, when he expeced value of a jump is posiive, he jump helps he invesor by inroducing a posiive skew o reurns. However, he jump also increases reurn volailiy. Wheher he opimal sock invesmen increases or decreases depends on wheher he skewness effec or he volailiy effec dominaes. The asymmery occurs because downward jumps end o inroduce a negaive skew, which ends o bring he invesor closer o he solvency boundary and he associaed higher marginal uiliy. Figure 8 shows how he opimal rading boundaries vary as we change J as before, bu insead Figure 8 Opimal Trading Boundaries as a Funcion of J wih 1 = 20% we assume ha he afer-crash volailiy is 20% in he illiquid regime. Again we see he nonmonoonic ransacion boundaries, bu now hey are almos always separaed and he NT region is lower in he illiquid regime han in Figure 7, which is driven by he worsened invesmen opporuniy se in he illiquid regime. Ineresingly, in he liquid regime he NT region in Figure 8 is quie similar o ha in Figure 7 excep for large negaive jump sizes. This occurs because even hough he invesor knows he will hold less of he risky asse in he illiquid regime following a marke crash, he also knows ha a marke crash will already make him hold less of he risky asse even wihou any rading, and hus he required ransacion cos paymen may be small when he regime swiches. Therefore, i is less cosly o hold more of he risky asse in he liquid regime wih a larger expeced jump size Changes in Volailiy of he Log Jump Size Figure 9 shows how he opimal ransacion region varies wih he uncondiional log jump size volailiy J (wih he same expeced reurn in boh regimes). When J ges large, he ransacion boundaries generally go down because he increase in volailiy makes he sock less aracive How Oher Parameers Affec he Opimal Trading Boundaries and Hedging Demands We show how oher parameers affec he opimal rading boundaries in he op secion of Table 2. The baseline row corresponds o he baseline case and he oher rows correspond o a change in he saed parameer alone from he baseline case. Consisen wih Liu and Loewensein (2002), as he expeced invesmen horizon 1/ decreases, he invesor invess Figure 9 Opimal Trading Boundaries as a Funcion of J (0) (0) (1) (1) (0) (1) (1) 0.2 (0) Uncondiional expeced log jump size J Noe. This figure shows how he opimal rading boundaries vary wih J for parameers J = 00666, 0 = 01190, 0 = 1 = 007, r = 001, = 004, = 5, 0 = 0, 1 = 09367, 0 = 1 = 0, 0 = 05%, 1 = 25%, R = 003, R = 003, U 0 = 1 D1 = 1 D0 = 0, and 2 U1 = Uncondiional log jump size volailiy J Noe. This figure shows how he opimal rading boundaries vary wih J for parameers J = 00259, 0 = 1 = 01190, 0 = 1 = 007, r = 001, = 004, = 5, 0 = 0, 1 = 09367, 0 = 1 = 0, 0 = 05%, 1 = 25%, R = 003, R = 003, U 0 = 1 D1 = 1 D 2 0 = 0, and 1 = U 1

13 Liu and Loewensein: Marke Crashes, Correlaed Illiquidiy, and Porfolio Choice Managemen Science 59(3), pp , 2013 INFORMS 727 Table 2 Boundaries and Hedging Demands as Oher Parameer Values Change Wih TC Wihou TC Parameers Baseline = r = = = = = 01%, = 025% 1 = = Percenage hedging demand Baseline = r = = = = = 01%, = 025% Noe. TC, ransacion coss. less in he sock o reduce he impac of ransacion coss. As expeced, wih a decrease in ineres rae, risk aversion, volailiy, he correlaion beween he large downward jump and swiching ino he illiquid regime (0), or an increase in expeced reurn, sock invesmen increases. Wih a decrease in he correlaion beween he large upward jump and swiching ino he liquid regime (1), sock invesmen decreases. Similar o he effec of increasing he ransacion cos rae in he illiquid regime, he NT regions widen in boh regimes when he liquidiy in he liquid regime increases. The boom secion of Table 2 examines how hedging demands vary wih parameer values. I is difficul o come up wih a precise measure of hedging demands in a model wih ransacion coss and jumps. However, a reasonable way o measure hedging demands is o compare he opimal porfolio policy when changes in he invesmen opporuniy se are correlaed wih he jumps o ha when he correlaion is zero (i.e., 1 D0 = U 1 1 = 0 = 1 = 0). The boom par of Table 2 repors he percenage difference beween hese porfolios. As expeced, o hedge agains changes in he invesmen opporuniy se from regime shifs, he invesors increase (decrease) sock invesmen when regime swiching ino a beer (worse) invesmen opporuniy se is possible. I is ineresing o noe ha he magniude of he changes in he ransacion boundaries is much higher han hose in he no-ransacion-cos case. Figure Cerainy Equivalen Wealh Loss as a Funcion of Illiquid regime volailiy (1) Two years One year Noes. This figure shows how he cerainy equivalen wealh loss from using wrong esimaes as a fracion of he iniial wealh varies wih 1 for rue parameers J = 00259, J = 00666, 0 = 01190, 0 = 1 = 007, r = 001, = 004, = 5, 0 = 0, 1 = 09367, 0 = 1 = 0, 0 = 05%, 1 = 25%, R = 003, R = 003, U 0 = 1 D1 = 1 D0 = 0, 2 U 2 0 = U = 01003, U1 = 00370, 2 U 1 1 = 00633, M = 01620, and D0 = 1 D1 = 2 D = The wrong esimaes are D0 = D, and D0 = D Cerainy Equivalen Wealh Loss from Misesimaion of he Correlaion Beween Marke Crash and Marke Illiquidiy So far we have been focusing on he analysis of he opimal rading sraegies. Nex, as an example, we show he economic significance of correcly aking ino accoun he correlaion beween marke crash and marke illiquidiy. Specifically, suppose an invesor underesimaes he correlaion beween marke crashes and marke illiquidiy o be 0.1, alhough he rue correlaion is 1, and adops he opimal rading sraegy ha is based on he wrong esimae. We compue he cerainy equivalen wealh loss as a fracion of his iniial wealh from his misesimaion. Figure 10 plos his loss agains he volailiy in he illiquid regime for wo cases, one wih expeced illiquid regime duraion of one year and he oher wo years. Figure 10 shows ha misesimaion of he correlaion is cosly o he invesor. For example, he equivalen wealh loss can be as high as 2% for he one-year case and abou 36% for he wo-year case. This finding indicaes he economic imporance of correcly aking ino accoun he correlaion beween marke crashes and marke illiquidiy. 6. Conclusions In his paper, we develop a racable workhorse model of opimal porfolio choice wih marke crashes and correlaed changes in he invesmen opporuniy se (e.g., higher illiquidiy, greaer volailiy). Our analysis demonsraes ha he presence of hese risks can be an imporan facor in deermining an opimal