Caroline HILLAIRET 1, Cody HYNDMAN 2, Ying JIAO 3 and Renjie WANG 2. Introduction

Transcription

1 ESAM: PROCEEDNGS AND SURVEYS, June 27, Vol. 56, p S. Crépey, M. Jeanblanc and A. Nikeghbali Ediors RADNG AGANS DSORDERLY LQUDAON OF A LARGE POSON UNDER ASYMMERC NFORMAON AND MARKE MPAC Caroline HLLARE, Cody HYNDMAN 2, Ying JAO 3 and Renjie WANG 2 Absrac. We consider rading agains a hedge fund or large rader ha mus liquidae a large posiion in a risky asse if he marke price of he asse crosses a cerain hreshold. Liquidaion occurs in a disorderly manner and negaively impacs he marke price of he asse. We consider he perspecive of small invesors whose rades do no induce marke impac and who possess differen levels of informaion abou he liquidaion rigger mechanism and he marke impac. We classify hese marke paricipans ino hree ypes: fully informed, parially informed and uninformed invesors. We consider he porfolio opimizaion problems and compare he opimal rading and wealh processes for he hree classes of invesors heoreically and by numerical illusraions. nroducion here is a large lieraure on insider rading, asymmeric informaion, and marke manipulaion rading sraegies including seminal works by 2,7,5,6,2. hese works generally assume ha an insider is aemping o influence a price by, or profi from, he release of poenially false informaion known o he insider. hese sudies also generally break marke paricipans ino noise raders, sandard informaional raders, and informed raders. he exisence of arbirage sraegies, price equilibrium, or specific marke manipulaion sraegies are he primary concerns of hese early works. Oher papers dealing wih insider informaion which quanify he value of insider informaion hrough he maximizaion of agens wealh or uiliy include 4, 5,,. More recenly liquidiy modeling has become an inense area of sudy. Marke micro-srucure and limi order books presen one approach o modelling liquidiy based on rading mechanisms. Models ha specify he price impac of rades as exogenously deermined and depending on he size of ransacions consiue anoher srand of he lieraure. Boh approaches rea problems associaed wih he fac ha rading large posiions impacs marke prices. A good overview of liquidiy models can be found in 4. he modeling of marke micro-srucure and he opimal liquidaion of large posiions has also been sudied exensively and an overview of hese opics can be found in. o he bes of our knowledge, among works dealing wih asymmeric informaion, only few papers concern he marke impac of liquidaion risk. n paricular, 6 sudies opimal liquidaion problems of an insider. n conras o he exising lieraure we are concerned wih disorderly, raher han opimal, liquidaion and he poin of view of marke paricipans oher han he large rader or hedge fund liquidaing he posiion. n paricular, we are ineresed in he following quesion: is i possible for a marke paricipan o profi from CRES, Ensae, Universié Paris Saclay, 3 av Pierre Larousse, Malakoff, France; caroline.hillaire@ensae.fr 2 Deparmen of Mahemaics and Saisics, Concordia Universiy, 455 Boulevard de Maisonneuve Oues, Monréal, QC H3G M8, Canada; cody.hyndman@concordia.ca & renjie.wang@concordia.ca 3 SFA, Universié Lyon, 5 avenue ony Garnier, 697 Lyon, France; ying.jiao@univ-lyon.fr c EDP Sciences, SMA 27 Aricle published online by EDP Sciences and available a hp:// or hps://doi.org/.5/proc/275642

2 ESAM: PROCEEDNGS AND SURVEYS 43 he knowledge ha anoher marke paricipan, wih large posiions in a sock or derivaive, will be forced o liquidae some or all of is posiion if he price crosses a cerain hreshold? here is ample evidence from financial markes concerning he imporance of liquidiy risks. For example, consider a hedge fund wih a large posiion in naural gas fuures conracs, such as Amaranh Advisors LLC in 26, and macro-economic or weaher evens conribue o an unexpeced adverse change in he price. n his case he fund may be forced o unwind is posiions in a disorderly fashion, which would have a furher marke impac on he price. Oher examples include he case of Long erm Capial Managemen L.P. LCM in 998 and numerous firms during he financial crisis of We assume ha liquidaion occurs immediaely when he marke price his he liquidaion rigger level and has a emporary impac on he asse price, by he marke price is depressed away from he fundamenal value, and gradually dissipaes. We model he emporary marke impac by a funcion wih parameers ha conrol he impac speed and magniude. Oher marke paricipans may have differen levels of informaion abou he liquidaion rigger mechanism and he liquidaion impac. We aim o find he opimal rading sraegy ha maximizes an invesor s erminal uiliy of wealh under differen ypes of informaion ha are accessible o paricular marke paricipans. n he sandard informaion case an uninformed marke paricipan is no aware of he liquidaion rigger mechanism. hey believe and ac, erroneously when liquidaion occurs, as if he marke price is equal o he fundamenal asse price. n he parial informaion case an insider or informed marke paricipan knows he level a which he hedge fund will be forced o liquidae he posiion bu does no have informaion abou he liquidaion volume which deermines he price impac. n he full informaion case he insider has complee informaion abou he liquidaion hreshold and he price impac. Cerain marke paricipans may have access o his ype of informaion owing o heir posiion, couner-pary saus, echnology, or knowledge of he marke. he fully informed invesor s perfec informaion represens one exreme which may be unobainable in pracice. However, we shall show numerically in he power-uiliy case ha he opimal sraegy for he parially informed invesor is quie close o ha of he fully informed invesor. he remainder of he paper is organized as follows. Secion ses up he framework of our model. Secion 2 solves he porfolio opimizaion problem for a fully informed invesor and gives he explici expression of he opimal expeced uiliy in case of log uiliy. Secions 3 and 4 explore he opimizaion problems for uninformed and parially informed invesors, respecively. Secion 5 presens some numerical resuls. Secion 6 concludes and an appendix conains echnical resuls and proofs... Asse price and liquidaion impac. he Marke Model Fix a probabiliy space Ω, A, P equipped wih a reference filraion F F saisfying he usual condiions, wih W, an F, P-Brownian moion. Le > be a finie horizon ime. n our model, we assume ha marke paricipans may inves in a riskless asse and a risky asse. Wihou loss of generaliy we suppose ha he ineres rae of he riskless asse is zero. he fundamenal value of he risky asse is modelled by a Black-Scholes diffusion: ds S d + dw,, and are supposed o be consans, and >. We consider a hedge fund which holds a large long posiion in he risky asse over he invesmen horizon,. n normal circumsances, his posiion could be held unil ime. However, according o risk managemen policies, exchange rules, or regulaory requiremens, he long posiion mus be liquidaed in cerain circumsances. n his paper, we assume ha he liquidaion will be riggered when he marke price of he risky asse passes below a pre-deermined level. Before liquidaion, he marke price, denoed by S M, is equal o he fundamenal value S. So he liquidaion ime is defined as he firs passage ime of a fixed consan hreshold αs α,, by he marke price process S M, i.e., : inf, S M αs inf, S αs 2

3 44 ESAM: PROCEEDNGS AND SURVEYS mpac funcion mpac funcion θ.5, K.5 θ., K Θ.5 Θ 2. K.2 K imeyear imeyear Figure. mpac funcion wih 2 parameers Figure 2. mpac funcion wih 4 parameers wih he convenion inf. We noe ha is an F-sopping ime. n he simples case he scenario described corresponds o a margin call ha canno be covered resuling in he liquidaion, in full or in par, of he posiion. he marke price of he risky asse will be influenced by liquidaion. Since he number of shares of he risky asse o be sold is very large in comparison o he average volume raded in a shor ime period, immediae liquidaion would have a emporary impac on he marke price which would be driven down away from he fundamenal price afer liquidaion. We denoe by S u he marke price of he risky asse a ime afer he liquidaion ime u. Suppose ha i is given as S u g u; Θ, KS, u, 3 g is an impac funcion and Θ and K are parameers which will be made precise laer. We noe ha he mahemaical characerizaion of marke impac is a very complicaed problem, and we refer he ineresed reader o 2 for deails. n his paper, inspired by 22, we characerize he emporary influence of liquidaion on marke by he impac funcion g of he form g; Θ, K K Θ e Θ 4 Θ and K are posiive parameers wih Θ conrolling he speed of he marke impac and K represening he magniude of he marke impac. n paricular, we assume ha Θ is a posiive random variable and K is a random variable valued in,, boh of which are independen of F and wih join probabiliy densiy funcion ϕ,, i.e. PΘ dθ, K dk ϕθ, kdθdk. Figure illusraes he impac funcion 4 wih K. and wo differen realized values of Θ. Clearly he shape of he impac funcion wih Θ.5 is seeper han wih Θ.. We noe ha for each fixed scenario ω, he funcion g aains is minimum value Kω a Θω. Also, we observe ha he funcion g firs declines from and hen rises back and converges o, which characerizes he marke impac of liquidaion wih ime evoluion. For realized values K. and Θ. i would ake. year, which is approximaely 25 rading days, for he asse price o reach he minimum value K S afer liquidaion occurs. he marke impac in he firs rading day afer liquidaion is g 25 ;.,. %. herefore, he parameer Θ needs o be small o more accuraely reflec he impac of disorderly liquidaion. n Secion 5 we presen some numerical resuls which use a raher large Θ ha guaranees beer accuracy of he numerical resuls, bu hese could be improved by applying oher numerical echniques for smaller values of Θ.

4 ESAM: PROCEEDNGS AND SURVEYS 45 Remark.. is naural o consider a jump effec for he price impac of liquidaion. n our model, by 3, he price before and jus afer liquidaion saisfies he relaion S S. However, we can approximae downward jumps of asse prices afer liquidaion by choosing small values of Θ in he smooh funcion g. Furher, our model allows us o consider he siuaion ha liquidaion by he large rader may have no long-erm informaional conen. he emporary impac on he marke price decays as liquidiy providers reurn o he marke and oher marke paricipans realize ha here may be no informaion abou he fundamenal value of he risky asse conveyed by he hedge fund s disorderly liquidaion. A possible exension is o consider a modified impac funcion wih addiional parameers and flexibiliy. For example, le g; Θ, Θ 2, K, K 2 K +K 2 Θ e Θ < Θ, K K2+Θ2 Θ Θ 2 +Θ2 Θ Θ e 2 Θ. he impac funcion given by 5 incorporaes boh permanen and emporary marke impacs wih K and K 2 conrolling he magniude of permanen and emporary marke impacs respecively. he parameers Θ and Θ 2 deermine boh he deviaion and reversal speed see Figure 2. Moreover, a long-erm ime scale, he impac funcion can come back o a differen level oher han. For simpliciy, we will use he impac funcion given by 4 in his paper and suppose he parameers Θ and K o be random variables. Considering he marke price of he asse o be equal o he fundamenal value before he liquidaion ime and o be he impaced asse price afer liquidaion, we have ha he marke price is given as 5 S M < S + S S and S are given by and 3 respecively. Moreover, for any u, he dynamics of he process S u saisfies he SDE ds u S u u, Θ, Kd + dw, u, Θ, K g u; Θ, K g u; Θ, K +. u Remark.2. he process S u, u is adaped wih respec o he filraion F Θ, K which is he iniial enlargemen of F by he random variables Θ, K. As we suppose Θ, K is independen of F, he F, P-Brownian moion W is also a F Θ, K, P-Brownian moion see e.g. 7, Secion 5.9. hus he marke price process of he risky asse, denoed as S M S M,, saisfies he SDE ds M S M M Θ, Kd + dw 6 M Θ, K < +, Θ, K. 7 We noe ha he marke price admis a regime change a he liquidaion ime, in paricular on he drif erm. We give an illusraive example as below. Example.3. Suppose ha he fundamenal value process is given by he Black-Scholes model wih parameers S M 8,.7,.2, α.9, Θ., K.. Figure 3 shows ha liquidaion riggers a downward jump of he drif erm. Aferward he drif erm firs rises quickly and hen declines gradually back o he original drif erm. Correspondingly, Figure 4 shows he sample marke price processes of he asse subjec o liquidaion impac compared wih he fundamenal value process.

5 46 ESAM: PROCEEDNGS AND SURVEYS asse price wihou marke impac asse marke price under marke impac liquidaion barrier -.2 drif erm before liquidaion drif erm afer liquidaion ime ime Figure 3. Drif Figure 4. Asse price S M.2. he opimal invesmen problem Our objecive is o consider he opimal invesmen problem from he perspecive of invesors who rade in he marke for he risky asse subjec o price impac from disorderly liquidaion of he hedge fund s posiion. For simpliciy we assume hese agens may rade in he marke for he risky asse wihou ransacion coss. We consider fully informed invesors, parially informed invesors and uninformed invesors. We suppose ha all invesors have access o he marke price of he risky asse S M bu heir knowledge of he liquidaion and price impac are differen. We furher assume ha all he invesors know he values of he parameers and. Fully informed invesors observe he marke price and are assumed o have complee knowledge of he mechanism of liquidaion and he price impac funcion. Hence hey know, in mahemaical erms, he liquidaion rigger level α, he impac funcion g, and he values of he random variables Θ and K when liquidaion occurs. herefore, fully informed invesors have complee knowledge of he dynamics of he marke price process, ogeher wih he informaion of he price impac. Parially informed invesors are also able o observe he marke price and know he liquidaion rigger level α, herefore, he liquidaion ime is also observable for hem. However, parially informed invesors do no have complee informaion abou he price impac funcion. We suppose he parially informed invesors know he funcional form of he price impac funcion g. However, we assume he parially informed invesors only know he disribuions of Θ and K bu no he realized value ha is necessary o have full knowledge of he price impac of liquidaion. Uninformed invesors are no aware of he liquidaion rigger mechanism. hey erroneously believe he marke price process follows he Black-Scholes dynamics wihou price impac. herefore, hey behave under incorrec assumpions, or a misspecificaion of he marke model, which leads hem o ac like he Meron invesor. Considering such uninformed invesors allows us o quanify he value of informaion abou he liquidaion barrier and price impac, compared o a Meron-ype invesor. We denoe by F S F S he naural filraion generaed by he marke price process S M. Since he marke price coincides wih he fundamenal value process S before liquidaion, he liquidaion ime, which is an F-sopping ime, is also an F S -sopping ime. We summarize he knowledge of he various invesors in he following assumpion. Assumpion.4. All invesors observe he marke price of he risky asse and know he values of he parameers and. n addiion cerain marke paricipans possess addiional informaion: i he observable informaion for fully informed invesors is modeled by he filraion G 2 F S Θ, K F Θ, K,

6 ESAM: PROCEEDNGS AND SURVEYS 47 hey furher know he liquidaion barrier α, as well as he form of he impac funcion g. ii he observable informaion for parially informed invesors is modeled by he filraion G F S, hey furher know he liquidaion barrier α, he form of he impac funcion g, and he disribuion of Θ, K. iii o compare wih he above wo ypes of insiders, we consider uninformed invesors who ac as Meron-ype invesors, erroneously considering Black-Scholes dynamics wih consan over he enire period,. hey have no informaion abou he liquidaion mechanism. Furher, hey do no updae heir knowledge of he drif process afer. Remark.5. he common informaion o hree ypes invesors are represened by he public filraion F S since he marke price of he risky asse is observable o all invesors. Assumpion.4 implies ha parially informed invesors know he he law of M Θ, K. his is similar o he weak informaion case of 8. he essenial differences among hese hree ypes of invesors lie in heir knowledge on he drif erm M Θ, K defined in 7. Fully informed invesors are able o compleely observe he drif erm. Parially informed invesors parially observe he drif erm, corresponding o he case of parial observaions considered by 9. Parially informed invesors may obain an esimae of he drif erm which is adaped o heir observaion process using filering heory. Uninformed invesors do no have any informaion abou he liquidaion mechanism and marke impac which causes hem o erroneously specify he drif erm as. ha is, uninformed invesors believe ha he marke prices follow he Black-Scholes dynamics. f he uninformed invesor reaed he drif of as an unobservable process he could perhaps apply filering heory o improve his invesmen decisions even wihou knowing anyhing abou he liquidaion mechanism or marke impac funcion. However, in his paper we shall only consider he case of Assumpion.4, ha is of uninformed invesors who esimae he drif a he beginning of he period and do no updae i, since from heir own view poin no liquidaion even happened during he period,. he uninformed invesors are mainly considered as a benchmark for comparison wih he Meron model. We shall sudy he porfolio opimizaion problem for hree ypes of invesors in he remainder of his paper under logarihmic and power uiliy. 2. Fully informed invesors Fully informed invesors choose heir rading sraegy o adjus he porfolio of asses according o heir informaion accessibiliy. As discussed in Secion fully informed invesors know he realized values of he random variables Θ and K. he invesmen sraegy is characerized by a G 2 -predicable process π 2 which represens he proporion of wealh invesed in he risky asse. he admissible sraegy se A 2 is a collecion of π 2 such ha, for any θ, k, +,, π 2 M θ, k d + π 2 2 d <. 8 he risk aversion of he invesors is modeled by classic uiliy funcions U defined on, ha are sricly increasing, sricly concave, wih coninuous derivaive U x on,, and saisfying lim U x + x + lim U x. x We define he G 2 -maringale measure Q by he likelihood process L : dq exp dp G 2 M v Θ, K M dw v v Θ, K dv. 9

7 48 ESAM: PROCEEDNGS AND SURVEYS As menioned in Remark.2, W is a G 2, P-Brownian moion. By Girsanov s heorem, he process W Q defined as W Q M v Θ, K W + dv is an G 2, Q-Brownian moion and he dynamics of he asse price S M under Q may be wrien as ds M S M dw Q. By aking a sraegy π 2 A 2, he wealh process wih iniial endowmen X G 2 evolves as dx 2 X 2 π 2 M Θ, Kd + dw, ha is X 2 X + π v 2 Sv M dwv Q. 2 he fully informed invesor s objecive is o maximize her expeced uiliy of erminal wealh or V 2 : sup E U π 2 A 2 X 2 V 2 Θ, K : ess sup E U π 2 Θ,K A 2 X 2 G 2 G 2 Θ, K. he link beween he opimizaion problems 3 and 4 is given by 4; if he supremum in 4 is aained by some sraegy in A 2, hen he ω-wise opimum is also a soluion o 3. As Θ, K is independen of F, a maringale represenaion heorem holds for G 2, Q-local maringale, hus we adop he sandard "maringale approach" see 8 o solve he uiliy opimizaion problem 4. We may consider he following saic opimizaion problem V X 2 X 2 sup X 2 V E U X 2 G 2 X + π2 v S M v dw Q v, π 2 A 2. he opimizaion problem 5 can be solved by using he mehod of Lagrange mulipliers see 4, Proposiion 4.5. he opimal erminal wealh is given by 2 ΛL, 6 U and he G 2 -measurable random variable Λ is deermined by E Q ΛL G 2 X. 7 n order o find he opimal sraegy ˆπ 2 one should provide he dynamics of he opimal wealh process E Q 2 G2. 8 his assumpion can be relaxed ino a densiy Jacod hypohesis, using hen he resul of 3, Proposiion 4.6 for a maringale represenaion heorem.

8 ESAM: PROCEEDNGS AND SURVEYS 49 Since X 2, is a G 2, Q-maringale, here exiss a G 2 -adaped process J such ha Subsiuing 7 ino 9 we have 2 E Q 2 2 G 2 X + + J v dwv Q. 9 Comparing 2 wih 2, we obain he opimal sraegy J v dw Q v. 2 ˆπ 2 J Noice ha he opimal sraegy ˆπ 2, involves he process J which is implicily deermined by he maringale represenaion as in 9. o obain an explici expression for he opimal sraegy, we will consider power and logarihmic uiliies in he following subsecions Power uiliy We firs consider he power uiliy Ux xp p, < p <. 6-7 we obain he opimal erminal wealh Using he fac ha x x/p and by 2 X E L p p G 2 L p 2 L is given by 9. he following proposiion gives hen he opimal expeced uiliy as well as he opimal sraegy: Proposiion 2.. For power uiliy Ux xp p, < p <, he opimal expeced uiliy is and he opimal sraegy is given by ˆπ 2 V 2 Θ, K X p E L p p p G 2 22 p < ˆπ 2,b + ˆπ 2,a,, 23 ˆπ 2,b p 2 + ZH,,, 24 H ˆπ 2,a wih H, Z H saisfying he following linear BSDE, Θ, K,, 25 p2 p M H + v Θ, K 2 2 p 2 2 H v + pm v Θ, K p ZH v dv Zv H dw v. 26

9 5 ESAM: PROCEEDNGS AND SURVEYS Proof. Following 9 we find he explici expression for he opimal sraegy, by compuing he dynamics of he opimal wealh process. Applying he absrac Bayes formula o 8, we obain Subsiuing 9 and 2 ino 27 we have Defining 2 X EL p p G 2 L X EL p p G 2 L X L p EL p 2 p G E H : Eexp he opimal wealh process wries as 2 E L p E exp exp E Q 2 G2 L E p G 2 2 L G p M v Θ, K p dw p M v Θ, K 2 v + 2 p 2 dv p M v Θ, K p dw v + p M v Θ, K 2 2 p 2 dv p M v Θ, K p dw p M v Θ, K 2 v + 2 p 2 dv G 2 G 2 G 2 2 X H L p H. 28 n order o find he dynamics of H,, we firs remark ha M : H D, is a G 2, P-maringale, p M v Θ, K D : exp p dw p M v Θ, K 2 v + 2 p 2 dv. 29 By he maringale represenaion heorem here exiss a G 2 -adaped process Z M such ha From equaion 29 M M + M v Z M v dw v. d p 2 M Θ, K 2 D D 2 p 2 2 pm Θ, K 2 2 p 2 d pm Θ, K p dw which leads o he following dynamics for he process H, dh H p 2 M Θ, K 2 2 p 2 2 pm Θ, K 2 2 p 2 pm Θ, K p ZM d + Z M pm Θ, K dw. p Denoing Z H : H Z M pm Θ,K p and using he erminal condiion H, H, saisfies he following BSDE p M H + v Θ, K 2 2 p 2 2 H v + pm v Θ, K p ZH v dv Zv H dw v. 3

10 ESAM: PROCEEDNGS AND SURVEYS 5 hus he dynamics of he opimal wealh process, using 28, are d 2 2 M Θ, K 2 p 2 + M Θ, KZ H H d + 2 M Θ, K p + ZH dw 3 H ha leads o he opimal sraegy by comparing 3 wih ˆπ 2 M Θ, K p 2 + ZH H. 32 We decompose he ime horizon, ino wo random ime inervals, and,. On he random inerval,, he fully informed invesor observes he drif erm M hus he BSDE 3 can be solved explicily on, : H exp p v, Θ, K 2 2 p 2 2 dv, 33 Z H. 34 Recalling 7 and using we may decompose he opimal sraegy in 32 ino wo pars: ˆπ 2 < ˆπ 2,b + ˆπ 2,a ˆπ 2,b p 2 + ZH,,, H ˆπ 2,a, Θ, K,,. p2 he opimal sraegy afer liquidaion in 25 is essenially a Meron-ype sraegy. he par before liquidaion in 24 is he sum of a Meron sraegy and an exra componen 2 which is deermined by he soluion of he BSDE 26. is hard o obain a closed-form soluion for he BSDE 26, however, we may solve he BSDE 26 numerically which will be discussed in Secion 5. We nex consider he case of logarihmic uiliy for he fully informed invesor Logarihmic uiliy n his secion we consider he logarihmic uiliy Ux lnx. Using he fac ha x x and by 6-7 we obain he opimal erminal wealh 2 X L 35 L is given by 9. he opimal expeced uiliy is V 2 Θ, K lnx E lnl. 2 his exra erm is called "hedging demand for parameer risk" by Björk e al. 9.

11 52 ESAM: PROCEEDNGS AND SURVEYS Applying he absrac Bayes formula o 8 and using 35, we obain whose dynamics is given by 9 as 2 E Q 2 G2 E 2 L L G 2 X L d 2 2 M Θ, K 2 2 d + M Θ, K dw 2 M Θ, K dw Q. 36 Comparing 36 wih we obain he opimal sraegy ˆπ 2 M Θ, K 2. Recalling 7 we may decompose he opimal sraegy ino wo pars ˆπ 2 < ˆπ 2,b + ˆπ 2,a ˆπ 2,b,,, 37 2 ˆπ 2,a, Θ, K 2,,. he opimal rading sraegy for he fully informed invesor is composed of wo Meron sraegies before and afer-liquidaion. Accordingly we decompose he opimal wealh process as 2 2 < 2,b + 2,a 2,b and 2,a saisfy he following SDEs d d 2,b 2,a 2,b ˆπ 2,b d + dw,,, 38 2,a ˆπ 2,a, Θ, Kd + dw,,. 39 hen we decompose he expeced uiliy of erminal wealh ino wo pars depending on if liquidaion occurs before or afer ime : V 2 2,b Θ, K E > ln G 2 + E 2,a ln G 2. 4 he wo condiional expecaions in 4 are calculaed in Lemma A. and A.2 respecively. Combining hose lemmas we obain he following resul.

12 ESAM: PROCEEDNGS AND SURVEYS 53 Proposiion 2.2. he opimal log expeced uiliy for fully informed invesors is V 2 Θ, K N + 2 exp 2xx 2y + exp y 2π 3 2 x N lnx + 2 exp 2π h 2, Θ, Kd 2y x2 dxdy h 2 ; θ, k : ln X v, θ, k dv. n he nex secion we consider he opimizaion problem for he parially informed invesors. 3. Parially informed invesors he porfolio sraegy for parially informed invesors is supposed o be G -adaped and denoed by π,. he wealh process evolves as π dx X π M Θ, Kd + dw,. 4 Similar o 8, he admissible sraegy se A is a collecion of π such ha, for any θ, k, +,, π M θ, k d + π 2 d <. he porfolio opimizaion problem for parially informed invesors is V sup E U X. 42 π A Noe ha he opimizaion problem 4-42 is he case of parial observaions since he drif erm in 4 is no G -adaped. Following 9 we firs reduce he opimizaion problem of parial observaion o he case of complee observaion. Recall ha he probabiliy measure Q is defined as dq dp G 2 wih he densiy process L given by 9 which is a G 2, P-maringale. We nex define he filered esimae of he drif M Θ, K, based on he observaion of he marke price, by M E M Θ, K G. We define he innovaions process W by L d W dw + M Θ, K M d,. 43

13 54 ESAM: PROCEEDNGS AND SURVEYS By 9, Lemma 4. we know W is a sandard G, P-Brownian moion. hen we may rewrie 6 as ds M S M M d + d W, and he wealh process X as dx X π M d + d W, wih iniial wealh x, +. Now he dynamics of he wealh process X is wihin he framework of a full observaion model since M is G -adaped. Similar o he case of fully informed invesors, he opimizaion problem 42 can be solved by he maringale approach. Proposiion 3.. ihe opimal erminal wealh of a parially informed invesors, wih uiliy funcion U and U is given by λ L. he Lagrange muliplier λ is deermined by he budge consrain E Q λ L x and L is he densiy of he risk neural probabiliy measure Q for he filraion G defined by L d Q dp G exp M v d W M 2 v v 2 2 dv wih M E M Θ, K G and he innovaion process W given by 43 is a G, P-Brownian moion. ii he filered drif esimae M can be compued as,, M M θ,k exp M v θ,k dw Q v M v θ,k dv ϕθ,kdθdk,,. exp M v θ,k dwv Q M v θ,k2 2 2 dv ϕθ,kdθdk Proof. i We define he process L EL G and begin by proving ha i equals he righ-hand side of 44. By rewriing L in 9 as L exp and noing ha he process /L saisfies he equaion M v Θ, K dwv Q M + v Θ, K dv, + L M v Θ, K we have by aking he condiional expecaion of 45 ha E Q G + E Q L M v Θ, K 44 dwv Q, 45 L v dwv Q G L v. 46

14 ESAM: PROCEEDNGS AND SURVEYS 55 By 23, heorem 5.4, we have E Q M v Θ, K dwv Q G L v E Q M v Θ, K Using he Bayes formula 7, Proposiion.7..5 and 9, we have respecively G dwv Q. 47 L v E Q L G EL G 48 and E Q M v Θ, K G dwv Q L v Subsiuing 48 and 49 ino 46 and 47 we obain EL G + which implies L M v exp dw v Q + Combining and 43 we have Subsiuing 5 ino 5 we find L exp dw Q E M v Θ, K G EL G dw v Q. 49 E M v Θ, K G EL G dw v Q d W + M M v dv. 5 d. 5 M v d W M 2 v v 2 2 dv, 52 which is a G, P-maringale, and we define he risk neural probabiliy measure Q by d Q dp G L. By he fac ha W is a G, P-Brownian moion and he Girsanov s heorem, he process W Q defined as W Q W M v + dv, is a G, Q-Brownian moion. Following he same procedure as in Secion 2 we find he opimal erminal wealh given by λ L, U and he Lagrange muliplier λ is deermined by E λ L L x.

15 56 ESAM: PROCEEDNGS AND SURVEYS ii Recall ha dp L dq G 2 exp is G 2, Q-maringale. By Bayes formula, we have M E M Θ, K G E Q M Θ, KL G E Q L G E Q E Q E Q M Θ, KL G 2 E Q L G M Θ, KL G E Q L G E Q M Θ, K exp E Q exp M v Θ, K dwv Q G M v Θ,K dwv Q M v Θ,K dwv Q M v Θ, K dv M v Θ,K2 2 2 M v Θ,K2 2 dv 2 dv G G Since he measure Q coincides wih P on G 2 Θ, K, he disribuion of Θ, K under Q is idenical o he one under P. Recall ha he Brownian moion W Q is independen of Θ, K we have M M θ, k exp exp M v θ,k dwv Q M v θ,k dwv Q M v θ,k2 2 2 M v θ,k2 2 dv 2. dv ϕθ, kdθdk. 53 ϕθ, kdθdk For < we have M due o he fac ha M. Following 2 here exis a maringale represenaion heorem wih respec o he G, Q-Brownian moion W Q. Similar o he case of fully informed invesors, he opimal sraegy π relies on he maringale represenaion heorem. For a general uiliy funcion, he opimal sraegy π does no have explici expression. n he nex subsecions, we will consider power and logarihmic uiliies. 3.. Power uiliy We firs consider he power uiliy Ux xp p, < p <. he opimal erminal wealh a is given by E x L L p p p L is given by equaion 44. he opimal expeced uiliy is V x p E L p p p. 54 p Similar o he case of fully informed invesors, we may decompose he opimal sraegy ˆπ ino wo pars: ˆπ < ˆπ,b + ˆπ,a.

16 ESAM: PROCEEDNGS AND SURVEYS 57 Following a similar procedure as in Secion 2. we obain H, Z H saisfies he linear BSDE ˆπ,b p 2 + Z H H,,, ˆπ,a,,, p2 p M 2 H + v 2 p 2 H 2 v + p M v p Z H v dv Z H v d W v. 55 We will discuss he numerical soluion of BSDE 55 in Secion Log uiliy n his secion we consider he logarihmic uiliy Ux lnx. he opimal erminal wealh a is given by he opimal expeced uiliy is he opimal invesmen process ˆπ is given by ˆπ x L. V lnx E ln L. < ˆπ,b + ˆπ,a ˆπ,b,,, 56 2 ˆπ,a,,. 2 We decompose he opimal wealh process ino before and afer liquidaion pars as <,b +,a,b and,a saisfy he following SDEs d d,b,a,b ˆπ,b d + dw,,, a d + d W,,.,a ˆπ,a hen we decompose he expeced uiliy of erminal wealh V ino wo pars depending on if liquidaion occurs before or afer ime : V E < ln,b + E ln,a. 57

17 58 ESAM: PROCEEDNGS AND SURVEYS Comparing 37 and 56, we know parially informed invesors holds he same opimal sraegy as he fully informed invesor before liquidaion. he opimal erminal wealh for parially and fully informed invesors are idenical if no liquidaion occurs before, ha is E < ln,b E < ln 2,b. hus he firs expecaion in 57 has been calculaed in Lemma A. and he oher expecaion is calculaed in Lemma A.3. Combining hose lemmas we obain he following resul. Proposiion 3.2. he opimal log expeced uiliy for he fully informed invesor is V N exp 2 N lnx + 2xx 2y + exp y 2π 3 2 x y x2 dxdy 2 exp 2π h d h : ln x M v 2 2 dv. We will consider he opimizaion problem for he uninformed invesors. 4. Uninformed invesors 2 he uninformed invesors erroneously believe he marke price of he asse follows a Black-Scholes dynamics wih consan. ha is, uninformed invesors ac as Meron invesors. o compare wih he fully informed and parially informed invesors, we shall consider boh he power uiliy and logarihmic uiliy in he following secions. 4.. Power Uiliy We firs consider he power uiliy, i.e. Ux xp p. he uninformed invesors adop he Meron sraegy ˆπ p However, he marke price process of he asse is given by 6. herefore, corresponding o he sub-opimal sraegy given by 58, he wealh process is wrien as <,b +,a b and,a are given by d d,b,a,b ˆπ d + dw,,,,a ˆπ, Θ, Kd + dw,,.

18 ESAM: PROCEEDNGS AND SURVEYS 59 We nex compue he expeced uiliy of final wealh EU using he invesmen sraegy given by 58. We decompose EU ino wo pars depending on wheher or no liquidaion occurs before ime E U E > U,b + E U he wo expecaions in 59 are compued in Lemma A.4 and A.5 respecively.,a. 59 Proposiion 4.. he expeced power uiliy of an uninformed invesor who follows he subopimal sraegy 58 is E U xp p 2 p exp 2 p 2 N + p 2 2 exp p 2 N + p 2 exp 2π l, θ, kϕθddθdk l, θ, k xp p exp p p 2 2 p p v, θ, k p 2 dv. We nex consider he same problem for he uniformed invesor under logarihmic uiliy Logarihmic Uiliy n case of logarihmic uiliy, uninformed invesors adop he Meron sraegy ˆπ 2. 6 We denoe by he wealh process for uninformed invesors as holding he sub-opimal sraegy ˆπ given by 6. Similar o he case of power uiliy we calculae he expecaion EU using he decomposiion Eln E > ln,b,a + E ln. 6 Comparing 37 and 6, we know uninformed invesors hold he same opimal sraegy as he fully informed invesors before liquidaion. he erminal wealh for uninformed and fully informed invesors are idenical if no liquidaion occurs before, ha is E < ln,b E < ln 2,b. hus he firs expecaion in 6 has been calculaed in Lemma A. and he oher expecaion is calculaed in Lemma A.6.

19 6 ESAM: PROCEEDNGS AND SURVEYS Proposiion 4.2. he expeced log uiliy of an uniformed invesor who follows he subopimal invesmen sraegy 6 is Eln N + 2 exp 2xx 2y + exp y 2π 3 2 x N lnx + 2 exp 2π h, θ, k : ln x We nex presen some numerical resuls. 5. Numerical resuls 2y x2 dxdy h, θ, kϕθ, kddθdk 2 v, θ, k 2 n his secion we illusrae numerical resuls of he opimizaion problem for he hree ypes of invesors. We se he parameers.7,.2 and he iniial value S 8. We le he invesmen horizon. he liquidaion rigger level is chosen as α.9. he sochasic processes are discreized using an Euler scheme wih M 25 seps and ime inervals of lengh 25. he number of simulaions is N 5. We suppose he disribuion of Θ, K is uniform on.5,.5.2,.8. he iniial wealh is assumed o be x 8. he power uiliy funcion is specified as Ux 2x Filered esimae of he drif he ime horizon, is discreized equally as < < < M. For m M we denoe by M m Θ, K he discreized approximaion of M Θ, K a ime m. For m M, we denoe by W m he incremen of he Brownian moion over he ime inerval m, m+. he approximaion of he incremen of he G 2, Q-Brownian moion is Wm Q W m + M m Θ,K. We approximae he filered drif esimae in 53 a ime m by ˆ M m M m θ, k exp exp i m i m M i θ,k 2 2 dv. M θ,k i W i + M Θ,K i M i θ,k 2 2 ϕθ, kdθdk 2 2 W i + M Θ,K i M θ,k i 2 ϕθ, kdθdk We use he Mone-Carlo mehod o esimae he inegral in 62. Suppose he number of simulaions is N. For n N, we denoe by θ n, k n he realized value of he random variable Θ, K in he nh simulaion. We esimae ˆ M m in 62 by he sample mean M m θ n, k n exp M θ n,k n i W i + M Θ,K i M θ n,k n i M n N i m m 2. exp M θ n,k n i W i + M Θ,K i M θ n,k n i 2 2 n N m m

20 ESAM: PROCEEDNGS AND SURVEYS 6.4 Filer esimae compared wih realized drif filer esimae of he drif realized drif ime Figure 5. Filer esimae of he drif compared wih he realized drif n Figure 5 we illusrae a sample filer esimae M compared wih he drif erm M Θ, K in a specific scenario he realized value of he liquidaion random variables are Θ, K.,.5. From Figure 5 we noe ha he filered esimae of he drif is very close o he realized drif. his resul suggess ha knowing he funcional form of he marke impac is more relevan han he acual realizaion of Θ, K Opimal sraegy for power uiliy n his secion we illusrae he opimal sraegies for fully and parially informed invesors in case of power uiliy by solving he relaed BSDE numerically. We skip he discussion of log uiliy since he opimal sraegies are simply he "myopic" Meron sraegy. n case of fully informed invesors, we approximae he BSDE 3 by he following discreized BSDE H m+ H M. H m p M m θ, k 2 2 p 2 H 2 m + pm m θ, k p Z H m + Z H m W m, m < M, 63 he BSDE can be solved using he following recursive scheme see 3 64 Z H m E H m+ W m G 2 m, 65 H m E H m+ G 2 m + pm m θ, k p pm m θ, k 2 2 p 2 2 Z H m. 66 We esimae he condiional expecaion in 65 and 66 by he Mone-Carlo regression approach proposed by 3. Noe ha he marke price process S M is no Markovian wih respec o G 2, P. We define he running minimum process S M infsv M v and noe ha he pair S M, S M is Markovian wih respec o G 2, P. Hence we may choose he regression basis funcions:, x, x 2, y, y 2 and xy. By he regression mehod of 3 he condiional expecaions in 65 and 66 can be esimaed by c + c 2 S M αs + c 3 S M αs 2 + c 4 S M αs + c 5 S M αs 2 + c 6 S M αs S M αs

21 62 ESAM: PROCEEDNGS AND SURVEYS Asse marke price over, asse marke price liquidaion barrier ime Opimal sraegy for fully and parially informed invesors over, Full informed invesor Parially informed invesor ime Figure 6. Approximaed opimal sraegy for fully and parially informed invesors over, for some coefficiens c i, i 6. We approximae he opimal sraegy for fully informed invesor ˆπ 2 by π 2,b as follows π 2 m p 2 + Z H m H, m M. m Following a similar procedure we may solve he relaed BSDE for parially informed invesors and obain he approximae opimal sraegy. Figure 6 illusraes he approximaed opimal sraegies for fully and parially invesors respecively corresponding o one sample pah of he risky asse price liquidaion occurs well before he erminal ime. n paricular for he pah of he asse price in Figure 6 liquidaion occurs a ime.54. Before liquidaion he wo sraegies are indisinguishable due o he scale. We plo he opimal sraegies before liquidaion in Figure 7 and noe ha here is some racking error before liquidaion. his difference may be due o he fac ha he before liquidaion sraegy of boh invesors conains a componen which depends on he soluion of a BSDE, which is accomplished backward in ime, and in paricular depends recursively on he filered drif esimae for he parially informed invesor. Hence, owing o racking error ypical o filering problems some errors may be propogaed o he before liquidaion sraegy hrough he numerical soluion procedure for he associaed BSDE. able presens he approximaed opimal sraegies for fully and parially invesors a imes before liquidaion corresponding o Figure 7. Figure 8 illusraes he approximaed opimal sraegies for fully and parially invesors respecively corresponding o a realized pah of he asse price ha does no induce liquidaion. n paricular, he opimal rading sraegies of he fully informed and parially informed invesors appear almos idenical. We also observe

22 ESAM: PROCEEDNGS AND SURVEYS Asse marke price before liquidaion ime Opimal sraegy for fully and parially informed invesors before liquidaion Fully informed invesors Parially informed invesors ime Figure 7. Approximaed opimal sraegy for fully and parially informed invesors before liquidaion m S M m π m π 2 m able. Approximaed opimal sraegies before liquidaion a general endancy for he opimal sraegies o decrease he posiion in he sock as is price moves oward he liquidaion barrier and increase he posiion in he sock as he price moves away from he liquidaion barrier. However, as he ime o he end of he invesmen horizon shorens and he probabiliy of liquidaion appears less likely he overall rend o increase he posiion in he sock, oward he level of he Meron sraegy, dominaes Opimal expeced uiliy n his subsecion we implemen he Mone-Carlo mehod o find he opimal expeced power and log uiliies. n case of uninformed invesors, since he "opimal" sraegy is simply he Meron sraegy, we may approximae he wealh process X direcly using he Euler scheme. For m M and n N, we denoe by X,n m he realized wealh for uninformed invesors a ime m in he n-h simulaion. he expeced uiliy EUX is approximaed by he sample mean V N n N UX,n M. he sandard error of he sample mean is SE N N n N UX,n M V 2.

23 64 ESAM: PROCEEDNGS AND SURVEYS asse marke price liquidaion barrier Asse marke price over, ime Opimal sraegy for fully and parially informed invesors over, Full informed invesor Parially informed invesor ime Figure 8. Approximaed opimal sraegy for fully and parially informed invesors wihou liquidaion he relaive sandard error of he sample mean is RSE SE / V. he 95% confidence inerval esimae of he sample mean is V.96 SE, V +.96 SE. his simulaion scheme also applies o he log uiliy for fully and parially informed invesors. However, in case of he power uiliy for fully and parially informed invesors we canno approximae he wealh process direcly since he opimal sraegies are no explicily deermined. Alhough we can firs approximae he opimal sraegies by solving he relaed BSDE, his would increase he size of simulaion error. nsead we simulae he likelihood process L in 9 and L in 52 since he opimal expeced power uiliies are funcionals of L and L given by 22 and 54 respecively. For insance, in case of power uiliy for fully informed invesors, we denoe he discreized realizaion of L in he n-h simulaion by L n m for m M and n N. he expecaion EL p p is esimaed by he sample mean ξ N n N Ln M p p. he sandard error of he sample mean is SE 2 N N n N L n M p p ξ 2. he relaive sandard error of he sample mean is RSE 2 SE 2 / ξ. he 95% confidence inerval esimae of he sample mean is ξ.96 SE 2, ξ +.96 SE 2. By 22 he opimal expeced uiliy for fully informed invesors is esimaed by V 2 xp p ξ p. he 95% confidence inerval esimae of opimal expeced uiliy is xp p ξ.96 SE 2 p x, p p ξ +.96 SE 2 p. A similar scheme can be applied o he case of power uiliy for parially informed invesors.

24 ESAM: PROCEEDNGS AND SURVEYS 65 Numerical evaluaion Expeced uiliies Relaive 95% esimaed Sample mean sandard error confidence inerval Fully informed , Parially informed , Uninformed , able 2. Numerical evaluaion of opimal power uiliies for hree ypes of invesors Numerical evaluaion Expeced uiliies Relaive 95% esimaed Sample Mean sandard error confidence inerval Fully informed , Parially informed , Uninformed , able 3. Numerical evaluaion of opimal log uiliies for hree ypes of invesors We presen he numerical resuls on he opimal expeced uiliies for he hree ypes of invesors in he able 2 and able 3 for power and log uiliies respecively. As should be expeced here exiss cerain gaps among he opimal expeced uiliies of differen ypes of invesors. We may inerpre hose gaps as he value of informaion asymmery. he resuls are more pronounced in he case of power uiliy han in he case of power uiliy. Neverheless, in boh cases here are saisically significan differences in opimal expeced wealh given ha he confidence inervals do no overlap. n he power uiliy case he opimal sraegy of he parially informed invesor is very close o ha of he fully informed invesor. However, he inabiliy o fully capure he poenial gains from rading agains liquidaion, owning o he need o esimae he drif and he racking error, leads o a significanly lower opimal expeced uiliy. 6. Conclusion n his paper, we characerize he marke impac of liquidaion by a funcion of cerain form. We consider he porfolio opimizaion problem for hree ypes of invesors wih differen level of informaion abou he liquidaion rigger mechanism and he marke impac. n case of logarihmic uiliy, we find he closed-form opimal sraegy for all hree ypes of invesors. n he case of power uiliy i is no as sraighforward o find he closed-form opimal sraegy for he parially informed invesors. Finally we presen some numerical resuls using Mone-Carlo simulaion mehod. hese resuls indicae ha here is significan value, in erms of opimal expeced uiliy, of increased informaion abou he opporuniy o rade opimally agains an invesor who may need o liquidae a large posiion in a disorderly fashion. here are several possible direcions for improving he model. We can use more realisic models of marke impac or he barrier ha may depend on marke, regulaory, or macro-economic variables. For parial insiders, he occupaion ime below he liquidaion hreshold is a random variable raher han a known consan as in case of full insiders. We plan o incorporae permanen price impac ino he liquidaion impac funcion generalizing he emporary price impac funcion. We shall explore he effec of differen liquidaion impac funcions on he opimal rading sraegies and uiliy of erminal wealh of uninformed, parially informed, and compleely informed invesors. Since cerain marke paricipans possess differen marke informaion i is naural o discuss he value of informaion in erms of porfolio uiliy. he resuls of fuure research can also inform financial and operaional risk managemen processes and regulaions for cerain agens and rading aciviies including shor-selling prohibiions, buying consrains, or derivaives marke paricipaion.

25 66 ESAM: PROCEEDNGS AND SURVEYS Acknowledgemens he auhors hank he nsiu de finance srucurée e des insrumens dérivés de Monréal FSD for funding his research. We are also graeful o he anonymous referee for helpful commens and suggesions. Caroline Hillaire acknowledges suppor from nvesissemens d Avenir ANR--DEX-3/Labex Ecodec/ANR-- LABX-47. Ying Jiao hanks BCMR Peking Universiy for hospialiy. References Frédéric Abergel, Jean-Philippe Bouchaud, hierry Foucaul, Charles-Alber Lehalle, and Mahieu Rosenbaum. Marke microsrucure: confroning many viewpoins. John Wiley & Sons, Franklin Allen and Douglas Gale. Sock-price manipulaion. Review of Financial Sudies, 53:53 529, Jürgen Amendinger. Maringale represenaion heorems for iniially enlarged filraions. Sochasic Processes and heir Applicaions, 89: 6, 2. 4 Jürgen Amendinger, Dirk Becherer, and Marin Schweizer. A moneary value for iniial informaion in porfolio opimizaion. Finance and Sochasics, 7:29 46, Jürgen Amendinger, Peer mkeller, and Marin Schweizer. Addiional logarihmic uiliy of an insider. Sochasic Processes and heir Applicaions, 752: , Sefan Ankirchner, Chrisophee Blanche-Scallie, and Anne Eyraud-Loisel. Opimal liquidaion wih direcional views and addiional informaion. Working Paper: hp://hal. archives-ouveres. fr/hal , Kerry Back. nsider rading in coninuous ime. Review of Financial Sudies, 53:387 49, Fabrice Baudoin. Modeling anicipaions on financial markes. n Paris-Princeon Lecures on Mahemaical Finance 22, volume 84 of Lecure Noes in Mah., pages Springer, omas Björk, Mark HA Davis, and Camilla Landén. Opimal invesmen under parial informaion. Mahemaical Mehods of Operaions Research, 72:37 399, 2. Rober J. Ellio, Helyee Geman, and Bob M. Korkie. Porfolio opimizaion and coningen claim pricing wih differenial informaion. Sochasics and Sochasics Repors, 63-4:85 23, 997. Rober J Ellio and Monique Jeanblanc. ncomplee markes wih jumps and informed agens. Mahemaical Mehods of Operaions Research, 53: , Masaoshi Fujisaki, Gopinah Kallianpur, and Hiroshi Kunia. Sochasic differenial equaions for he non linear filering problem. Osaka J. Mah, 59:9 4, Emmanuel Gobe, Jean-Philippe Lemor, and Xavier Warin. A regression-based mone carlo mehod o solve backward sochasic differenial equaions. Annals of Applied Probabiliy, 53: , Selim Gökay, Alexandre F Roch, and H Mee Soner. Liquidiy models in coninuous and discree ime. n G. Di Nunno and B. Øksendal, ediors, Advanced Mahemaical Mehods for Finance, pages Springer, 2. 5 Rober A Jarrow. Marke manipulaion, bubbles, corners, and shor squeezes. Journal of Financial and Quaniaive Analysis, 273:3 336, Rober A Jarrow. Derivaive securiy markes, marke manipulaion, and opion pricing heory. Journal of Financial and Quaniaive Analysis, 292:24 26, Monique Jeanblanc, Marc Yor, and Marc Chesney. Mahemaical mehods for financial markes. Springer, oannis Karazas and Seven E Shreve. Mehods of mahemaical finance. Springer, oannis Karazas and Xiaoliang Zhao. Bayesian adapive porfolio opimizaion. Preprin, Columbia Universiy, Rober Kissell and Moron Glanz. Opimal rading sraegies: quaniaive approaches for managing marke impac and rading risk. Amacom, Alber S Kyle. Coninuous aucions and insider rading. Economerica, 526:35 335, Jingya Li, Adam Mezler, and R. Mark Reesor. A coningen capial bond sudy: Shor-selling incenives near conversion o equiy. Working Paper, Rober Lipser and Alber N Shiryaev. Saisics of random processes:. general heory, volume 5. Springer, 23. A. Appendix n he appendix we provide echnical lemmas and proofs which allow us o easily jusify he main resuls.

26 ESAM: PROCEEDNGS AND SURVEYS 67 Lemma A.. E > ln N 2,b G 2 lnx exp 2 2 N + 2 2xx 2y + y exp 2π 3 2 x y x2 dxdy 2 N x 2π x e u2 2 du is he cumulaive disribuion funcion of a sandard normal random variable. Proof. By we have S S exp 2 + W. 67 Define B 2 + W and B infb v v. Recalling he definiion of in 2 we find Le κ 2. From 7 we know and PB dx, B dy x>y y< 2x 2y 2π 3 P > N On he oher hand, by 38 we know 2,b > B >. 68 expκx 2 κ2 2 2y x2 dxdy lnα exp 2 lnα N X exp ˆπ 2,b 2 ˆπ2,b 2 + ˆπ 2,b W X exp 2 + W X exp B Using 68 and 7 we compue E > ln 2,b G 2 E > lnx + B G P > lnx E ˆB > B 72 since Θ, K is independen of F and X is G 2 -measurable. Finally we apply 69 and 7 o 72 o obain he resul.

27 68 ESAM: PROCEEDNGS AND SURVEYS Lemma A.2. Proof. Le in 67 we have Using he fac S αs we find By 38 and 74 we compue Solving 39 we obain 2,a E ln G 2 exp 2π h 2, Θ, Kd h 2, θ, k : ln X v, θ, k dv. 73 2,b S S exp W. W X exp ˆπ 2,b 2 ˆπ2,b ˆπ b W 2 X exp X exp W ,a 2,b exp 2,b exp ˆπ v 2,a Θ, K v, Θ, K 2 ˆπ2,a v, Θ, K dv + v Θ, K 2 2 dv + ˆπ 2,a v Θ, KdW v v, Θ, K dw v. 76 Using 75 and 76 we compue E E E E ln E 2,a ln G 2 2,b + ln 2,b + v, Θ, K 2 ln X v, Θ, K dw v G dv + v, Θ, K dv + v, Θ, K v, Θ, K dv dw v, G 2 Recall ha Θ, K is independen o F and ha from 7, Sec ha he densiy of is P d exp 2π d. 77 G 2. G 2

28 ESAM: PROCEEDNGS AND SURVEYS 69 Using 77, and he definiion of he funcion h 2, θ, k in 73, we obain he resul. Lemma A.3. E ln,a α ln exp 2π h d h : ln x M v 2 2 dv. 78 Proof. Similar o he proof of Lemma A.2, we find he erminal wealh We compue E E E E,a ln E x exp,a,a M v dv ln x ln x ln x M v dv + M v dv + M v dv if liquidaion occurs before M v d W v M v d W v M v d W v. Using he densiy of given in 77 and he definiion of he funcion h in 78, we obain he resul. Lemma A.4. p E,b > p xp p 2 p exp 2 p 2 N + p 2 2 exp p 2 N + p 2. Proof. he proof is basically he same as ha of Lemma A. excep using he power uiliy funcion insead of log uiliy funcion. We compue p E,b > p p 2 p exp 2 p 2 xp y 2x 2y 2π 3 exp p 2 x 2 p 2 2 2y x2 2 dxdy. 79