Non-parametric liquidity-adjusted VaR model: a stochastic programming approach

Transcription

1 Article Emmauel Fragière Profeor, Haute École de Getio de Geève, ad Lecturer, Uiverity of Bath Jace Godzio Profeor, School of Mathematic, Uiverity of Ediburgh liquidity-adjuted VaR model: a tochatic programmig approach il S. Tuchchmid Profeor, Haute École de Getio de Geève Qu Zhag School of Mathematic, Uiverity of Ediburgh Abtract Thi paper propoe a Stochatic Programmig (SP) approach for the calculatio of the liquidity-adjuted Value-at-Ri (LVaR). The model preeted i thi paper offer a alterative to Almgre ad Chri mea-variace approach (999 ad 000). I thi reearch, a twotage tochatic programmig model i developed with the itetio of derivig the optimal tradig trategie that repod dyamically to a give maret ituatio. The ample path approach i adopted for ceario geeratio. The ceario are thu repreeted by a collectio of imulated ample path rather tha the tree tructure uually employed i tochatic programmig. Coequetly, the SP LVaR preeted i thi paper ca be coidered a a o-parametric approach, which i i cotrat to Almgre ad Chri parametric olutio. Iitially, a et of umerical experimet idicate that the LVaR figure are quite imilar for both approache whe all the uderlyig fiacial aumptio are idetical. Followig thi aity chec, a ecod et of umerical experimet how how the radome of the differet type (i.e., bid ad a pread) ca be eaily icorporated ito the problem due to the tochatic programmig formulatio ad how optimal ad adaptive tradig trategie ca be derived through a two-tage tructure (i.e., a recoure problem). Hece, the reult preeted i thi paper allow for the itroductio of ew dimeioalitie ito the computatio of LVaR by icorporatig differet maret coditio. 09

2 liquidity-adjuted VaR model: a tochatic programmig approach Developed over the lat couple of decade, Value-at-Ri (VaR) model have bee widely ued a the mai maret ri maagemet tool i the fiacial world [Jorio (006)]. VaR etimate the lielihood of a portfolio lo caued by ormal maret movemet over a give period of time. However, VaR fail to tae ito coideratio the maret liquidity impact. It etimate i quite ofte baed o mid-price ad the aumptio that traactio do ot affect maret price. everthele, large tradig bloc might impact price, ad tradig activity i alway cotly. To overcome thee problem, ome reearcher have propoed the calculatio of liquidity adjuted VaR (LVaR) [Dowd (998)]. Differig from the covetioal VaR, LVaR tae both the ize of the iitial holdig poitio ad liquidity impact ito accout. The liquidity impact i commoly ubcategorized ito exogeou ad edogeou illiquidity factor. The former i ormally meaured by the bid-a pread, ad the latter i expreed a the price movemet caued by maret traactio [Bagia et al. (999)]. From thi perpective, LVaR ca be ee a a complemetary tool for ri maager who eed to etimate maret ri expoure ad are uwillig to diregard the liquidity impact. Bagia et al. (999) propoed a imple but practical olutio that i directly derived from the covetioal VaR model i which a illiquidity factor i expreed a the bid-a pread. Although thi approach avoid may complicated calculatio, it fail to tae ito coideratio edogeou illiquidity factor. Hece, liquidity ri ad LVaR are uderetimated. A more promiig olutio for LVaR etimatio tem from the derivatio of optimal tradig trategie a uggeted by Almgre ad Chri (999 ad 000). I their model, Almgre ad Chri adopted the permaet ad temporary maret impact mechaim from Holthaue et al. wor (987) ad aumed liear fuctio for both of them. By exterally ettig a ale completio period, they derived a optimal tradig trategy defied a the trategy with the miimum variace of traactio cot, or of hortfall, for a give level of expected traactio cot. Or iverely, a trategy that ha the lowet level of expected traactio cot for a give level of variace. With the ormal ditributio ad the mea ad variace of traactio cot, LVaR ca alo be determied ad miimized to derive optimal tradig trategie. I thi ettig, LVaR ca be udertood a the p th percetile poible lo that a tradig poitio ca ecouter whe liquidity effect are icorporated ito the ri meaure computatio. Later o, Almgre (003) exteded thi model by uig a cotiuou-time approximatio, ad alo itroduced a o-liear ad tochatic temporary maret impact fuctio. Aother alterative i the liquidity dicout approach preeted by Jarrow ad Subramaia (997 ad 00). Similar to Almgre ad Chri approach (999 ad 000), the liquidity dicout approach require that the ale completio period be give a a exogeou factor. The optimal tradig trategy i the derived by maximizig a ivetor expected utility of coumptio. ote that both approache require exterally ettig a fixed horizo for liquidatio. Aimig to overcome thi problem, Hiata ad Yamai (000) exteded Almgre ad Chri approach by aumig a cotat peed of ale ad by uig cotiuou approximatio. They could derive a cloed-form aalytical olutio for the optimal holdig period. I thi ettig, the ale completio time thu become a edogeou variable. Yet, Hiata ad Yamai model relie o the trog aumptio of a cotat peed of ale. Krohmal ad Uryaev (006) argued that the olutio offered by Almgre ad Chri ad that of Jarrow ad Subramaia were uable to dyamically repod to chage i maret coditio. Therefore, they uggeted a tochatic dyamic programmig method ad derived a optimal tradig trategy by maximizig the expected tream of cah flow. Uder their framewor, the optimal tradig trategy become highly dyamic a it ca repod to maret coditio at each time tep. Aother methodology that icorporate thee dyamic ito a optimal tradig trategy i that of Bertima ad Lo (998). They applied a dyamic programmig approach to the optimal liquidatio problem. Aalytical expreio of the dyamic optimal executio trategie are derived by miimizig the expected tradig cot over a fixed time horizo. I thi paper, we preet a ew framewor for the calculatio of o-parametric LVaR by uig tochatic programmig (SP) techique. Over the pat few year, tochatic programmig ha grow ito a mature methodology ued to approach deciio maig problem i ucertai cotext. The mai advatage of SP i it ability to better tacle optimizatio problem uder coditio of ucertaity over time. Due to the fat developmet of computig power, it ha bee ued to olve large cale optimizatio problem. Therefore, we believe it i a promiig methodology for LVaR modelig. The SP approach preeted i thi paper i exteded from Almgre ad Chri framewor (999 ad 000). The ample path approach i adopted for ceario geeratio, rather tha the ceario tree tructure uually employed i SP. The ceario et i repreeted by a collectio of imulated ample path. Differig from Almgre ad Chri parametric formulatio of LVaR, we preet a o-parametric formulatio for LVaR. Both exogeou ad edogeou illiquidity factor are tae ito accout. The former i meaured by the bid-a pread, ad the latter i expreed by liear maret impact fuctio, which are related to the quatity of ale. The model i thi paper i built i a dicrete-time maer, ad the holdig period i required to be determied exterally. The permaet ad temporary maret impact mechaim propoed by Holthaue et al. (987) i adopted to formulate the maret impact, ad both permaet ad temporary maret impact are aumed a liear fuctio. 0 The joural of fiacial traformatio Godzio ad Grothey (006) howed i their reearch that they could olve a qua- dratic fiacial plaig problem exceedig 0 9 deciio variable by applyig a tructure-exploitig parallel primal-dual iterior-poit olver.

3 liquidity-adjuted VaR model: a tochatic programmig approach Stochatic programmig LVaR model Thi paper propoe a SP approach to etimate the o-parametric LVaR, which i baed o Almgre ad Chri mea-variace approach (999 ad 000). While their model ha bee how to be a iteretig methodology for the calculatio of LVaR ad ha a huge potetial i practice, the optimal tradig trategie derived by their model fail to repod dyamically to the maret ituatio, a they rely o a cloed-form or tatic framewor. For itace, if a icreaig tred i oberved i the maret price, ivetor may decide to low their liquidatio proce. If, o the other had, uexpected maret hoc occur, ivetor may decide to adjut their tradig trategy ad to peed up the completio of their ale. Thee maret ituatio ca be imulated ad icorporated ito ceario. Clearly, ay cloed form olutio caot deal with thi type of ucertaity i uch a dyamic maer. The LVaR formulatio i Almgre ad Chri model i baed o the mea-variace framewor; thu, it ca be coidered a a parametric approach. I cotrat, the LVaR formulatio preeted i thi paper i oparametric ad allow for the icorporatio of variou dyamic i the liquidatio proce. Thu, we propoe a ew framewor for LVaR modelig. Almgre ad Chri mea-variace model Accordig to Almgre ad Chri framewor, a holdig period T i required to be et exterally. The, thi holdig period i divided ito iterval of equal legth (t = T/). The tradig trategy i defied a the quatity of hare old i each time iterval, which i deoted by a lit of,,,,, where i the umber of hare that the trader pla to ell i the th iterval. Accordigly, the quatity of hare that the trader pla to hold at time t = τ i deoted by x. Suppoe a trader ha a poitio X that eed to be liquidated before time T, the we have: = x, X = ad = x = X =, = 0,...,. j j j= j= + Price dyamic i Almgre ad Chri model are formulated a a arithmetic radom wal a follow: = + + µ () S S g where S i the equilibrium price after a ale, m ad are the drift ad volatility of the aet price, repectively, ad x i a radom umber that follow a tadard ormal ditributio (0, ). The lat term, g( /t), decribe the permaet maret impact from a ale. The actual ale price i calculated by ubtractig the temporary impact, h( /t), from the equilibrium price: S = S h () Accordig to Almgre ad Chri framewor (999 ad 000), both g( /t) ad h( /t) are aumed to be liear fuctio: g = (3) ad h = + (4) where g ad h are the permaet ad temporary maret impact coefficiet, repectively, ad e deote the bid-a pread. They are all aumed to be cotat. Baed o the previouly preeted equatio, the formula for the actual ale price i derived a: S = S + + µ 0 j j= j= I II III A you ca ee from thi formula, the actual ale price ca be decompoed ito three part. Part I i the price radom wal, which decribe the price dyamic without ay maret impact. Part II ad III are the price declie caued by the permaet ad temporary maret impact, repectively. The the total proceed ca be calculated by ummig the ale value over the etire holdig period: total proceed = S = XS0+ x + µ x ( X x) X = = = = = = XS0+ x + µ x X X = = = Coequetly, liquidatio cot (LC) 3 ca be derived by ubtractig the total actual ale proceed from the trader iitial holdig value, that i: LC = XS0 S = x µ x + X + X + (7). = = = = Almgre ad Chri derive the formulae for the mea ad variace of the liquidatio cot a: E LC X x X [ ] = µ + + ad [ ] V LC = = = (5) (6) (8) = x (9) Fially, they formulate the LVaR by uig the parametric approach with the mea ad variace of the LC a: [ ] [ ] LVaR = E LC + V LC (0) cl where cl deote the cofidece level for the LVaR etimatio, ad α cl i the correpodig percetile of the tadard ormal ditributio. A expreed, LVaR meaure a poible lo with a give poitio while taig ito coideratio both maret ri coditio ad liquidity effect. For the etimatio of temporary ad permaet maret impact coefficiet, Almgre ad Chri did ot propoe a pecific method. They aumed that: for the temporary maret impact, tradig each % of the daily volume icur price depreio of oe bida pread, ad for the permaet maret impact, tradig 0% of the daily volume will have a igificat impact o price, ad icur price depreio of oe bid-a pread. Sice thi paper focue o the LVaR modelig, ot the etimatio of maret impact coefficiet, Almgre ad Chri imple aumptio i adopted for all the umerical experimet i thi paper. 3 I Almgre ad Chri paper, thi cot i referred a the traactio cot. However, the traactio cot i commoly ow a the fee ivolved for participatig i the maret, uch a the commiio to the broer. Therefore, i order to avoid ay cofuio, it i amed liquidatio cot i thi paper.

4 liquidity-adjuted VaR model: a tochatic programmig approach The optimal tradig trategy could be derived by miimizig the LVaR. The mathematical programmig formulatio of thi optimizatio problem i thu writte a: mi [ ] + V[ LC] t. x = = 0,..., E LC X = j j= + = cl,..., 0. Baed o thi brief itroductio to Almgre ad Chri mea-variace approach, we ca thereo proceed to preet the SP approach to LVaR modelig. Stochatic programmig traformatio I tochatic programmig, ucertaity i modeled with ceario that are geerated by uig available iformatio to approximate future coditio. Before coductig the SP traformatio, we eed to briefly itroduce the ceario geeratio techique ued i thi paper. The liquidatio proce of ivetor poitio i a multi-period problem. The mot commoly ued techique i to model the evolutio of tochatic parameter with multiomial ceario tree, a how i Figure (a). However, the ue of ceario tree tructure ofte lead to coiderable computatioal difficulty, epecially whe dealig with large cale practical problem. I the ceario tree tructure, the ucertaitie are repreeted by the brache that are geerated from each ode. Icreaig the umber of brache per ode ca improve the quality of the approximatio of the ucertaity. However, it caue a expoetial growth i the umber of ode. Ideed, i order to approximate the future value of the ucertai parameter with a ufficiet degree of accuracy, the reultig ceario tree could be of a huge ize. It i commoly ow a the cure of dimeioality [Bellma (957)]. It i a igificat obtacle for dyamic or tochatic optimizatio problem. A alterative method to overcome thi problem i to imulate a collectio of ample path to reveal the future ucertaity a how i Figure (b). Each imulated path repreet a ceario. Thee ample path ca be geerated by uig Mote Carlo imulatio, hitorical imulatio, or boottrappig. There have bee everal iteretig paper regardig the applicatio of the ample path method i tochatic programmig [Hibii (000), Krohmal ad Uryaev (006)]. Uig ample path i advatageou becaue icreaig the umber of path to achieve a better approximatio caue the umber of ode to icreae liearly rather tha expoetially. Thi advatage i alo preet whe the time period i icreaed. The umber of ode icreae liearly with the ample path method ad expoetially with ceario tree tructure. Let =be a collectio of ample path {( C, 0 C,,,,,,,,, ),, C C C Sc}, = = where C, repreet the iformatio about relevat parameter. I Almgre ad Chri model (999 ad 000), we hould recall that the oly radome coidered i maret price. Hece, to et a poit of compario betwee their reult ad the reult from the SP approach, we firt aume the oly radome that i coidered i the ample path will come from the maret price compoet,. Yet, thi retrictive aumptio ca clearly be eaily relaxed, ad radome ca be added to other parameter, uch a the bid-a pread or the temporary ad permaet maret impact coefficiet 4. Uder the SP framewor, the tradig trategy i o loger a vector but a two dimeioal matrix trategy=,,,,,,, Sc Sc,, Sc firt tage ecod tage (a) ceario tree (b) ample path where, i the quatity of hare old i th iterval o path, i the idex of ceario, ad Sc i the umber of ceario. Thi i a two tage SP problem., ( =,...,Sc) are the firt tage variable, ad, ( =,, ad =,...,Sc) are the ecod tage variable. Due to the oaticipativity i the firt tage, the firt tage variable mut be loced: = = Sc.,,,... Time period Figure Sceario geeratio Time period For the actual ale price formulatio, recall Equatio (5). Taig ito accout the ceario ad replacig part I with, (i.e., the The joural of fiacial traformatio 4 A exteio i preeted below.

5 liquidity-adjuted VaR model: a tochatic programmig approach aet price without maret impact i th iterval for each ceario), the actual ale price i reformulated a: S = Sˆ,, j, j=, A we ow have the ale price formulatio, the total ale proceed correpodig to each ceario i aturally obtaied by ummig up the ale proceed over the etire et of iterval: ˆ total proceed = S = S =,,,,,, j, j = = = ˆ S X X,,, = =, (). () ario ad calculate the liquidatio cot for that pecific ceario. That i to ay, we ubtitute the optimal tradig trategy matrix ito the liquidatio cot formula (Equatio [3]), ad calculate LC, which i a vector idexed by. 3 Sort the vector LC, ad fid the value of the α th percetile LC, i.e., the α%-lvar. The mot commoly ued α value i 95 ad 99. umerical experimet I A previouly metioed, we firt coducted a aity chec. Thi ectio detail the umerical experimet for both the SP model ad the Almgre ad Chri mea-variace model with the retrictio of radome o oe compoet oly, i.e., the pure maret price. Coequetly, the liquidatio cot uder ceario i derived by ubtractig the correpodig total ale proceed from the trader iitial holdig value, that i: LC XS S XS X X S (3) ˆ = 0,, = 0 + +,, +, = = = The determiitic equivalet formulatio of thi SP problem with oaticipativity cotrait i: mi, Sc p = LC t. X = =,..., Sc, =,..., 0 =,..., Sc,, = =,... Sc,, where p i the probability of ceario. Sice the ceario are obtaied by the Mote Carlo imulatio, they are thu equally probable with p = / Sc. The reultig problem i a quadratic optimizatio oe. The objective, the expected value of the liquidatio cot, i a quadratic fuctio, ad all cotrait are liear. JP Morga toc data wa collected for the umerical experimet. The holdig period, T, wa et to be 5 day, ad we elected the time iterval to be 0.5 day. Thu, the total umber of ale,, wa 0. The electio of the holdig period ad time iterval wa arbitrary. For the price ample path geeratio, the Mote Carlo imulatio wa applied. The tochatic evolutio of the price wa aumed to follow a geometric Browia motio: Sˆ ˆ = S exp µ + (4) Uder Almgre ad Chri mea-variace framewor, maret price wa aumed to follow a arithmetic radom wal becaue it i ultimately rather difficult to derive a cloed-form olutio baed o a aumptio of geometric Browia motio. Yet, with Mote Carlo imulatio, formulatig the price evolutio uder differet aumptio create o iue related to the uderlyig ditributio that could geerate price ad retur. Sice the geometric radom wal i the mot commoly ued aumptio for price tochatic procee, it i ued i thi paper eve though differece betwee thee two radom wal are almot egligible over a hort period of time. LVaR formulatio Depedig o the et of aumptio, the calculatio methodology, ad their ue, two differet type of VaR uually exit, i.e., the parametric VaR ad the o-parametric VaR. The ame categorizatio obviouly applie to LVaR. The LVaR etimated by Almgre ad Chri (999 ad 000) i parametric a how i Equatio [0]. I thi paper, we have to rely o a o-parametric formulatio becaue it tem from the SP olutio that we have adopted. More preciely, the calculatio procedure i a follow: Solve the tochatic optimizatio problem tated above ad obtai the optimal tradig trategy matrix. Apply the optimal tradig trategie to the correpodig ce- 0,000 ample path were geerated by uig the Mote Carlo imulatio. The imulated price form a 0-by-0,000 matrix. The iitial price i Thee imulated ample path are diplayed i Figure. Five differet iitial holdig were choe for the umerical experimet with the aim of obervig how the iitial poitio affected the LVaR etimatio. The LVaR were calculated with the mot commoly ee cofidece level of 95% ad 99%. The reult are ummarized i Table. The umerical reult how that the computed by the SP model are lightly lower tha thoe computed by Almgre ad 3

6 liquidity-adjuted VaR model: a tochatic programmig approach Price Price ceario (imulated radom wal Chri model at both the 95% ad 99% cofidece level. Alo, a expected, the umerical reult how that the LVaR etimate icreae whe the iitial holdig icreae. Thi i true for both Almgre ad Chri model ad the SP model. A previouly tated, a the iitial holdig become larger, the maret impact become troger whe a trader liquidate hi poitio. Coequetly, the LVaR will icreae a well. Thi i clearly a characteritic that ditiguihe the LVaR from the traditioal VaR Figure Simulated price ceario Time (day) Iitial holdig (hare) % per hare ratio 99% per hare ratio LVaR 95% LVaR per hare LVaR 99% LVaR per hare Almgre ad Chri Mea-Variace Model.45E E+05.00E E E % 3.8%.68%.57%.47%.863E E E E E % 4.36% 3.68% 3.56% 3.46% Stochatic Programmig Model.90E E E E E %.86%.6%.03%.9%.8E E+05.34E E+04.49E % 4.% 3.56% 3.4% 3.3% Table umerical reult ummary *LVaR per hare = LVaR/Iitial holdig; = LVaR per hare/iitial price SP LVaR are lower tha Almgre ad Chri LVaR becaue the SP model optimal tradig trategie ca dyamically adapt to the maret ituatio. Thi fit ivetor actual tradig behavior i the maret, a they will adjut their tradig pla accordig to the maret eviromet. Therefore, the SP model ca provide more precie LVaR etimate due to the characteritic of the SP model adaptive tradig trategie. Geeralizatio of the tochatic programmig LVaR model A imple tochatic programmig LVaR model that wa traformed from Almgre ad Chri mea-variace model wa preeted above i order to compare with other model dicued (i.e., both two LVaR approache were ued uder the ame ettig). Let u ow exted the aalyi ad how ome of the advatage provided by the SP approach. Cotrary to Almgre ad Chri model that aume that both the bid-a pread ad maret impact coefficiet are cotat, we geeralize the SP LVaR model by relaxig thi aumptio ad treatig thee two compoet a radom variable. By icorporatig radome i the bid-a pread ad both the permaet ad temporary maret impact coefficiet, the formula of the actual ale price eed to be rewritte a: S = Sˆ,, j, j,, j=,, (5) For the formulatio of e,, we employ a tadardizatio proce. Sice bid-a pread ted to be proportioal to aet price, pat obervatio may ot accurately reflect the curret variatio. Bagia et al. (999) uggeted calculatig a relative bid-a pread that i equal to the bid-a pread divided by the mid-price. By employig thi calculatio, the bid-a pread i expreed a a proportio of the aet price; thu, the curret bid-a pread variatio i eitive to the curret aet price rather tha pat obervatio. The relative bid-a pread, a a ormalizig device, ca improve the accuracy of the bid-a pread variatio etimatio. The bid-a pread i thu formulated a: = S ˆ (6),,, where i the relative bid-a pread at time t, o path. Recall the ample path et 4 The joural of fiacial traformatio

7 liquidity-adjuted VaR model: a tochatic programmig approach {( C, 0 C,,,,,,,,, ),, C C C Sc}. = = By icorporatig the radome ito the relative bid-a pread ad maret impact coefficiet, C, i exteded to C ( S ˆ,,, ) =.,,,,, I other word, i the geeralized SP model, each ode o the imulated ample path cotai iformatio for the aet price, the relative bid-a pread, ad the permaet ad temporary maret impact coefficiet. A we ow have the ew ale price formulatio, the formula for liquidatio cot uder ceario (a how i Equatio [3]) i rewritte a: ˆ ˆ,, LC = XS0, S, = XS0 S,, S,,,,, j, (7) j = = = The determiitic equivalet formulatio ad the LVaR calculatio procedure are the ame a how above. By geeralizig the SP model, more parameter are icorporated i the ample path et, which lead to a more accurate approximatio of future ucertaitie. umerical experimet II Thi ectio report the umerical experimet for the geeralized SP LVaR model preeted above. We ue the ame dataet that wa ued for aforemetioed umerical experimet. The holdig period ad time iterval remai idetical, i.e., 5 day ad half a day, repectively. Iitial holdig (hare) LVaR 95% LVaR per hare LVaR 99% LVaR per hare Table umerical reult.084e E E E E %.5%.07%.98%.89%.6E E+05.3E E+04.43E % 3.9% 3.48% 3.40% 3.30% I Figure 3, we ca ee that the computed by the SP model with the icorporatio of radome ito the bid-a pread ad the maret impact coefficiet are lightly lower tha thoe computed by the SP model with the cotat bid-a pread ad maret impact coefficiet. Whe the iitial holdig i mall, icorporatig thee ew radom variable doe ot caue a igificat chage to the LVaR etimate. However, whe the iitial holdig i large, the differece are ubtatial. For itace, whe the iitial holdig i,000,000, icorporatig radome reduce the 95% from 3.4% to.87% ad the 99% from 4.80% to 4.30%. The mai reao for thee differece mut lie i the way the optimal tradig trategie that are derived by the SP model repod to the variatio of the bid-a pread ad maret impact coefficiet. For example, if we aume that the bid-a pread i cotat, the For the ample path geeratio of the relative bid-a pread, ad the permaet ad temporary maret impact, we aumed that they followed idepedet logormal ditributio ad imulated each of them imply a a white oie: = exp µ +,, = exp µ +,, = exp µ +,, (8) (9) (0) where m ad are the mea ad tadard deviatio, repectively, of the three radom variable (i.e., e, g ad h). Oce agai 0,000 ample path were geerated by uig the Mote Carlo imulatio for each parameter. The LVaR at the 95% ad 99% cofidece level were computed for the ame five iitial holdig ceario employed above. The reult are ummarized i Table. 5,0% 4,5% 4,0% 3,5% 3,0%,5%,0%,5% Iitial holdig (hare) 95% with icorporatio of the variatio of pread ad maret impact coefficiet 95% with cotat pread ad maret impact coefficiet 99% with icorporatio of the variatio of pread ad maret impact coefficiet 99% with cotat pread ad maret impact coefficiet Figure 3: compario 5

8 liquidity-adjuted VaR model: a tochatic programmig approach lo caued by the pread i the whole liquidatio proce i e X/ for each ceario a how i Equatio (3) (e i the mea value of the bid-a pread). With the icorporatio of radome ito the bid-a pread, the optimal tradig trategie are adjuted i accordace with it variatio. Whe the pread i high, the optimal tradig trategy may ugget ellig le. O the cotrary, whe it i low, the optimal tradig trategy may ugget ellig more. Therefore, the average lo caued by the pread ca be expected to be lower tha e X/. Stated otherwie, the SP model optimal tradig trategie ca tae advatage of chage by actig i a flexible ad timely maer. Alo ote that itroducig the calculatio of the relative bid-a pread ad the Mote Carlo imulatio itelf ca caue certai differece. However, the effect are preumably mall. Fially, it i worthwhile metioig that icorporatig radome ito the bid-a pread ad maret impact coefficiet withi the Almgre ad Chri model will defiitely elarge the reultig LVaR etimate. Ideed, it would add ew variace term to the variace of the liquidatio cot, ice the variatio of parameter are repreeted by their variace. Thi would lead to the icreae of the LVaR etimate. The SP olutio ad it umerical experimet idicate that if ucertaity i hadled well, it doe ot ecearily caue a icreae i the LVaR etimate. It highlight the tregth of the SP approach, which provide adaptive trategie (or recoure trategie ). Moreover, addig ew radom variable i the model doe ot icreae the difficulty of the problem due to the o-parametric ature of the SP LVaR. Referece Almgre, R. ad. Chri, 999, Value uder liquidatio, Ri,, 6-63 Almgre, R. ad. Chri, 000, Optimal executio of portfolio traactio, Joural of Ri, 3, 5-39 Almgre, R., 003, Optimal executio with oliear impact fuctio ad tradigehaced ri, Applied Mathematical Fiace, 0, -50 Bagia, A., F. Diebold, T. Schuerma ad J. Stroughair, 999, Liquidity o the outide, Ri,, Bertima, D., ad A. Lo, 998, Optimal cotrol of executio cot, Joural of Fiacial Maret,, -50 Bellma, R.E., 957, Dyamic programmig, Priceto Uiverity Pre, Priceto, ew Jerey Dowd K., 998, Beyod Value at Ri: the ew ciece of ri maagemet, Wiley ad So, ew Yor Godzio, J. ad A. Grothey, 006, Solvig oliear fiacial plaig problem with 09 deciio variable o maively parallel architecture, i Cotatio, M., ad C.A. Brebbia (ed.), Computatioal fiace ad it applicatio II, WIT traactio o modellig ad imulatio, 43, WIT Pre, Southampto Hibii,., 000, Multi-period tochatic programmig model for dyamic aet allocatio, Proceedig of the 3t ISCIE iteratioal ympoium o tochatic ytem theory ad it applicatio, 37-4 Hiata, Y. ad Y. Yamai, 000, Reearch toward the practical applicatio of liquidity ri evaluatio method, Moetary ad Ecoomic Studie, 83-8 Holthaue, R. W., R. W. Leftwich ad D. Mayer, 987, The effect of large bloc traactio o ecurity price: a cro-ectioal aalyi, Joural of Fiacial Ecoomic, 9, 987, Holthaue, R. W., R. W. Leftwich ad D. Mayer, 990, Large-bloc traactio, the peed of repoe, ad temporary ad permaet toc-price effect, Joural of Fiacial Ecoomic, 6, 990, 7-95 Jarrow, R. A., ad A. Subramaia, 997, Moppig up liquidity, Ri, 0, Jarrow, R. A., ad A. Subramaia, 00, The liquidity dicout, Mathematical Fiace, :4, Jorio P., 006, Value at Ri: the ew bechmar for maagig fiacial ri, 3rd Ed., McGraw-Hill, ew Yor Krohmal, P., ad S. Uryaev, 006, A ample-path approach to optimal poitio liquidatio, Aal of Operatio Reearch, Publihed Olie, ovember, -33 Cocluio Thi paper preet a tochatic programmig approach for LVaR modelig, which i exteded from Almgre ad Chri meavariace approach. I cotrat to their approach, the optimal tradig trategie are derived by miimizig the expected liquidatio cot. Thu, the SP trategie dyamically adapt to ew maret ituatio. Thi i the tregth of SP i the cotext of deciio maig uder ucertaity. Aother cotributio from thi paper i the o-parametric formulatio of the SP LVaR. It cotrat with the LVaR modelig methodologie that quite ofte rely o parametric approache. Overall, the umerical reult idicate that the two approache are ot idetical. Ideed, the LVaR computed uig the SP model i thi paper are lower tha thoe computed by Almgre ad Chri model. Yet, LVaR modelig till remai i it ifacy, epecially whe uig SP i thi cotext. 6 The joural of fiacial traformatio