Inrnaonal Confrnc on Appld Mahmac, Smulaon and Modllng (AMSM 6 Eulr-Maruyama Appromaon for Man-vrng gm Swchng CE Proc ung u* and Dan Wu Dparmn of Mahmac, Chna Jlang Unvry, Hangzhou, Chna * Corrpondng auhor dffrn pcfcaon uch a wchng n man, wchng n varanc, and wchng n boh man and varanc Ang and mmrmann [5] alo how ha rgm-wchng modl can capur h ylzd bhavor of many a rurn Abrac h man-rvrng conan lacy of varanc (CE proc wh rgm wchng on of h mo uccful connuou-m modl of h hor rm ra, volaly, and ohr fnancal quan Howvr, mo SDE wh Marovan wchng do no hav plc oluon h papr oban h Eulr-Maruyama approma oluon for man-rvrng gm Swchng CE proc and provd a dald proof of h convrgnc of h EM approma oluon o h ac oluon In h papr, w nvga h Eulr-Maruyama approma oluon of a ochac dffrnal quaon, whr w gnralz h man-rvrng CE proc by rplacng h conan paramr wh h corrpondng paramr modulad by a connuou-m, fn-a, Marov chan h papr oban h Eulr-Maruyama approma oluon for man-rvrng gm Swchng CE proc and provd a dald proof of h rong convrgnc of h EM approma oluon o h ac oluon Kyword-CE proc; man-rvrng; rgm wchng; Eulr-Maruyama; Lpchz condon I INODUCION h papr organzd a follow In Scon II, w dvlop a man-rvrng CE proc wh rgm wchng h Eulr-Maruyama approma oluon provdd n Scon III In Scon I, w provd a dald proof of h rong convrgnc of h EM approma oluon o h ac oluon Concluon gvn n Scon Opon prcng on of h mo mporan rarch fld n fnancal conomc from boh praccal and horcal pon of vw h wor of Blac and Schol [] and Mron [] lad h foundaon of h rarch fld and movad mporan rarch n opon prcng hory, mahmacal modl and compuaonal chnqu h Blac-ScholMron formula on of h mporan produc of conomc rarch of h la cnury and ha bn wdly adopd by radr, analy, nvor and ohr fnanc rarchr II W l (W, F, {F } ³, P b a compl probably pac Dp populary, h Blac-Schol-Mron formula no whou flaw I ha bn documnd n many ud n mprcal fnanc ha h omrc Brownan Moon (BM aumd n h Blac-Schol- Mron modl do no provd a ralc dcrpon for h bhavor of a prc dynamc On of ubu h CE modl, whch orgnally nroducd by Co [3] and Co and o [] Many mprcal ud hav bn conducd n h lraur o jufy h u of h CE modl, for nanc, MndozaArraga and Lny [5], ua, Da and Nun [6], Largunho, Da and Braumann [7], haoor, angman and Bhuruh [8] wh a flraon F ³ afyng h uual condon, upon whch all ochac proc ar dfnd L b a fn-a connuou-m Marov chan ang valu among dffrn a, whr h oal numbr of a condrd n h conomy Each a rprn a parcular rgm and labld by an ngr bwn and Hnc h a pac of gvn by Á : {,,,} whch can b ud o modl facor of h conomy Hr h Marov chan aumd o b obrvabl and rv a a proy for om ognou conomc facor uch a DP and oc prc ndc On mgh nrpr h a of a dffrn ag of a bun cycl By nrprng h a of h Marov chan a dffrn ag of a bun cycl, on could uppo ha 5 and ha a, a,, and a 5 rprn panon, pa,, and rcovry, rpcvly o oban h ranon probabl of h Marov chan, w nd o pcfy gnraor mar Q For ay Marovan rgm-wchng modl hav drawn a gnfcan amoun of anon n rcn yar du o hr ably o nclud h nflunc of macroconomc facor on ndvdual a prc dynamc[9-3] hr ar ubanal mprcal vdnc n uppor of h nc of rgm wchng ffc on oc mar rurn and dfaul probabl Ung h CSP oc mar rurn ovr h prod 99-989, Schallr and Nordn [] dmonra ha hr compllng vdnc of rgm wchng n US oc mar rurn and h vdnc for wchng robu o 6 h auhor - Publhd by Alan Pr MEAN-EEIN EIME SWICHIN CE POCESS 88
poon, w aum ha a conan gnraor Q ( q j gvn Clarly raghforward o nd h framwor o h ca of m varyng gnraor From Marov chan hory, h lmn n h mar Q afy: q j q ³ f ¹ j ; j q and q - q for ach,, j¹ j Aum ha h Marov chan ( a any m > n a rgm ÎÁ hn afr a prod of m D, h Marov chan Y +D may ay n rgm wh probably Y P P (, or jump o any ohr rgm j ÎÁ wh probably (, j, whr h on-p ranon probabl P (, j of h Marov chan ( ar gvn by q D ìï, j p (, j í ï q q D j ï( -, j ¹ -q ïî L W ( b a andard Brownan moon dfnd on h probably pac ( W, F, { F}, P W condr h followng ³ rgm-wchng man-rvrng CE proc b d a ( b - d + dw(, ³, ( ( ( ( wh nal valu Y y and Y ( rprn an undrlyng varabl (for ampl, h ochac nr ra or dfaul nny, a dno h pd of man rvron, b dno h long rm man of h varabl, and h volaly coffcn a, b and ar pov ral numb dpndn on h Marov chan Y (, ndcang ha hy can a dffrn valu n dffrn rgm Snc h undrlyng varabl Y ( manly ud o modl ochac volaly or nr ra or an a prc, crcal ha Y ( wll nvr bcom ngav Mao al [] dcu analycal propr whn b and how ha for gvn any nal daa Y y > and ÎÁ, h oluon Y ( of ( wll rman pov wh probably, namly Y ( > for all ³ almo urly, f on of h followng wo condon hold: < b ; b and ab for all ÎÁ A Lmma h coffcn of ( afy h local Lpchz condon for gvn nal valu Y y >,, for vry ngr >, hr a pov conan L uch ha for all ÎÁ, and ho, y wh Î [, ] and y Î [, ], and b b ab ( --ab ( - y L - y, - y L - y, ( and hu hr a unqu local oluon o quaon ( h followng horm rval h nc of h pov oluon B Lmma For any gvn nal valu Y y >, a, b and > for all ÎÁ, hr a unqu pov global oluon Y ( o Eq ( on ³ III HE EULE-MAUYAMA APPOIMAE SOLUION o dfn h Eulr-Maruyama approma oluon, w wll nd h followng lmma C Lmma 3 vn a p z D >, l D for,,, hn a dcr-m Marov chan wh h onp ranon probably mar q D ìï, j p (, j í ï q q D j ï( -, j ¹ -q ïî hn h dcr-m Marov chan can b mulad a follow: Compu h ranon-probably mar P (, j ; L and gnra a random numbr whch unformly drbud n [, ] L ¹ and dfn - - ìï, p (, j p (, j j j ï < í - ï, p (, j ïî j 3 nra ndpndnly a nw random numbr whch agan unformly drbud n [, ] and hn L ¹ and dfn - - ìï, p (, j p (, j j j ï < í - ï, p (, j ïî j pang h procdur, a rajcory of,,,, can b gnrad h procdur can b carrd ou ndpndnly o oban mor rajcor 89
Afr planng how o mula h dcr-m Marov chan, w can now dfn h EM approma oluon for ( vn a p z D > Compu h dcr appromaon y» Y D by ng y Y, and y y + a ( b - y D + y D W, (3 + b whr D W W - W L ( + D D y ( y, (, Î[ D,( + D,,,,, ( and dfn h connuou EM approma oluon by (5 ( ( ( ( b y y + a b - y d+ y dw (, No ha Y y D y D, ha, Y ( and y ( concd wh h dcr approma oluon a h grd pon I CONEENCE OF HE EM APPOIMAE SOLUION h followng horm dcrb h convrgnc n probably of h EM oluon o h ac oluon undr h local Lpchz condon A horm For Y ( n ( and y ( n (5, lm(up Y ( y ( nprobably D -, B Proof W dvd h whol proof no hr p C Sp For uffcnly larg, dfn h mlar oppng m nf{ ³ Y ³ } Applyng h gnralzd Io formula o a C (( y yld h + ( E ( Y ( ( y E L ( Y ( ] d funcon nf{ Y (, Y ³, ÎÁ } W can drv ha { } P [ ( y + ] (6 D Sp For uffcnly larg, dfn h mlar oppng m h nf{ ³ y ³ } Ung (5 and applyng h gnralzd Io formula o (( y dfnd n ( yld ( E (( y h ( y ( h h - -3 + E [ a ( b -y(( q y( - g y( ( ( b - 3 - + y ( (- q y ( + 6 g y ( ( ( ( j + q((] y d j Snc L ( y K[ + ( y], arrangng h rm on h rgh-hand d by plu-and-mnu chnqu w oban ha E (( y h ( h h ( y KE [ ( ] y d h - - KE [ q ( y ( y ( g ( y ( y ( ] d h - KE [( q q y ( g g y ] d ( ( h - - E a ( b y[ q ( y y - g - ( ( h - -3 E a ( b y[ ( q q y ( g g y ] d ( ( h b - - 3 3 - - ( [ (( ( 6 (( ( ] ( ( ( q g h b - 3 - E y( [ ( q - q ( y + 6( g -g ( y ] d ( ( ( ( h - - E q [ q ( y ( y ( g (( y y ( ] d j ( j j j h - E ( q q ( q y g y d ( j j j j j + + + - - + - - - - j ( y y ] d + E y - y - y + y -y d + - + - + By condon ( w hav (7 Ung h ronwall nqualy, w oban E ( Y( h [ ( y + ], ( for om pov numbr K L 9
h - - [ q ( (- ( + g ( ( - ( ] h - - E [ Q y - y + y - y ] d h E y y y y d C E y( - y( d C E y( h -y( h d C ( E y h y h d - Smlarly, w can g h - - E a ( b -y([ q (( y -y( - g (( y -y( ] d ( ( h - - ABE [ Q y( - y( - y( - y( ] d C E h y( - y( d ( h h C E y -y d W can how, n h am way a abov, ha h - [( q - q + ( g -g ] ( ( h - [ Q + ] { ¹ } N - + C E[ I ( ] d { ¹ } E y y d Snc E y y I d EI [ ] I P ( ¹ { ¹ } { } { } j j¹ (8 I ( q ( - + o( - (ma(-q D + o( D whr C ( a cnan dpndn on bu ndpndn of D Smlarly w can how ha E y - y C D, ( h ( h Subung h no (9 yld ha E (( y h ( h + + D + D + 3 ( y C ( o K E y d, By h ronwall nqualy, E ( y( h [ ( y + + C ( D + o( D ] ( h 3 E Sp 3 In h am way a (6 wa oband, w can hn how ha { } P h [ ( y + + C ( ( D + o ( D ] 3 Now, l dî, (, b arbrarly mall S W compu W - ³ { w : up y y d} P( W P( WÇ { ³ } + P( < P( WÇ { ³ } + P( q < + P( r < C ( o [ ( y ] D + D + + d + C ( D + o( D 3 Subung h no (8 gv callng ha uffcnly larg for a, w can choo h - [( q - q + ( g -g ] ( ( ( ( E y y d C ( (ma( -q D + o( D W can mlarly ma h ohr rm on h rgh-hand d of (7 o g ha and hn choo [ ( y + ] D uffcnly mall for E ( h ( h h (( y h ( y + + E ( y d + C ( (ma(-q D + o( D, h + C E y -y d (9 C D + D + D + D, o oban ( o ( C3 ( ( o ( d 9
P W P w y - y ³ d { : up } h prov h aron (5 CONCLUSION In h papr w oban h Eulr-Maruyama approma oluon for man-rvrng gm Swchng CE proc and provd a dald proof of h rong convrgnc of h EM approma oluon o h ac oluon ACKNOWLEDMEN h auhor would l o han h Youh Projc of Naonal Socal Scnc Foundaon (NoCL, h Youh Projc n Human and Socal Scnc arch of h Mnry of Educaon of Chna (NoYJC79, and h Naonal Naural Scnc Foundaon of Chna (No7733 for fnancal uppor EFEENCES [] F Blac, M Schol h prcng of opon and corpora labl, Journal of Polcal Economy, vol 8,973, pp 637 659 [] Mron hory of raonal opon prcng, Bll Journal of Economc and Managmn Scnc, vol,973, pp 83 [3] J Co, No on Opon Prcng I: Conan Elacy of aranc Dffuon, Sanford unvry, Worng papr, 975 [] J Co, S o h valuaon of opon for alrnav ochac proc, Journal of Fnancal Economc, vol 3, 976, pp 5 66 [5] Mndoza-Arraga, Lny Prcng quy dfaul wap undr h jump-o-dfaul ndd CE modl Fnanc and Sochac, vol 5,, pp 53-5 [6] ua J P, Da J C, Nun J P Prcng and ac hdgng of Amrcan-yl opon undr h jump o dfaul ndd CE modl Journal of Banng & Fnanc, vol 37, 3, pp 59-7 [7] Largunho M, Da J C, Braumann C A On h compuaon of opon prc and r undr h CE modl Quanav Fnanc, vol 3, 3, pp 97-97 [8] haoor N, angman D Y, Bhuruh M A nw fourh-ordr numrcal chm for opon prcng undr h CE modl Appld Mahmac Lr, vol 6, 3, pp 6-6 [9] Ang A, Bar gm wch n nr ra Journal of Bun and Economc Sac, vol,, pp 63-8 [] Boyl P, Dravam, Prcng oc opon undr rgm wchng Inuranc: Mahmac and Economc, vol, 7, pp 67-8 [] Su, K, Erlwn, C, Mamon, S h prcng of crd dfaul wap undr a marov-modulad mron rucural modl Norh Amrcan Acuaral Journal,vol, 8, pp 8-6 [] Yao, D D, Zhang, Q, Zhou, Y A rgm-wchng modl for Europan opon In Houmn Yan, Yn, and Zhang, Q (Ed, Sochac proc, opmzaon, and conrol hory: applcaon n fnancal ngnrng, quung nwor, and manufacurng ym, 6, pp 8-3 Brln: Sprngr [3] Maalaou Chun, O, Donn,, Fran_co, P Crd prad chang whn wchng rgm Journal of Fnancal and Quanav Analy,, forhcomng [] Schallr H, Nordn S gm wchng n oc mar rurn Appld Fnancal Economc, vol 7, 997, pp 77-9 [5] Ang A, mmrmann A gm chang and fnancal mar Annual vw of Fnancal Economc, vol,, pp 33-337 9