Valuing Credit Derivatives Using Gaussian Quadrature : A Stochastic Volatility Framework

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1 alung Cr rvav Ung Gauan uaraur : A Sochac olaly Framwork Nabl AHANI * Fr vron: Jun h vron: January 3 Conac normaon Nabl ahan HC Monréal parmn o Fnanc Oc Chmn la Cô-San-Cahrn Monral ubc Canaa H3 A7 Phon : Fax : mal : nabl.ahan@hc.ca

2 alung Cr rvav Ung Gauan uaraur : A Sochac olaly Framwork Abrac h papr propo m-clo-orm oluon o valu rvav on man-rvrng a. W conr a vry gnral man-rvrng proc or h unrlyng a an wo ochac volaly proc: h Squar-Roo proc an h Ornn-Uhlnbck proc. For boh mol w rv m-clo-orm oluon or Characrc Funcon n whch w n o olv mpl Ornary rnal quaon an hn nvr hm o rcovr h cumulav probabl ung h Gauan-Lagurr quaraur rul. A bnchmark w u our mol o valu uropan Call opon whn Black- Schol 973 rprn conan volaly an no man-rvron Longa-Schwarz 995 rprn conan volaly an man-rvron Hon 993 an Zhu rprn ochac volaly an no man-rvron ramwork. h comparon how ha w only n polynomal wh mall gr or convrgnc an accuracy. In whn appl o our proc rprn ochac volaly an man-rvron h Gauan-Lagurr rul vry cn an vry accura. W alo how ha h manrvron coul hav a larg mpac on opon prc vn hough h rngh o h rvron mall. A applcaon w valu cr pra opon cap loor an wap. ywor : Man-rvron Sochac olaly Gauan uaraur Invr Fourr ranorm Fynman-ac horm Cr Spra Opon Cap Floor Swap. JL Clacaon : G3 C63

3 Som a uch a nr ra cr pra an om commo ar hown o xhb man-rvron aur. Many papr prc rvav on h a unr conan volaly aumpon. Cox Ingroll an Ro 985 propo a gnral qulbrum mol an rv a quar-roo nr ra mol an h coun bon prcng ormula a wll a h bon opon prc. Longa an Schwarz 995 how ha h log-cr pra ar man-rvrng. Aumng ha hy coul b moll wh a Gauan proc hy rv a mpl clo-orm oluon or uropan opon. Schwarz 997 propo a manrvrng proc or commo n orr o prc om rvav. Bu all h mol hav h common pon ha h volaly aum non-ochac whch a rong mplcaon ha om om mprcal aur lk lpokurc rbuon an la o unralc opon prc. In h way ubanal progr ha bn ma n vlopng mor ralc opon prcng mol by ncorporang ochac volaly an ump. Hon 993 prc uropan opon on ock bon an currnc unr a quar-roo volaly proc. In h am way Bakh Cao an Chn 997 combn ochac volaly an ump o h mprcal prormanc o om alrnav opon prcng mol. Schöbl an Zhu 998 an Zhu propo a mor lgan mho o rv opon prc unr om volaly mol uch a h quar-roo an h Ornn-Uhlnbck. Whl all h papr propo mpl an ay-o-u clo-orm oluon o non-man-rvrng a rvav unr ochac volaly aumpon hr a ll lraur or manrvrng a whn ochac volaly ramwork. Aumng a quar-roo volaly an ack 977 nr ra proc Fong an ack 99 vlop h unamnal paral rnal quaon or nr ra conngn clam bu rv a clo-orm oluon only or coun bon. h oluon

4 rqur a havy compuaon o h conlun hyprgomrc uncon whn h complx numbr algbra. o ovrr h culy Slby an Srcklan 995 propo a r oluon or h coun bon prc ha vry cn. In anohr papr 997 hy alo vlop a Mon Carlo valuaon o ohr nr ra rvav unr h Fong an ack 99 mol. In a cr-m ramwork ahan propo a clo-orm valuaon ormula or cr pra opon unr GARCH a a gnralzaon o Longa an Schwarz 995 an Hon an Nan mol. H alo how ha h GARCH u ha h quar-roo man-rvrng proc on o h wo mol u n h currn papr a a connuou-m lm. In h papr w propo o rv prcng ormula or opon on man-rvrng a whn wo ochac volaly ramwork h quar-roo an h Ornn- Uhlnbck proc. In h w gnralz h Longa an Schwarz 995 conan volaly mol an h work on by Hon 993 an Zhu by ncorporang a man-rvrng componn. Our work alo xn h Fong an ack 99 mol nc w propo a m-analyc valuaon ramwork or om rvav on gnral man-rvrng a na o ung Mon Carlo mulaon a on by Slby an Srcklan 997. Mon Carlo mho may n a larg numbr o pah mulaon whch mak m-analyc valuaon whn pobl much mor cn. For boh ochac volaly mol conr n h papr w rv m-cloorm characrc uncon n whch w only n o olv mpl ornary rnal quaon O. In h quar-roo ca vn hough w rv compl clo-orm oluon nvolv h Whakr uncon ha ar rla o h conlun hyprgomrc uncon w how ha a numrcal roluon o h O prov u wh a accura valu a h xac on bu n much l m. h u o h ac ha whn on al 3

5 wh complx uncon ha coul only b compu approxmaly a a r xpanon lk Whakr an h conlun hyprgomrc uncon vn wh om mahmacal owar uch a Mapl or Mahmaca on ha o ac larg m compuaon an uually ovrlow rror. Onc h characrc uncon rv n a m-clo-orm way w u h nvr Fourr ranorm chnqu o g h aoca cumulav probabl by a numrcal ngraon ba on h Gau-Lagurr quaraur rul. h Gauan ngraon chnqu wa prov vry cn an accura n many papr among whch Ba 996 who prc currncy opon whn a ochac volaly an ump ramwork an Sullvan who propo an approxmaon o Amrcan Pu opon. In our ramwork ung om bnchmark uch a h Black an Schol 973 h Longa an Schwarz 995 h Hon 993 an h Zhu mol h Gau- Lagurr quaraur rul hown o b vry accura an convrgn o h ru prc vn wh mall polynomal gr. A applcaon w valu cr pra opon cap loor an wap. h conrbuon o h work wool. W propo a m-analyc procur o prc rvav on vry gnral man-rvrng unrlyng a unr ochac volaly aumpon. W alo how ha a mall man-rvron cocn.g. or cr pra o orr. coul hav a larg mpac on opon prc up o % - 4%. h nx con prn a gnral man-rvrng ramwork an how o compu h characrc uncon. Scon II rv m-clo-orm oluon or characrc uncon unr boh h quar-roo an h Ornn-Uhlnbck volaly aumpon. Scon III prn h numrcal ngraon procur ung h Gauan quaraur rul 4

6 o rcovr h cumulav probabl. Scon I valu om cr pra rvav an hr Grk a parcular applcaon. Scon prn om rul on convrgnc an cncy. Scon I wll conclu. 5

7 I Gnral man-rvrng ramwork an characrc uncon W conr a mor gnral mol or h unrlyng a gvn unr h horcal maur P by : b Z µ whr coul b nong a log-ock prc or a log-currncy or any gnral man-rvrng proc lk log-cr pra a n Longa an Schwarz 995 an ahan. For h volaly w conr h ollowng gnral uon : a θ a b' Z κ whr a b an b ar ral-valu uncon o h quar volaly an wll b pc lar. h paramr µ κ an θ ar conan. Z an Z ar corrla Brownan moon unr h horcal maur P. W aum ha h volaly rk-prmum proporonal o a a n Hon 993 an ha h rk-prmum or h unrlyng a proporonal o b uch ha h wo uon unr a rk-nural maur bcom : [ b ] b W γ µ 3 a [ κ π a ] b' W κθ 4 whr γ an π no h un rk-prmum an W an W ar corrla Brownan moon unr. o valu opon-lk rvav w mu compu h ollowng yp o xpcaon unr a m : xp r > ln 5 an xp r > ln 6 6

8 In orr o oban mplr xpron or h xpcaon w conr wo probably maur an quvaln o an n by hr Raon-Nkoym rvav : g xp r xp r 7 g xp r xp r 8 mply h o-call -orwar maur. quaon 5 an 7 gv : r > xp r > ln ln 9 an quaon 6 an 8 gv : xp r ln > ln P > whr P no h zro-coupon bon maurng a. W alo n h characrc uncon o h proc unr an by : [ xp ] or xpr unr h rk-nural maur h characrc uncon bcom : [ g xp ] xp r xp xp r 7

9 an bcom : [ g xp ] xp r xp xp r 3 h xpron wll b rv pnnly on h rk-r ra pccaon. Howvr w n an acualz characrc uncon by : xp r xp 4 w can mply quaon an 3 a : 5 6 Unr our ochac volaly an man-rvrng mol w wll how ha h characrc uncon can b xpr a log-lnar combnaon o om uncon ha olv mpl O. o rcovr h cumulav probabl n quaon 9 an w apply h Fourr nvron horm nall an Suar 977 o oban : ln π 7 > R ln π 8 > R 8

10 9 whr R no h ral par o a complx numbr. h ngral ar wll-n Appnx C an convrgn. Alhough hy canno b compu analycally w wll u numrcal chnqu uch a Gauan quaraur o o. h nx con wll conr wo rn ochac volaly mol by choong appropra a an b uncon an rv hr characrc uncon. II Sochac volaly mol II. Squar-roo man-rvrng mol In h ubcon w gnralz Hon 993 mol by ncorporang a manrvrng unrlyng a. h mol gvn unr h rk-nural maur by : W γ µ 9 W λ κθ whr W W. For all mol w aum a conan rk-r ra no by r. h characrc uncon can b xpr by or al Appnx A. : xp xp ε ε κθ µ whr ε γ λ ε xp xp

11 Pung la obvouly o Zhu quaon. h xpcaon rm wll b compu ung Fynman-ac horm a gvn n araza an Shrv 99 Appnx. I w n h uncon F a : F xp{ ε } xp ε 3 hn by Fynman-ac horm w hav ha F mu ay h ollowng paral rnal quaon P : F F F xp ε κθ λ F ε F 4 An w aum ha F log-lnar an gvn by : [ C ] F xp 5 hn an C ay h ollowng O : ' λ ε ε 6 an C ' κθ C 7 Solvng h O wll gv h unqu oluon o h P n quaon 4 an hn o h acualz characrc uncon. Alhough h O 6 o Rcca yp w on hav mpl analyc oluon lk n Hon 993 an Zhu. Bcau o h

12 man-rvrng aur h uncon ε o xponnal yp whl npnn o h m varabl n Zhu. Howvr h r-gr O can b olv aly ung numrcal mho uch a Rung-ua ormula or Aam-Bahorh-Moulon mho. For al abou h mho orman an Prnc 98 Shampn 994 or Shampn an Goron 975. h O 6-7 hav analyc oluon gvn n Appnx A. ha nvolv h Whakr uncon. hy n much mor m abou m on avrag o b compu or larg valu o or quvalnly a n n quaon 4-6 han olv numrcally or h am orr o accuracy h rlav rror abou -8. h may b u o h ac ha h conlun hyprgomrc uncon approxma by a r xpanon n all mahmacal owar an h calculaon coul b low or larg npu valu on coul alo ac ovrlow rror a wll. h r xpanon or h Fong an ack 99 coun bon prc n Slby an Srcklan 995 cn bcau alway qual o. h acualz characrc uncon or h quar-roo man-rvrng mol hn gvn by : xp r xp r xp xp µ [ C ] κθ 8 Now ha w can valua numrcally xpron lk an or all pobl valu o w ar abl o compu cumulav probabl unr maur

13 an by nvrng h Fourr ranorm whch la o valua h ngral gvn n quaon 7 an 8. h wll b on by h Gauan quaraur rul ung Lagurr polynomal. Scon III wll crb h mho an how how appl o our mol. II. Ornn-Uhlnbck man-rvrng mol A. Rk-prmum or h unrlyng a proporonal o h quar volaly In h ubcon w u an Ornn-Uhlnbck mol or h volaly an a man-rvrng unrlyng a. h mol gvn unr h rk-nural maur by : µ γ W 9 κθ λ W 3 whr W W. W can wr h characrc uncon a or al Appnx B : xp xp η3 µ η η 3 whr λ η γ xp κθ η xp η 3 xp 3 Agan pung la o Zhu quaon. n h uncon G a :

14 3 G 3 xp η η η 33 By Fynman-ac horm w hav ha v G mu ay h ollowng P : 3 xp η η η λ κθ G G G G G 34 Aum ha G log-lnar an gvn by : xp C G 35 hn an C ay h ollowng O : η η λ 3 ' 36 ' η κθ λ 37 an ' C C κθ 38 Alhough h Rcca-yp O 36 ha an xac analyc oluon Appnx B. h O o no hav clo-orm oluon. A cu arlr all h O wll

15 4 b olv numrcally. h acualz characrc uncon or h Ornn- Uhlnbck man-rvrng mol hn gvn by : xp xp xp xp µ C r r 39 B. Rk-prmum or h unrlyng a proporonal o h volaly I h rk-prmum or h unrlyng a proporonal o h volaly na o quar h mol quaon bcom : W γ µ 4 W λ κθ 4 an or h characrc uncon w oban wh h am calculaon a bor : 3 xp xp ω ω ω µ 4 whr

16 ω λ xp κθ ω γ xp ω 3 xp 43 Ung Fynman-ac horm a bor gv or h acualz characrc uncon : r xp r xp µ xp xp C 44 whr uncon an C olv h am O n quaon by rplacng η η an η 3 rpcvly by ω ω an ω 3. III Numrcal ngraon ung Gauan quaraur In gnral a quaraur rul approxmaon allow o ma an ngral o a uncon g ovr a gvn nrval wh a lnar combnaon o uncon valu n h nrval [ b]... n a. Ar pcyng a o abca... n ω h ngral approxma by : b n w g a g an hr corrponng wgh ω 45 whr w a wgh uncon o b pc pnnly on h rul whch u. h abca an h wgh ar pc uch ha h approxmaon xac or any gvn 5

17 polynomal uncon wh a maxmum gr. h hgh gr call h orr o h quaraur rul. Whl rul uch a rapzoal an Smpon pcy a o qually pac abca an choo h wgh o maxmz h orr Gauan rul rmn boh abca an wgh o maxmz h orr. For n abca an n wgh h hgh orr n. Furhrmor n many u Gauan rul ar hown o convrg ar han h clac rapzoal an Smpon rul an gv grar accuracy vn or mall n Sullvan. h Gau-Lagurr quaraur rul ovr h nrval [ [ wgh uncon : w xp whl h abca an h wgh olv h ollowng n quaon : ha h ollowng q q q q ω xp ω xp... ω xp xp n n n 46 or q n. h abca an wgh can alo b rmn ung om propr o Lagurr polynomal. hy ar abula n Abramowz an Sgun 968. h nx ubcon gv a br ovrvw o h polynomal an how how o pcy h rul. h on ar wll apply h quaraur rul o nvr our mol characrc uncon o rcovr cumulav probabl. III. A br ovrvw o Lagurr polynomal h n-h Lagurr polynomal n by : n n n [ xp ]... n L n xp 47 n n! n!! 6

18 hy hav many characrc among whch h orhonormaly wh rpc o h wgh uncon : n p xp Ln L p δ np 48 n p whr δ np h ronckr ymbol. W can alo proo ha h n-h Lagurr polynomal ha xacly n ral zro ovr h nrval ] [. h zro ar h abca... n n or h Gau-Lagurr quaraur rul o orr n. h aoca wgh... n ω ar hn gvn by : ω... n 49 n [ L ] n In orr o apply h ngraon mho o our mol w n o moy h wgh o ak no accoun h uncon o b ngra. Rcall ha w mu valu h yp o ngral : g xp n [ xp g ] ω xp g 5 Whl n ncra h um convrg o h ru valu h uncon xp g a om aumpon a cu n av an Rabnowz 984. III. Rcovrng cumulav probabl For a x orr n w up h abca... n xp... n an h mo wgh ω a hown bor. W alo choo all h mol paramr a wll a 7

19 h m h maury h rk h nal valu or h unrlyng a an pnng on h mol h nal valu or h volaly or h quar volaly. For vry abca or quvalnly a n arlr by or by w olv h O n quaon 6-7 or h quar-roo mol an quaon or h Ornn-Uhlnbck mol o g h uncon valu an C n n con II or vry... n. W can hn compu h acualz characrc uncon valu a h n pon an h cumulav probabl can hn b approxma by : > ln π π n R ω xp R xp { ln } 5 an > ln π π n R ω xp R xp { ln } 5 horcally a n bcom larg h approxmaon convrg o h ru probably valu. A wll b hown wh many valuaon xampl a a convrgnc an a goo accuracy can b achv vn wh mall n. 8

20 I Cr pra opon cap loor an wap valuaon I. Cr pra opon A cr pra opon gv h rgh o buy or ll h cr a h rk prc unl or a h xpraon a pnnly h opon Amrcan or uropan. On coul buy a cr pra opon or hgng cr rk xpour agan up or own movmn n a cr valu a wll a or pculav purpo. For an xhauv cr rvav ovrvw Howar 995. Mor pccally nong h maury a by an h rk by unr h mol u n arlr con h uropan Call prmum gvn by : > ln > ln Call 53 whr h acualz characrc uncon an h cumulav probabl ar n a bor by : an xp r xp 54 ln π 55 > R ln π 56 > R h uropan Pu can b rv ung h Call-Pu pary : < ln < ln Pu 57 9

21 In orr o hg opon agan chang n h unrlyng a an n h volaly w n o rv h Grk. h calculu al ar gvn n Appnx. For boh ochac volaly mol h la gvn by : ln Call la > 58 an h Gamma by : R π la la Gamma 59 h rvaon o h ga pn on h ochac volaly mol u. For h quarroo mol h ga gvn by : > R R ln π π Call ga 6 an or h Ornn-Uhlnbck mol by :

22 > R R ln π π Call ga 6 I. Cr pra Cap an Floor A cr pra cap or loor prov h rgh o g payo a proc a call h r a. A ach r a h cap/loor payo h am a or a call/pu. In h h cap/loor can b n a a qunc o many call/pu call capl or loorl. h Fgur blow how h r a an h aoca payo or a cr pra cap wh a maury an rn rk prc corrponng o h n pro. h cap/loor prmum hn qual o h um o h corrponng capl/loorl prma. h cap prmum gvn by : > > n n Cap ln ln 6 { } xp { } xp Cap n { } n n xp

23 whr > ln an ln > ar n or ach r a a or h Call n quaon h loor prmum can alo b valu by : Floor n n < ln < ln 63 a wll a h la by : Cap n > ln 64 I.3 Cr pra Swap A cr pra wap an oblgaon o g payo a proc a call h r a. A ach r a h wap payo h am a or a orwar conrac. In h h wap can b n a a qunc o many orwar conrac call wapl. h Fgur blow how h r a an h aoca payo or a cr pra wap wh a maury an a rk prc. xp{ } { } xp xp{ n } Swap n h wap valu hn qual o h rnc bwn h cap an h loor valu wh h am rk prc a h r a. By analogy wh nr ra wap w can rv h valu o h rk prc whch mak h wap valu a h bgnnng qual o :

24 n n 65 whr ar n or ach r a a n quaon 54. 3

25 mprcal rul on convrgnc an h mpac o man-rvron o a h accuracy an h cncy o our procur w prc many uropan call opon whn rn ramwork Black an Schol 973 B&S hrar Longa an Schwarz 995 L&S hrar an Zhu h Hon 993 mol on o h mol u by Zhu. For rn paramr w ry o convrg o h xac ru prc. h ru prc or B&S an L&S ramwork ar gvn by h corrponng mpl ormula whl or Zhu ramwork w u a Malab roun or numrcal ngraon. h abl o 4 prn h rul or rn maur rn rk an rn mol paramr. All h applcaon how ha a goo accuracy achv vn wh mall quaraur rul orr bwn an 5 pnng on whch mol u an whn whch ramwork. W alo n ha h rlav prcng rror ar vry mall an convrg o. h Fgur o 4 prn h rlav rror. Noc ha or Zhu ramwork h quar-roo mol convrg ar han h Ornn-Uhlnbck mol. h cncy an h accuracy o our m-analyc procur whn h xac ramwork ar ll ru or our man-rvrng ramwork. In or boh h quar-roo an h Ornn-Uhlnbck mol h ru aympoc prc h aympoc prc coul b compu wh a Mon Carlo mulaon aan or mall quaraur rul orr bwn an 5. h abl 5 an 6 prn h convrgnc an h Fgur 5 an 6 prn h rlav rror wh rpc o h aympoc prc. Agan h convrgnc ar or h quar-roo mol han or h Ornn-Uhlnbck mol. W alo how ha h man-rvron coul hav a larg mpac on opon prc an ha ru vn hough h rngh o h rvron mall. h rul prn n abl 4

26 7 an 8 how ha wh a mall man-rvron cocn or cr pra abou. h rlav mpac wh rpc o h no man-rvron opon prc.. bwn % an 4% pnng on h maury. 5

27 I Concluon In h papr w propo m-analyc prcng ormula or rvav on manrvrng a whn wo ochac volaly ramwork. In h w gnralz Longa an Schwarz 995 by makng h volaly ochac an Hon 993 an Zhu mol by ncorporang a man-rvrng componn n h unrlyng a uon. Our work alo xn h Fong an ack 99 mol nc w valu opon on gnral man-rvrng unrlyng a m-analycally. Howvr ang a man-rvrng aur n our mol only allow gng a mclo-orm characrc uncon n h n ha w propo o olv om O wh numrcal mho lk Rung-ua ormula or Aam-Bahorh-Moulon mho. In our work h numrcal roluon vry accura an ak much l m han h xac compuaon nc analyc oluon whn hy x nvolv complx algbra wh Whakr an h conlun hyprgomrc uncon. h u o numrcal ngraon mho lk h Gauan-Lagurr quaraur rul o nvr h characrc uncon ncary o rcovr cumulav probabl an hn o prc rvav. Wh om prcng applcaon whn rn ramwork a Black an Schol 973 Longa an Schwarz 995 Hon 993 an Zhu hown ha a goo accuracy achv vn wh mall quaraur rul orr an ha h rlav prcng rror ar vry mall an convrgn o. h rul alo apply o our man-rvrng ramwork h prc convrg o aympoc prc an h rlav rror ar mall an n o. A parcular applcaon o our gnral valuaon mol w rv m-cloorm prcng ormula or cr-pra uropan opon cap loor an wap. W alo how an nrng aur o rvav prc on man-rvrng a. W n ha h 6

28 mpac o mall rvron cocn on h prc coul b vry larg. h nng prov ha h prcng o rvav on man-rvrng unrlyng a vry nv o h rngh o h rvron an ha ha o b akn no accoun. h combnaon o numrcal roluon o O wh numrcal ngraon ung Gauan-Lagurr quaraur rul prov xrmly accura valuaon o cr rvav an may o wll or rvav on ohr man-rvrng unrlyng a lk nr ra an commo. 7

29 Rrnc Abramowz M. an I. Sgun 968. Hanbook o Mahmacal Funcon 5 h on ovr Publcaon Inc. Nw York. Ba. 996 Jump an Sochac olaly: xchang Ra Proc Implc n uch Mark Opon h Rvw o Fnancal Su ol Bakh G. C. Cao an Z. Chn 997 mprcal Prormanc o Alrnav Opon Prcng Mol h Journal o Fnanc ol Black F. an M. Schol 973 h aluaon o Opon an Corpora Labl Journal o Polcal conomy ol Clwlow L. J. an C. R. Srcklan 997 Mon Carlo aluaon o Inr Ra rvav unr Sochac olaly h Journal o Fx Incom ol Cox J. J. Ingroll an S. Ro 985 A hory o h rm Srucur o Inr Ra conomrca ol av P. an P. Rabnowz 984. Mho o Numrcal Ingraon n Acamc Pr Inc. Calorna. on orman J. R. an P. J. Prnc 98 A amly o mb Rung-ua ormula Journal o Compuaon an Appl Mahmac ol uan J.C. 995 h GARCH Opon Prcng Mol Mahmacal Fnanc ol Fllr W An Inroucon o Probably hory an I Applcaon ol Wly & Son Nw York. Fong H.G. an O. A. ack 99 Inr Ra olaly a a Sochac Facor Workng papr Gor Aoca. Hon S.L. 993 A Clo-Form Soluon or Opon wh Sochac olaly wh Applcaon o Bon an Currncy Opon h Rvw o Fnancal Su ol Hon S.L. an S. Nan A Clo-Form GARCH Opon aluaon Mol h Rvw o Fnancal Su ol

30 Howar. 995 An Inroucon o Cr rvav rvav uarrly ol nall M. an A. Suar 977. h Avanc hory o Sac ol Macmllan Publhng Co. Inc. Nw York. araza I. an S.A. Shrv 99. Brownan Moon an Sochac Calculu Sprngr rlag Nw York. Longa F.A. an.s. Schwarz 995 alung Cr rvav h Journal o Fx Incom ol Schöbl R. an J. Zhu 998 Sochac olaly Wh an Ornn-Uhlnbck Proc : An xnon Workng papr brhar-arl-unvrä übngn. Schwarz.S. 997 h Sochac Bhavor o Commoy Prc: Implcaon or aluaon an Hgng h Journal o Fnanc ol Slby M. J. P. an C. R. Srcklan 995 Compung h Fong an ack Pur coun Bon Prc Formula h Journal o Fx Incom ol Shampn L. F Numrcal Soluon o Ornary rnal quaon Chapman & Hall Nw York. Shampn L. F. an M.. Goron 975. Compur Soluon o Ornary rnal quaon: h Inal alu Problm W. H. Frman San Francco. Sullvan M.A. alung Amrcan Pu Opon Ung Gauan uaraur h Rvw o Fnancal Su ol ahan N. Cr Spra Opon aluaon unr GARCH Workng papr Rk Managmn Char HC Monral. ack O.A. 977 An qulbrum Characrzaon o h rm Srucur Journal o Fnancal conomc ol Zhu J. Moular Prcng o Opon Workng papr brhar-arl-unvrä übngn. 9

31 abl abl : Convrgnc o B&S AM call Maury 3 monh 6 monh 9 monh yar B&S prc n abl prn h rul o h valuaon o an a-h-mony call whn Black an Schol 973 ramwork boh wh h B&S analyc ormula an wh our numrcal procur. h opon paramr ar ln r. 5 an. 4. o mach B&S ramwork h mol paramr ar µ r γ. 5 λ κ an θ. 3

32 abl : Convrgnc o L&S AM call Maury 3 monh 6 monh 9 monh yar L&S prc n abl prn h rul o h valuaon o an a-h-mony call whn Longa an Schwarz 995 ramwork boh wh h L&S analyc ormula an wh our numrcal procur. h opon paramr ar ln.. r. 5 an. 4. o mach L&S ramwork h mol paramr ar µ.. 5 γ λ κ an θ. 3

33 abl 3 : Convrgnc o Zhu Squar-Roo AM call Maury 3 monh 6 monh 9 monh yar Zhu prc n abl 3 prn h rul o h valuaon o an a-h-mony call whn Zhu ramwork boh wh h Zhu quar-roo m-analyc ormula an wh our numrcal procur. h opon paramr ar ln r. 5 an. 4. o mach Zhu ramwork h mol paramr ar µ r γ λ 4 κ 4 an θ. 6. 3

34 abl 4 : Convrgnc o Zhu Ornn-Uhlnbck AM call Maury 3 monh 6 monh 9 monh yar Zhu prc n abl 4 prn h rul o h valuaon o an a-h-mony call whn Zhu ramwork boh wh h Zhu Ornn-Uhlnbck m-analyc ormula an wh our numrcal procur. h opon paramr ar ln r. 5 an.. o mach Zhu ramwork h mol paramr ar µ r γ λ 4 κ 4 an θ

35 abl 5 : Prc o an AM call unr h Squar-Roo Man-Rvrng mol Maury 3 monh 6 monh 9 monh yar n abl 5 prn h rul o h valuaon o an a-h-mony call whn our quar-roo man-rvrng ramwork. h opon paramr ar ln.. r. 5 an. 4. h mol paramr ar µ. 3. γ. 5. λ κ an θ

36 abl 6 : Prc o an AM call unr h Ornn-Uhlnbck Man-Rvrng mol Maury 3 monh 6 monh 9 monh yar n abl 6 prn h rul o h valuaon o an a-h-mony call whn our Ornn- Uhlnbck man-rvrng ramwork. h opon paramr ar ln.. r. 5 an.. h mol paramr ar µ. 3. γ.5. λ κ an θ

37 abl 7 : Impac o h man-rvron on h Call prc unr h Squar-Roo Man-Rvrng mol Maury ln.8 ln. ln % % 5% 6% 34% 38% 4% 44% 3 45% 5% 54% 56% % 5% 8% % % 8% 3% 35% 3 9% 37% 4% 46% % 7% % % 8% 4% 8% % 3 % % 5% 3% abl 7 prn h mpac o h man-rvron cocn whn our quar-roo manrvrng ramwork. h rlav rnc compu wh rpc o h no manrvron prc... h opon paramr ar. r. 5 an. 4. h mol paramr ar µ. 3 γ. 5. λ κ an θ.5. 36

38 abl 8 : Impac o h man-rvron on h Call prc unr h Ornn-Uhlnbck Man-Rvrng mol Maury ln.8 ln. ln % 7% 3% 34% 38% 46% 5% 54% 3 5% 58% 63% 66% % 8% % 4% 3% 3% 36% 4% 3 3% 4% 47% 5% % 7% % 3% 8% 4% 9% 3% 3 % % 6% 3% abl 8 prn h mpac o h man-rvron cocn whn our Ornn- Uhlnbck man-rvrng ramwork. h rlav rnc compu wh rpc o h no man-rvron prc... h opon paramr ar. r. 5 an. 4. h mol paramr ar µ. 3 γ. 5. λ κ an θ

39 Fgur Fgur : B&S call prcng rlav rror 8-5 AM call rlav rror IM call rlav rror OM call rlav rror Rlav rror uaraur rul orr Fgur how h rlav prcng rror o a call whn Black an Schol B&S ramwork. h ru prc gvn by B&S analyc ormula. h unrlyng a valu ar ln ln an ln9. h opon paramr ar.5 r. 5 an. 4. o mach B&S ramwork h mol paramr ar µ r γ. 5 λ κ an θ. 38

40 Fgur : L&S Call prcng rlav rror -4 AM call rlav rror IM call rlav rror OM call rlav rror Rlav rror uaraur rul orr Fgur how h rlav prcng rror o a call whn Longa an Schwarz L&S ramwork. h ru prc gvn by L&S analyc ormula. h unrlyng a valu ar ln. ln. an ln.8. h opon paramr ar.. 5 r. 5 an. 4. o mach L&S ramwork h mol paramr ar µ.. 5 γ λ κ an θ. 39

41 Fgur 3 : Zhu quar-roo Call prcng rlav rror 3-4 AM call rlav rror IM call rlav rror OM call rlav rror Rlav rror uaraur rul orr Fgur 3 how h rlav prcng rror o a call whn Zhu ramwork. h ru prc gvn by Zhu quar-roo m-analyc ormula. h unrlyng a valu ar ln ln an ln8. h opon paramr ar. 5 r.5 an. 4. o mach Zhu ramwork h mol paramr ar µ. 5 γ λ 4 κ 4 an θ. 6. 4

42 Fgur 4 : Zhu Ornn-Uhlnbck Call prcng rlav rror - AM call rlav rror IM call rlav rror OM call rlav rror Rlav rror uaraur rul orr Fgur 4 how h rlav prcng rror o a call whn Zhu ramwork. h ru prc gvn by Zhu Ornn-Uhlnbck m-analyc ormula. h unrlyng a valu ar ln ln an ln8. h opon paramr ar r.5 an.. o mach Zhu ramwork h mol paramr ar µ. 5 γ λ 4 κ 4 an θ. 6. o kp h am cal h ou-o-h-mony prcng rlav rror v by 5. 4

43 Fgur 5 : Squar-roo Man-Rvrng Call prcng rlav rror 5-3 AM call rlav rror IM call rlav rror OM call rlav rror -3 Rlav rror uaraur rul orr Fgur 5 how h rlav prcng rror o a call whn our quar-roo man-rvrng ramwork. h ru prc h aympoc prc. h unrlyng a valu ar ln. ln.5 an ln.8. h opon paramr ar..5 r. 5 an. 4. h mol paramr ar µ. 3 γ.5. λ κ an θ. 5. 4

44 Fgur 6 : Ornn-Uhlnbck Man-Rvrng Call prcng rlav rror 35-3 AM call rlav rror IM call rlav rror OM call rlav rror Rlav rror uaraur rul orr Fgur 6 how h rlav prcng rror o a call whn our Ornn-Uhlnbck manrvrng ramwork. h ru prc h aympoc prc. h unrlyng a valu ar ln. ln.5 an ln.8. h opon paramr ar. r. 5 an.. h mol paramr ar µ. 3 γ.5. λ κ an θ

45 44 Appnx A. : rvaon o h quar-roo man-rvrng characrc uncon h mol gvn unr h rk-nural maur by : W γ µ W λ κθ whr W W. I w n Y by Io lmma w hav : W Y γ µ Solvng h S gv : W Y Y γ µ W Y Y γ µ h proc hn xpr a : W γ µ So w hav ha : W xp xp xp xp γ µ Snc W an W ar corrla w can wr : W W W whr W an W ar uncorrla Brownan moon. W hn oban :

46 45 W W xp xp xp xp γ µ I W W W xp xp xp xp γ µ Snc W an W ar npnn W W I h quaon bcom : W W xp xp xp xp γ µ W xp xp xp γ µ A h ag w no n h parcular quar-roo pccaon o h volaly uon. h quaon wll b alo val or h Ornn-Uhlnbck volaly uon. For h quar-roo mol by Io lmma w can wr or h quar volaly :

47 46 W λ κθ Ingrang h S an r-arrangng la o : W λ κθ W hn oban : xp xp xp xp λ κθ γ µ xp xp γ λ κθ µ W can rwr h quaon a : xp xp ε ε κθ µ whr ε γ λ ε xp xp

48 47 n h uncon F by : { } F xp xp ε ε hn by Fynman-ac horm w hav ha F mu ay h ollowng P : F F F F F xp ε ε λ κθ Rplacng h m varabl by w can rwr h P a whou ambguy w kp h am uncon F : F F F F F xp ε ε λ κθ I w aum ha F log-lnar an gvn by : [ ] xp C F whr C ε w hav : [ ] ' ' F F F F F C F

49 48 h P or F bcom : ' ' ε λ κθ C Ar r-arrangng a a polynomal o w uc h O a by : ε ε λ ' an by C : ' C C κθ h acualz characrc uncon hn gvn by : [ ] xp xp xp xp κθ µ C r r

50 49 Appnx A. : xac roluon o h O a by an C n h quar-roo ramwork h O a by h uncon an C ar : ε λ ' an ' C C κθ Makng h raonal or Rcca-yp O ollowng ranormaon : U xp la o h ollowng lnar homognou con-orr O : ' ' ' ε λ U U U Unr h ranormaon w rcovr h orgnal uncon an C mply by : ln ' κθ U C U U A urhr ubuon xp U z ruc h O o : ' ' ' z z z z z λ

51 whr z ε. W hn only n o olv or h uncon z. Sowar lk Mapl gv h oluon o h O n rm o pcal uncon known a h Whakr uncon. h uncon ar rla o h wll-known conlun hyprgomrc uncon Abramowz an Sgun 968. h oluon U gvn by : U Axp M a b c B xp W a b c whr M. an W. ar rpcvly h WhakrM an h WhakrW uncon an : γ λ a c λ b λ Conan A an B ar rmn by wrng own ha an C. 5

52 5 Appnx B. : rvaon o h Ornn-Uhlnbck man-rvrng characrc uncon h mol gvn unr h rk-nural maur by : W γ µ W λ κθ whr W W. From Appnx A. w can wr h characrc uncon a : W xp xp xp γ µ For h Ornn-Uhlnbck mol w can olv or h volaly : W λ κθ Ingrang h S an r-arrangng la o : W λ κθ By Io lmma w alo hav : [ ] an

53 5 [ ] W λ κθ [ ] κθ λ h characrc uncon hn gvn by : xp xp γ λ κθ µ W can rwr h quaon a : 3 xp xp η η η µ whr η κθ η γ λ η xp xp xp 3

54 53 n h uncon G a : G 3 xp η η η By Fynman-ac horm w hav ha v G mu ay h ollowng P : 3 xp η η η λ κθ G G G G G Rplacng h m varabl by a bor w can rwr h P a whou ambguy w kp h am uncon G : 3 xp η η η λ κθ G G G G G Aumng ha G log-lnar an gvn by : xp C G whr 3 C η la o : [ ] [ ] ' ' ' G G G G G G C G

55 54 h P a by G bcom : [ ] [ ] [ ] ' ' ' η η λ κθ C Ar r-arrangng a a polynomal o w uc h O a by : η η λ 3 ' by : ' η κθ λ an by C : ' C C κθ h acualz characrc uncon hn gvn by : xp xp xp xp µ C r r

56 55 Appnx B. : xac roluon o h O a by n h Ornn-Uhlnbck ramwork h O a by h uncon : η η λ 3 ' A al n Appnx A. w u wo ranormaon bor gng h xac oluon. Makng h r ranormaon : U xp an h urhr ubuon xp U z la o : xp xp c b a W B c b a M A U whr agan M. an W. ar rpcvly h WhakrM an h WhakrW uncon an : λ λ λ γ c b a Conan A an B ar rmn by wrng own ha U an ' U. W rcovr h uncon by : ' U U

57 o ha nhr nor C coul b xpr n a clo-orm way w u a raonal chnqu o olv lnar r-gr O o n : κθ λ λ U λ U λ U h xpron coul no b mpl urhr. 56

58 Appnx C : Proo o h wll-nn o h ngran For h purpo o ngraon w mu proo h wll-nn o h uncon ovr h nrval o ngraon pcally aroun h ponal ngular. In parcular w hav o valu h wo ngral : an R R whr a characrc uncon whch o cla [ [ C n by : xp r xp { } Ung a aylor xpanon aroun h wo ngran n rpcvly o : R ' ln r r ln an R ' ln r r xp xp { } { } ln 57

59 58 Appnx : Fynman-ac horm araza an Shrv 99 Unr om rgulary aumpon w uppo ha F R ] R :[ o cla C R ] [ an a h Cauchy problm : k F F h g A F F whr A h con orr rnal opraor hn F unqu an am h ochac rprnaon : h g h k F xp xp

60 59 Appnx : rvaon o h Grk Rcall ha h Call prmum gvn by : ln ln Call > > For mplcy w no by an ln > by. W n o compu h ollowng rvav : R π R π W hn can wr : Call Call la By a ormal chang o varabl w can how ha : an w can uc h ormula or h la a gvn n h man x.

61 6 n h Gamma by : la Gamma Ung h am calculu on o rv h la la o : R π la la Gamma For h ga conr r h quar-roo mol : Call ga W compu h ollowng rvav : R π R π

62 6 W now can wr or h ga : ga R R π π For h Ornn-Uhlnbck mol n h ga by : Call ga h ollowng rvav ar n : [ ] R π R π

63 6 h ga can b wrn a : ga R R π π

64 Foono * Ph.. cana n Fnanc a h Écol HC Monréal. Fnancal uppor wa prov by h Rk Managmn Char h FCAR an h Socal Scnc an Human Rarch Councl o Canaa. I woul lk o hank my proor Gorg onn Jan-Guy Smonao an Pr Chrorn or hr commn. I am alo graul o h parcpan a h Brown Bag mnar a HC Monréal a h n CIRPÉ annual conrnc a Monral an a h MF conrnc a Syny or hlpul cuon. 63