BSWithJump Model And Pricing Of Quanto CDS With FX Devaluation Risk

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1 MPRA Mnich Prsonal RPEc Archiv BSWihmp Mol An Pricing Of Qano CDS Wih FX Dvalaion Risk Rachi EL-Mohammai Bank Of Amrical Mrrill Lynch Ocobr 9 Onlin a hps:mpra.b.ni-mnchn.478 MPRA Papr No. 478, pos 8. Novmbr 3:4 UC

2 BSWihmp BSWihmp Mol An Pricing Of Qano CDS Wih FX Dvalaion Risk RACHID EL-MOHAMMADI BAN OF AMERICA MERRILL LYNCH Ocobr 9 Absrac W prsn a nw mol for pricing qano CDS whr h FX col b srongly pnn on h cri rfrnc. h mol assms lognormal hazar ra an rminisic FX al volailiy whr h FX spo can jmp a im of fal of h cri rfrnc. W prsn h mol, h calibraion algorihm, an h qano CDS pricing. Dircor in h Analyics qan am h viws xprss ar h ahor own an no ncssarily BAN Of AMERICA MERRILL LYNCH. I hank pariclarly Abrrahman abach, Philipp Ballan, n ng an Pankaj hamb for hir vry hlpfl commns, arik El-Yobi, Alx Lipon, an Lif Anrson for hir hlpfl iscssion an commns. All rrors ar min.

3 BSWihmp abl of conns Inrocion...3. BSWihmp mol...3. Dfiniions an noaions...3. Molling h fal innsiy: LN mol Molling Emrging mark FX wih jmp a fal im...3 Pricing FX call opions...4. Simplificaion of h fal par...5. Pricing of FX opions in cas of rminisic cri Mol Calibraion LN Calibraion Calibraion of BSWihmp o AM FX opions Pricing qano srvival probabiliis an qano CDS Pricing Qano srvival probabiliis Pricing qano CDS Exampl: Qano CDS... 5 Conclsion an possibl xnsions of h mol...3

4 BSWihmp Inrocion In his papr, w prsn a nw mol for pricing qano CDS whr h FX is srongly corrla o h cri nam. Exampl: CDS on BRAZIL sa wih USD as naral crrncy b pai in ral (BRAZIL crrncy). W wol lik h mol o ak ino accon h FX valaion risk a im of fal. By aing h jmp of h FX a im of fal, h FX ynamic is mor ralisic han h simpl BS mol an h mol is procing richr rm srcrs of qano CDS crvs. h jmp siz will allow s o conrol h shor a qano CDS whil h volailiy of h innsiy an h corrlaion bwn h innsiy an h FX will conrol h rm srcr of qano CDS. h jmp paramr is vry simpl o mark by h rars as i rprsns on mins h raio of al crrncy CDS o omsic crrncy (ypically USD) CDS for shor a mariis. h firs scion scribs h ynamic of h mrging mark FX whr w assm consan jmp siz an sochasic innsiy. h scon scion scribs h pricing of FX opions wihin h mol. h hir scion scribs h calibraion of LN mol (lognormal cri innsiy) an h calibraion of h FX volailiy o h rm srcr of AM volailiis. In h forh scion, w giv som xampls of how qano CDS pns on h varios mol paramrs. In h las scion, w giv a conclsion an prsn possibl xnsions of BSWihmp mol.. BSWihmp mol. Dfiniions an noaions N is a non-homognos Poisson procss wih innsiy. is h firs im whr h Poisson procss jmp or fal im. S is h FX spo whr is omsic crrncy (ypically a G7) an is h forign crrncy (ypically: an mrging mark crrncy). In all h papr, w assm ha h inrs ras ar rminisic. W no B,, an B, rspcivly h omsic zro copon an al crrncy zro copon rspcivly wih mariy.. Molling h fal innsiy: LN mol W sppos ha h innsiy follows a lognormal procss wih consan volailiy an consan man rvrsion. s Ws Z In pracic, w always s h man rvrsion o zro. For simpliciy w no an.3 Molling Emrging mark FX wih jmp a fal im W assm ha h FX spo h ynamic: 3

5 BSWihmp S S fx fx r r W N Z fx W, W h procss Z is a Gassian procss scrib in h prvios scion. is a consan bwn an. In cas of rminisic cri, h spo procss follows a lognormal ynamic bfor an afr h fal im. h FX spo can jmp only onc: a im of fal. W sppos ha h inrs ras ar rminisic. L s calcla h ynamic of ln S fx fx ln S r r W ln N I follows ha h sock procss is givn by: Whr N xp s S X xp ln N s s s s X s X Z B, fx fx fx X S xp, W B h sock procss is h proc of h forwar a coninos maringal an a isconinos maringal. Pricing FX call opions In orr o pric a call opion wihin his mol w n o spara h calclaion ino wo cass: fal bfor mariy an no fal bfor mariy. C, B, E S, B E S Q, B E S h FX spo coniionally on fal occrring a whr I follows ha xp s is: S S X s 4

6 BSWihmp Cf C sr Q ss ss C, B, E X Q ss ss B, E X Cf, Csr, is h fal par of h call pric is h srvival par of h call pric.. Simplificaion of h fal par h fal par of h call pric maks h calibraion of h FX volailiy sing forwar PDEs mor ifficl bcas i pns on h whol pah of h innsiy an h srvival probabiliis. W will work o C f o mak i pns only on h rminal vals of markovian procsss. W prform a chang in C h xprssion of Cf f bcoms: v s v s X s Q Cf, B, E X v v k s s k s ss x an h fncion Ax W fin h procss Y X v v k I follows ha: Q Q Cf, B, E X AB, E X A D D s s L s calcla A(x): x x Ax v v kx kx v k v k x k v k x k x v v k v k x x k k x k x k k 5

7 BSWihmp By rplacing x by in A, w fin D D B E X Q X B, E, Q X s s By rplacing x by in A, w fin D D B E Y X Q ss, Y Q ss Y B, E Y Groping or rms, w riv h following xprssion for h call pric: Q C, B, E X X Q X B, E X X Q ss B, E Y B Q ss, E Y Y ss, Y Q B E Y Y W grop h call opion rms in wo pars whr h firs par can b calcla sing a clos form solion. cf Q,, f C B E X Q s,, s Cncf B E gy h firs par of h call opion formla pns only on h rminal val of h lognormal procss X an h srvival val of a fncion of h rminal valy. his form will allow s o calibra h FX volailiy mor asily han h original formla. 6

8 BSWihmp. Pricing of FX opions in cas of rminisic cri In cas of rminisic cri, h FX opion pric is givn by a clos form solion. In h prvios scion, w hav prov ha a call opion pric is h sm of fiv rms: C, C, C, C, C, C, Calclaion of C, an C3, C, is givn by: Q C, B, E X X Q B, F E M, X B Q X B, F Q F M B, Q X,, M B S N B N C is similar oc 3,, F ln, h only iffrnc is h forwars ar iffrn: Q ss C3, B, E Y Y B S N g B N g ss F ln g g ss s s,,.. Calclaion of C, an C4, C is givn by:, 7

9 BSWihmp F Q,, C B F E M X B * E B F, F xp Q X, F xp W F M, xp Q X F, xp B F N Whr 3 is givn by: Similarly o, 3 xp F ln F ln C C, h qaniy 4, ss ss F Q 4,, Y C B F E M Whr g 3 is givn by: ss ss F B, F xp N g3..3 Calclaion of C g h qaniy C 3 ss F ln 5, is a BS yp formla an givn by: 5, 3 8

10 BSWihmp C5 B E Y Q ss,, Whr f an f ar givn by: 3 Mol Calibraion ss s s B, S N f B, N f f f ss F ln f 3. LN Calibraion h calibraion of h mol consiss on calibraing h fncion o h rm srcr of srvival probabiliis. s W fin h grn fnciong, E s Z. W know ha his grn fncion is solion of h Fokkr-plank qaion G G G G, f Givn a fin schl (xampl: wkly),,.., n, an h grn fncion a im i, w look for ii, (val of h fncion bwn i an i ) ha will vrify h qaion: i ss i ss G, i Q, i E E Z i i i i i i, i i i i, i Onc w calcla ii, sing a roo finr algorihm (Nwon for xampl), w calcla h grn fncion a im i by propagaing h forwar PDE from i o i. W rpa hs wo sps nil w calibra h srvival probabiliy p o h final mariy f. W s Cranck-Nicholson PDE schm for h forwar PDE. W hav wo nmrical paramrs ha allow s o conrol h calibraion accracy: h nmbr of sas an h nmbr of sps. W rcommn sing 4 for h nmbr of sas an h maximm of 4 an 5* f f (wkly sps) for h nmbr of sps, whr is h las calibraion mariy. hs paramrs nsr a vry accra calibraion, vn for xrm CDS crvs an mol paramrs. 3. Calibraion of BSWihmp o AM FX opions W assm ha w ar givn a lognormal innsiy mol calibra o srvival probabiliis. In his scion, w will scrib h calibraion of FX volailiy o AM impli volailiis. 9

11 BSWihmp In h prvios scion, w riv h following xprssion for h call opion pric: Q Q s,,, s C B E g X B E g Y W will focs only on h calclaion of h scon rm as h firs on can asily b calcla sing a nmrical ingraion sinc X is a lognormal procss. B, Y W fin h procss Z ln Y S B, Y h ynamic of Z is givn by Y fx fx fx Z W W fin h grn fncion Y Q s,, s Y Y G E Z Z. is h Dirac fncion. h grn fncion G is solion o of h Fokkr-Plank qaion: G fx G G fx G fx G Y Y G Y Y Y,, G h call pric a mariy can b calcla asily if w know h srvival join pf of h FX spo an h innsiy a. W sppos ha h volailiy fncion fx is pic wis consan fncion hrfor; w calibra i sing a roo finr algorihm by propagaing (forwar) h grn fncion from oay o. o nsr a goo calibraion of h mol o shor-rm FX opions an long-rm FX opions, w s wo forwar PDEs: h firs on o calibra h shor a FX opion p o mariy an a scon on o calibra h FX opions from o h las calibraion mariy if. 4 Pricing qano srvival probabiliis an qano CDS 4. Pricing Qano srvival probabiliis L s calcla h al crrncy srvival probabiliy or h qano srvival probabiliy: B, S Q Q, E SB, Q E M xp N Q E M xp xp Q E M xp

12 BSWihmp fx fx fx M is an xponnial maringal xp M W W can s ha h qano srvival probabiliy is similar o h qano srvival probabiliy wih no BS mol xcp ha h innsiy is mlipli by a cofficin. By oing a chang of nmrair, w concl ha h qano srvival probabiliy is:, xp M Q Q E h innsiy is lognormal nr h omsic masr an says lognormal nr h nw masr wih h sam volailiy an man rvrsion b iffrn fncion. h innsiy of fal nr h al crrncy is a LN mol wih a fncion givn by h formla: fx xp h rm srcr of qano srvival probabiliy can b asily calcla sing h sam forwar PDE on h grn fncion fin in h LN calibraion scion. If h corrlaion bwn h FX an cri is w can s ha h raio al crrncy CDS o h omsic CDS is approximaly. his is r for vry shor a mariis vn if h corrlaion is no. 4. Pricing qano CDS Onc w calibra h mol, w can pric h qano srvival probabiliis sing a forwar PDE as xplain in prvios scion. h pricing of qano CDS is h sraighforwar givn h qano srvival probabiliis as w sppos rminisic inrs ras. h al crrncy qano CDS ar no liqi b w can g som qoaions from brokrs. h al crrncy CDS is sally qo as prcnag of h USD nomina CDS. 4.3 Exampl: Qano CDS W will show via an xampl how h qano CDS pns on iffrn mol paramrs. W chos an arbirary Mxican corpora CDS, which is qo in USD. W wol lik o s how h al crrncy (MXN) CDS pns on varios mol paramrs. W rprsn all h rsls as h raio of qano CDS o h USD CDS Mark Daa h CDS is givn by (qoaion crrncy: USD) is Ma y 3y 3y 5y 7y y CDS h rcovr is 4% h USDMXN AM volailiy is givn by: Ma w w m m 3m 6m 9m y y 3y 4y 5y 7y y volailiis 4.% 4.5% 6.% 6.% 6.% 6.% 6.% 6.% 6.% 6.% 6.3% 6.5% 6.5% 6.5%

13 BSWihmp 4.3. Impac of cri volailiy an FX-cri corrlaion W sppos ha h jmp siz is (no jmp). Blow w show wo graphs: h firs on shows how h raio qcscs pns on h cri volailiy an h scon on how i pns on h corrlaion FX-cri. PEMEX Qano CDS ovr CDS: CDS mp=, corrl=-95% % % 8% vol.35 6% vol.7 vol.4 4% % % mariy W can s from h firs graph ha highr cri volailiy givs s a lowr qcscs if h corrlaion is ngaiv an highr if h corrlaion is posiiv. PEMEX Qano CDS ovr CDS: CDS mp=, vol=.4 4% % % qcscs 8% 6% corrl -.3 vol -.7 corrl -.95 corrl.3 4% % % mariy W can s from h scon graph ha givn cri volailiy a highr corrlaion (absol val) givs s a lowr qcscs if h corrlaion is ngaiv an highr if h corrlaion is posiiv. W can s from hs wo graphs ha h long-rm qcscs vary in a rlaivly larg rang whn h cri volailiy an h FX-cri corrlaion vary. Howvr, h shor-rm raio says always clos o % Impac of h mp siz W can s from h graph blow ha h mol can gnra varios vals for h raio qcscs in h shor-rm an long-rm whn w vary h jmp siz, h cri volailiy, an h FX-cri volailiy. h jmp siz conrols h ovrall lvl of qcscs whil h cri volailiy an h corrlaion FX-cri conrols h rm srcr of qcscs.

14 BSWihmp PEMEX Qano CDS ovr CDS: CDS vol=.7, corrl=-.95 % % 8% jmp 6% jmp -5% jmp -5% 4% % % mariy 5 Conclsion an possibl xnsions of h mol In his papr, w hav prsn a nw mol ha ak ino accon h FX valaion risk. W hav prsn how o calibra h mol sing forwar PDE an, h pricing of qano CDS in his framwork. h mol col b s vn if h FX is no srongly link o h cri rfrnc (In his cas, w can s h jmp siz o ). In his papr, w hav spcifi a consan jmp siz, b all h formlas an calibraions say almos h sam if w s a ranom jmp siz. W prfrr o s a consan jmp bcas h qano CDS (linar payoff on h FX) os no pns a lo on h varianc of h jmp siz. his mol is convnin for pricing linar FX srcrs which knock o a im of fal (lik qano CDS) b in orr o pric mor xoic srcrs, w n o a a al volailiy anor sochasic volailiy componn in orr o calibra h FX smil an no only h AM volailiy. his wol b h sbjc of coming rsarch. Anohr inrsing xnsion of h mol is pricing of qano FD whr w n o ak ino accon h iffrn pnncis of h consins of h FD bask an h FX. Hnc, w n o spcify a iffrn jmp siz o ach cri nam. 3

15 BSWihmp REFERENCES [].R.Bilcki, M.anblanc, M.Rkowski, Pricing An raing cri fal swaps in a hazar procss mol. Dcmbr-7 [] M.anblanc, Y.L.Cam, Rc form molling for cri risk. 4

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