Online appendices from The xva Challenge by Jon Gregory. APPENDIX 14A: Deriving the standard CVA formula.

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1 Onlne appendces fro he xa Challenge by Jon Gregory APPNDX 4A: Dervng he sandard CA forla We wsh o fnd an expresson for he rsky vale of a need se of dervaves posons wh a ax ary dae Denoe he rsk-free vale crren MM of he relevan posons as and he defal e of he conerpary as hen s s wll denoe he fre nceran MM acconng for dsconng effecs here are wo cases o consder: Conerpary does no defal before n hs case he rsky poson s eqvalen o he rsk-free poson and we wre he correspondng payoff as: Where s he ndcor fncon denong defal hs akes he vale f defal has no occrred before or a e and zero oherwse 2 Conerpary does defal before n hs case he payoff consss of wo ers he vale of he poson ha wold be pad before he defal e all cash flows before defal wll sll be pad by he conerpary pls he payoff a defal Cashflows pad p o he defal e Defal payoff Here f he MM of he rade a he defal e s posve hen he nson wll receve a recovery fracon of he rsk-free vale of he dervaves posons whls f s negave hen hey wll sll have o sele hs aon Hence he defal payoff a e s: where x n x0 and y ax y0 Png he above payoffs ogeher we have he followng expresson for he vale of he rsky poson nder he rsk-neral easre: Copyrgh 205 Jon Gregory

2 Onlne appendces fro he xa Challenge by Jon Gregory Copyrgh 205 Jon Gregory 2 he above expresson s general b no especally sefl or even nsghfl However re-arrangng and sng he relaonshp x x x we oban: Now realsng ha we can cobne wo ers snce we have: nally snce we have: he above eqaon s crcal snce defnes he rsky vale of a neng se of dervaves posons wh respec o he rsk-free vale he relevan er s ofen known as CA cred vale adjsen s an adjsen o he rsk-free vale of he posons whn he neng se o accon for conerpary rsk: CA CA Noe ha we ade he asspon ha he fre MM vale s ncldes dsconng for noaonal splcy f we drop hs asspon he above forla wll nclde dsconng: CA where s s he vale of he oney arke accon a e s

3 Onlne appendces fro he xa Challenge by Jon Gregory APPNDX 4B: CA copaon deals o derve he classc CA forla we can hen wre: CA * * Where s he ean or expeced recovery vale We se o denoe: * hs s a crcal pon n he analyss as he above saeen reqres he exposre a a fre dae knowng ha defal of he conerpary has occrred a ha dae * gnorng wrong-way rsk sple ses whch we do fro now on Snce he expecaon n he above eqaon s over all es before he fnal ary we can negrae over all possble defal es We oban: CA B d where B s he rsk-free dscon facor and s he clave defal probably for he conerpary probably of no defal as descrbed n Appendx 0B We recognse he dsconed expeced exposre calclaed nder he rsk-neral easre denoed by d B Assng ha he defal probables are deernsc we have: CA d d nally we cold cope he above eqaon va soe negraon schee sch as: CA d where we have perods gven by [ 0 ] As long as s reasonably large hen hs wll be a good approxaon Wh frher splfyng asspons one can oban a sple expresson for CA lnked o he cred spread of he conerpary o do hs we have o work wh he non-dsconed expeced exposre as herefore sar fro he followng CA forla: As descrbed n Chaper 2 hs can be done f s coped sng he -forward easre Copyrgh 205 Jon Gregory 3

4 Onlne appendces fro he xa Challenge by Jon Gregory Copyrgh 205 Jon Gregory 4 d B CA

5 Onlne appendces fro he xa Challenge by Jon Gregory APPNDX 4C: Approxae CA forla Sppose ha we approxae he ndsconed expeced exposre er as a fxed known aon whch wold obvosly be he P he fxed P defned n Chaper 8 wold os obvosly be coped fro he averaged over e for exaple: P d j j Clearly he approxaon wll be a good one f he relaonshp beween P defal probably and ndeed dscon facors s reasonably hoogeneos hrogh e or oher cancellaon effecs coe no play Usng hs approach he CA s: CA B d P ecallng Appendx 0B we can see hs s sply he vale of CDS proecon on a noonal eqally he P Hence we have he followng approxaon gvng a rnnng CA e expressed as a spread: CA P Spread We can se he rsky anny forlas Appendx 0B o conver hs o an p-fron vale Copyrgh 205 Jon Gregory 5

6 Onlne appendces fro he xa Challenge by Jon Gregory APPNDX 4D: CA forla for a long opon poson n hs case we have a splfcaon snce he exposre of he long opon poson can never be negave: CA opon opon B where opon s he pfron pre for he opon hs eans ha he vale of he rsky opon can be calclaed as: opon CA opon opon opon opon opon opon Wh zero recovery we have sply ha he rsky pre s he rsk-free vale lpled by he srvval probably over he lfe of he opon Copyrgh 205 Jon Gregory 6

7 Onlne appendces fro he xa Challenge by Jon Gregory Copyrgh 205 Jon Gregory 7 APPNDX 4: ncreenal CA forla o calclae ncreenal CA we need o qanfy he change before and afer added a new rade : CA CA We herefore sply need o se he ncreenal n he sandard CA forla

8 Onlne appendces fro he xa Challenge by Jon Gregory Copyrgh 205 Jon Gregory 8 APPNDX 4: Dervng he blaeral CA forla We wsh o fnd an expresson for he rsky vale of a need se of dervaves posons wh a ax ary dae as n Appendx 2A b nder he asspon ha he nson concerned ay also defal n addon o her conerpary Denong he defal e of he nson as her recovery vale as and followng he noaon and logc n Appendx 2A we now have he followng cases we denoe he frs-o-defal e of he nson and conerpary as n Neher conerpary nor nson defals before n hs case he rsky poson s eqvalen o he rsk-free poson and we wre he correspondng payoff as: 2 Conerpary defals frs and also before e hs s he defal payoff as n Appendx 7A: 3 nson defals frs and also before e hs s an addonal er copared wh he nlaeral CA case and corresponds o he nson self defalng f hey owe oney o her conerpary negave MM hen hey wll pay only a recovery fracon of hs whls f he conerpary owes he oney posve MM hen hey wll sll receve hs Hence he payoff s he oppose of case 2 above: 4 f eher he nson or conerpary does defal hen all cashflows pror o he frs-o-defal dae wll be pad Png he above payoffs ogeher we have he followng expresson for he vale of he rsky poson: Slarly o Appendx 7A we splfy he above expresson as:

9 Onlne appendces fro he xa Challenge by Jon Gregory Copyrgh 205 Jon Gregory 9 nally obanng: We can denfy he BCA blaeral CA er as beng: BCA nally nder he slar asspons of no wrong-way rsk and of no slaneos defal beween he defal of he nson and s conerpary we wold have a forla analogos o ha derved n Appendx 2B for copng BCA: ds S B ds S B BCA An obvos approxaon o cope hs forla sng he dsconng and N wold hen be: d d S N S BCA More deals on hese calclaons and dscsson on ncorporang dependency beween he defal of he nson and he conerpary can be fond n Gregory 2009a A sple spread based approxaon wold be:

10 Onlne appendces fro he xa Challenge by Jon Gregory where CA P Spread N Spread Spread represens he cred spread of he nson heselves Copyrgh 205 Jon Gregory 0