Online appendices from Counterparty Risk and Credit Value Adjustment a continuing challenge for global financial markets by Jon Gregory

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1 Onlne appendces fro Counerpary sk and Cred alue Adusen a connung challenge for global fnancal arkes by Jon Gregory APPNDX A: Dervng he sandard CA forula We wsh o fnd an expresson for he rsky value of a need se of dervaves posons wh a axu aury dae Denoe he rsk-free value curren MM of he relevan posons as and he defaul e of he counerpary as hen s s wll denoe he fuure unceran MM accounng for dscounng effecs here are wo cases o consder: Counerpary does no defaul before n hs case he rsky poson s equvalen o he rsk-free poson and we wre he correspondng payoff as: Where s he ndcor funcon denong defaul hs akes he value f defaul has no occurred before or a e and zero oherwse Counerpary does defaul before n hs case he payoff consss of wo ers he value of he poson ha would be pad before he defaul e all cash flows before defaul wll sll be pad by he counerpary plus he payoff a defaul Cashflows pad up o he defaul e Defaul payoff Here f he MM of he rade a he defaul e s posve hen he nsuon wll receve a recovery fracon of he rsk-free value of he dervaves posons whls f s negave hen hey wll sll have o sele hs aoun Hence he defaul payoff a e s: where x n x0 and y ax y0 Pung he above payoffs ogeher we have he followng expresson for he value of he rsky poson under he rsk-neural easure:

2 Onlne appendces fro Counerpary sk and Cred alue Adusen a connung challenge for global fnancal arkes by Jon Gregory he above expresson s general bu no especally useful or even nsghful However re-arrangng and usng he relaonshp x x x we oban: Now realsng ha we can cobne wo ers snce we have: nally snce we have: he above equaon s crucal snce defnes he rsky value of a neng se of dervaves posons wh respec o he rsk-free value he relevan er s ofen known as CA cred value adusen s an adusen o he rsk-free value of he posons whn he neng se o accoun for counerpary rsk: CA CA Noe ha we ade he assupon ha he fuure MM value s ncludes dscounng for noaonal splcy f we drop hs assupon he above forula wll nclude dscounng: CA where s s he value of he oney arke accoun a e s

3 Onlne appendces fro Counerpary sk and Cred alue Adusen a connung challenge for global fnancal arkes by Jon Gregory APPNDX B: Copuaon of he CA forula and sple spread-based approxaon o derve he classc CA forula we can hen wre: CA u u Where s he ean or expeced recovery value We use u o denoe: u u u hs s a crcal pon n he analyss as he above saeen requres he exposure a a fuure dae u knowng ha defaul of he counerpary has occurred a ha dae u gnorng wrong-way rsk sple ses u u whch we do fro now on Snce he expecaon n he above equaon s over all es before he fnal aury we can negrae over all possble defaul es We oban: CA B u u d u where B u s he rsk-free dscoun facor and u s he cuulave defaul probably for he counerpary probably of no defaul as descrbed n Appendx 0B We recognse he dscouned expeced exposure calculaed under he rsk-neural easure denoed by d u B u u Assung ha he defaul probables are deernsc we have: CA d u d u nally we could copue he above equaon va soe negraon schee such as: CA d where we have perods gven by [ 0 ] As long as s reasonably large hen hs wll be a good approxaon Wh furher splfyng assupons one can oban a sple expresson for CA lnked o he cred spread of he counerpary o do hs we have o work wh he 3

4 Onlne appendces fro Counerpary sk and Cred alue Adusen a connung challenge for global fnancal arkes by Jon Gregory non-dscouned expeced exposure forula: as herefore sar fro he followng CA CA u B u d u Suppose ha we approxae he undscouned expeced exposure er u as a fxed known aoun whch would obvously be he P he fxed P defned n Chaper 8 would os obvously be copued fro he averaged over e for exaple: P u du Clearly he approxaon wll be a good one f he relaonshp beween P defaul probably and ndeed dscoun facors s reasonably hoogeneous hrough e or oher cancellaon effecs coe no play Usng hs approach he CA s: CA B u d u P ecallng Appendx 0B we can see hs s sply he value of CDS proecon on a noonal equally he P Hence we have he followng approxaon gvng a runnng CA e expressed as a spread: CA P Spread We can use he rsky annuy forulas Appendx 0B o conver hs o an up-fron value As descrbed n Chaper he can be done f s copued usng he -forward easure 4

5 Onlne appendces fro Counerpary sk and Cred alue Adusen a connung challenge for global fnancal arkes by Jon Gregory APPNDX C: CA forula for an opon poson n hs case we have a splfcaon snce he exposure of he long opon poson can never be negave: CA opon opon B where opon s he upfron preu for he opon hs eans ha he value of he rsky opon can be calculaed as: opon CA opon opon opon opon opon opon Wh zero recovery we have sply ha he rsky preu s he rsk-free value ulpled by he survval probably over he lfe of he opon 5

6 Onlne appendces fro Counerpary sk and Cred alue Adusen a connung challenge for global fnancal arkes by Jon Gregory APPNDX D: Se-analycal calculaon of he CA for a swap As noed by Sorensen and Boller 994 he CA of a swap poson can be wren as: CA swap swapon ; where swapon ; s s he value oday of he reverse swap wh aury dae and exercse dae he nuon s ha he counerpary has he opon o defaul a any pon n he fuure and herefore cancel he rade execue he reverse poson he values of hese swapons are weghed by he relevan defaul probables and recovery s aken no accoun No only s hs forula useful for analycal calculaons s also que nuve for explanng CA An neres rae swapon can be prced n a odfed Black-Scholes fraework va he forula: d X d D d X d D payer swapon recever swapon d log / X 05 S d S Where s he forward rae of he swap X s he srke rae he fxed swap of he underlyng swap s he swap rae volaly s he aury of he swapon S he e horzon of neres wll be he aury of he underlyng swap he exposure of he swap wll be defned by he neracon beween wo facors: he swapon payoff d X and he underlyng swap duraon anny d D hese quanes respecvely ncrease and decrease onooncally over e he overall swapon value herefore peaks soewhere n-beween as llusraed n gure 6 and Spreadshee 6

7 Onlne appendces fro Counerpary sk and Cred alue Adusen a connung challenge for global fnancal arkes by Jon Gregory 7 APPNDX : ncreenal CA forula o calculae ncreenal CA we need o quanfy he change before and afer added a new rade : CA CA We herefore sply need o use he ncreenal n he sandard CA forula