(使用股票价格计算违约机率)

20081017paperreportR96072黃源鱗20.6UsingEquityPricestoEstimateDefaultProbabilities(使用股票價格計算違約機率)Moreup-to-dateThevalueoftheequityattimeTasET=max(VT-D,0)ThisshowthattheequityisacalloptionSotheBlack-ScholesformulagivesthevalueoftheequitytodayasE0=V0N(d1)-De-rTN(d2)----(20.3)whered1=ln(V0/D)+(r+σv2/2)Tσv√Td2=d1-σv√TTherisk-neutraldefaultprobabilityisN(-d2)(seemsN(d2)liketheliveprobability(當VTD))TocaculateN(-d2),weneedV0,σ0butweonlyknowσE、E0andequation(20.3)FromIto’sLemma,wecangetσEE0=N(d1)σVV0----(20.4)dV/V=uVdt+σVdZV---(1)dE/E=uEdt+σEdZE---(2)-dE=EvdV+½Evv(dV)2+Etdt-dE=(½EvvσV2V2+σVVEv+Et)dt+σVVEvdZV----(3)由(2)(3)比照係數-σEE0dZE=σVV0EvdZV=σVV0N(d1)dZVσEE0=N(d1)σVV0(設dZE=dZV)Ito’sLemaWecangetV0,σ0byequations(20.3)and(20.4)**TosolveF(x,y)=0andG(x,y)=0.wecanusetheSolverroutineinExceltofindthevaluesofxandythatminimize[F(x,y)]2+[G(x,y)]2*seealsothekeyword“Merton’sModel”20.7CreditRiskinDerivativesTransactions(衍生性金融商品交易的信用風險)BecausetheclaimthatwillbemadeintheeventofadefaultismoreuncertainWecandistinguishthreesituations:1.Contractisaliability(負債)2.Contractisanasset3.ContractcanbecomeeitheranassetoraliabilityExample1.ashortoptionposition2.alongoptionposition3.aforwardcontractAdjustingDerivatives’ValuationsforCounterpartyDefaultRiskTheexpectedlossatti:qi(1-R)E[max(fi,0)]-Σuivi---(20.5)fi:thevalueofthederivativetothefinancialinstitutionqi:therisk-neutraldefaultprobabilityR:recoveryrateui:qi(1-R)vi:thevaluetodayoftheinstrumentIncase1.fiisalwaysnegative,sotheexpectedlossiszeroIncase2.themax(fi,0)ifalwaysfi.viisthepresentvalueoffi,italwaysequalsf020.8CreditRiskMitigation(減緩信用風險)Netting(類似貨品抵押)假如一家公司原持有+10,+30,-25的契約當對方倒閉,此契約價值變-10,-30,+25若是沒有此條約,則損失會計為-40,但若是有此條約,則損失會便-40+25=-15Collateralization(類似保證金)當契約價值隨市價改變時,受益方需給另一方現值和原值的價差.(ex:$10-$10.5itcanaskfor$0.5ofcollateral)DowngradeTriggers(降級觸發)當對方信用等級評比下降到某種等級,可以規定馬上以市價直接清算掉此契約,不用等到到期日