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Analyzing Credit Risk with Equity Prices<br>

Explore using equity prices to estimate default probabilities. Learn about risk-neutral default probabilities and credit risk mitigation techniques. Understand calculations and adjustments for counterparty default risk.<br>

josephgordon
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Analyzing Credit Risk with Equity Prices<br>

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  1. 20081017 paper report R96072 黃源鱗

  2. 20.6 Using Equity Prices to Estimate Default Probabilities(使用股票價格計算違約機率) • More up-to-date • The value of the equity at time T as ET = max(VT-D,0) • This show that the equity is a call option

  3. So the Black-Scholes formula gives the value of the equity today as E0=V0N(d1) - De-rTN(d2) ---- (20.3) where d1= ln(V0/D)+(r+σv2/2)T σv√T d2= d1 - σv√T

  4. The risk-neutral default probability is N(-d2) (seems N (d2) like the live probability (當VT>D)) • To caculate N(-d2), we need V0 , σ0 but we only know σE 、E0 and equation(20.3) • From Ito’s Lemma , we can get σEE0 = N(d1) σVV0 ----(20.4)

  5. Ito’s Lema dV / V = uVdt+ σVdZV ---(1) dE / E = uEdt+ σEdZE ---(2) -> dE = EvdV+ ½ Evv(dV)2 + Etdt -> dE = (½ EvvσV2V2 + σVVEv + Et)dt+σVVEvdZV ----(3) 由(2)(3)比照係數 -> σEE0dZE = σVV0 EvdZV =σVV0 N(d1) dZV  σEE0 = N(d1) σVV0(設dZE = dZV )

  6. We can getV0 , σ0 by equations (20.3) and (20.4) * • * To solve F(x,y)=0 and G(x,y)=0. we can use the Solver routine in Excel to find the values of x and y that minimize [F(x,y)]2 + [G(x,y)]2 • * see also the keyword “ Merton’s Model”

  7. 20.7 Credit Risk in Derivatives Transactions (衍生性金融商品交易的信用風險) • Because the claim that will be made in the event of a default is more uncertain • We can distinguish three situations: • 1. Contract is a liability (負債) • 2. Contract is an asset • 3. Contract can become either an asset or a liability

  8. Example • 1. a short option position • 2. a long option position • 3. a forward contract

  9. Adjusting Derivatives’ Valuations for Counterparty Default Risk • The expected loss at ti: qi(1-R)E[max(fi,0)] -> Σuivi ---(20.5) fi:the value of the derivative to the financial institution qi:the risk-neutral default probability R:recovery rate ui :qi(1-R) vi :the value today of the instrument

  10. In case 1. fi is always negative , so the expected loss is zero • In case 2. the max(fi ,0) if always fi . vi is the present value of fi, it always equals f0

  11. 20.8 Credit Risk Mitigation (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

  12. 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 -> $10.5 it can ask for $0.5 of collateral ) • Downgrade Triggers (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

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