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 E T = max(V T -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 E 0 =V 0 N(d 1 ) - D e -rT N(d 2 ) ---- (20.3) where d1= ln(V 0 /D)+(r+σ v 2 /2)T σ v √ T d2= d1 - σ v √ T

4.  The risk-neutral default probability is N(-d 2 ) (seems N (d 2 ) like the live probability ( 當 V T >D))  To caculate N(-d 2 ), we need V 0, σ 0 but we only know σ E 、 E 0 and equation(20.3)  From Ito ’ s Lemma, we can get σ E E 0 = N(d 1 ) σ V V 0 ----(20.4)

5. dV / V = u V dt+ σ V dZ V ---(1) dE / E = u E dt+ σ E dZ E ---(2) -> dE = E v dV+ ½ E vv (dV) 2 + E t dt -> dE = ( ½ E vv σ V 2 V 2 + σ V VE v + E t )dt + σ V VE v dZ V ----(3) 由 (2)(3) 比照係數 -> σ E E 0 dZ E = σ V V 0 E v dZ V = σ V V 0 N(d 1 ) dZ V  σ E E 0 = N(d 1 ) σ V V 0 ( 設 dZ E = dZ V ) Ito ’ s Lema

6.  We can get V 0, σ 0 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 t i : q i (1-R)E[max(f i,0)] -> Σu i v i ---(20.5) f i :the value of the derivative to the financial institution q i :the risk-neutral default probability R:recovery rate u i :q i (1-R) v i :the value today of the instrument

10.  In case 1. f i is always negative, so the expected loss is zero  In case 2. the max(f i,0) if always f i. v i is the present value of f i, it always equals f 0

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