1、報告者：陳博宇隨機利率下風險模型之衡量探討架構I.Longstaff&SchwartzII.鍾明章鍾明章III.xtensionFrancis A.Longstaff,Eduardo S.SchwartzI.A Simple Approach to Valuing Risky Fixed and Floating Rate Debt 11.Introduction 2.The Model3.Empirical Analysis4.ConcludingINTRODUCTIONThe articles developed a simple approach to valuing risky cor
2、porate debt that incorporates both default and interest risk.The model provides a number of interesting new insights about pricing and hedging corporate debt securities.THE MODEL-ASSUMPTION11 1.Let V designate the total value of the assets of the firm.dV=uVdt+VdZ is a constant and Z standard Wiener
3、process.2.Let r denote the short-term riskless interest whereis a22rate.dr=()dt+dZ ,and are constants and Z is also a standard Wiener process.3.The value of the firm is independent of the capital structure of the firm.4.Frwhere ollowing Black and Cox(1976),we assume there is a threshold value K for
4、the firm at which financial distress occurs.（1）（2）THE MODEL-ASSUMPTION5.If a reorganization occurs during the life of a security,the security holder receives 1-w times the face value of the security at maturity.6.We assume perfect,frictionless markets in which securities trade in continuous time.THE
5、 MODEL-ASSUMPTION222vvvrrrvrT From assum ption 6,w e can define the price H(V,r,T)of anyderivative security w ith payoff at tim e T contingent on the valuesof V and r.V H+V H+H+rV H+(-)H-rH=H22 w e also 222323use the value of a riskless discount bond D(r,T)is give by the V asicek(1977)m odel D(r,T)=
6、exp(A(T)-C(T)r),w here A(T)=(-)T+(-)(exp(-T)-1)2 -(4)(exp(-2T)-1),1-exp(-T)C(T)=.（3）（4）2122ttt0123ttttt0123ttttt0t()(,)(,)(,)(,)dB=rBdtself-financing trading strategy=(,)U()=H(V,r,t)+V+B t,T+BdU()=(dH(V,tttBttrtdrr dtdZdVrVdtVdZdBt Trdtr t T dZrdtp r t T dZBt T 假設有一個123ttttr,t)+dV+dB(t,T)+dB等式等式(3)的
7、推導的推導22tVVVrrrrV222ttVV1VVtrr2rr11dH(V,r,t)=H(V,r,t)dt+H(V,r,t)dV+H(V,r,t)(dV)+H(V,r,t)dr+H(V,r,t)(dr)22 +H(V,r,t)(dr)(dV)11 =Hdt+rVH dt+VH dZ+VH dt+(-r)Hdt+HdZ+Hdt22 rV222ttVtrVVrrrVV1r2 +H dt11 =H+rVH+(-r)H+VH+H+VH dt22 +VH dZ+HdZ 等式等式(3)的推導的推導By Ito lemma0123ttttt012012ttttttttt012012ttttttt0t(H(
8、V,r,t)+V+B(t,T)=-B(H(V,r,t)+V+B(t,T)dU()=(dH(V,r,t)+dV+dB(t,T)-rBdtB =(dH(V,r,t)+dV+d(B(t,T)-(H(V,r,t)+V+B(t,T)rdt-1 設令我們要222ttVtrVVrrrV112V1r2t tt1tt212trt2tt ttt11dU()=(-H-rVH-(-r)H-V H-H-VH)dt22 -VH dZ-HdZ+rVdt+VdZ+B(t,T)rdt +B(t,T)p(r,t,T)dZ+rHdt-rVdt-rB(t,T)dt=0 複製它等式等式(3)的推導的推導222ttvtrVVrrrVt1
9、2tV1trtr222tttVttrtVVt2rrtVrtt11-H-rVH-(-r)H-V H-H-VH+rHdt22+(V-VH)dZ+(B(t,T)p(r,t,T)-H)dZ=01H(V,r,t)+rVH(V,r,t)+(-r)H(V,r,t)+V H(V,r,t)21+H(V,r,t)+VH(V,r,t)-rH(V,2 t*222*VVrrr*tr,t)=0when t=T-tV H(V,r,t)+VH(V,r,t)+H(V,r,t)+(-r)H(V,r,t)22-rH(V,r,t)=H(V,r,t)等式等式(3)的推導的推導 0000002202222202()(1)12(,)ttt
10、ttatauaututatauatattttatauttuattatautttdra br dtdWrerabe due dWE rerabe duerb eVar rEee dWeeEeduarN Vasicek 零息債券價格推導零息債券價格推導00002()002()20(0,)exp()()(),(0)()(0)()(0)(0)()(1)(),()Tuutuauassautattua u tasasssua u tasDTEr duX urbdX taX tdWXrbX ueXe dWE X uXeXEX u dueaCov X tX ueEe dWe dWeeds 2()2()12ta
11、 u ta u teeaVasicek 零息債券價格推導零息債券價格推導atatttsuasuattttttttttteeatadudseeadudssXuXCovdudssXEsXuXEuXEdssXEdssXduuXEduuXEdssXduuXCovduuXVar2320 0)(2)(20 00 00000000432212)(),()()()()()()()(,)()(00(0)000223000()(1)(0)()2342(0,)exp1exp2ttua tttuatattuttuuEr duEX ub durbeb taVarr duVarX u duateeaDtEr duEr d
12、uVarr du Vasicek 零息債券價格推導零息債券價格推導THE MODEL-VALUING Fixed-rate debtFloating-rate debt 目前我們只討論fixed-rate debt的評價,因為Floating-rate debt對利率較不敏感 THE MODEL-VALUING FIXED-RATE DEBTTni11i=1Let P(V,r,T)denote the price of a risky discount bond with maturity data T.The payoff function:1-wI.VX:.K P(X,r,T)=D(r,T
13、)-wD(r,T)Q(X,r,T)where Q(X,r,T,n)=q,q=N(a),i-1iijijij=1ijiT-ln X-M(,T)n q=N(a)-q N(b),i=2,3,.,n.a=,iTS()njTiTM(,T)-M(,T)nn b=,iTjTS()-S()nn無風險的零息債券Vasicek的D(r,T)2222232322222-Tt-t22232-T-t3()()2 -(+)(1-exp(-t)+()(1-exp(-2 t)2-M(t,T)=(-)t2r +(+)e(e-1)+(-+)(1-e)2 -()e(1-e)2S tt tt22ttt1T2tt22tt1-marti
14、ngale measuredr=(-r)dt+dZdlnV=(r-)dt+VdZ2 forward martingale measure Pdr=(-r-C(t,T)dt+dZdlnX=(r-(t,T)dt+dZ2C 在在2222-(-u)u2uu200022u-(-2u)00d(e r)=e dr+r e dt =e(-r-C(t,T)+r)dt+dZ =e(-C(t,T)dt+edZ(1-e)r()e-r(0)=e du-edu+edZ =e-1-e du+edu+e u2022-2 uu220022-2u2220dZ =e-1-e-1+eedu+edZ =(-)e-1+ee-1+edZ2
15、 22-22-u20r()=r(0)e+(-)(1-e)+ee-e2 +eedZTTT22-uu-Tu-uT022000TuTTT2-us-(T-u)21000002222-T-23223lnX-lnX=redu+(-)(1-e)du+e(e-e)du2 +ee dZ(s)du-du-(1-e)du+dZ2r =(-+)1-e+(-)T+(+)e22 TTTuT2-T-T-us213000TuT-us21000(e-1)-e(1-e)+ee dZ(s)du+dZ2 =M(T,T)+ee dZ(s)du+dZTT-(T-t)T02100lnX=lnX+M(T,T)+(1-e)dZ+dZt00-l
16、nX-M(t,T)M(,T)-M(t,T)N()=q(0,|lnX,0)N()dS(t)S(t)-S()i-1iijijj=1iN(a)=q+q N(b)i=2,3,.,niTTq=q(0,|lnX,0)nnBy Fubinis TheoremBy Buonocore,Nobile,and Ricciardi(1987)1.(R的影響)2.用X作為違約指標.不同的對credit spread的影響.R的credit spread的影響.對credit spread的影響.對credit spread的影響EMPIRICAL ANALYSISThe regression equation is
17、given by S=a+b Y+cI+where S:credit spread.Y:the change in the 30-year Treasury yield.I:return on the appropriate equCONCLUDING 1.The approach can be applied directly to value risky debt when there are many coupon payment dates or when the capital structure of the firm is very complex.2.The important
18、 insightsCorrelation of a firms assets with changes in the level of the interest rate can have significant effect on the value of risky fixed income securities.The term structure of credit spreads can have a variety of different shapes.3.Empirical result Credit spreads are negatively related to the
19、level of interest rates.Difference in credit spreads across industries and sectors appear to be related to difference in correlations between equity returns and changes in the interest rate.鐘明璋隨機利率下信用風險之衡量隨機利率下信用風險之衡量使用創新之立體樹狀模型使用創新之立體樹狀模型1.目的2.方法 3.比較 目的運用正交化的方法簡化資產和利率的聯合機率運算運用樹狀架構的數值方法求出有違約風險的公司債券
20、價值近似解利用信用利差來觀察利率與公司信用風險之間的關聯非線性誤差非線性誤差是選擇權價值函數所引起的。門檻選擇權的報酬函數是根據在到期日公司資產有無碰到門檻值而定。STAIR TREE方法運用正交化的方法將具有相關性的公司價值和隨機利率轉換成彼此獨立的新變數運用新變數建造數值樹狀模型資產部位：BTT Tree利率部位：二元樹並將二獨立的隨機變數的機率相乘，求得聯合機率計算其資產與利率的價值，最後導出其風險性債券的期初價值，並計算出其信用風險溢酬（credit spread）正交化公司資產標GMB:利率Vasicek:其中 ，與 獨立1ZttdVrVdtVd22121()Z()(1)()1ttt
21、tttdra b r dtda b r dtdBdZa b r dtdBdZ2211ZtdZdBdtdB1Zd正交化將利率和資產表示成矩陣形式並試找到來正交化資產與利率2210ln2Z1()ttttttdVdBrdtdrda br 1201 11001ZXtYtdXdBdtdYd22-1-101 222221222222-1-1ln10111-1-201Z11()00()ln()1()121-111ln()2tttttttttdVdBrdtdrda brra brV tdr tdV tr12110011()10101ttrXttttXd BdtdZuudYd BdtdXdZu22ln()/(0
22、)1()()1()(0)()(1)ttttXttVtVXr tYXVtVer tYX示意圖 X:Y:：BTT treeY:二元樹示意圖Default Barrier利用樹狀結構的選擇權評價方法可求出具有違約風險的債券價值另外經由 求出風險利差，就可以觀察其信用風險與其相關因子的關係。1(,)(0,)ln()*(,)P x r TC STTDD r T 與LONGSTAFF&SCHWARTZ的模型比較X與LONGSTAFF&SCHWARTZ的模型比較W與LONGSTAFF&SCHWARTZ的模型比較R與LONGSTAFF&SCHWARTZ的模型比較與LONGSTAFF&SCHWARTZ的模型比較
23、與LONGSTAFF&SCHWARTZ的模型比較多討論公司資產發生jump的情況與LONGSTAFF&SCHWARTZ的模型比較多討論稅盾的影響陳博宇隨機利率下信用風險之衡量隨機利率下信用風險之衡量使用使用Hull-White Model目前的想法鐘明璋（2007）EDFPM的隨機利率是運用Vasicek model下的二元樹去模擬。運用Hull-White model(no-arbitrage model)的二階段三元樹去模擬。建構出立體二維樹後，求出信用風險溢酬與其經濟參數做比較。參照Eric Briys&Francois de Varenne(1997)所定的隨機threshold運用實驗方法去支持發現的論點Thank for your attention
