Pricing of CDO s Based on the Multivariate Wang Transform*

Transcription

1 Prcg of DO s Based o he Mulvarae Wag Trasform* ASTIN 2009 Helsk 02 Jue 2009 Masaak Kma Tokyo Meropola versy/ Kyoo versy Emal: kma@mu.ac.p hp:// * Jo Work wh Sh-ch Moomya ad Yoch Suzuk red Prcg orporao P Tokyo Japa 0

2 Pla of my Talk Prcg Prcples for Isurace Rsk Esscher rasform Wag Trasform Eeso o he Mulvarae Seg The Prcg of DO s Image of DO Syhec DO Sadard Model Proposed Models for he Prcg of DO s Numercal Eamples

3 Premum Prcples for Isurace Rsk I he acuaral leraure a popular mehod for he prcg of facal ad surace rsks amog ohers s δ δ he Esscher rasform: π E[ e ]/ E[e ] where δ > 0 sads for rsk adusme. Recely Wag 2002 ASTIN proposed a prcg mehod based o he followg rasformao: [ δ ] δ > The dsoro descrbes rsk adusme. 0 Noe: Isurace markes are complee ad ehb fa-aled dsrbuos as for DO markes.

4 Recall ha Esscher: Wag: δ E[ e ] π δ > 0 δ E[e ] π d ; [ δ ] Ma drawback of he Esscher rasform for he praccal use he dscree-me seg s ha he MG E[e δ ] mus es couerpar of Novkov s codo. The Wag rasform has a mer hs aspec. Also Wag 2002 ASTIN showed ha he rasform s he oly rasform amog he famly of dsoros ha ca recover APM ad he Black-Scholes formula for opos.

5 The Buhlma's equlbrum prcg model 980 ASTIN: - pure rsk-echage ecoomy - epoeal uly wh dsc rsk averso de - age faces a rsk of poeal loss He derved he followg equlbrum prcg formula for rsk Y where Z e π Y E[ ηy ] η Z E[e ] Z Z s he aggregae rsk ad > 0 s he rsk averso de of he represeave age he marke. The Esscher rasform ca be reduced from he Buhlma's formula by assumg Y << Z whece has a soud ecoomc erpreao.

6 I ca be show ha he Wag rasform s he same as he Esscher rasform for ormally dsrbued rsks. Moreover Wag 2003 ASTIN showed ha hs rasform ca be derved from Buhlma's formula eve for geeral rsk uder some assumpos o Z. Hece he Wag rasform also has a soud ecoomc erpreao. However acuaral prcg formulas cludg he Esscher ad Wag are o lear yeldg arbrage opporues. π a by aπ bπ Y Kma 2006 ASTIN developed a mulvarae Wag rasform based o he Buhlma's equlbrum prcg formula.

7 The mulvarae Wag rasform y Σ K K [ k The ormal case: y ] k σ Z w y k k :D of N 0 Σ Rsk averso de of he represeave age I hs case we have σ w σ k kk σ Z ad μ / σ σ Z K y K y y [ Wag chages μ μ Z μ Z] / σ rsk premum

8 Prcg of Traches DO L D A Noe: We assume ha he recovery s zero. M D A { L < A} D L { A L < D} B 0 e r s ds : Moey Marke Accou Remag Prcpal a me for he rache V T cm MT E d 0 B B T The prese value of cash flows arsg from he rache.

9 Sadard Model: Oe-acor Gaussa opula Model I geeral Assumed o be cosa L M N M δ N { τ } We eed o model 2 τ τ K τ he boom-up seg. osder he Mero s srucural model: τ logv where V sads for he frm value. So sead of modelg he correlaed τ τ K τ should be easer o model he correlaed 2 2 K

10 Take K o be lae varables such ha: 2 2 uder where has he D H has G ad has K By defo { τ } { } K Supposg for all o be calbraed Eample The dusry sadard model: s follow a ormal. The K s also ormal ad τ Eample 2 Hull ad Whe 2004: s follow a dsrbuo. The K eeds o be evaluaed umercally

11 Z N 2 0 ~ Z : d K Σ Σ K K K 2 As for redmercs we sar wh assumg ha uder P : follow N0 : Aggregae rsk hage of Measures from P o } { } { τ Apply he mulvarae Wag. rsk-adused log-frm values By defo : Σ K

12 2 } { } { q τ odoal o he oe-facor model we have : Σ K rsk averso de : o D of uder τ's : margal D of uder τ The rsk premum s embedded he margal defaul D ha s calbraed from marke quoes for DS s u u u q d φ : he sadard model f By depedece follows ha Derved from Mero s srucural model ad Buhlma s prcple whece has a soud ecoomc erpreao.

13 Rsk-Adused Gaussa opula Model I he sadard model he correlao s he oly free parameer ha ca be calbraed from marke quoes. The correlao should be hough of as he rsk premum. However he wha s he defaul correlao? Noe: I he Gaussa model he chage of measures does o chage he varace-covarace srucure. The correlao parameers should be esmaed uder P ad he rsk premum s calbraed from marke quoes uder ac: The DO marke s segmeed o raches accordg o vesors prefereces agas rsks.

14 Whe evaluag he rache we use : D D Σ K rsk premum for DS of ame : margal D of uder τ Parameer esmao ad calbrao Σ ; D D d K K The rsk premum for rache wh deachme D where ε 2 se he smple regresso o esmae : calbraed from marke quoes for DS s D : calbraed from marke quoes for he DO rache Assume:

15 年 3 月 Prces of Tra Traches sce 2006/ 年 月 2006 年 2 月 2007 年 月 2007 年 2 月 2007 年 3 月 2007 年 4 月 2007 年 5 月 2007 年 6 月 2007 年 7 月 2007 年 8 月 2007 年 9 月 2007 年 0 月 2007 年 月 2007 年 2 月 2008 年 月 2008 年 2 月 2006 年 9 月 2006 年 0 月

16 Averaged orrelao uder P sce 2006/9 相関 平均 推移 系列 2006 年 月 2006 年 2 月 2007 年 月 2007 年 2 月 2007 年 3 月 2007 年 4 月 2007 年 5 月 2007 年 6 月 2007 年 7 月 2007 年 8 月 2007 年 9 月 2007 年 0 月 2007 年 月 2007 年 2 月 2008 年 月 2008 年 2 月 2008 年 3 月 2006 年 9 月 2006 年 0 月

17 albraed Rsk Premum urves sce 2007/6/ Traso of urve2007_ _ % 6% 9% 2% 22% 2007_ _ _ _ _0 2007_ 2007_2 2008_0 2008_ _ Deachme Po

18 Rsk-Adused opula Model The rsk-adused Gaussa model ca f perfecly marke quoes for all raches of sadard DO s by calbrag he rsk adusme parameers D Noe: We have erpreed D as he rsk averso de for rache D of he represeave age he marke. I may be more plausble o assume ha s creasg D We cosder e.g. he case Also facal markes ofe ehb fa-aled dsrbuos. D D a b log D Iroduce he copula framework Easy o calculae

19 0 ] [ > θ θ α Y E Y G α where Y > 0 ad for some G. Eeso by Kma ad Muromach 2008 IME I parcular Y mples he Wag rasform. : rsk premum ] [ ] [ : P Y G E Y ν θ ν θ χ ν /ν 2 Y Also whe we have where deoes he D of a o-ceral dsrbuo. : P δ ν 0 ] [ > θ θ θ Y G E Y However agas our epecao we ca prove Therefore followg he Wag s dea we propose Z k k k k w y y y σ ν Σ ] [ K K

20 D β β ν Σ : 2 τ τ τ K The Proposed Model: or However hs formula volves a double egral. Adop he followg appromao Prcg of DO s based o he 2-parameer Wag Trasform uder D Y Y 2 2 : / ν χ ν ν ν Assume Defe 2 u u Y u u δ ν δ ξ 2 : } { } { D P q τ δ ν

21 albrao Resuls A-.DJ Tra 5-year Ide TrachesER 2004/8/23 Traches 0-3% 3-6% 6-9% 9-2% 2-22% RMSE Marke md Prce 25.5% Bd/ask spread.3% Jump-dffuso eses 25.0% Pure dffuso eses 30.0% Gaussa copula 27.4% RL Gaussa copula 25.3% Double- copula 24.0% Rsk-Adused opula 25.8% ed parameers : μ ldeachme Po

22 A-2.DJ Tra 5-year Ide TrachesER 2005/2/5 Traches 0-3% 3-6% 6-9% 9-2% 2-22% RMSE Marke md Prce 26.3% Bd/ask spread 0.6% Jump-dffuso eses 28.7% Pure dffuso eses 32.5% Gaussa copula 34.6% RL Gaussa copula 27.0% Double- copula 29.8% Rsk-Adused opula 26.5% ed parameers : μ ldeachme Po /8/23 D 2005/2/5

23 ocludg Remarks. We showed ha corary o he crcsm he oe-facor Gaussa copula model s cosse wh Buhlma's ecoomc premum prcple whece has a soud ecoomc erpreao. 2. Based o hs fdg we developed a alerave wh he Buhlma's framework. Namely we roduce he rsk averso de for each rache o be calbraed whle keepg he correlao srucure as gve uder he acual probably measure; we also apply he Sude copula. 3. Numercal epermes reveal ha our model provde a beer f ha he esg models he leraure. 22

24 Thak You for Your Paece