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# BasketCDSPres - Structural Models of Credit Spectral...

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Structural Models of Credit Spectral Element Methods Structural Models of Credit: A spectral Element Approach. Pierre Garreau Financial Mathematics Seminar Florida State University November 17, 2011

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Structural Models of Credit Spectral Element Methods 2 3 4 5 6 7 8 9 10 01/01/05 01/01/06 01/01/07 01/01/08 01/01/09 01/01/10 01/01/11 01/01/12 Baa Aaa 20Y TBill Figure: 20-Year Aaa and Baa bond spreads.
Structural Models of Credit Spectral Element Methods Definition A credit spread is the premium paid to insure the risk of default of the underlying entity, i.e. it is the difference between the risk free rate, r t , and the yield on the corporate bond, y t . B t = e - r t t D t = Le - y t t s t = ln D t / LB t t

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Structural Models of Credit Spectral Element Methods 0 1 2 3 4 5 6 08/13 08/27 09/10 09/24 10/08 10/22 11/05 11/19 MF Global Figure: MF Global stock price.
Structural Models of Credit Spectral Element Methods Asset Liabilities V t S t D t Table: Modigliani-Miller theorem

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Structural Models of Credit Spectral Element Methods t s t s t s t s t (1 R ) t 0 = 0 τ t N = T . . .
Structural Models of Credit Spectral Element Methods DL = E bracketleftBig (1 R )1 τ< T e - integraltext τ 0 r s ds bracketrightBig t s t s t s t s t (1 R ) t 0 = 0 τ t N = T . . . FL = E bracketleftBigg N summationdisplay i =1 s t 1 τ> t i e - integraltext t i 0 r s ds bracketrightBigg

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Structural Models of Credit Spectral Element Methods DL = (1 R ) integraldisplay T 0 B (0 , s ) d P ( τ s ) t s t s t s t s t (1 R ) t 0 = 0 τ t N = T . . . FL = N summationdisplay i =1 s t P ( τ > t i ) B (0 , t i )
Structural Models of Credit Spectral Element Methods DL = (1 R ) parenleftBigg 1 e - rT P ( τ > T ) r integraldisplay T 0 P ( τ > s ) e - rs ds parenrightBigg t s t s t s t s t (1 R ) t 0 = 0 τ t N = T . . . FL = N summationdisplay i =1 s t P ( τ > t i ) e - rt i

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Structural Models of Credit Spectral Element Methods s = (1 R ) parenleftBig 1 e - rT P ( τ > T ) r integraltext T 0 P ( τ > s ) e - rs ds parenrightBig N i =1 t P ( τ > t i ) e - rt i
Structural Models of Credit Spectral Element Methods Default time τ = inf { t : min 0 s t V s D } P ( τ > t ) = P (min 0 s t V s > D ) = E bracketleftbig 1 min 0 s t V s > D bracketrightbig D T = R T . 1 { τ T } + D . 1 { τ> T } 4 6 8 10 12 14 16 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Figure: V t = V 0 e μ t + σ W t + N t i =0 Y i

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Structural Models of Credit Spectral Element Methods Default time τ = T . 1 { V T D } + 1 { V T > D } P ( τ > T ) = P ( V T > D ) D T = V T . 1 { V T D } + D . 1 { V T > D } 4 6 8 10 12 14 16 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Figure: V t = V 0 e μ t + σ W t + N t i =0 Y i
Structural Models of Credit Spectral Element Methods

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BasketCDSPres - Structural Models of Credit Spectral...

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