Credit Risk Models Based on Time Changed Brownian Motion - ICMS

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<strong>Risk</strong> Neutral DynamicsSuppose W t is a BM. Note1 eX tis a σ(W )-martingale if X t = x + W t − t/2;2 eX t∧t∗ is a σ(W )-martingale for any σ(W )-stopping time t ∗ (byOptional Sampling);Tom Hurd (McMaster) Time Changed Brownian Motion ICMS 2007 19 / 20
<strong>Risk</strong> Neutral DynamicsSuppose W t is a BM. Note1 eX tis a σ(W )-martingale if X t = x + W t − t/2;2 eX t∧t∗ is a σ(W )-martingale for any σ(W )-stopping time t ∗ (byOptional Sampling);3 eL (2)t is a F t -martingale, where L (2)t := X Gt∧t ∗ is the time-changedstopped martingale.Tom Hurd (McMaster) Time Changed Brownian Motion ICMS 2007 19 / 20
Page 1 and 2: Credit Ris
Page 3 and 4: Toronto: the place to be in 2010 !W
Page 9 and 10: Two outstanding issues in credit ri
Page 11 and 12: Black-Cox (1976)A time-consistent m
Page 13 and 14: Other Approaches1 Reduced form mode
Page 15 and 16: Jump-Diffusion Default Mode
Page 17 and 18: First passage times for time-change
Page 19 and 20: Example1 Take L0 = 1;2 Take b = 0.0
Page 21 and 22: NumericsYield Spread0.35$L 0=0.3$0.
Page 23 and 24: Computing the Joint Default Distrib
Page 25: Risk Neutral Dynam
Page 29 and 30: Risk Neutral Dynam
Page 31 and 32: Equity-Credit Comp
Page 33 and 34: Conclusion1 By taking Lt = X Gt wit

Credit Risk Models Based on Time Changed Brownian Motion - ICMS

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