Credit Risk Models Based on Time Changed Brownian Motion - ICMS
Credit Risk Models Based on Time Changed Brownian Motion - ICMS
Credit Risk Models Based on Time Changed Brownian Motion - ICMS
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<str<strong>on</strong>g>Risk</str<strong>on</strong>g> 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 ∗ (byOpti<strong>on</strong>al Sampling);3 eL (2)t is a F t -martingale, where L (2)t := X Gt∧t ∗ is the time-changedstopped martingale.4 Therefore e −rt S t = D 0 (e L(2) t5 The model is arbitrage-free!− 1) is a F t -martingale;6 St first hits zero at precisely t (2) , and remains zero thereafter.Tom Hurd (McMaster) <strong>Time</strong> <strong>Changed</strong> <strong>Brownian</strong> Moti<strong>on</strong> <strong>ICMS</strong> 2007 19 / 20
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