Basics of Credit Risk - Universität Hohenheim

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Investment Banking and Capital Markets – Universität Hohenheim Investment Banking and Capital Markets A Multi-Step Model ◮ The value of the claim in this case is ◮ thus, we arrive at V risky i−1 V0(m) = = (1 − pi)(V risky i mX i=1 S(i)V cf i + mX i=1 + V cf i ) + piV default i (S(i − 1) − S(i))V default i (28) (29) which is the price of a central pricing relation for credit-risky instruments with multiple cash-flows, you can value regular bonds, zero coupon bonds, credit derivatives such as credit default swaps, and even portfolio derivatives such as collateralised debt obligations. ◮ survival function and the recovery model necessary ◮ if V default is independent of the time, the second term of (29) will collapse to (1 − S(m))V default Chair for Banking and Finance Winter term 2009 Slide 14
Investment Banking and Capital Markets – Universität Hohenheim Investment Banking and Capital Markets Default Probability in a Continuous-Time Approach ◮ τ ∗ : stochastic variable as the time until default ◮ Ft(τ) = P(τ ∗ ≤ τ) St(τ) = P(τ ∗ > τ) with ◮ Ft(0) = 0, lim Ft(τ) = 1, Ft(τ + ∆τ) ≥ Ft(τ) ∀ ∆τ > 0, and St(τ) = τ→∞ 1 − Ft(τ) ◮ The default probability of the claim in the time frame τ to τ + ∆τ is Ft(τ + ∆τ) = P(τ ∗ ≤ τ + ∆τ) which is equivalent to = P(τ ∗ ≤ τ) + [1 − P(τ | {z } Ft (τ) ∗ ≤ τ)] | {z } St (τ) · P(τ ∗ ≤ τ + ∆τ|τ ∗ > τ) | {z } probability for τ ∗ ∈[τ,τ+∆τ] St(τ) − St(τ + ∆τ) = St(τ) · P(τ ∗ ≤ τ + ∆τ|τ ∗ > τ) (30) Chair for Banking and Finance Winter term 2009 Slide 15
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Basics of Credit Risk - Universität Hohenheim

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