The simplest way to visualize such models is to suppose that � evolves

# The simplest way to visualize such models is to

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The simplest way to visualize such models is to suppose that λ evolves according to a trinomial tree. A standard binomial tree model is somewhat unrealistic because it implies the intensity grows exponentially on average, whereas it is more likely to be mean-reverting. At node ( n, j ) on the tree, the intensity is denoted λ n,j . Default occurs at the end of the period between ( n Δ t, ( n + 1)Δ t ) with (risk-neutral) probability λ n,j Δ t . This is motivated by the fact that for the simple case where τ is exponential with parameter λ (constant), we have IP Q { τ > t + Δ t | τ > t } = e - λ Δ t 1 λ Δ t, where the approximation is good for small Δ t . In other words, the probability of default over the time period ( t, t t ] given survival up till time t is approximately 1 (1 λ Δ t ) = λ Δ t . To price a zero-coupon defaultable bond with no recovery on default, we initialize the end of the tree V N,j = 1 j = N, N + 1 , · · · , 0 , 1 , · · · , N. Then the value is found backwards via V n,j = e - r Δ t [ λ n,j Δ t × 0 + (1 λ n,j Δ t )( q u V n +1 ,j +1 + q m V n +1 ,j + q d V n +1 ,j - 1 )] , with analogous formulas for the other two branching cases. We point out that the probabil- ities depend on j . To incorporate (deterministic) recovery, the 0 in the previous algorithm is adjusted appropriately. As well as corporate bonds, there is a (erstwhile) very liquid market in Credit Default Swaps (CDSs), which provide protection against a single firm’s default event (an insurance payout), in return for a yield above default-free rates. CDS quotes are commonly used to infer the market’s (risk-neutral) probability of the firm defaulting. 80
• Fall '11
• COULON
• Variance, Probability theory, Trigraph, Credit default swap, Wiener process

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