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Jhammer
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Posts: 37
Joined: February 24th, 2014, 8:05 pm

Credit Modeling in Interest Rate Tree

February 9th, 2016, 5:42 pm

Let say you have to use a pricing tree for valuation of a defaultable callable (or putable) bond. Short interest rate is assumed to follow HW one-factor model. Hazard rate follows a similar HW one-factor model correlated with short rate. Recovery rate is constant. This framework is discussed in "A TREE IMPLEMENTATION OF A CREDIT SPREAD MODEL FOR CREDITDERIVATIVES" by PHILIPP J. SCHONBUCHER and a trinomial pricing three is calibrated. In this paper, an additive credit spread tree is calibrated to model the stochastic hazard rate. Also, an absorbing default state is added to model default.Here is my question:Why credit is needed to be modeled in two different places? i.e., additive spread to interest rate + absorbing default state.Shouldn't we ALWAYS discount by using risk-free rate? Wouldn't the absorbing default state be enough?Hope this thread attracts some responses!
 
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cenperro
Posts: 31
Joined: October 7th, 2014, 1:33 pm

Credit Modeling in Interest Rate Tree

February 10th, 2016, 3:37 pm

I might not be fully understanding your question but, while the default absorbing state is needed for any default payment (e.g. maybe a Recovery in your callable bond case) you need to 'discount' all you alive payoffs taking into account the 'survival probability'... Bear in mind that, between 2 time nodes in your tree) any credit contingent payoff (X_T) is calculated as: Numeraire_(T-1) x E[1_{tau>T} X_T / Numeraire(T)] + Numeraire_(T-1) x E[1_{tau<T} R/Numeraire(T)] (Sorry for the crappy notation...first is the alive payment and the second is the default one, the one coming from the absorbing state)How do you 'discount' (calculate that expectation in the tree) without modelling those hazard rates and using them in the 2 places?This way, in your i,j node you will have a value of your short rate r_i and your hazard rate (h_j) and from moving backward the payoff from T_k+1 to T_k you need to bring back your default payment (the absorbing state) but also the alive payment (that only happens if alive, so 'discounted' also with the hazard rate)hope this helps.
 
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Jhammer
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Posts: 37
Joined: February 24th, 2014, 8:05 pm

Credit Modeling in Interest Rate Tree

February 10th, 2016, 3:54 pm

Thanks for this: "Numeraire_(T-1) x E[1_{tau>T} X_T / Numeraire(T)] + Numeraire_(T-1) x E[1_{tau<T} R/Numeraire(T)] "So we need to incorporate both 'survival' and 'default' states in pricing the instrument, which are two disjoint events.
 
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Jhammer
Topic Author
Posts: 37
Joined: February 24th, 2014, 8:05 pm

Credit Modeling in Interest Rate Tree

February 10th, 2016, 6:12 pm

I was mixing up transition probabilities with survival probabilities; and hazard rate dynamics appear in both..
 
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cenperro
Posts: 31
Joined: October 7th, 2014, 1:33 pm

Credit Modeling in Interest Rate Tree

February 10th, 2016, 6:21 pm

no worries..we all have been there :-)
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