# Modeling Term Structures of Defaultable Bonds

Modeling Term Structures of Def aultable Bonds Duffie and Singleton (1999) Presented by Xicheng Jiang Outline Introduction of defaultable claims modeling Consider alternative recovery methods

Valuation of defaultable bonds Review For a contingent claim at , given its real-world contd return : Using the equivalent martingale approach: If the risk-free rate is random process (this is the case in most fixed-in come modelling)

Hazard Rate Survival function: , which is decreasing Default probability: Conditional default probability: Density of conditional default probability: Hazard rate: As a result, the conditional default probability in a short time interval can be written as

Intuition Short rate process and equivalent martingale measure Let denotes the hazard rate for default at time Let denotes the expected fractional loss in market value if default we re to occur at time , conditional on The initial market value of the defaultable claim to is where the default-adjusted short-rate process

Need to be proven under both discrete and continuous settings Defaultable Claims in Discrete Space Let denotes the dollar amount of recovery given default at time . Whats the mar ket value of an asset , given future recovery given default and future value given no default? Recursively solving forward...

Defaultable Claims in Discrete Space Suppose we adapt RMV (recovery of market value) assumption here, i.e., take t he RN expected recovery as a fraction of RN expected survival contingent market value. Substitute it into the expression: Where Or

Defaultable Claims in Discrete Space Why this representation is good? If we assume that and are exogenous process, we can just model for the defaultable bonds, instead of , using single- or multifactor model s uch as CIR or Vasicek, or HJM model. State Dependence is accommodated, i.e., and may be correlated wit h each other, with , with economic cycle...

If the exogeneity is violated, we must find other methods. (For exampl e, market value of recovery is fixed..) Defaultable Claims in Continuous Space Contingent claim : random variable and stopping time where is paid. is measurable. The ex-dividend price process for is given by: Defaultable claim : is the obligation of issuer to pay at . defines the s

topping time when the issuer defaults and is recovered. Actual claim generated by such a defaultable claim is defined by: Defaultable Claims in Continuous Space Note that is random by nature since we dont know when the issuer defaults. We model by setting a variable From the definition of hazard rate, we know that the instantaneous co

nditional default probability can be written as . However, in this case, a defaultable claim can only default once. Once it defaults the probabi lity will become one and will never change. To model this, we rewrite the probability as After adding a demean process , we can get Defaultable Claims in Continuous Space The payoff at default is also random. It is modeled as

where is the price of the claim just before default Key assumption is that this is predictable by the information up to , i. e., Defaultable Claims in Continuous Space We know that if we discounted the gain from an asset by the short-rat e process , the gain process must be a martingale under . The discounted gain process is defined by:

This is a martingale. What should we do to get ? Itos Lemma. Let and use the fact that , we can get Defaultable Claims in Continuous Space Given the terminal boundary condition , we can get: where We can see another advantage of this model. In its final form, we can ge

t rid of , and . That being said, we dont need to model the characteristi cs of the defaultable claim. Instead, only by considering the non-default able contingent claim and changing the discount rate can get the final a nswer. Some Special Cases Continuous-time Markov formulation: Assume a state variable proces s that is Markovian

solves a SDE: solves the backward Kolmogorov PDE: with boundary condition Some Special Cases Price-dependent expected loss rate: Corresponding PDE can be treated numerically Uncertainty about recovery:

where is a bounded, measurable random variable With this change, the pricing formula still applys. Defaultable Bonds: Recovery and valuation Consider the following recovery methods: Under RT, the computational burden of directly computing can be su

bstantial. Time of default, the joint -conditional distributions of for all between and plays a computationally challenging role in determinin g Defaultable Bonds: Recovery and valuation RMV: RMV vs RFV: RMV matched to the legal structure of swap contract bette r. RMV model is more convenient for corporate bonds because we can j

ust apply standard default-free term-structure modelling techniques. R FV, on the other hands, is more realistic when absolute priority applies. Is there a significant difference between RMV and RFV model? For simplicity, we take , a constant. We model and by where is square root diffusions under Defaultable Bonds: Recovery and valuation

Under RMV assumption: where Under RFV assumption: where Defaultable Bonds: Recovery and valuation

Calibrate the RMV and RFV model: Bonds with fixed ten-year par-coupon spreads. (known ) Fixed and are modelled by several square-root diffusion processes Minimizing the error between model estimated bond prices and real bond prices through changing the parameters of and . Compute the mean implied intensity

Noncallable Corporate Bonds Note that the hazard rate process and the fractional loss enter the di scount rate in the product from Knowledge of defaultable bond prices before default alone is not suffi cient to separately identify and If one has prices of undefaulted junior and senior bonds of the same i ssuer, along with the prices of the Treasury bonds, we can extract and , thus can infer .

We can just model jointly the dynamic properties of and the short sp read Noncallable Corporate Bonds Case 1: Square root diffusion model of Dai and Singleton (1998) proposes the most flexible affine term stru cture model

where is a 3*3 matrix with positive diagonal and nonpositive off- dia gonal elements; in ; is 3*3 diagonal matrix with diagonal elements , a nd Noncallable Corporate Bonds Duffie (1999) considered the special case in which and , so could take on negative values and depend only on the first two state variables. He also assumed that is diagonal (, and are independent)

However, the only means of introducing negative correlation among and is to allow for negative , which means the hazard rate may take o n negative values. Within this correlated square-root model of (, ), one cannot simultane ously have a nonnegative hazard rate process and negatively correlate d and without having one or more or negative. Noncallable Corporate Bonds

Case 2: More flexible correlation structure for (, ) We assume that where is a 3*3 matrix with positive diagonal and nonpositive off- dia gonal elements; in ; and Noncallable Corporate Bonds All of , , are strictly positive

Dai and Singleton show that in this case the most flexible and admissi ble affine term structure has: The short-spread rate is strictly positive. At the same time, the signs o f and are unconstrained, so the third state variable may have increm ents that are negatively correlated with the first two. Valuation of defaultable callable bonds

During the time window of callability, we have the recursive pricing fo rmula Outside the callability window, Valuation of defaultable callable bonds In a more continuous time context, let denote the set of feasible call

policies. The market price at is: where This equation can be solved by a discrete algorithm similar to the equ ations in the previous slide. More (in the paper) Defaultable HJM model

Pricing Credit Derivatives Etc. Take home message The initial market value of the defaultable claim to is where the default-adjusted short-rate process All financial products with defaultable nature can be modeled in this way.

If we assume that and are exogenous process, we can just model the process for the defaultable bonds instead of . My remarks (pros) Very detailed. A variety of models regards to defaultable claims/bonds under differe nt assumptions are given. These models can be directly applied in the market using market data

(calibration). Some brief comparisons of models are given. My remarks (cons) Too much theoretical stuffs, should give more calibration (I mean to b e theoretical is good, but it is always better to give some empirical res ults) More like a encyclopedia instead of a paper. (always introduce a mode

l and say please refer to some other papers. I think a paper should f ocus one or a few models and dig deeper). Without a clear conclusion. (which model is good or bad under which conditions?) Thank you

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