Extended-Gaussian Term Structure Models and Credit Risk Applications

This paper presents three factor "Extended Gaussian" term struc- ture models (EGM) to price default-free and defaultable bonds. To price default-free bonds EGM assume that the instantaneous interest rate is a possibly non-linear but monotonic function of three latent factors that follow correlated Gaussian processes. The bond pricing equation can be solved conveniently through separation of variables and finite difference methods. The merits of EGM are hetero-schedastic yields, unrestricted correlation between factors and the absence of the admissibility restric- tions that affect canonical affine models. Unlike quadratic term structure models, EGM are amenable to maximum likelihood estimation, since ob- served yields are sufficient statistics to infer the latent factors. Empirical evidence from US Treasury yields shows that EGM fit observed yields quite well and are estimable. EGM are of even greater interest to price fixed and floating rate defaultable bonds. A reduced form, a credit rating based and a structural credit risk valuation model are presented: these credit risk models are EGM and their common merit is that bond pricing remains tractable through separation of variables even if interest rate risk and credit risk are arbitrarily correlatedbond pricing, Gaussian term structure models, Vasicek model, separation of variables, finite difference method, reduced form, credit risk model, credit ratings model, structural model.

Corporate Bond Valuation with Both Expected and Unexpected Default

This paper presents three variants of a tractable structural model in which default may take place both expectedly and unexpectedly. The model has the merit of predicting realistically high short term credit spreads. Closed form solutions are provided for corporate bonds (and default swaps) when interest rates are constant or stochastic and when the bond recovery value is exogenous or endogenous to the model. The analysis suggests that, in order for the observed short term yield spreads on high grade corporate bonds to be compensation for credit risk, bond holders must believe that a dramatic sudden plunge in the firm's assets value is possible, even if extremely unlikely.Corporate bond valuation: Structural model; Unexpected default; Short term credit spreads; endogenous bond recovery value; plunge of assets value

A Two Factor Black-Karasinski Credit Default Swap Pricing Model (forthcoming in the Icfai Journal of Derivatives Markets, Vol IV, No 4, October 2007; all copyrights rest with the Icfai University Press)

This paper presents, estimates and tests a reduced form sovereign credit default swap (CDS) pricing model where the default intensity is driven by two latent Black-Karasinski-type processes. CDS pricing re- quires finite difference numerical solutions, but parameter estimation is still feasible. Evidence from a sample of sovereign CDS rates shows the good empirical performance of the model and that a second stochastic factor driving the default intensity is statistically significant. Surprisingly the evidence fails to support the view that the risk associated with the dynamics of the default intensity is priced. For all countries the bulk of variations of the default intensity are explained by just one factor. As a by-product, a viable methodology for maximum likelihood estimation of pricing models with two latent factors is provided despite the fact that the pricing requires numerical solutions through finite difference methods.sovereign CDS pricing, reduced-form credit risk model, Black-Karasinski, implicit .nite di¤erence method, maximum likelihood estimation.

The Target Rate and Term Structure of Interest Rates

This paper presents a tractable bond valuation model, which further develops the approach proposed by Piazzesi (2005). The short term inter-bank interest rate is equal to the target rate set by the central bank plus a spread. Bond yields are driven by the intensities that determine the probabilities that the central bank may raise or cut the target interest rate. Unlike in Piazzesi (2005), negative intensities have a convenient interpretation and do not complicate estimation, and two accurate approximations to the bond pricing equation provide new closed form solutions for discount bond prices that require no numerical integration. Unlike in Piazzesi the target interest rate can be constrained to be non-negative. Yields, especially long term ones, decrease when the central bank is expected to decide more frequent and/or larger average future changes in the target interest rate. The model lends itself to easy calibration and estimation.Bond valuation, target interest rate, closed form solution, yield curve, central banker's meeting

Book Values and Market Values of Equity and Debt

This paper propses a contingent claims model to value a firm's debt and equity as functions of observable book values appearing in published financial statements. Equity fair value critically depends on expected earnings, equity book value and earnings volatility, because of the options to default or to voluntarily liquidate the firms. Debt value increases in earnings volability in the proximity of default. Default is triggered by the erosion of equity due to negative earnings. Debt and equity values are materially affected by the strength of the mean reversion of profitability. Voluntary liquidation before default may be optimal and it entails that a sudden sharp decline in profitability can be less detrimental to creditors than a slower but persistent one.Book values, mean reverting return on assets, equity valuation, debt valuation, default option, structural models, voluntary liquidation.

Valuation of Put Options on Leveraged Equity

This paper presents new closed form solutions for the valuation of European put options and of "down-an-in" barrier options written on leveraged equity. Unlike in past literature (Toft and Prucyk, 1997) and in keeping with empirical evidence, the model allows equity to retain value even after the firm's default and reorganisation. This stylised fact can significantly alter the valuation of equity put and "down-and-in" options as bankruptcy costs, bargaining power of equity holders, debt maturity and other firm parameters change. The value of "in-the-money" puts often decreases in the firm's assets volatility. The model can produce a variety of realistic implied equity volatility "skews".Equity put options; Leveraged equity; Default and reorganisation; Barrier options; "down-and-in" options

An Extended Structural Credit Risk Model (forthcoming in the Icfai Journal of Financial Risk Management; all copyrights rest with the Icfai University Press)

This paper presents an extended structural credit risk model that pro- vides closed form solutions for fixed and floating coupon bonds and credit default swaps. This structural model is an "extended" one in the following sense. It allows for the default free term structure to be driven by the a multi-factor Gaussian model, rather than by a single factor one. Expected default occurs as a latent diffusion process first hits the default barrier, but the diffusion process is not the value of the firm's assets. Default can be "expected" or "unexpected". Liquidity risk is correlated with credit risk. It is not necessary to disentangle the risk of unexpected default from liquidity risk. A tractable and accurate recovery assumption is proposed.structural credit risk model, Vasicek model, Gaussian term structure model, bond pricing, credit default swap pricing, unexpected default, liquidity risk.

Quadratic Term Structure Models in Discrete Time

This paper extends the results on quadratic term structure models in continuos time to the discrete time setting. The continuos time setting can be seen as a special case of the discrete time one. Recursive closed form solutions for zero coupon bonds are provided even in the presence of multiple correlated underlying factors. Pricing bond options requires simple integration. Model parameters may well be time dependent without scuppering such tractability. Model estimation does not require a restrictive choice of the market price of risk. The model can also be used for pricing credit risk and is particularly useful when the factors are or depend on periodically released macroeconomic data or corporate financial reports.Quadratic term structure model, discrete time, bond valuation, recursive solution, bond option

Valuation of the Firm's Liabilities when Equity Holders are also Creditors

This paper presents a tractable structural model whereby controlling equity holders are also among the creditors of the firm. As the firm approaches distress, equity holders can depauperate the firm and expropriate other creditors by repaying their credit before bankruptcy. The bankruptcy court's right to revoke such repayment protects arm's length creditors, reduces the cost of borrowing and induces equity holders to anticipate repayment of their credit. Equity holders decide repayment neither too early nor too late, so as to reduce the risk of repayment revocation by the bankruptcy court. Similar conclusions apply to the preferential repayment of bank loans personally guaranteed by equity holders. The analysis also suggests that callable bearer bonds may be more valuable to equity holders than to other creditors.equity holders's credit, debt repayment, assets liquidation, revocatoria, debt valuation, default, structural model

Convertible Subordinated Debt Valuation and "Conversion in Distress"

This paper presents new formulae for the valuation of convertible debt and shows how it can be rational for convertible holders to convert not only when the debtor's equity value increases, ut also when the debtor approaches distress. Even if debt cannot be enegotiated, "conversion in distress" averts costly bankruptcy. If ankruptcy costs are high, neglecting "conversion in distress" may ntail a significant undervaluation of subordinated convertibles. Conversion in distress" makes convertible debt less sensitive than on-convertible debt to the recovery value of assets in bankruptcy. So onvertible financing can reduce the cost of borrowing when lenders are symmetrically informed about the debtor's assets recovery value.Subordinated convertible debt; Default; Bankruptcy costs; "Conversion in distress"