Merton model

In this chapter, we have considered both equilibrium and arbitrage-free interest-rate models. These are one-factor Gaussian models of the term structure of interest rates. We saw that in order to specify a term structure model, the respective authors described the dynamics of the price process, and that this was then used to price a zero-coupon bond. The short-rate that is modelled is assumed to be a risk-free interest rate, and once this is modelled, we can derive the forward rate and the yield of a zero-coupon bond, as well as its price. So, it is possible to model the entire forward rate curve as a function of the current short-rate only, in the Vasicek and Cox-Ingersoll-Ross models, among others. Both the Vasicek and Merton models assume constant parameters, and because of equal probabilities of forward rates and the assumption of a normal distribution, they can, xmder certain conditions relating to the level of the standard deviation, produce negative forward rates. 

Tudela, M., Young, G., 2003. A Merton-model approach to assessing the default risk of UK public companies. Bank of England Working Paper, No. 194, June. 

Some practitioners argue that Merton models are more appropriate than reduced form models when pricing default swaps on high-yield bonds, due to the higher correlation of high-yield bonds with the underlying equity of the issuer firm. 

Kamaknra ntilizes an ROC Accuracy Ratio to evaluate the performance of models. This ratio is designed to determine which family of models is best by determining if a model correctly identifies defaults and nondefaults in a sample of historical observations. A score of 100 would be a perfect score. According to Kamakura s analysis, the Jarrow-Chava reduced form model received a score of 92.74. This compares to scores in the 80 s for various Merton models. 

According to Merton (1973), advantages in science can accumulate in two ways by addition or multiplication. These models are succinctly summarized by Zuckerman (1977, p. 60) ... 

B. M. and Merton, B., Nematodes as Biological Models, Academic Press, New York, 1980 Emmons, S.W., Mechanisms of C. elegans development. Cell 51, 881-883, 1987 Blumenthal, T. and Thomas, J., Cis and trans... 

In Merton s (1971) model, the interest-rate process is assumed to be a generalised Weiner process, described by Equation (3.31) ... 

The Vasicek, Cox-Ingersoll-Ross, Hull-White and other models incorporate mean reversion. As the time to maturity increases and as it approaches infinity, the forward rates converge to a point at the long-run mean reversion level of the current short-rate. This is the limiting level of the forward rate and is a function of the volatility of the current short-rate. As the time to maturity approaches zero, the short-term forward rate converges to the same level as the instantaneous short-rate. In the Merton and Vasicek models, the mean of the short-rate over the maturity period T is assumed to be constant. The same constant for the mean, or the drift of the interest rate, is described in the Ho-Lee model, but not the extended Vasicek or Hull-White model. 

Merton, R., 1971. Optimum consumption and portfolio rules in a continuous time model. J. Econ. Theory 3, 373 13. 

As shown in previous sections, the credit spread on a corporate bond takes into account its expected default loss. Structural approaches are based on the option pricing theory of Black Scholes and the value of debt depends on the value of the underlying asset. The determination of yield spread is based on the firm value in which the default risk is found as an option to the shareholders. Other models proposed by Black and Cox (1976), Longstaff and Schwartz (1995) and others try to overcome the limitation of the Merton s model, like the default event at maturity only and the inclusion of a default threshold. This class of models is also known as first passage models . 

Merton s model is one of the most important models of credit risk. Merton (1974) and Black and Scholes (1973) proposed a model to assess the credit risk recalling the concept of capital structure, according to Modigliani and Miller s theorem (1958, 1963). According to the Black and Scholes s assumptions, at basis of the model the critical ones are two ... 

Black and Cox (1976) proposed a modified version of the Merton s model by considering three main elements ... 

Finally, Merton s model assumes that when the value of firm is lower than the value of debt, equity holders sell assets in order to fulfill their... 

Extending the work of Black and Cox, Longstaff and Schwartz (1995) developed a new approach to value both, fixed and floating defaultable bonds. Like Black and Cox (1976), the model proposed overcomes the assumption at basis of the Merton s model in which the defaults occur when the firm exhausts its assets only and adding a value barrier by which the firm defaults. The assumptions at basis of this model are two ... 

The Web site associated with this book contains an Excel spreadsheet demonstrating Merton s model. The reader may use the spreadsheet to value credit risky bonds using this model and also to determine the embedded equity value. Details of how to access the Web site are contained in the preface. 

Hull, M., Nelken, 1., White, A., 2004. Merton s model, credit risk, and volatility skews. J. Credit Risk 1 (1), 3-28. 

Sundaresan, S., 2013. A review of Merton s model of the firm s capital structure with its wide applications. Annu. Rev. Financ. Ecoil 5, 21-41. 

The theory of options was developed in the assumption of market equilibrium. The first option pricing model was proposed by Black and Scholes (1973) and then by Merton (1973), in which they did not consider dividend payments. Authors as Schwartz (1975) include dividend payments into valuation model and also consider the possibility of exercising the option before the maturity... 

Structural models are characterized by modeling the firm s value in order to provide the probability of a firm default. The Black-Scholes-Merton option pricing framework is the foundation of the structural... 

Merton applied option pricing techniques to the valuation of corporate debt." By extension the pricing of credit derivatives based on corporate debt may in some circumstances be treated as an option on debt, which is therefore analogous to an option on an option model. 

Structural-. The structural models are based on Robert Merton s approach, utilizing accounting information. ... 

Hybrid Reduced Form and Structural. These models combine the Merton structnral approach with Jarrow s variables. ... 

Merton, R. 1993. Continuous Time Finance C.h. 11). Oxford Blackwell. Rebonato, R. 1996. Interest Rate Option Models. Chichesten John Wiley Sons. 

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