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2018 | OriginalPaper | Buchkapitel

5. The Genesis of Credit-Risk Modelling

verfasst von : David Jamieson Bolder

Erschienen in: Credit-Risk Modelling

Verlag: Springer International Publishing

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Abstract

The path-breaking work, Merton (1974, Journal of Finance, 29, 449–470), not only addressed a number of important asset-pricing and corporate-finance questions, but was also the genesis of the field of credit-risk modelling. This chapter focuses exclusively on this approach. Not only would it be an injustice to ignore this still-pertinent model, but it offers a range of useful insights into the class of threshold models. The Merton (1974, Journal of Finance, 29, 449–470) framework was conceived and developed in a continuous-time, mathematical-finance setting. To address this complicating factor, a significant amount of effort is allocated to the basic intuition, notation, and mathematical structure leading to a motivating discussion regarding the notion of geometric Brownian motion. Armed with this detail, the chapter proceeds to investigate two possible implementations of Merton (1974, Journal of Finance, 29, 449–470)’s model, which we term the indirect and direct approaches. The indirect approach will turn out to be quite familiar, whereas the direct method requires a significant amount of heavy lifting for its implementation. As in previous chapters, parameter calibration options are explored and both methods are applied to practical portfolio examples.

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Vorheriges Kapitel Threshold Models

Nächstes Kapitel A Regulatory Perspective

See Sundaresan (2013) for a useful overview of the range of applications stemming from Merton’s capital-structure model.

As a useful example, the Standard & Poor’s rating agency in S&P (2011, Paragraph 20) states further that “the most important step in analyzing the creditworthiness of a corporate or governmental obligor is gauging the resources available to it for fulfilling its obligations.”

Naturally, the firm’s debt does not typically mature all on a single date, but this is a useful approximation of reality.

The first-passage time is a special case of a more general probabilistic concept termed a stopping time.

If it is extremely long, say 100 years, then the probability of default grows dramatically and the exercise becomes less meaningful.

See Karatzas and Shreve (1998), Oksendal (1995), and Heunis (2011) for much more information and rigour on Brownian motion and stochastic differential equations.

Readers not familiar with this area can look to the references for more background. Neftci (1996) is also a highly-recommended, gentle and intuitive introduction to this area of finance.

It is, in fact, a nested sequence of σ-algebras. This is an important concept in measure theory; see Royden (1988) for more rigour and background.

In reality, it is closer to -22%, but the rough approximation still holds.

One suggested quantity has the following form

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \delta_t \approx \frac{1-\frac{K}{A_{t}}}{\sigma\sqrt{T-t}}, \end{array} \end{aligned} $$

(5.14)

which is computed using a number of approximations. In particular, one assumes that $\ln \left (\frac {K}{A_t}\right )\approx \frac {A_t - K}{A_t}$, $\mu (T-t)\approx \frac {A_T-A_t}{A_t}$, σ ²(T − t) ≈ 0, and $\frac {A_T}{A_t}\approx 0$.

Please see Harrison and Kreps (1979) and Harrison and Pliska (1981) for the foundational work underlying these ideas.

Duffie and Singleton (2003, Section 3.2) actually refer to this approach as the Black-Scholes-Merton default model.

Karatzas and Shreve (1998) is the standard reference in this area, whether one is a financial analyst, mathematical physicist or electrical engineer.

The solution is easily verified by differentiation,

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \frac{dX_T}{dT} &\displaystyle =&\displaystyle \mu \underbrace{X_t e^{\mu(T-t)}}_{X_T},\\ dX_T = \mu X_T dT. \end{array} \end{aligned} $$

(5.25)

This holds, of course, for any value of T and coincides with equation 5.22.

There may be other possibilities, but these seem to logically incorporate the two main perspectives.

This can be done directly or one might employ an external firm such as KMV or Kamakura corporation to do this on your behalf.

The firm also presumably takes its asset volatility into account when taking its capital structure choices.

If not, one can always use the Box-Muller method. See, for example, Fishman (1995, Chapter 3).

More generally, one may also use the so-called singular value decomposition; this useful alternative, for large and potentially corrupted systems, is outlined in Appendix A.

For more information on the Cholesky decomposition, see Golub and Loan (2012) or Press et al. (1992, Chapter 11).

Geske (1977), for example, is the first in a series of papers that seeks to incorporate a greater degree of realism in the Merton (1974) setting.

In other words, the sum of two Gaussian random variables is also Gaussian. This convenient property does not hold for all random variables.

The derivation of this expression in found in Appendix B.

Tracing out the history of this idea is also a bit challenging. Hull et al. (2005) make reference to an original paper, Jones et al. (1984), which appears to sketch out the basic idea for the first time.

Although, generally when one talks about an option delta, it is the first partial derivative of the option price with respect to the underlying stock value, not the firm’s assets.

Again, the option gamma is usually the second-order sensitivity to movement in the underlying equity price.

The reason is obvious: it involves some fairly disagreeable computations.

From an econometric perspective, you also could interpret $\frac {\partial E_n}{\partial A}$ as a regression coefficient. In practice, it is not particularly easy to estimate since the A _t outcomes are not observable.

It is also consistent with our previous assumption of assuming that the equity and asset return share a common diffusion term.

See, for more information on these models, Bolder (2001, 2006) and many of the excellent references included in it.

See Bolder (2006) for some practical discussion and relative performance of these two modelling frameworks.

See Diebold and Rudebusch (2013, Chapter 3) for more details.

See, for example, Sundaresan (2013) and Hull et al. (2005) for a description of alternative applications of the Merton (1974) model.

Black, F., & Scholes, M. S. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–654.CrossRef

Bolder, D. J. (2001). Affine term-structure models: Theory and implementation. Bank of Canada: Working Paper 2001–15.

Bolder, D. J. (2006). Modelling term-structure dynamics for portfolio analysis: A practitioner’s perspective. Bank of Canada: Working Paper 2006–48.

Crosbie, P. J., & Bohn, J. R. (2002). Modeling default risk. KMV.

de Servigny, A., & Renault, O. (2002). Default correlation: Empirical evidence. Standard and Poors, Risk Solutions.

Diebold, F. X., & Li, C. (2003). Forecasting the term structure of government bond yields. University of Pennsylvania Working Paper.CrossRef

Diebold, F. X., & Rudebusch, G. D. (2013). Yield curve modeling and forecasting: The dynamic Nelson-Siegel approach. Princeton, NJ: Princeton University Press.CrossRef

Duffie, D., & Singleton, K. (2003). Credit risk (1st edn.). Princeton: Princeton University Press.

Durrett, R. (1996). Probability: Theory and examples (2nd edn.). Belmont, CA: Duxbury Press.

Fishman, G. S. (1995). Monte Carlo: Concepts, algorithms, and applications. 175 Fifth Avenue, New York, NY: Springer-Verlag. Springer series in operations research.

Geske, R. (1977). The valuation of compound options. Journal of Financial Economics, 7, 63–81.CrossRef

Golub, G. H., & Loan, C. F. V. (2012). Matrix computations. Baltimore, Maryland: The John Hopkins University Press.

Harrison, J., & Kreps, D. (1979). Martingales and arbitrage in multiperiod security markets. Journal of Economic Theory, 20, 381–408.CrossRef

Harrison, J., & Pliska, S. (1981). Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes and Their Applications, 11, 215–260.CrossRef

Heunis, A. J. (2011). Notes on stochastic calculus. University of Waterloo.

Hull, J. C., Nelken, I., & White, A. D. (2005). Merton’s model, credit risk, and volatility skews. Journal of Credit Risk, 1, 3–27.CrossRef

Jeanblanc, M. (2002). Credit risk. Université d’Evry.

Jones, E. P., Mason, S. P., & Rosenfeld, E. (1984). Contingent claims analysis of corporate capital structures: An empirical investigation. The Journal of Finance, 39(3), 611–625.CrossRef

Karatzas, I., & Shreve, S. E. (1991). Brownian motion and stochastic calculus (2nd edn.). Berlin: Springer-Verlag.

Karatzas, I., & Shreve, S. E. (1998). Methods of mathematical finance. New York, NY: Springer-Verlag.

Merton, R. (1974). On the pricing of corporate debt: The risk structure of interest rates. Journal of Finance, 29, 449–470.

Neftci, S. N. (1996). An introduction to the mathematics of financial derivatives. San Diego, CA: Academic Press.

Oksendal, B. K. (1995). Stochastic differential equations (4th edn.). Berlin: Springer-Verlag.CrossRef

Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. (1992). Numerical recipes in C: The art of scientific computing (2nd edn.). Trumpington Street, Cambridge: Cambridge University Press.

Royden, H. L. (1988). Real analysis. Englewood Cliffs, NJ: Prentice-Hall Inc.

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Sundaresan, S. (2013). A review of merton’s model of the firm’s capital structure with its wide applications. Annual Review of Financial Economics, 1–21.

Titel: The Genesis of Credit-Risk Modelling
verfasst von: David Jamieson Bolder
Verlag: Springer International Publishing
Buch: Credit-Risk Modelling
Print ISBN: 978-3-319-94687-0

Electronic ISBN: 978-3-319-94688-7

Copyright-Jahr: 2018
DOI: https://doi.org/10.1007/978-3-319-94688-7_5