Option pricing models as insolvency forecasting methods

Option pricing models as a failure prediction method call by Black , Scholes (1973) and Merton founded (1973) option pricing theory to buy from the in the price and volatility of listed securities implicit information contained on the probability of insolvency of companies to close.

The Black-Scholes-Merton model

Preliminary remarks

The primary field of application of the option price theory developed by Black and Scholes (1973) and Merton (1973) in the early 1970s is the formal analytical valuation of options . Options are conditional forward transactions in which the buyer acquires the right from the seller ( writer , subscriber) to allocate a certain amount of a good ( underlying ) during a specified period ( American option ) or at a certain point in time ( European option ) to purchase ( purchase option , "call") or to sell ( sell option , "put") payment terms specified in advance . The underlying assets include stocks , indices , foreign currencies , interest rates , bonds , commodities , food and even other options. Physical delivery of the underlyings at the time of exercise is often not desired or (for example in the case of indices or interest rates) even technically impossible. In these cases, a cash settlement is made based on the current market values .

With the Merton (1974) model, the option price approach was also used for the first time to determine the probability of default by companies. As with bond spread-based approaches , Merton's option pricing model is motivated to provide a theoretical foundation for utilizing the information implicit in capital market prices. In contrast to the bond-spread-based approaches, however, the Merton model is not based on the observation of the bond market , but of the stock market . In particular, due to the higher liquidity , stock market data is generally considered more reliable than bond market data. Significantly more companies have listed equity capital than they have listed debt .

The assumptions of the model

The basic idea of ​​the Merton approach can be described as follows: Assume that the assets of a company consist exclusively of a listed security, the market price of which follows a stochastic process over time with the following properties: ${\displaystyle V_{A}}$

Formula F1: ... geometric Brownian motion ${\displaystyle dV_{A}=\mu _{A}\cdot V_{A}\cdot dt+\sigma _{A}\cdot V_{A}\cdot dz}$

Formula F2: ${\displaystyle {\frac {dV_{A}}{V_{A}}}=\mu _{A}\cdot dt+\sigma _{A}\cdot dz}$

${\displaystyle V_{A}}$ ... value of the asset,

${\displaystyle \mu _{A}}$... expected growth of per time unit, ${\displaystyle V_{A}}$

${\displaystyle \sigma _{A}}$... standard deviation of each time unit, ${\displaystyle V_{A}}$

${\displaystyle dz}$... standard Gauss-Wiener process

It is also assumed that the company has taken out third-party liabilities and must repay them at the time . If the value of the assets , at the time ( lower) than to use liabilities of the company, the company is insolvent and therefore falls to the creditors , since the non subsequent payment duty of owners of the company have no incentive to creditors for the shortfall from to reimburse their private assets. A personal bond of the owners to the company or other subjective differences in assessment due to different preferences , information or expectations is excluded. ${\displaystyle t_{E}}$ ${\displaystyle V_{A}^{tE}}$ ${\displaystyle t_{E}}$ ${\displaystyle B}$ ${\displaystyle V_{A}^{tE}-B}$

The value of the asset at the time is calculated as follows: ${\displaystyle t_{E}}$

Formula F3: with ${\displaystyle V_{A}^{t_{E}}=V_{A}^{t_{0}}\cdot \mathrm {e} ^{\left(\mu _{A}-{\frac {1}{2}}\cdot \sigma _{A}^{2}\right)\cdot \Delta t+\sigma _{A}\cdot {\sqrt {\Delta t}}\cdot \epsilon }}$

Formula F4: ${\displaystyle \Delta t\equiv t_{E}-t_{0}}$

${\displaystyle V_{A}^{t_{E}},V_{A}^{t_{0}}}$- value of the asset (asset) at the time or , ${\displaystyle t_{E}}$ ${\displaystyle t_{0}}$

${\displaystyle \epsilon }$- standard normally distributed random variable

The following figure illustrates the basic idea of ​​the option price approach for determining the default probability of companies:

Determination of the default probability of companies in the context of the option price approach, sketch

From the perspective of the value of is asset in unknown, but can be by a random variable with known distribution ( lognormal distribution ) and known distribution parameters model (see the figure above). The probability of failure PD results formally as follows: ${\displaystyle t_{0}}$ ${\displaystyle V_{A}}$ ${\displaystyle t_{E}}$

Formula F5: ${\displaystyle {\mathit {PD}}=p\left(V_{A}^{t_{E}}<B\right)}$

with PD - probability of failure

${\displaystyle B}$- at the time repayable liabilities of the company (including interest portion) ${\displaystyle t_{E}}$

Formula F6: By changing it results ${\displaystyle {\mathit {PD}}=p\left(V_{A}^{t_{0}}\cdot \mathrm {e} ^{\left(\mu _{A}-{\frac {1}{2}}\cdot \sigma _{A}^{2}\right)\cdot \Delta t+\sigma _{A}\cdot {\sqrt {\Delta t}}\cdot \epsilon }<B\right)}$

Formula F7: and thus ${\displaystyle {\mathit {PD}}=p\left(\left(\mu _{A}-{\frac {1}{2}}\cdot \sigma _{A}^{2}\right)\cdot \Delta t+\sigma _{A}\cdot {\sqrt {\Delta t}}\cdot \epsilon <\ln \left({\frac {B}{V_{A}^{t_{0}}}}\right)\right)}$

Formula F8: ${\displaystyle {\mathit {PD}}=p{\Biggl (}\epsilon <{\frac {\ln \left({\frac {B}{V_{A}^{t_{0}}}}\right)-\left(\mu _{A}-{\frac {1}{2}}\cdot \sigma _{A}^{2}\right)\cdot \Delta t}{\sigma _{A}\cdot {\sqrt {\Delta t}}}}{\Biggr )}}$

Formula F9: ("distance to default") ${\displaystyle {\mathit {DD}}\equiv -{\frac {\ln \left({\frac {B}{V_{A}^{t_{0}}}}\right)-\left(\mu _{A}-{\frac {1}{2}}\cdot \sigma _{A}^{2}\right)\cdot \Delta t}{\sigma _{A}\cdot {\sqrt {\Delta t}}}}}$

Formula F10: ${\displaystyle {\mathit {PD}}=\Phi (-{\mathit {DD}})}$

with - distribution function of the standard normal distribution ${\displaystyle \Phi }$

DD - The variable DD (distance-to-default) indicates, for certain parameter constellations, approximately the distance between the asset value and the “insolvency point”, the amount of third-party liabilities, measured in standard deviations of the asset value.

The company's probability of default thus be in a function of the observable quantities , , and the unobservable variables and determine. The (arbitrage-free) market value of the company's equity and its volatility at the time can also be determined using this approach: ${\displaystyle t_{0}}$ ${\displaystyle V_{A}^{t0}}$ ${\displaystyle \Delta t}$ ${\displaystyle B}$ ${\displaystyle \mu _{A}}$ ${\displaystyle \sigma _{A}}$ ${\displaystyle t_{0}}$

Formula F11: with ${\displaystyle V_{E}^{t_{0}}=V_{A}^{t_{0}}\cdot \Phi (d_{1})-B\cdot \mathrm {e} ^{-r\cdot \Delta t}\cdot \Phi (d_{2})}$

FORMULA F12: and ${\displaystyle d_{1}\equiv {\frac {\ln \left({\frac {V_{A}^{t_{0}}}{B}}\right)+\left(r+{\frac {1}{2}}\cdot \sigma _{A}^{2}\right)\cdot \Delta t}{\sigma _{A}\cdot {\sqrt {\Delta t}}}}}$

Formula F13: ${\displaystyle d_{2}\equiv d_{1}-\sigma _{A}\cdot {\sqrt {\Delta t}}}$

with r - risk-free interest rate

The following applies to the volatility of equity:

Formula F14: ${\displaystyle \sigma _{E}=\sigma _{A}\cdot {\frac {V_{A}}{V_{E}}}\cdot \Phi (d_{1})}$

Empirical findings on the suitability of the Black-Scholes-Merton model for forecasting corporate insolvencies

In contrast to the asset volatility , the volatility of the market value of equity is therefore not a constant and the market value of equity therefore follows not a geometric Brownian motion. The theoretically secured scope of the model thus extends to the insolvency forecast (the presumably empty set) of passive and unsecured hedge funds . Whether the market price valuation is "efficient", i. H. Whether the market price of the asset or its volatility are justified in any fundamental sense does not matter here - it is only important that the market price development within the forecast period can be well described by the assumed geometric Brownian movement with known parameters. When the option pricing model is used in practice, it is also applied to any other listed company whose assets do not (entirely) consist of securities listed on the capital market. With regard to the aspect of valuation, the opposite approach is taken: the asset value, asset volatility and asset drift are estimated from the observable market value of equity, the book value of debt and equity volatility. However, the "asset" itself is neither observable nor tradable. In this case, asset volatility and asset drift thus describe the alleged changes in an unobservable, random variable. Even with a restriction to listed companies with extremely simple, "theory-compatible" capital structures , a comprehensive empirical investigation of the Merton model and various model variants based on it on the basis of capital market data from 1974-2001 yielded the following findings: ${\displaystyle \sigma _{A}}$ ${\displaystyle \sigma _{E}}$

• The probabilities of failure estimated with the different models differ considerably at the individual level.
• The estimated failure probabilities were mostly either implausibly low or implausibly high.
• Almost always implausibly high spreads resulted when using the various model variants, for example in the context of the stock market turbulence in October 1987 and the following months. The median default probabilities determined for this period for speculative bonds were 60% - 80% p. a. and 30% - 40% p. a. for investment-worthy bonds .
• The size of the estimation error varies systematically with the variables leverage and (estimated) asset volatility.

Adjustments to the Black-Scholes-Merton model by Moody's-KMV

Similar to the theory-based bond spread models, the stock price-based models also exhibit considerable mis- calibration and behavior that is unstable over time. These problems can be remedied by means of suitable calibrations, such as those implemented in the context of the Moody's KMV EDF model. With the KMV model, when applied to listed companies, insolvency forecasts of a very high forecast quality can be created that even exceed the forecast quality of agency ratings . However, this is also possible with bond spread-based approaches - and even with empirical-statistical methods based on annual financial statements .

The core of the calibrations of the KMV model concerns the use of an empirical allocation function of the distance-to-default quotient to the probability of failure (PD) instead of the normal distribution to be used according to formula F20 . Instead of the relationship shown in formula F5, the "insolvency point" is only considered to be reached according to the KMV model when the (unobservable) asset value falls below the short-term plus 50% (instead of 100%) of the long-term liabilities. In addition, the asset volatility determined (among other things) on the basis of equity volatility using numerical approximation methods is still "adjusted" for specific sectors, countries and sizes.

The aim of these interventions is to improve the forecast quality of the model and to calibrate the failure probabilities estimated by the model. At the same time, however, the model loses its theoretical foundation. The formal analytical substructure of the Merton model thus ultimately only serves as a justification for forming a quotient from a (however defined) net inventory value and a (however defined) risk variable, which is then calibrated to empirical default data. This approach is so unspecific and flexible that it can also be used, for example, to forecast the insolvency of state debtors on the basis of bond market data or for non-listed companies on the basis of annual financial statements. In the latter case, however, from a theoretical point of view, no advantages over conventional empirical-statistical methods are to be expected, since no individual capital market-based inventory and risk information is available for non-listed companies, but must first be estimated on the basis of the same data that is also used in the framework conventional, year-end-based financial ratio methods can be used. These negative expectations have been confirmed in empirical studies.

literature

• Basel Committee on Banking Supervision (Ed.) (2000): "Credit Ratings and Complementary Sources of Credit Quality Information" (PDF; 885 kB), Working Paper # 3, 2000
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• Hering, T. (1999): "Financial enterprise valuation", approved habilitation thesis, University of Greifswald, Deutscher Universitätsverlag, Wiesbaden, ISBN 978-3824470440 , 1999
• Karmann, A. , Maltritz, D. (2003): "Sovereign Risk in a Structural Approach, Evaluating Sovereign Ability-to-Pay and Probability of Default" (PDF; 628 kB), in Dresden Discussion Paper Series in Economics, No. 7/03, 2003
• Kealhofer, S. (2003): "Quantifying Credit Risk I: Default Prediction", in Financial Analysts Journal, Jan./ Feb. 2003, pp. 30-44, 2003
• McQuown, JA (1993): "A Comment on Market vs. Accounting-Based Measures of Default Risk" ( Memento of April 5, 2004 in the Internet Archive ), KMV Working Paper, 1993
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• Soberhart, JR, Stein, RM, Mikityanska, V., Li, L. (2000): "Moody's Public Firm Risk Model: A Hybrid Approach to Modeling Short Term Default Risk", Moody's Investors Service , Rating Methodology, Report # 53853, 03/2000
• Shiller, RJ (2003): "From Efficient Market Theory to Behavioral Finance" , in Journal of Economic Perspectives, Vol. 17 (1), pp. 83-104, 2003
• Stein, RM, Kocagil, A. E, Bohn, J., Akhavein, J. (2003): "Systematic and Idiosyncratic Risk in Middle-Market Default Prediction: A Study of the Performance of the RiskCalc ™ and PFM ™ Models" ( Memento from December 15, 2005 in the Internet Archive ), Moody's Investors Service , Special Comment, Report # 77261, 02/2003

Individual evidence

1. ↑ This article is based on Bemmann (2007, section 2.3.3.3)
2. ↑ See Hull (2002, p. 6ff.)
3. ↑ see Eom, Helwege, Huang (2003) and Crosbie, Bohn (2003, p. 20).
4. ↑ see Kealhofer (2003, p. 42)
5. ↑ See Merton (1974, p. 450) for a detailed presentation and discussion of further technical assumptions regarding efficient capital markets and properties of the risk-free interest rate .
6. ↑ This presentation of the relationship between assets and securities is based on Kealhofer (2003, p. 31). In the original article by Merton (1974) no assumptions are made about the composition of the company's assets, instead the term “assets” is used in the sense of “ company value” (“value of the firm”, see ibid., P. 450). At the same time, however, the existence of a security is assumed whose (dividend-adjusted) market price should always correspond to the “company value”. This representation is not only unrealistic because there are practically no such asset securities (except for the Kealhofer's case), but also because the existence of an objectively correct procedure for determining “company values” is assumed (without specifying this) and it is assumed that the market prices always correspond exactly to this value. In Kealhofer's presentation, this contradiction does not apply, since the market value determines the asset value - and not the other way around.
7. ↑ In the practice of company valuation, innumerable fundamentally different valuation approaches can be identified, which usually lead to different results, see Günther (1997, p. 73ff.), Schierenbeck, Lister (2002, p. 77ff.), Fernández (2004) . Herring (1999, p 1) summarizes the basic dilemma of business valuation pointedly as follows: "1. [there is] the need to economically assess uncertain cash flows and 2. [there is] the impossibility of obtaining the necessary information. ”See also the study by Chan, Karceski, Lakonishok (2003). The authors show that the future corporate earnings growth rates, which are central to company valuation, are neither persistent nor predictable .
8. ↑ see Merton (1974, p. 450, own notation, exclusion of inter-period dividend payments)
9. ↑ In the terminology of option price theory, the owners of the company who are not required to make additional payments make use of their “put option right” in this case. Instead of paying monetary units to creditors , they transfer the company's assets to them as their value is less than . ${\displaystyle B}$ ${\displaystyle V_{A}^{tE}}$ ${\displaystyle B}$
10. ↑ see Crosbie, Bohn (2003, p. 17)
11. ↑ see Crosbie, Bohn (2003, p. 17)
12. ↑ See Crosbie, Bohn (2003, p. 18). In the KMV variant of the option price model presented below, the distance-to-default parameter is therefore also defined as follows: see Falkenstein, Boral, Carty (2000, p. 21, own notation). This representation corresponds only loosely to formula F9, neglecting the right term in the denominator of formula F9, which implies an asset drift of zero and assumes that the asset variance is very small in relation to and under the approximation.The latter approximation is only for very highly indebted companies well met. ${\displaystyle DD_{KMV}\equiv {\frac {V_{A}^{t_{0}}-B}{V_{A}^{t_{0}}\cdot \sigma _{A}\cdot {\sqrt {t}}}}}$ ${\displaystyle \mu _{A}}$ ${\displaystyle \sigma _{A}^{2}\cdot \Delta t}$ ${\displaystyle \ln(B/V_{A}^{t_{0}})}$ ${\displaystyle -\ln {\Biggl (}{\frac {B}{V_{A}^{t_{0}}}}{\Biggr )}\cong {\frac {V_{A}^{t_{0}}-B}{V_{A}^{t_{0}}}}}$
13. ↑ Merton (1974, Formula 12, own notation)
14. ↑ see Crouhy, Galai, Mark (2001, p. 88 and the literature cited there)
15. ↑ ... or to financial institutions that are allowed to take on outside liabilities, invest all of their assets in risky securities, do not shift assets within the forecast period and do not hedge against price fluctuations.
16. ↑ For example, Shiller (2003) argues that the capital markets are characterized by excess volatility, which can only be explained by the market participants' irrational expectations . See ibid., P. 85: "[T] the fundamental principle of optimal forecasting is that the forecast must be less variable than the variable forecasted."
17. ↑ Even equity volatility cannot be directly observed and can at best be estimated on the basis of historical observations of the realized return on equity. However, it is problematic that - as can be seen from the above explanations, equity volatility is not a constant, but depends on the unobservable asset value that changes over time and its constant, but likewise unobservable, volatility.
18. ↑ The endogenisation the capital structure, the model would complexity enormously increase. If the model allows the issue of new equity , the equity at time t0 must be viewed as an option on an option on an option, etc., see Eom, Helwege, Huang (2003)
19. ↑ see Eom, Helwege, Huang (2003)
20. ↑ Soberhart et al. (2000, p. 9): “Unfortunately, the original Merton model does not correlate well with observed market behavior. [...]. In fact, in order to obtain spreads of similar magnitude to the observed values, the volatility of the assets needs to be set to unrealistic values ​​[…] More concerning, the values ​​for the firm's assets and volatility implied from equity prices often disagree with those obtained from bond prices. ”
21. ↑ see Delianedis, Geske (1998, p. 26, Figure 1)
22. ↑ see Kealhofer (2003, p. 30). The KMV model is used by over 120 credit institutions worldwide, see Blochwitz, Liebig, Nyberg (2000, p. 18). According to the Basel Committee (2000, p. 86), the KMV-EDF [Kealhofer-McQuown-VASICEK, expected default frequency model is the “best-known option price model” of all.
23. ↑ McQuown (1993, p. 9): “[EDF] is consistently better than S&P ratings in discrimination and lead-time. And the difficult task of adjusting ordinal default ranks to the period of the credit cycle is not left entirely to the intuition of the user. " On the basis of a non-parametric study, Kealhofer (2003, p. 36ff.) Also shows that the KMV failure forecasts not only in themselves have a higher forecast quality than the ratings from S&P, but also that the ratings from S&P do not provide any additional information for improvement who can contribute to KMV failure forecasts.
24. ↑ For a comparison of financial ratio models based on annual financial statements and agency ratings, see the studies by Carey, Hrycay (2001) and Fons, Viswanathan (2004).
25. ↑ Crosbie, Bohn (2003, p. 14): “Moreover, the usual assumptions of normal or lognormal distributions cannot be used. For default measurement, the likelihood of large adverse changes in the relationship of asset value to the firm's default point is critical to the accurate determination of the default probability [...] Consequently, MKMV first measures the distance-to-default as the number of standard deviations the asset value is away from default and then uses empirical data to determine the corresponding default probability. " See Basler Committee (2000, p. 115) and Kealhofer (2003, p. 32).
26. ↑ Crosbie, Bohn (2003, p. 7): “In our study of defaults, we have found that in general firms do not default when their asset value reaches the book value of their total liabilities. While some firms certainly default at this point, many continue to trade and service their debts. The long-term nature of some of their liabilities provides these firms with some breathing space. We have found that the default point, the asset value at which the firm will default, generally lies somewhere between total liabilities and current, or short-term, liabilities. "
27. ↑ Crosbie, Bohn (2003, p. 17): “In addition, the asset volatility derived above is combined in a Bayesian manner with country, industry and size averages to produce a more predictive estimate of the firm's asset volatility.”
28. ↑ see Karmann, Maltritz (2003)
29. ↑ see Blochwitz, Liebig, Nyberg (2000, p. 20)
30. ↑ Blochwitz, Liebig, Nyberg (2000, p. 18)
31. ↑ In a comparative study, the Moody's KMV method (Private Firm Model), a non-parametric empirical-statistical method ( Moody's RiskCalc ) based on the total sample used, but also inferior in all examined subgroups, is inferior to it, see Stein et al. (2003, p. 11ff.). Furthermore, it was not possible to find a combination of both rating procedures that would lead to a significant improvement in the RiskCalc insolvency prognosis, see Stein et al. (2003, p. 22). Nevertheless, with reference to a non-parametric study, the authors assume that the PFM model could provide additional information not contained in RiskCalc, see ibid., P. 18f. Test statistics are not given and the results presented graphically leave room for interpretation.