Risk-neutral and actual default probabilities with an endogenous bankruptcy jump-diffusion model. (English) Zbl 1131.91354
Summary: This paper focuses on historical and risk-neutral default probabilities in a structural model, when the firm assets dynamics are modeled by a double exponential jump diffusion process. Relying on the H. Leland [J. Finance 49, 1213–1252 (1994)] or H. E. Leland and K. B. Toft [“Optimal capital structure, endogenous bankruptcy, and the term structure of credit spreads”, ibid. 51, No. 3, 987–1019 (1996; doi:10.1111/j.1540-6261.1996.tb02714.x)] endogenous structural approaches, as formalized by B. Hilberink and L.C.G. Rogers [Finance Stoch. 6, No. 2, 237–263 (2002; Zbl 1002.91019)], this article gives a coherent construction of historical default probabilities. The risk-neutral world where evolve the firm assets, modeled by a class of geometric Lévy processes, is constructed based on the Esscher measure, yielding useful and new analytical relations between historical and risk-neutral probabilities. We do a complete numerical analysis of the predictions of our framework, and compare these predictions with actual data. In particular, this new framework displays an enhanced predictive power w.r.t. current Gaussian endogenous structural models.
MSC:
| 91B30 | Risk theory, insurance (MSC2010) |
| 60J99 | Markov processes |
Keywords:
cumulative default probability; structural model; jump-diffusion; endogenous capital structure; Esscher transform; Kou processesCitations:
Zbl 1002.91019References:
| [1] | Altman, E. (1968). Financial ratios, discriminant analysis and the prediction of corporate bankruptcy. Journal of Finance, 23(3), 589–609. · doi:10.2307/2978933 |
| [2] | Black, F., & Cox, J. (1976). Valuing corporate securities: Some effects of bond indenture provisions. Journal of Finance, 31(2), 351–367. · doi:10.2307/2326607 |
| [3] | Chen, N., & Kou, S. G. (2005). Credit spreads, optimal capital structure, and implied volatility with endogenous default and jump risk. Working paper. · Zbl 1168.91379 |
| [4] | Collin-Dufresne, P., & Goldstein, R. (2001). Do credit spreads reflect stationary leverage ratios. Journal of Finance, 56(5), 1929–1958. · doi:10.1111/0022-1082.00395 |
| [5] | Crouhy, M., Galai, D., & Mark, R.(2000). A comparative analysis of current credit risk models. Journal of Banking and Finance, 24, 59–117. · doi:10.1016/S0378-4266(99)00053-9 |
| [6] | Dao, T. B., & Jeanblanc, M. (2005). A double exponential structural jump diffusion model with endogenous default barrier. Working paper. |
| [7] | Duffie, D., & Singleton, K. (1999). Modeling term structures of defaultable bonds. Review of Financial Studies, 12, 687–720. · doi:10.1093/rfs/12.4.687 |
| [8] | Eom, Y. H., Helwege, S., & Huang, J. (2003). Structural models of corporate bond pricing: An empirical analysis. Review of Financial Studies, 17(2), 499–544. · doi:10.1093/rfs/hhg053 |
| [9] | Fujiwara, T., & Miyahara, Y. (2003). The minimal entropy martingale measures for geometric Lévy processes. Finance and Stochastics, 7, 509–531 · Zbl 1035.60040 · doi:10.1007/s007800200097 |
| [10] | Gerber, H., & Shiu, E. S. (1994). Option pricing by Esscher transforms. Transactions of the Society of Actuaries, 46(1), 99–191. |
| [11] | Hilberink, B., & Rogers L. (2002). Optimal capital structure and endogenous default. Finance and Stochastics, 6(2), 237–263. · Zbl 1002.91019 · doi:10.1007/s007800100058 |
| [12] | Huang, J.-Z., & Huang, M. (2003). How much of the corporate-treasury yield spread is due to credit risk. Working paper. |
| [13] | Hull, J. C., Predescu, M., & White, A. (2005). Bond prices, default probabilities, and risk premiums. Journal of Credit Risk, 1(2), 53–60. |
| [14] | Jarrow, R. A., Lando, D., & Turnbull, S. M. (1997). A Markov model for the term structure of credit risk spreads. Review of Financial Studies, 10, 481–523. · doi:10.1093/rfs/10.2.481 |
| [15] | Jarrow, R. A., & Protter, P. (2004). Structural versus reduced form models: A new information based perspective. Journal of Investment Management, 2(2), 1–10. |
| [16] | Kou, S. (2002). A jump diffusion model for option pricing. Management Science, 48, 1086–1101. · Zbl 1216.91039 · doi:10.1287/mnsc.48.8.1086.166 |
| [17] | Kou, S. G., Petrella, G., & Wang, H. (2005). Pricing path dependent options with jump risk via Laplace transforms. Kyoto Economic Review, 76(1), 1–23. |
| [18] | Kou, S. G., & Wang, H. (2003). First passage times of a jump diffusion process. Advances in Applied Probability, 35, 504–531. · Zbl 1040.60057 · doi:10.1239/aap/1059486822 |
| [19] | Kou, S. G., & Wang, H. (2004). Option pricing under a double exponential jump diffusion model. Management Science, 50(9), 1178–1192. · doi:10.1287/mnsc.1030.0163 |
| [20] | Leland, H. (1994a). Corporate debt value, bond covenants, and optimal capital structure. Journal of Finance, 49, 1213–1252. · doi:10.2307/2329184 |
| [21] | Leland, H. (1994b). Bond prices, yield spreads, and optimal capital structure with default risk. Working paper no. 240, IBER, University of California, Berkeley. |
| [22] | Leland, H. (2004). Predictions of default probabilities in structural models of debt. Journal of Investment Management, 2(2), 5–20. |
| [23] | Leland, H., & Toft, K. (1996). Optimal capital structure, endogenous bankruptcy, and the term structure of credit spreads. Journal of Finance, 51(3), 987–1019. · doi:10.2307/2329229 |
| [24] | Longstaff, F. A., Mithal, S., & Neis, E. (2005). Corporate yield spreads: Default risk or liquidity? New evidence from the credit default swap market. Journal of Finance, 60(5), 2213–2253. · doi:10.1111/j.1540-6261.2005.00797.x |
| [25] | Longstaff, F., & Schwartz, E. (1995). A simple approach to valuing risky fixed and floating rate debt. Journal of Finance, 50, 789–819. · doi:10.2307/2329288 |
| [26] | Madan, D., Unal, H. (1998). Pricing the risks of default. The Review of Derivatives Research, 2, 121–160. · Zbl 1274.91426 |
| [27] | Merton, R. (1974). On the pricing of corporate debt: The risk structure of interest rates. Journal of Finance, 29, 449–470 · doi:10.2307/2978814 |
| [28] | Miyahara, Y. (2001). [Geometric Lévy Process & MEMM] pricing model and related estimation problems. Asia-Pacific Financial Markets, 8(1), 45–60. · Zbl 1070.91012 · doi:10.1023/A:1011445109763 |
| [29] | Miyahara, Y. (2004). A note on Esscher transformed martingale measures for geometric Lévy processes. Discussion papers in Economics, Nagoya City University, No. 379, pp. 1–14. |
| [30] | Miyahara, Y. (2005). Martingale measures for geometric Lévy process models. Discussion papers in Economics, Nagoya City University, No. 431, pp. 1–14. |
| [31] | Modigliani, F., & Miller, M. (1958). The cost of capital, corporation finance and the theory of investment. American Economic Review, 48(3), 261–297. |
| [32] | Moody’s Investors Service (1997). Historical default rates of corporate bond issuers, 1920–1996. |
| [33] | Nielsen, L., Saa-Requejo, J., & Santa-Clara, P. (1993). Default risk and interest rate risk: the term structure of default spreads. Working paper. |
| [34] | Scherer, M. (2005). A structural credit risk model based on a jump diffusion. Working paper. |
| [35] | Usabel, M. (1999). Calculating multivariate ruin probabilities via Gaver-Stehfest inversion technique. Insurance: Mathematics and Economics, 25(2), 133–142. · Zbl 1028.91561 · doi:10.1016/S0167-6687(99)00029-3 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
