The time evolution of the value of a firm is commonly modeled by a linear, scalar stochastic differential equation (SDE) of the type where the coefficient in the drift term denotes the (exogenous) stochastic short term interest rate and is the given volatility of the value process. In turn, the dynamics of the short term interest rate are modeled by a scalar SDE. It is shown that exhibits a lognormal distribution when is a normal/Gaussian process defined by a common variety of narrow sense linear SDEs. The results can be applied to different financial situations where modeling value of the firm is critical. For example, with the context of the structural models, using this result one can readily compute the probability of default of a firm.
KeywordsAlue of FirmLognormalInterest Rate Process
D. Hackbarth, C. A. Hennessy and H. E. Leland, “Can the Trade-Off Theory Explain Debt Structure?” Review of Financial Studies, Vol. 20, No. 1, 2007, pp. 1389-1428. doi:10.1093/revfin/hhl047
H. E. Leland and K. B. Toft, “Optimal Capital Structure, Endogenous Bankruptcy, and the Term Structure of Credit Spreads,” Journal of Finance, Vol. 51, No. 3, 1996, pp. 987-1019. doi:10.2307/2329229
H. Qi, “Credit Spread by a Modified Leland-Toft Model,” FMA Meetings, Orlando, 18-20 October 2007, unpublished.
A. Saha-Bubna and E. Barrett, “Credit Derivatives Show Surge,” Wall Street Journal, Vol. 255, No. 98, 2007, pp. c1.
D. Duffie, D. Filipovic and W. Schachermayer, “Affine Processes and Applications in Finance,” The Annals of Probability, Vol. 13, No. 3, 2003, pp. 984-1053. doi:10.1214/aoap/1060202833
D. Duffie and K. J. Singleton, “Credit Risk: Pricing, Measurement, and Management,” Princeton University Press, Princeton, 2003.
D. Lamberton and B. Lapeyre, “Introduction to Stochastic Calculus applied to Finance,” English Edition, Translated by N. Ra-beau and F. Mantion, 2nd Edition, Chapman & Hall, London, 2007.
L. Arnold, “Stochastic Differential Equations: Theory and Applications,” John Wiley & Sons, New York, 1974.
V. V. Acharya and J. N. Carpenter, “Corporate Bond Valuation and Hedging with Stochastic Interest Rate and Endogenous Bankruptcy,” The Review of Financial Studies, Vol. 15, No. 5, 2002, pp. 1355-1383. doi:10.1093/rfs/15.5.1355
R. C. Merton, “On the Pricing of Corporate Debt: The Risk Structure of Interest Rates,” Journal of Finance, Vol. 29, No. 3, 1974, pp. 449-470. doi:10.2307/2978814
F. A. Longstaff and E. S. Schwartz, “A Simple Approach to Valuing Risky Fixed and Floating Rate Debt,” Journal of Finance, Vol. 50, No. 3, 1995, pp. 789-819. doi:10.2307/2329288
A. J. G. Cairns, “Interest Rate Models: An Introduction,” Princeton University Press, Prince-ton, 2004.
N. Privault, “An Elementary Introduction to Stochastic Interest Rate Modeling,” World Scientific, Singapore, 2008.
R. C. Merton, “Theory of Rational Option Pricing,” Bell Journal of Economics and Management Science, Vol. 4, No. 1, 1973, pp. 141-183. doi:10.2307/3003143
O. Vasicek, “An Equilibrium Characterization of the Term Structure,” Journal of Financial Economics, Vol. 37, 1977, pp. 339-348.
S. Y. Ho and S. B. Lee, “Term Structure Move-ments and Pricing Interest Rate Contingent Claims,” Journal of Finance, Vol. 41, No. 5, 1986, pp. 1011-1029. doi:10.2307/2328161
J. C. Hull and A. D. White, “Pricing Interest Rate Derivative Securities,” Review of Financial Studies, Vol. 3, No. 4, 1990, pp. 573-592. doi:10.1093/rfs/3.4.573
D. Heath, R. Jarrow and A. Morton, “Bond Pricing and Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation,” Econometrica, Vol. 60, No. 1, 1992, pp. 77-105. doi:10.2307/2951677
P. E. Kloeden and E. Platen, “Numerical Solution of Stochastic Differential Equations,” Springer-Verlag, New York, 1992.
D. R. Bielecki and M. Rutkowski, “Credit Risk: Modeling, Valuation and Hedging,” Springer, New York, 2002.
R. A. Jarrow, D. Lando and S. M. Turnbull, “A Markov Model for the Term Structure of Credit Risk Spreads,” Review of Financial Studies, Vol. 10, No. 2, 1997, pp. 481-523. doi:10.1093/rfs/10.2.481
S. E. Shreve, “Stochastic Calculus Models for Finance,” Springer, New York, 2004.
J. C. Hull, “Options, Futures and other Derivatives,” 4th Edition, Prentice Hall, Englewood Cliff, 2000.
M. U. Dothan, “On the Term Structure of Interest Rates,” Journal of Financial Economics, Vol. 62, No. 6, 1978, pp. 59-69. doi:10.1016/0304-405X(78)90020-X
I. Karat-zas and S. E. Shreve, “Brownian Motion and Stochastic Calcu-lus,” Springer-Verlag, New York, 1991. doi:10.1007/978-1-4612-0949-2
J. Cox, J. Ingersoll and S. Ross, “A Theory of the Term Structure of Interest Rates,” Econometrica, Vol. 53, No. 2, 1985, pp. 385-407. doi:10.2307/1911242
N. Pearson and T. S. Sun, “An Em-pirical Examination of the Cox-Ingersoll-Ross Model of Term Structure of Interest Rates Using the Method Maximum Likeli-hood,” Journal of Finance, Vol. 54, 1994, pp. 929-959.
F. Black, E. Derman and W. Toy, “A One-factor Model of Interest Rates and Its Application to Treasury Bond Options,” Financial Analyst’s Journal, Vol. 46, No. 1, 1990, pp. 33-39. doi:10.2469/faj.v46.n1.33
F. Black and P. Karasinski, “Bond and Option Pricing When Short Rates Are Log-Normal,” Financial Analysis Journal, Vol. 47, No. 4, 1991, pp. 52-59. doi:10.2469/faj.v47.n4.52
T. Mikosch, “Elementary Sto-chastic Calculus with Finance in View,” World Scientific, Singapore, 1999.
T. C. Gard, “Introduction to Stochastic Differential Equations,” Marcel Dekker Inc., New York, 1988.
H. H. Kuo, “Introduction to Stochastic Integration,” Springer, New York, 2006.
N. L. Johnson, S. Kotz and N. Balakrishnan, “Continuous Univariate Distributions,” 2nd Edition, John Wiley & Sons, Inc., New York, 1994.
K. Gie-secke, “Credit Risk Modeling and Valuation: An Introduction,” In: D. Shimko, Ed., Credit Risk: Models and Management, Riskbooks, London, 2004, pp. 1-67.
R. Elliott and P. Kopp, “Mathematics of Financial Markets,” 2nd Edition, Springer-Verlag, New York, 2005
A. N. Shiryaev, “Essentials of Stochastic Finance: Facts, Models, Theory,” World Scientific, New York, 1999. doi:10.1142/9789812385192
B. Oksendal, “Stochastic Differential Equations: An Introduction with Applications,” 6th Edition, Springer Verlag, New York, 2003.
M. Brennan and E. Schwartz, “A Continuous-Time Approach to the Pricing of Bonds,” Journal of Banking and Finance, Vol. 3, No. 4, 1979, pp. 133-155. doi:10.1016/0378-4266(79)90011-6
T. Vorst, “Prices and Hedge Ratios of Average Exchange Rate Options,” Interna-tional Review of Financial Analysis, Vol. 1, No. 3, 1992, pp. 179-193. doi:10.1016/1057-5219(92)90003-M
M. Yor, “On Some Exponential Functionals of Brownian Motion,” Advances in Applied Probability, Vol. 24, No. 3, 1992, pp. 509-531. doi:10.2307/1427477
H. Geman and M. Yor, “Bessel Processes, Asian Options, and Perpetuities,” Mathematical Finance, Vol. 3, No. 4, 1993, pp. 349-375. doi:10.1111/j.1467-9965.1993.tb00092.x
D. C. Shimko, N. Tejima and D. Van Deventer, “The Pricing of Risky Debt When Interest Rates Are Stochastic,” Journal of Fixed Income, Vol. 3, No. 2, 1993, pp. 58- 65. doi:10.3905/jfi.1993.408084