Problema de calibración de mercado y estructura implícita del modelo de bonos de Black-Cox // Market Calibration Problem and the Implied Structure of the Black-Cox Bond Model

Autores/as

  • Nikolay Sukhomlin Universidad Autónoma de Santo Domingo (República Dominicana) CEREGMIA, Université des Antilles et de la Guyane (France)
  • Lisette Santana Grupo de Investigación en Econofísica Universidad Autónoma de Santo Domingo

Palabras clave:

Modelo de Black-Cox, volatilidad implícita, arbitraje, Black-Cox model, implied volatility, arbitrage

Resumen

El principal resultado de este artículo consiste en la resolución del problema inverso del modelo de Black-Cox (1976), usando el método propuesto por Sukhomlin (2007). Se parte del enfoque retrógrado (backward) para obtener una expresión exacta de la volatilidad implícita en función de parámetros cuantificables con datos de mercado y de variables conocidas. Se descubre la existencia de dos valores de la volatilidad para un activo subyacente en el modelo referido, lo que indica que las asunciones tradicionales no lo definen de manera unívoca. Se encuentra la causa de que el modelo de Black-Cox contenga dos valores de la volatilidad. Además, se lleva a cabo una simulación, a fin de verificar, numéricamente, que la expresión obtenida para la volatilidad es la inversión de la fórmula que representa la probabilidad de que la firma no alcance un nivel de insolvencia antes del tiempo de madurez de la deuda. Finalmente, se resuelve el problema de calibración de mercado desde el punto de vista directo (forward), encontrándose una expresión que resulta de mayor utilidad para los agentes de mercado.

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The main result of this paper consists in the resolution of the inverse problem for the Black-Cox (1976) model, using the method proposed by Sukhomlin (2007). Based on the backward approach, we obtain an exact expression of the implied volatility expressed as a function of quantifiable market parameters and known variables. We discover the existence of two values of the volatility for an underlying asset, in the referred model, which means that the model’s traditional assumptions do not define it univocally. We find the cause that the Black-Cox model contains two values of the volatility. Besides, we carry out a simulation in order to verify, numerically, that our volatility expression is in fact the inversion of the formula that represents the probability that the firm has not reached the reorganization boundary before the debt expires. Finally, we solve the market calibration problem from the forward approach, finding an expression that is more useful for market agents.

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Citas

Black, F., Cox, J. (1976). “Valuing Corporate Securities: Some Effects of Bond Indenture Provisions”, The Journal of Finance, 31, pp. 351–367.

Black, F., Scholes, M. (1973). “The Pricing of Options and Corporate Liabilities”, Journal of Political Economy, 81, pp. 637–654.

Brenner, M., Subrahmanyam, M. (1988). “A Simple Formula to Compute the Implied Standard Deviation”, Financial Analysts Journal, 5, pp. 80–83.

Brigo, D., Tarenghi, M. (2004), “Credit Default Swap Calibration and Equity Swap Valuation under Counterparty Risk with a Tractable Structural Model”, in Proceedings of the FEA, 2004, Conference at MIT, Cambridge, Massachusetts.

Brigo, M., Morini, M. (2006), “Credit Default Swap Calibration with tractable structural models under uncertain credit quality, Risk Magazine, 2006, April issue.

Bruche, M. (2006). “Estimating Structural Models of Corporate Bond Prices”, Centro de Estudios Monetarios y Financieros (CEMFI), Working Paper, Madrid, n. 0610.

Cané de Estrada, M, Cortina, E, Ferro Fontan, C, Di Fiori, J.(2005) “Pricing of defaultable bonds with log-normal spread: development of the model and an application to Argentinean and Brazilian bonds during the Argentine crisis”, Review of Derivatives Research 8(1): 40–60.

Cathcart, L., El-Jahel, L. (1998), “Valuation of defaultable bonds”, Journal of Fixed Income, 8 (1), pp. 65–78.

Chambers, D., Nawalkha, S. (2001). “An improved approach to computing implied volatility”, The Financial Review, 38, pp. 89–100.

Chance, D.M. (1996). “A generalized simple formula to compute the implied volatility”, Financial Review 31(4), pp. 859–867.

Chen, R., Hu, S., Pan, G. (2006). “Default Prediction of various structural models”, Working Paper, Fordham University, NY.

Corrado, C., Miller, T. (1996). “A Note on a Simple, Accurate Formula to Compute Implied Standard Deviations”, Journal of Banking and Finance, 20, pp. 595–603.

Collin-Dufresne, P., Goldstain, R. (2001). “Do Credit Spreads Reflect Stationary Leverage Ratios?”, Journal of Finance, 56, pp.1929–1957.

Eom, Y., Helwege, J., Huang, J. (2004). “Structural Models of Corporate Bond Pricing: An Empirical Analysis”, The Review of Financial Studies, 17 (2), pp. 499–544.

Fouque, J., Papanicolaou, G., Sircar, R., Solna, K. (2004). “Maturity Cycles in Implied Volatility”, Finance & Stochastics, 8 (4), pp. 451–477.

Fujita T., Ishizaka M. (2002), “An application of new barrier options (Edokko options) for pricing bonds with credit risk”, Hitotsubashi Journal of Commerce Management, 37 (1), pp. 17–23.

Geske, R. (1977). “The Valuation of Corporate Liabilities as Compound Options”, Journal of Finance and Quantitative Analysis, 12, pp. 541–552.

Hein, T., Hofmann B. (2003) “On the nature of ill-posedness of an inverse problem arising in option pricing”, Inverse Problems, 19, pp. 1319–1338.

Hull,J., White, A. (1990). “Pricing interest-rate derivative securities”, The Review of Financial Studies, 3(4), pp. 573–592 (reprinted in Options: Recent Developments in Theory and Practice, v.2, 1992, pp. 160–180).

Isengildina-Massa, O., Curtis, C., Bridges, W., Nian, M. (2007). “Accuracy of Implied Volatility Approximations Using “Nearest-to-the-Money M” Option Premiums”, Paper presented at the Southern Agricultural Economics Association Meetings Mobile, AL, February, 2007.

Ishizaka, M., Takaoka, K. (2003). “On the pricing of defaultable bonds using the framework of barrier options”, Asia-Pacific Financial Markets, 10, pp. 151–162.

Kelly, M.A. (2006). “Faster Implied Volatilities via the Implicit Function Theorem”, The Financial Review, 41, pp. 589–597.

Kraft, H., Steffensen, M. (2007). “Bankruptcy, Counterparty Risk, and Contagion”. Review of Finance, 11(2), pp. 209–252.

Lauren, J.P., Leissen, D. (1998). “Building a consistent pricing model from observed option prices”. Stanford University, Hoover Institution, Working Paper No. B-443.

Leland, H. (1994). “Corporate Debt Value, Bond Covenants and Optimal Capital Structure”, Journal of Finance, 49, pp. 1213–1252.

Leland, H., Toft, K. (1996). “Optimal Capital Structure, Endogenous Bankruptcy and the Term Structure of Credit Spreads”, Journal of Finance, 51, pp. 987–1019.

Longstaff, F., Schwartz, E. (1995). “A Simple Approach to Valuing Risky Fixed and Floating Rate Debt”, The Journal of Finance, 50, pp. 789–820.

Merton, R. (1974). “On the pricing of corporate debts: the risk structure of interest rates”, Journal of Finance, 29, pp. 449–470.

Minqiang, L. (2008). “Approximate Inversion of the Black-Scholes Formula Using Rational Functions”, European Journal of Operational Research, 185 (2), pp. 743–759.

Nardon, M. (2005). “Valuing defaultable bonds: an excursion time approach”, Finance 0511015, EconWPA.

Sukhomlin, N. (2007). “The Black-Scholes type financial models and the arbitrage opportunities”, Revista de Matemática: Teoría y Aplicaciones, 14 (1), pp. 1–6.

Sukhomlin, N., Jacquinot, Ph. (2007). “Solution Exacte du Problème Inverse du Valorisation des Options dans le Cadre du Model de Black et Scholes”, Paper published in “ Hyper Articles en Ligne”. Archives ouvertes. France (hal-00144781, version 1).

Wong, H., Li, K. (2004). “On Bias of Testing Merton’s Model”, Proceeding of International Association of Science and Technology for Development (IASTED), Conference on Financial Engineering and Applications, 9 pp. Alberta, Canada: ACTA Press, 8th Nov., 2004.

Publicado

2016-11-04

Cómo citar

Sukhomlin, N., & Santana, L. (2016). Problema de calibración de mercado y estructura implícita del modelo de bonos de Black-Cox // Market Calibration Problem and the Implied Structure of the Black-Cox Bond Model. Revista De Métodos Cuantitativos Para La Economía Y La Empresa, 10, Páginas 73 a 98. Recuperado a partir de https://www.upo.es/revistas/index.php/RevMetCuant/article/view/2166

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