Valuation of embedded options in non-marketable callable bonds: a new numerical approach
Abstract
The issue of how to price options embedded in callable bonds has attracted a lot of interest over the years. The usual bond valuation methods rely on yield curves, risk premium, and other parameters to estimate interest rates used in discounted cash flow calculations. The option to retire the bond is, however, neglected in the standard pricing models, causing a systematic overvaluation of callable bonds. In the event of a decline in interest rates, investors are exposed to the risk of a lower return on investment than indicated by the yield to maturity. We propose a novel approach to valuing the risk that the issuer will use the right to buy back the bond at a specific call price. While prior models are focused on valuing marketable callable bonds, we deliver a unique approach to valuing bonds with an embedded European option (or a multiple option) that are traded solely through private transactions. These can typically be characterized by the lack of historical records on transaction prices. The modular character of calculation we propose allows us to take into account additional information, such as probable behaviour of the issuer, available opportunities for achieving alternative earnings or different estimates in terms of interest rate development.
Keyword : non-marketable callable bonds, options embedded in bonds, embedded options valuation, option pricing, bond pricing, loss of interest income, interest rate volatility, probability of call
How to Cite
Skalický, R., Zinecker, M., Balcerzak, A. P., Pietrzak, M. B., & Rogalska, E. . (2022). Valuation of embedded options in non-marketable callable bonds: a new numerical approach. Technological and Economic Development of Economy, 28(4), 1115–1136. https://doi.org/10.3846/tede.2022.17060
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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Bodie, Z., & Taggart Jr, R. A. (1978). Future investment opportunities and the value of the call provision on a bond. Journal of Finance, 33(4), 1187–1200. https://doi.org/10.1111/j.1540-6261.1978.tb02056.x
Boyce, W. M., & Kalotay, A. J. (1979). Tax differentials and callable bonds. Journal of Finance, 34(4), 825–838. https://doi.org/10.1111/j.1540-6261.1979.tb03438.x
Brennan, M. J., & Schwartz, E. S. (1977). Savings bonds, retractable bonds and callable bonds. Journal of Financial Economics, 5(1), 67–88. https://doi.org/10.1016/0304-405X(77)90030-7
Bukalska, E. (2020). Are companies managed by overconfident CEO financially constraint? Investment-cash flow sensitivity approach. Equilibrium. Quarterly Journal of Economics and Economic Policy, 15(1), 107–131. https://doi.org/10.24136/eq.2020.006
Büttler, H.-J. (1995). Evaluation of callable bonds: Finite difference methods, stability and accuracy. Economic Journal, 105(429), 374–384. https://doi.org/10.2307/2235497
Büttler, H.-J., & Waldvogel, J. (1996). Pricing callable bonds by means of green’s function. Mathematical Finance, 6(1), 53–88. https://doi.org/10.1111/j.1467-9965.1996.tb00112.x
Chang, B.-G., & Wu, K.-S. (2021). The nonlinear relationship between financial flexibility and enterprise risk-taking during the COVID-19 pandemic in Taiwan?s semiconductor industry. Oeconomia Copernicana, 12(2), 307–333. https://doi.org/10.24136/oc.2021.011
Cox, J. C., Ingersoll, J. E., & Ross, S. A. (1985). An intertemporal general equilibrium model of asset prices. Econometrica, 53(2), 363–384. https://doi.org/10.2307/1911241
Cox, J. C., Ingersoll, J. E., & Ross, S. A. (1985). A theory of the term structure of interest rates. Econometrica, 53(2), 385–407. https://doi.org/10.2307/1911242
Dai, Q., & Singleton, K. J. (2000). Specification analysis of affine term structure models. The Journal of Finance, 55(5), 1943–1978. https://doi.org/10.1111/0022-1082.00278
D’Halluin, Y., Forsyth, P. A., Vetzal, K. R., & Labahn, G. (2001). A numerical PDE approach for pricing callable bonds. Applied Mathematical Finance, 8(1), 49–77. https://doi.org/10.1080/13504860110046885
Duffie, D., & Singleton, K. J. (1999). Modeling term structures of defaultable bonds. Review of Financial Studies, 12(4), 687–720. https://doi.org/10.1093/rfs/12.4.687
François, P., & Pardo, S. (2015). Prepayment risk on callable bonds: Theory and test. Decisions in Economics and Finance, 38(2), 147–176. https://doi.org/10.1007/s10203-015-0162-0
Geske, R. (1977). The valuation of corporate liabilities as compound options. Journal of Financial and Quantitative Analysis, 12(4), 541–552. https://doi.org/10.2307/2330330
Goldberg, R. S., Ronn, E. I., & Xu, L. (2021). Valuation of callable/putable corporate bonds in a one-factor lognormal interest-rate model. Journal of Fixed Income, 31(1), 80–95. https://doi.org/10.3905/jfi.2021.1.111
Ho, T. S. Y., Lee, S.-B., & Son, K.-H. (1992). Term structure estimation and pricing of callable Treasury bonds. Review of Quantitative Finance and Accounting, 2(2), 127–143. https://doi.org/10.1007/BF00243798
Hull, J., & White, A. (1990). Pricing interest-rate-derivative securities. Review of Financial Studies, 3(4), 573–592. https://doi.org/10.1093/rfs/3.4.573
Hull, J., & White, A. (1993). One-factor interest-rate models and the valuation of interest-rate derivative securities. Journal of Financial and Quantitative Analysis, 28(2), 235–254. https://doi.org/10.2307/2331288
Jarrow, R., Li, H., Liu, S., & Wu, C. (2010). Reduced-form valuation of callable corporate bonds: Theory and evidence. Journal of Financial Economics, 95(2), 227–248. https://doi.org/10.1016/j.jfineco.2009.10.001
Kaczmarek, J., Alonso, S. L. N., Sokołowski, A., Fijorek, K., & Denkowska, S. (2021). Financial threat profiles of industrial enterprises in Poland. Oeconomia Copernicana, 12(2), 463–498. https://doi.org/10.24136/oc.2021.016
Kalotay, A. J., Williams, G. O., & Fabozzi, F. J. (1993). A model for valuing bonds and embedded options. Financial Analysts Journal, 49(3), 35–46. https://doi.org/10.2469/faj.v49.n3.35
Karas, M., & Režňáková, M. (2021). The role of financial constraint factors in predicting SME default. Equilibrium. Quarterly Journal of Economics and Economic Policy, 16(4), 859–883. https://doi.org/10.24136/eq.2021.032
Kliestik, T., Valaskova, K., Nica, E., Kovacova, M., & Lazaroiu, G. (2020). Advanced methods of earnings management: Monotonic trends and change-points under spotlight in the Visegrad countries. Oeconomia Copernicana, 11(2), 371–400. https://doi.org/10.24136/oc.2020.016
Kramoliš, J., & Dobeš, K. (2020). Debt as a financial risk factor in SMEs in the Czech Republic. Equilibrium. Quarterly Journal of Economics and Economic Policy, 15(1), 87–105. https://doi.org/10.24136/eq.2020.005
Lim, D., Li, L., & Linetsky, V. (2012). Evaluating callable and putable bonds: An eigenfunction expansion approach. Journal of Economic Dynamics & Control, 36(12), 1888–1908. https://doi.org/10.1016/j.jedc.2012.06.002
Marsh, T. A., & Rosenfeld, E. R. (1983). Stochastic processes for interest rates and equilibrium bond prices. Journal of Finance, 38(2), 635–646. https://doi.org/10.1111/j.1540-6261.1983.tb02275.x
Meluzín, T., Zinecker, M., Balcerzak, A. P., Pietrzak, M. B., & Doubravský, K. (2021). Institutional settings and their impact on the IPO activity: An exploratory study based on qualitative modelling. Acta Polytechnica Hungarica, 18(2), 215–235. https://doi.org/10.12700/APH.18.2.2021.2.12
Merton, R. C. (1974). On the pricing of corporate debt: The risk structure of interest rates. Journal of Finance, 29(2), 449–470. https://doi.org/10.1111/j.1540-6261.1974.tb03058.x
Myers, S. C. (1977). Determinants of corporate borrowing. Journal of Financial Economics, 5(2), 147–175. https://doi.org/10.1016/0304-405X(77)90015-0
Onuferová, E., Čabinová, V., & Dzurov Vargová, T. (2020). Analysis of modern methods for increasing and managing the financial prosperity of businesses in the context of performance: A case study of the tourism sector in Slovakia. Oeconomia Copernicana, 11(1), 95–116. https://doi.org/10.24136/oc.2020.004
Panova, E. (2020). Determinants of capital structure in Russian small and medium manufacturing enterprises. Equilibrium. Quarterly Journal of Economics and Economic Policy, 15(2), 361–375. https://doi.org/10.24136/eq.2020.017
Priem, R. (2021). An exploratory study on the impact of the COVID-19 confinement on the financial behavior of individual investors. Economics, Management, and Financial Markets, 16(3), 9–40. https://doi.org/10.22381/emfm16320211
Robbins, E. H., & Schatzberg, J. (1988). Callable bonds: A risk-reducing signalling mechanism – A reply. Journal of Finance, 43(4), 1067–1073. https://doi.org/10.1111/j.1540-6261.1988.tb02625.x
Schinckus, C., Nguyen, C. P., & Chong, F. H. L. (2021). Are Bitcoin and ether affected by strictly anonymous crypto-currencies? An exploratory study. Economics, Management, and Financial Markets, 16(4), 9–27. https://doi.org/10.22381/emfm16420211
Setianto, R. H., Sipayung, R. S., & Azman-Saini, W. N. W. (2022). Working capital financing and corporate profitability in the ASEAN region: The role of financial development. Entrepreneurial Business and Economics Review, 10(1), 51–64. https://doi.org/10.15678/EBER.2022.100104
Valaskova, K., Adamko, P., Frajtova Michalikova, K., & Macek, J. (2021a). Quo Vadis, earnings management? Analysis of manipulation determinants in Central European environment. Oeconomia Copernicana, 12(3), 631–669. https://doi.org/10.24136/oc.2021.021
Valaskova, K., Kliestik, T., & Gajdosikova, D. (2021b). Distinctive determinants of financial indebtedness: Evidence from Slovak and Czech enterprises. Equilibrium. Quarterly Journal of Economics and Economic Policy, 16(3), 639–659. https://doi.org/10.24136/eq.2021.023
Vasicek, O. (1977). An equilibrium characterization of the term structure. Journal of Financial Economics, 5(2), 177–188. https://doi.org/10.1016/0304-405X(77)90016-2
Xie, D. (2009). Theoretical and numerical valuation of callable bonds. International Journal of Business and Finance Research, 3(2), 71–82.
Xue, J. (2011). Pricing callable bonds (Project Report 2011:24). Uppsala University. Department of Mathematics. https://uu.diva-portal.org/smash/get/diva2:462176/FULLTEXT01.pdf
Zinecker, M., Skalická, M., Balcerzak, A. P., & Pietrzak, M. (2021a). Business angels in the Czech Republic: Characteristics and a classification with policy implications. Economic Research-Ekonomska Istraživanja. https://doi.org/10.1080/1331677X.2021.1890179
Zinecker, M., Skalická, M., Balcerzak, A. P., & Pietrzak, M. (2021b). Identifying the impact of external environment on business angel activity. Economic Research-Ekonomska Istraživanja. https://doi.org/10.1080/1331677X.2021.1888140
Barnea, A., Haugen, R. A., & Senbet, L. W. (1980). A rationale for debt maturity structure and call provisions in the agency theoretic framework. Journal of Finance, 35(5), 1223–1234. https://doi.org/10.1111/j.1540-6261.1980.tb02205.x
Ben-Ameur, H., Breton, M., Karoui, L., & L’ecuyer, P. (2007). A dynamic programming approach for pricing options embedded in bonds. Journal of Economic Dynamics & Control, 31(7), 2212–2233. https://doi.org/10.1016/j.jedc.2006.06.007
Black, F., & Cox, J. C. (1976). Valuing corporate securities: Some effects of bond indenture provisions. Journal of Finance, 31(2), 351–367. https://doi.org/10.1111/j.1540-6261.1976.tb01891.x
Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81(3), 637–654. https://doi.org/10.1086/260062
Blume, M. E., & Keim, D. B. (1988). The valuation of callable bonds. Rodney L White Center for Financial Research. https://repository.upenn.edu/fnce_papers/1
Bodie, Z., & Taggart Jr, R. A. (1978). Future investment opportunities and the value of the call provision on a bond. Journal of Finance, 33(4), 1187–1200. https://doi.org/10.1111/j.1540-6261.1978.tb02056.x
Boyce, W. M., & Kalotay, A. J. (1979). Tax differentials and callable bonds. Journal of Finance, 34(4), 825–838. https://doi.org/10.1111/j.1540-6261.1979.tb03438.x
Brennan, M. J., & Schwartz, E. S. (1977). Savings bonds, retractable bonds and callable bonds. Journal of Financial Economics, 5(1), 67–88. https://doi.org/10.1016/0304-405X(77)90030-7
Bukalska, E. (2020). Are companies managed by overconfident CEO financially constraint? Investment-cash flow sensitivity approach. Equilibrium. Quarterly Journal of Economics and Economic Policy, 15(1), 107–131. https://doi.org/10.24136/eq.2020.006
Büttler, H.-J. (1995). Evaluation of callable bonds: Finite difference methods, stability and accuracy. Economic Journal, 105(429), 374–384. https://doi.org/10.2307/2235497
Büttler, H.-J., & Waldvogel, J. (1996). Pricing callable bonds by means of green’s function. Mathematical Finance, 6(1), 53–88. https://doi.org/10.1111/j.1467-9965.1996.tb00112.x
Chang, B.-G., & Wu, K.-S. (2021). The nonlinear relationship between financial flexibility and enterprise risk-taking during the COVID-19 pandemic in Taiwan?s semiconductor industry. Oeconomia Copernicana, 12(2), 307–333. https://doi.org/10.24136/oc.2021.011
Cox, J. C., Ingersoll, J. E., & Ross, S. A. (1985). An intertemporal general equilibrium model of asset prices. Econometrica, 53(2), 363–384. https://doi.org/10.2307/1911241
Cox, J. C., Ingersoll, J. E., & Ross, S. A. (1985). A theory of the term structure of interest rates. Econometrica, 53(2), 385–407. https://doi.org/10.2307/1911242
Dai, Q., & Singleton, K. J. (2000). Specification analysis of affine term structure models. The Journal of Finance, 55(5), 1943–1978. https://doi.org/10.1111/0022-1082.00278
D’Halluin, Y., Forsyth, P. A., Vetzal, K. R., & Labahn, G. (2001). A numerical PDE approach for pricing callable bonds. Applied Mathematical Finance, 8(1), 49–77. https://doi.org/10.1080/13504860110046885
Duffie, D., & Singleton, K. J. (1999). Modeling term structures of defaultable bonds. Review of Financial Studies, 12(4), 687–720. https://doi.org/10.1093/rfs/12.4.687
François, P., & Pardo, S. (2015). Prepayment risk on callable bonds: Theory and test. Decisions in Economics and Finance, 38(2), 147–176. https://doi.org/10.1007/s10203-015-0162-0
Geske, R. (1977). The valuation of corporate liabilities as compound options. Journal of Financial and Quantitative Analysis, 12(4), 541–552. https://doi.org/10.2307/2330330
Goldberg, R. S., Ronn, E. I., & Xu, L. (2021). Valuation of callable/putable corporate bonds in a one-factor lognormal interest-rate model. Journal of Fixed Income, 31(1), 80–95. https://doi.org/10.3905/jfi.2021.1.111
Ho, T. S. Y., Lee, S.-B., & Son, K.-H. (1992). Term structure estimation and pricing of callable Treasury bonds. Review of Quantitative Finance and Accounting, 2(2), 127–143. https://doi.org/10.1007/BF00243798
Hull, J., & White, A. (1990). Pricing interest-rate-derivative securities. Review of Financial Studies, 3(4), 573–592. https://doi.org/10.1093/rfs/3.4.573
Hull, J., & White, A. (1993). One-factor interest-rate models and the valuation of interest-rate derivative securities. Journal of Financial and Quantitative Analysis, 28(2), 235–254. https://doi.org/10.2307/2331288
Jarrow, R., Li, H., Liu, S., & Wu, C. (2010). Reduced-form valuation of callable corporate bonds: Theory and evidence. Journal of Financial Economics, 95(2), 227–248. https://doi.org/10.1016/j.jfineco.2009.10.001
Kaczmarek, J., Alonso, S. L. N., Sokołowski, A., Fijorek, K., & Denkowska, S. (2021). Financial threat profiles of industrial enterprises in Poland. Oeconomia Copernicana, 12(2), 463–498. https://doi.org/10.24136/oc.2021.016
Kalotay, A. J., Williams, G. O., & Fabozzi, F. J. (1993). A model for valuing bonds and embedded options. Financial Analysts Journal, 49(3), 35–46. https://doi.org/10.2469/faj.v49.n3.35
Karas, M., & Režňáková, M. (2021). The role of financial constraint factors in predicting SME default. Equilibrium. Quarterly Journal of Economics and Economic Policy, 16(4), 859–883. https://doi.org/10.24136/eq.2021.032
Kliestik, T., Valaskova, K., Nica, E., Kovacova, M., & Lazaroiu, G. (2020). Advanced methods of earnings management: Monotonic trends and change-points under spotlight in the Visegrad countries. Oeconomia Copernicana, 11(2), 371–400. https://doi.org/10.24136/oc.2020.016
Kramoliš, J., & Dobeš, K. (2020). Debt as a financial risk factor in SMEs in the Czech Republic. Equilibrium. Quarterly Journal of Economics and Economic Policy, 15(1), 87–105. https://doi.org/10.24136/eq.2020.005
Lim, D., Li, L., & Linetsky, V. (2012). Evaluating callable and putable bonds: An eigenfunction expansion approach. Journal of Economic Dynamics & Control, 36(12), 1888–1908. https://doi.org/10.1016/j.jedc.2012.06.002
Marsh, T. A., & Rosenfeld, E. R. (1983). Stochastic processes for interest rates and equilibrium bond prices. Journal of Finance, 38(2), 635–646. https://doi.org/10.1111/j.1540-6261.1983.tb02275.x
Meluzín, T., Zinecker, M., Balcerzak, A. P., Pietrzak, M. B., & Doubravský, K. (2021). Institutional settings and their impact on the IPO activity: An exploratory study based on qualitative modelling. Acta Polytechnica Hungarica, 18(2), 215–235. https://doi.org/10.12700/APH.18.2.2021.2.12
Merton, R. C. (1974). On the pricing of corporate debt: The risk structure of interest rates. Journal of Finance, 29(2), 449–470. https://doi.org/10.1111/j.1540-6261.1974.tb03058.x
Myers, S. C. (1977). Determinants of corporate borrowing. Journal of Financial Economics, 5(2), 147–175. https://doi.org/10.1016/0304-405X(77)90015-0
Onuferová, E., Čabinová, V., & Dzurov Vargová, T. (2020). Analysis of modern methods for increasing and managing the financial prosperity of businesses in the context of performance: A case study of the tourism sector in Slovakia. Oeconomia Copernicana, 11(1), 95–116. https://doi.org/10.24136/oc.2020.004
Panova, E. (2020). Determinants of capital structure in Russian small and medium manufacturing enterprises. Equilibrium. Quarterly Journal of Economics and Economic Policy, 15(2), 361–375. https://doi.org/10.24136/eq.2020.017
Priem, R. (2021). An exploratory study on the impact of the COVID-19 confinement on the financial behavior of individual investors. Economics, Management, and Financial Markets, 16(3), 9–40. https://doi.org/10.22381/emfm16320211
Robbins, E. H., & Schatzberg, J. (1988). Callable bonds: A risk-reducing signalling mechanism – A reply. Journal of Finance, 43(4), 1067–1073. https://doi.org/10.1111/j.1540-6261.1988.tb02625.x
Schinckus, C., Nguyen, C. P., & Chong, F. H. L. (2021). Are Bitcoin and ether affected by strictly anonymous crypto-currencies? An exploratory study. Economics, Management, and Financial Markets, 16(4), 9–27. https://doi.org/10.22381/emfm16420211
Setianto, R. H., Sipayung, R. S., & Azman-Saini, W. N. W. (2022). Working capital financing and corporate profitability in the ASEAN region: The role of financial development. Entrepreneurial Business and Economics Review, 10(1), 51–64. https://doi.org/10.15678/EBER.2022.100104
Valaskova, K., Adamko, P., Frajtova Michalikova, K., & Macek, J. (2021a). Quo Vadis, earnings management? Analysis of manipulation determinants in Central European environment. Oeconomia Copernicana, 12(3), 631–669. https://doi.org/10.24136/oc.2021.021
Valaskova, K., Kliestik, T., & Gajdosikova, D. (2021b). Distinctive determinants of financial indebtedness: Evidence from Slovak and Czech enterprises. Equilibrium. Quarterly Journal of Economics and Economic Policy, 16(3), 639–659. https://doi.org/10.24136/eq.2021.023
Vasicek, O. (1977). An equilibrium characterization of the term structure. Journal of Financial Economics, 5(2), 177–188. https://doi.org/10.1016/0304-405X(77)90016-2
Xie, D. (2009). Theoretical and numerical valuation of callable bonds. International Journal of Business and Finance Research, 3(2), 71–82.
Xue, J. (2011). Pricing callable bonds (Project Report 2011:24). Uppsala University. Department of Mathematics. https://uu.diva-portal.org/smash/get/diva2:462176/FULLTEXT01.pdf
Zinecker, M., Skalická, M., Balcerzak, A. P., & Pietrzak, M. (2021a). Business angels in the Czech Republic: Characteristics and a classification with policy implications. Economic Research-Ekonomska Istraživanja. https://doi.org/10.1080/1331677X.2021.1890179
Zinecker, M., Skalická, M., Balcerzak, A. P., & Pietrzak, M. (2021b). Identifying the impact of external environment on business angel activity. Economic Research-Ekonomska Istraživanja. https://doi.org/10.1080/1331677X.2021.1888140