Abstract
Most empirical studies show that three factors are sufficient to explain all the relevant uncertainties inherent in option prices. In this paper, we consider a three-factor CIR model exhibiting unspanned stochastic volatility (USV), which means that it is impossible to fully hedge volatility risk with portfolios of bonds or swaps. The incompleteness of bond markets is necessary for the existence of USV. Restrictions on the model parameters are needed for incompleteness. We provide necessary and sufficient conditions for a three-factor CIR model that generates incomplete bond markets. Bond prices are exponential affine functions of only the two term-structure factors, independent of the unspanned factor. With our three-factor CIR model exhibiting USV, we derive the dynamic form of bond futures prices. By introducing the exponential solution of a transform and using the Fourier inversion theorem, we obtain a closed-form solution for the European zero-coupon option prices. The pricing method is efficient for taking into account the existence of unspanned stochastic volatility.
Similar content being viewed by others
References
Andersen, T. G., & Benzoni, L. (2010). Do bonds span volatility risk in the U.S. Treasury market? A specification test of affine term structure models. Journal of Finance, 65(2), 603–653. https://doi.org/10.1111/j.1540-6261.2009.01546.x
Backwell, A. (2021). Unspanned stochastic volatility from an empirical and practical perspective. Journal of Banking and Finance, 122, 1–14. https://doi.org/10.1016/j.jbankfin.2020.105993
Bakshi, G., Crosby, J., Gao, X., & Hansen, J. W. (2023). Treasury option returns and models with unspanned risks. Journal of Financial Economics, 150(3), 103736. https://doi.org/10.1016/j.jfineco.2023.103736
Bakshi, G., & Madan, D. (2000). Spanning and derivative-security valuation. Journal of Financial Economics, 55, 205–238. https://doi.org/10.1016/S0304-405X(99)00050-1
Bikbov, R., & Chernov, M. (2009). Unspanned stochastic volatility in affine models: evidence from eurodollar futures and options. Management Science, 55(8), 1292–1305. https://doi.org/10.1287/mnsc.1090.1020
Carr, P., Gabaix, X., & Wu, L. (2011). Linearity-generating process, unspanned stochastic volatility, and interest-rate option pricing. Working paper. https://doi.org/10.2139/ssrn.1789763.
Casassus, J., Collin-Dufresne, P., & Goldstein, R. S. (2005). Unspanned stochastic volatility and fixed income derivative pricing. Journal of Banking and Finance, 29(11), 2723–2749. https://doi.org/10.1016/j.jbankfin.2005.02.007
Collin-Dufresne, P., & Goldstein, R. S. (2002). Do bonds span the fixed income markets? Theory and evidence for unspanned stochastic volatility. Journal of Finance, 57(4), 1685–1730. https://doi.org/10.1111/1540-6261.00475
Collin-Dufresne, P., & Goldstein, R.S. (2003). Generalizing the affine framework to HJM and random field models. Working Paper. https://doi.org/10.2139/ssrn.410421.
Cox, J., Ingersoll, J., & Ross, R. (1981). The relation between forward prices and futures prices. Journal of Financial Economics, 36(9), 321–346. https://doi.org/10.1016/0304-405X(81)90002-7
Duffie, D., Pan, J., & Singleton, K. (2000). Transform analysis and option pricing for affine jump-diffusions. Econometrica, 68(6), 1343–1376. https://doi.org/10.1111/1468-0262.00164
Fan, R., Gupta, A., & Ritchken, P. (2003). Hedging in the possible presence of unspanned stochastic volatility: Evidence from swaption markets. Journal of Finance, 58(5), 2219–2248. https://doi.org/10.1111/1540-6261.00603
Filipović, D., Larsson, M., & Statti, F. (2019). Unspanned stochastic volatility in the multi-factor CIR model. Mathematical Finance, 29(3), 827–836. https://doi.org/10.1111/mafi.12193
Filipović, D., Larsson, M., & Trolle, A. B. (2017). Interest appendix for Linear-rational term structure models. Journal of Finance, 72(2), 655–704. https://doi.org/10.1111/jofi.12488
Heidari, M., & Wu, L. (2003). Are interest rate derivatives spanned by the term structure of interest rates? Journal of Fixed Income, 13(1), 75–86. https://doi.org/10.3905/jfi.2003.319347
Heston, S. L. (1993). A closed form solution for options with stochastic volatility. Review of Financial Studies, 6(2), 327–343. https://doi.org/10.1093/rfs/6.2.327
Jagannathan, R., Kaplin, A., & Sun, S. (2003). An evaluation of multi-factor CIR models using LIBOR, swap rates, and cap and swaption prices. Journal of Econometrics, 116(1–2), 113–146. https://doi.org/10.1016/S0304-4076(03)00105-2
Joslin, S. (2018). Can unspanned stochastic volatility models explain the cross section of bond volatilities? Management Science, 64(4), 1477–1973. https://doi.org/10.1287/mnsc.2016.2623
Li, H., & Zhao, F. (2006). Unspanned stochastic volatility: Evidence from hedging interest rate cap prices. Journal of Finance, 61(1), 341–378. https://doi.org/10.1111/j.1540-6261.2006.00838.x
Litterman, R., & Scheinkman, J. (1991). Common factors affecting bond returns. Journal of Fixed Income, 1(1), 54–61. https://doi.org/10.3905/jfi.1991.692347
Schöne, M. F., & Spinler, S. (2017). A four-factor stochastic volatility model of commodity prices. Review of Derivatives Research, 20, 135–165. https://doi.org/10.1007/s11147-016-9126-y
Thompson, S. (2008). Identifying term structure volatility from the LIBOR-swap curve. Review of Financial Studies, 21(3), 819–854. https://doi.org/10.1093/rfs/hhm082
Trolle, A. B., & Schwartz, E. S. (2009). Unspanned stochastic volatility and the pricing of commodity derivatives. Review of Financial Studies, 22(11), 4423–4461. https://doi.org/10.1093/rfs/hhp036
Acknowledgements
This work was partially supported by the National Natural Science Foundation of China (No.11871244) and the Fundamental Research Funds for the Central Universities, JLU.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors have no competing interests to declare that are relevant to the content of this article.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Han, Y., Zhang, F. Pricing fixed income derivatives under a three-factor CIR model with unspanned stochastic volatility. Rev Deriv Res 27, 37–53 (2024). https://doi.org/10.1007/s11147-023-09198-2
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11147-023-09198-2