The SAFEX-JIBAR Market Models

Author(s)
Victor Gumbo

Affiliation(s)

Department of Finance, National University of Science & Technology, Bulawayo, Zimbabwe.

Department of Finance, National University of Science & Technology, Bulawayo, Zimbabwe.

ABSTRACT

It is possible to construct an arbitrage-free interest rate model in which the LIBOR rates follow a log-normal process leading to Black-type pricing formulae for caps and floors. The key to their approach is to start directly with modeling observed market rates, LIBOR rates in this case, instead of instantaneous spot rates or forward rates. This model is known as the LIBOR Market Model. We formulate the SAFEX-JIBAR market model based on the fact that the forward JIBAR rates follow a log-normal process. Formulae of the Black-type are deduced.

It is possible to construct an arbitrage-free interest rate model in which the LIBOR rates follow a log-normal process leading to Black-type pricing formulae for caps and floors. The key to their approach is to start directly with modeling observed market rates, LIBOR rates in this case, instead of instantaneous spot rates or forward rates. This model is known as the LIBOR Market Model. We formulate the SAFEX-JIBAR market model based on the fact that the forward JIBAR rates follow a log-normal process. Formulae of the Black-type are deduced.

Cite this paper

V. Gumbo, "The SAFEX-JIBAR Market Models,"*Journal of Mathematical Finance*, Vol. 2 No. 4, 2012, pp. 321-326. doi: 10.4236/jmf.2012.24035.

V. Gumbo, "The SAFEX-JIBAR Market Models,"

References

[1] F. Black, “Pricing of Commodity Contracts,” Journal of Financial Economics, Vol. 3, No. 1-2, 1976, pp. 167-179. doi:10.1016/0304-405X(76)90024-6

[2] A. Brace, D. Gatarek and M. Musiela, “The Market Model of Interest rate Dynamics,” Mathematical Finance, Vol. 7, No. 2, 1997, pp. 127-155. doi:10.1111/1467-9965.00028

[3] K. Miltersen, K. Sandmann and D. Sondermann, “Closed Form Solutions for Term-Structure Derivatives with Log-normal Interest Rates,” Finance, Vol. 52, No. 1, 1997, pp. 407-430. doi:10.1111/j.1540-6261.1997.tb03823.x

[4] F. Jamshidian, “Libor and Swap Market Models and Measures,” Finance and Stochastics, Vol. 1, No. 4, 1997, pp. 293-330. doi:10.1007/s007800050026

[5] T. Bjork, “Arbitrage Theory in Continuous Time,” 2nd Edition, Oxford University Press, Oxford, 2004. doi:10.1093/0199271267.001.0001

[6] V. Gumbo, “The LIBOR Market Model and Its Application in the SAFEX-JIBAR Market,” Lap Publishing, 2011.

[7] J. Schoenmakers and B. Coffey, “LIBOR Rate Models, Related Derivatives and Model Calibration,” WIAS Preprint No. 480, 1999.

[8] R. Rebonato, “Modern Pricing of Interest Rate Derivatives—The LIBOR Market Model and Beyond”, Princeton University Press, Princeton, 2008.

[9] P. Spangenberg, “The Mechanics of Option-Styled Interest Rate Derivatives—caps and floors,” 1999. www.actsa.org.za/articles

[10] W. Drobetz, “Interest Rate Derivatives,” University of Basel and Otto Beisheim Graduate School of Management (WHU), Basel, 2002.

[11] A. Kuprianov, “Over-the-Counter Interest Rate Derivatives,” Federal Reserve Bank of Richmond, Richmond, 1998.

[12] M. Musiela and M. Rutkowski, “Martingale Methods in Financial Modelling,” Springer, New York, 1997.

[13] P. J?ckel and R. Rebonato, “Accurate and Optimal Calibration to Co-Terminal European Swaptions in a FRA-Based BGM Framework,” QUARC Paper, Royal Bank of Scotland Group, Edinburgh, 2000.

[14] D. Brigo and F. Mercurio, “Interest Rate Models: Theory and Practice,” Springer-Verlag, Berlin, 2001.

[15] D. Brigo, F. Mercurio and M. Morini, “The LIBOR Model Dynamics: Approximations, Calibration and Diagnostics,” European Journal of Operations Research, Vol. 163, No. 1, 2005, pp. 30-51. doi:10.1016/j.ejor.2003.12.004

[16] D. Brigo, F. Mercurio, C. Capitani, “On the joint calibration of the LIBOR market model to caps and swaptions market volatilities” July 2001 version.

[17] R. Bhar, C. Chiarella, H. Hung and W. J. Runggaldier, “The Volatility of the Instantaneous Spot Interest Rate Implied by Arbitrage Pricing—A Dynamic Bayesian Approach,” Automata, Vol. 42, No. 8, 2006, pp. 1381-1393. doi:10.1016/j.automatica.2005.12.027

[18] L. C. Rogers, “The Potential Approach to the Term Structure of Interest Rates and Foreign Exchange Rates,” Mathematical Finance, Vol. 7, No. 2, 1997, pp. 157-176. doi:10.1111/1467-9965.00029

[19] C. J. Hunter, P. Jackel and M. S. Joshi, “Drift Approximations in a Forward-Risk-Base LIBOR Market Model,” Market Model. Getting the Drift, Risk, Vol. 14, No. 7, 2001, pp. 81-84.

[20] M. S. Joshi and T. Jocken, “Bounding Bermudan Swaptions in a Swap-Rate Market Model,” Quantitative Finance, Vol. 2, No. 5, 2002, pp. 370-377.

[21] P. J?ckel, “Non-Recombining Trees for the Pricing of Interest rate Derivatives in the BGM/J Framework”, Working Paper, Quantitative Research Centre, Royal Bank of Scotland, Edinburgh, 2000.

[22] D. Duffie and K. J. Singleton, “An Econometric Model of Term Structure of Interest-Rate Swap Yields,” Finance, Vol. 52, No. 4, 1997, pp. 1287-1321. doi:10.1111/j.1540-6261.1997.tb01111.x

[23] F. De Jong, J. Driessen and A. Pelsser, “LIBOR and Swap Market Models for the Pricing of Interest Rate Derivatives: An empirical comparison,” Working Paper, Center for Economic Research, Tilburg University, Tilburg, 2000.

[24] P. Glasserman and N. Merener, “Numerical Solution of Jump-Diffusion LIBOR Market Models,” Finance & Stochastics, Vol. 7, No. 1, 2001, p. 1.

[25] P. Glasserman and N. Merener, “Cap and Swaption Approximations in LIBOR Market Models with Jumps,” Journal of Computational Finance, Vol. 7, No. 1, 2003, pp. 1-36.

[1] F. Black, “Pricing of Commodity Contracts,” Journal of Financial Economics, Vol. 3, No. 1-2, 1976, pp. 167-179. doi:10.1016/0304-405X(76)90024-6

[2] A. Brace, D. Gatarek and M. Musiela, “The Market Model of Interest rate Dynamics,” Mathematical Finance, Vol. 7, No. 2, 1997, pp. 127-155. doi:10.1111/1467-9965.00028

[3] K. Miltersen, K. Sandmann and D. Sondermann, “Closed Form Solutions for Term-Structure Derivatives with Log-normal Interest Rates,” Finance, Vol. 52, No. 1, 1997, pp. 407-430. doi:10.1111/j.1540-6261.1997.tb03823.x

[4] F. Jamshidian, “Libor and Swap Market Models and Measures,” Finance and Stochastics, Vol. 1, No. 4, 1997, pp. 293-330. doi:10.1007/s007800050026

[5] T. Bjork, “Arbitrage Theory in Continuous Time,” 2nd Edition, Oxford University Press, Oxford, 2004. doi:10.1093/0199271267.001.0001

[6] V. Gumbo, “The LIBOR Market Model and Its Application in the SAFEX-JIBAR Market,” Lap Publishing, 2011.

[7] J. Schoenmakers and B. Coffey, “LIBOR Rate Models, Related Derivatives and Model Calibration,” WIAS Preprint No. 480, 1999.

[8] R. Rebonato, “Modern Pricing of Interest Rate Derivatives—The LIBOR Market Model and Beyond”, Princeton University Press, Princeton, 2008.

[9] P. Spangenberg, “The Mechanics of Option-Styled Interest Rate Derivatives—caps and floors,” 1999. www.actsa.org.za/articles

[10] W. Drobetz, “Interest Rate Derivatives,” University of Basel and Otto Beisheim Graduate School of Management (WHU), Basel, 2002.

[11] A. Kuprianov, “Over-the-Counter Interest Rate Derivatives,” Federal Reserve Bank of Richmond, Richmond, 1998.

[12] M. Musiela and M. Rutkowski, “Martingale Methods in Financial Modelling,” Springer, New York, 1997.

[13] P. J?ckel and R. Rebonato, “Accurate and Optimal Calibration to Co-Terminal European Swaptions in a FRA-Based BGM Framework,” QUARC Paper, Royal Bank of Scotland Group, Edinburgh, 2000.

[14] D. Brigo and F. Mercurio, “Interest Rate Models: Theory and Practice,” Springer-Verlag, Berlin, 2001.

[15] D. Brigo, F. Mercurio and M. Morini, “The LIBOR Model Dynamics: Approximations, Calibration and Diagnostics,” European Journal of Operations Research, Vol. 163, No. 1, 2005, pp. 30-51. doi:10.1016/j.ejor.2003.12.004

[16] D. Brigo, F. Mercurio, C. Capitani, “On the joint calibration of the LIBOR market model to caps and swaptions market volatilities” July 2001 version.

[17] R. Bhar, C. Chiarella, H. Hung and W. J. Runggaldier, “The Volatility of the Instantaneous Spot Interest Rate Implied by Arbitrage Pricing—A Dynamic Bayesian Approach,” Automata, Vol. 42, No. 8, 2006, pp. 1381-1393. doi:10.1016/j.automatica.2005.12.027

[18] L. C. Rogers, “The Potential Approach to the Term Structure of Interest Rates and Foreign Exchange Rates,” Mathematical Finance, Vol. 7, No. 2, 1997, pp. 157-176. doi:10.1111/1467-9965.00029

[19] C. J. Hunter, P. Jackel and M. S. Joshi, “Drift Approximations in a Forward-Risk-Base LIBOR Market Model,” Market Model. Getting the Drift, Risk, Vol. 14, No. 7, 2001, pp. 81-84.

[20] M. S. Joshi and T. Jocken, “Bounding Bermudan Swaptions in a Swap-Rate Market Model,” Quantitative Finance, Vol. 2, No. 5, 2002, pp. 370-377.

[21] P. J?ckel, “Non-Recombining Trees for the Pricing of Interest rate Derivatives in the BGM/J Framework”, Working Paper, Quantitative Research Centre, Royal Bank of Scotland, Edinburgh, 2000.

[22] D. Duffie and K. J. Singleton, “An Econometric Model of Term Structure of Interest-Rate Swap Yields,” Finance, Vol. 52, No. 4, 1997, pp. 1287-1321. doi:10.1111/j.1540-6261.1997.tb01111.x

[23] F. De Jong, J. Driessen and A. Pelsser, “LIBOR and Swap Market Models for the Pricing of Interest Rate Derivatives: An empirical comparison,” Working Paper, Center for Economic Research, Tilburg University, Tilburg, 2000.

[24] P. Glasserman and N. Merener, “Numerical Solution of Jump-Diffusion LIBOR Market Models,” Finance & Stochastics, Vol. 7, No. 1, 2001, p. 1.

[25] P. Glasserman and N. Merener, “Cap and Swaption Approximations in LIBOR Market Models with Jumps,” Journal of Computational Finance, Vol. 7, No. 1, 2003, pp. 1-36.