# American Institute of Mathematical Sciences

January 2018, 5(1): 21-30. doi: 10.3934/jdg.2018003

## Pricing bond options in emerging markets: A case study

 1 Department of Legal Governance, Risk Management, and Compliance, AFISA SURA, L.A. De Herrera 1248 Torre 3 piso 16, Montevideo, Uruguay 2 Centro de Matemática, Facultad de Ciencias, Universidad de la Repúlbica, Iguá 4225, CP 11400, Montevideo, Uruguay

Received  January 2017 Revised  September 2017 Published  January 2018

We propose two methodologies to price sovereign bond options in emerging markets. The motivation is to provide hedging protection against price fluctuations, departing from the not liquid data provided by the stock exchange. Taking this into account, we first compute prices provided by the Jamshidian formula, when modeling the interest rate through Vasicek model, with parameters estimated with the help of the Kalman filter. The second methodology is the pricing strategy provided by the Black-Derman-Toy tree model. A numerical comparison is carried out. The first equilibrium approach provides parsimonious modeling, is less sensitive to daily changes and more robust, while the second non-arbitrage approach provides more fluctuating but also what can be considered more accurate option prices.

Citation: Guillermo Magnou, Ernesto Mordecki, Andrés Sosa. Pricing bond options in emerging markets: A case study. Journal of Dynamics & Games, 2018, 5 (1) : 21-30. doi: 10.3934/jdg.2018003
##### References:
 [1] F. Avalos and R. Moreno, Hedging in Derivatives Markets: The Experience of Chile, Quarterly Review, March 2013 -Bank for International Settlements, 2013. [2] OTC Interest Rate Derivatives Turnover in April 2016, Bank for International Settlements, 2016. Available from: http://www.bis.org/publ/rpfx16.htm. [3] F. Black, E. Derman and W. Toy, A one-factor model of interest rates and its application to treasury bond options, Financial Analysts Journal, 46 (1990), 24-32. doi: 10.2469/faj.v46.n1.33. [4] F. Black and P. Karasinski, Bond and Option pricing when Short rates are Log-normal, Financial Analysts Journal, (1991), 52-59. [5] D. Brigo and F. Mercurio, Interest Rate Models Theory and Practice with Smile, Inflation and Credit, 2$^{nd}$ edition, Springer-Verlag, New York, 2006. [6] R. Chen and L. Scott, Multi-factor cox-ingersoll-ross models of the term structure: Estimates and tests from a kalman filter model, The Journal of Real Estate Finance and Economics, 27 (2003), 143-172. [7] M. Choy and J. Cerna, Interrelación entre los mercados de derivados y el mercado de bonos soberanos del Perú y su impacto en las tasas de interés, [Working Paper -Banco Central de Reserva de Perú], (2012). [8] R. Dodd and S. Griffith-Jones, Brazil's Derivatives Markets: Hedging, Central Bank Intervention and Regulation, [Cepal Report], 2007. [9] D. Filipovic, Term-Structure Models, Springer Finance, Vienna, 2009. doi: 10.1007/978-3-540-68015-4. [10] A. Harvey, Forecasting, Structural Time Series Models and the Kalman Filter, Cambridge University Press, 1992. doi: 10.1017/CBO9781107049994. [11] D. Heath, R. Jarrow and A. Morton, Bond pricing and the term structure of interest rates: A new methodology, Econometrica, 60 (1992), 77-105. [12] T. Ho and S. Lee, Term structure movements and pricing interest rate contingent claims, Journal of Finance, 41 (1986), 1011-1029. doi: 10.1111/j.1540-6261.1986.tb02528.x. [13] J. Hull and A. White, One factor interest rate models and the valuation of interest rate derivative securities, Journal of Financial and Quantitative Analysis, 28 (1993), 235-254. doi: 10.2307/2331288. [14] D. Jamieson, Affine Term-Structure Models: Theory and Implementation, [Working Paper -Bank of Canada], 2001. [15] F. Jamshidian, An exact bond option formula, The Journal of Finance, 44 (1989), 205-209. doi: 10.1111/j.1540-6261.1989.tb02413.x. [16] R. Kalman, A new approach to linear filtering and prediction problems, Journal of Basic Engineering, 82 (1960), 35-45. doi: 10.1115/1.3662552. [17] G. Magnou, Opciones Financieras Sobre Bonos. Una Aplicación para el Mercado Uruguayo, Mg. thesis, Facultad de Ingeniería, Universidad de la República, 2015. [18] E. Mordecki and A. Sosa, Modeling the Uruguayan debt through gaussians models, Trends in Mathematical Economics. Springer Proceedings of Mathematics and Statistics Series, 1 (2016), 331-346. [19] C. Upper and M. Valli, Emerging derivatives markets?, International banking and financial market developments, (2016), 67-80. [20] O. Vasicek, An equilibrium characterization of the term structure, Journal of Financial Economics, 5 (1977), 177-188. doi: 10.1002/9781119186229.ch6.

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##### References:
 [1] F. Avalos and R. Moreno, Hedging in Derivatives Markets: The Experience of Chile, Quarterly Review, March 2013 -Bank for International Settlements, 2013. [2] OTC Interest Rate Derivatives Turnover in April 2016, Bank for International Settlements, 2016. Available from: http://www.bis.org/publ/rpfx16.htm. [3] F. Black, E. Derman and W. Toy, A one-factor model of interest rates and its application to treasury bond options, Financial Analysts Journal, 46 (1990), 24-32. doi: 10.2469/faj.v46.n1.33. [4] F. Black and P. Karasinski, Bond and Option pricing when Short rates are Log-normal, Financial Analysts Journal, (1991), 52-59. [5] D. Brigo and F. Mercurio, Interest Rate Models Theory and Practice with Smile, Inflation and Credit, 2$^{nd}$ edition, Springer-Verlag, New York, 2006. [6] R. Chen and L. Scott, Multi-factor cox-ingersoll-ross models of the term structure: Estimates and tests from a kalman filter model, The Journal of Real Estate Finance and Economics, 27 (2003), 143-172. [7] M. Choy and J. Cerna, Interrelación entre los mercados de derivados y el mercado de bonos soberanos del Perú y su impacto en las tasas de interés, [Working Paper -Banco Central de Reserva de Perú], (2012). [8] R. Dodd and S. Griffith-Jones, Brazil's Derivatives Markets: Hedging, Central Bank Intervention and Regulation, [Cepal Report], 2007. [9] D. Filipovic, Term-Structure Models, Springer Finance, Vienna, 2009. doi: 10.1007/978-3-540-68015-4. [10] A. Harvey, Forecasting, Structural Time Series Models and the Kalman Filter, Cambridge University Press, 1992. doi: 10.1017/CBO9781107049994. [11] D. Heath, R. Jarrow and A. Morton, Bond pricing and the term structure of interest rates: A new methodology, Econometrica, 60 (1992), 77-105. [12] T. Ho and S. Lee, Term structure movements and pricing interest rate contingent claims, Journal of Finance, 41 (1986), 1011-1029. doi: 10.1111/j.1540-6261.1986.tb02528.x. [13] J. Hull and A. White, One factor interest rate models and the valuation of interest rate derivative securities, Journal of Financial and Quantitative Analysis, 28 (1993), 235-254. doi: 10.2307/2331288. [14] D. Jamieson, Affine Term-Structure Models: Theory and Implementation, [Working Paper -Bank of Canada], 2001. [15] F. Jamshidian, An exact bond option formula, The Journal of Finance, 44 (1989), 205-209. doi: 10.1111/j.1540-6261.1989.tb02413.x. [16] R. Kalman, A new approach to linear filtering and prediction problems, Journal of Basic Engineering, 82 (1960), 35-45. doi: 10.1115/1.3662552. [17] G. Magnou, Opciones Financieras Sobre Bonos. Una Aplicación para el Mercado Uruguayo, Mg. thesis, Facultad de Ingeniería, Universidad de la República, 2015. [18] E. Mordecki and A. Sosa, Modeling the Uruguayan debt through gaussians models, Trends in Mathematical Economics. Springer Proceedings of Mathematics and Statistics Series, 1 (2016), 331-346. [19] C. Upper and M. Valli, Emerging derivatives markets?, International banking and financial market developments, (2016), 67-80. [20] O. Vasicek, An equilibrium characterization of the term structure, Journal of Financial Economics, 5 (1977), 177-188. doi: 10.1002/9781119186229.ch6.
Monthly returns of AFAPS. Year 2013 (in UYU).
Example of a four period binomial tree.
Option prices for zero coupon USD (left) and UI bonds (right). VK and BDT stand for Vasicek and Black-Derman-Toy models respectively
 USD 3 months option prices UI 3 months option prices Strike CALL PUT Strike CALL PUT VK BDT VK BDT VK BDT VK BDT $99.50$ $0.000$ $0.000$ $5.268$ $5.069$ $94.00$ $0.000$ $0.000$ $3.617$ $3.580$ $97.06$ $0.000$ $0.000$ $2.831$ $2.641$ $92.18$ $0.000$ $0.000$ $1.817$ $1.782$ $94.69$ $0.001$ $0.187$ $0.464$ $0.459$ $90.40$ $0.117$ $0.240$ $0.173$ $0.259$ $92.37$ $1.853$ $2.044$ $0.000$ $0.007$ $88.65$ $1.675$ $1.711$ $0.000$ $0.001$ $90.11$ $4.111$ $4.291$ $0.000$ $0.000$ $86.93$ $3.377$ $3.405$ $0.000$ $0.000$ USD 6 months option prices UI 6 months option prices Strike CALL PUT Strike CALL PUT VK BDT VK BDT VK BDT VK BDT $99.55$ $0.000$ $0.000$ $5.117$ $4.852$ $94.76$ $0.000$ $0.000$ $3.341$ $3.128$ $97.36$ $0.000$ $0.000$ $2.934$ $2.673$ $93.15$ $0.000$ $0.002$ $1.766$ $1.569$ $95.22$ $0.015$ $0.160$ $0.801$ $0.703$ $91.58$ $0.086$ $0.267$ $0.316$ $0.300$ $93.12$ $1.129$ $1.599$ $0.000$ $0.058$ $90.03$ $1.286$ $1.484$ $0.001$ $0.008$ $91.07$ $3.336$ $3.582$ $0.000$ $0.003$ $88.51$ $2.773$ $2.958$ $0.000$ $0.000$ USD 1 year option prices UI 1 year option prices Strike CALL PUT Strike CALL PUT VK BDT VK BDT VK BDT VK BDT $99.65$ $0.000$ $0.000$ $4.561$ $4.224$ $96.25$ $0.000$ $0.000$ $2.673$ $2.229$ $97.94$ $0.000$ $0.000$ $2.867$ $2.538$ $95.10$ $0.000$ $0.005$ $1.573$ $1.136$ $96.26$ $0.000$ $0.069$ $1.204$ $0.950$ $93.95$ $0.037$ $0.231$ $0.511$ $0.277$ $94.61$ $0.441$ $0.918$ $0.011$ $0.170$ $92.83$ $0.619$ $1.047$ $0.002$ $0.021$ $92.99$ $2.034$ $2.373$ $0.000$ $0.023$ $91.71$ $1.669$ $2.086$ $0.000$ $0.001$
 USD 3 months option prices UI 3 months option prices Strike CALL PUT Strike CALL PUT VK BDT VK BDT VK BDT VK BDT $99.50$ $0.000$ $0.000$ $5.268$ $5.069$ $94.00$ $0.000$ $0.000$ $3.617$ $3.580$ $97.06$ $0.000$ $0.000$ $2.831$ $2.641$ $92.18$ $0.000$ $0.000$ $1.817$ $1.782$ $94.69$ $0.001$ $0.187$ $0.464$ $0.459$ $90.40$ $0.117$ $0.240$ $0.173$ $0.259$ $92.37$ $1.853$ $2.044$ $0.000$ $0.007$ $88.65$ $1.675$ $1.711$ $0.000$ $0.001$ $90.11$ $4.111$ $4.291$ $0.000$ $0.000$ $86.93$ $3.377$ $3.405$ $0.000$ $0.000$ USD 6 months option prices UI 6 months option prices Strike CALL PUT Strike CALL PUT VK BDT VK BDT VK BDT VK BDT $99.55$ $0.000$ $0.000$ $5.117$ $4.852$ $94.76$ $0.000$ $0.000$ $3.341$ $3.128$ $97.36$ $0.000$ $0.000$ $2.934$ $2.673$ $93.15$ $0.000$ $0.002$ $1.766$ $1.569$ $95.22$ $0.015$ $0.160$ $0.801$ $0.703$ $91.58$ $0.086$ $0.267$ $0.316$ $0.300$ $93.12$ $1.129$ $1.599$ $0.000$ $0.058$ $90.03$ $1.286$ $1.484$ $0.001$ $0.008$ $91.07$ $3.336$ $3.582$ $0.000$ $0.003$ $88.51$ $2.773$ $2.958$ $0.000$ $0.000$ USD 1 year option prices UI 1 year option prices Strike CALL PUT Strike CALL PUT VK BDT VK BDT VK BDT VK BDT $99.65$ $0.000$ $0.000$ $4.561$ $4.224$ $96.25$ $0.000$ $0.000$ $2.673$ $2.229$ $97.94$ $0.000$ $0.000$ $2.867$ $2.538$ $95.10$ $0.000$ $0.005$ $1.573$ $1.136$ $96.26$ $0.000$ $0.069$ $1.204$ $0.950$ $93.95$ $0.037$ $0.231$ $0.511$ $0.277$ $94.61$ $0.441$ $0.918$ $0.011$ $0.170$ $92.83$ $0.619$ $1.047$ $0.002$ $0.021$ $92.99$ $2.034$ $2.373$ $0.000$ $0.023$ $91.71$ $1.669$ $2.086$ $0.000$ $0.001$
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