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## Equilibrium valuation of currency options with stochastic volatility and systemic co-jumps

 1 School of Finance, Nanjing Audit University, Key Laboratory of Financial Engineering, Nanjing Audit University, Nanjing 211815, China 2 School of Mathematics and Statistics, Ningbo University, Ningbo 315211, China 3 School of Statistics and Data Science, Nanjing Audit University, Key Laboratory of Financial Engineering, Nanjing Audit University, Nanjing 211815, China 4 School of Economics and Management, China University of Mining and Technology, Xuzhou 221116, China

*Corresponding author: Xiaonan Su

Received  October 2020 Revised  December 2021 Early access February 2022

Fund Project: This work was supported by Philosophy and Social Science Research in Colleges and Universities of Jiangsu Province (2021SJA0362), Open project of Jiangsu key laboratory of financial engineering (NSK2021-13, NSK2021-15), Applied Economics of Nanjing Audit University of the Priority Academic Program Development of Jiangsu Higher Education(Office of Jiangsu Provincial Peoples Government, No.[2018]87), Humanity and Social Science Youth Foundation of the Ministry of Education of China (18YJC910012), National Natural Science Foundation of China (71871120), the Major Natural Science Foundation of Jiangsu Higher Education Institutions (20KJA120002).

We consider the equilibrium valuation of currency options with stochastic volatility and systemic co-jumps under the setting of Lucas-type two country economy. Based on the stochastic volatility model in [2], we add an independent jump process and a co-jump process to model the money supply in each country. By solving a partial integro-differential equation (PIDE) for currency options, we can get a closed-form solution for a call currency option price. Compared with the option prices calculated by Monte Carlo method, we show the derived option pricing formula is efficient for practical use. The numerical results show that stochastic volatility and co-jumps have significant impacts on option prices and implied volatilities.

Citation: Yu Xing, Wei Wang, Xiaonan Su, Huawei Niu. Equilibrium valuation of currency options with stochastic volatility and systemic co-jumps. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022022
##### References:
 [1] T. G. Andersen, T. Bollerslev and D. Dobrev, No-arbitrage semi-martingale restrictions for continuous-time volatility models subject to leverage effects, jumps and i.i.d noise: Theory and testable distributional implications, J. Econometrics, 138 (2007), 125-180.  doi: 10.1016/j.jeconom.2006.05.018. [2] G. Bakshi and Z. Chen, Equilibrium valuation of foreign exchange claims, Journal of Finance, 52 (1997), 799-826. [3] G. Bakshi and D. Madan, Spanning and derivative-security valuation, Journal of Financial Economics, 55 (2000), 205-238. [4] O. E. Barndorff-Nielsen and N. Shephard, Econometrics of testing for jumps in financial economics using bipower variation, Journal of Financial Econometrics, 4 (2006), 1-30. [5] J. Barunik and L. Vacha, Do co-jumps impact correlations in currency markets?, Journal of Financial Markets, 37 (2018), 97-119. [6] S. Basak and M. Gallmeyer, Currency prices, the nominal exchange rate, and security prices in a Two country dynamic monetary equilibrium, Math. Finance, 9 (1999), 1-30.  doi: 10.1111/1467-9965.00061. [7] D. Bates, Jumps and stochastic volatility: Exchange rate processes implicit in deutsche mark options, Review of Financial Studies, 9 (1996), 69-107.  doi: 10.1093/rfs/9.1.69. [8] M. Bibinger and L. Winkelmann, Econometrics of co-jumps in high-frequency data with noise, J. Econometrics, 184 (2015), 361-378.  doi: 10.1016/j.jeconom.2014.10.004. [9] M. Cao, Systematic jump risks in a small open economy: Simultaneous equilibrium valuation of options on the market portfolio and the exchange rate, Journal of International Money and Finance, 20 (2001), 191-218.  doi: 10.1016/S0261-5606(00)00053-X. [10] P. Carr and L. Wu, Stochastic skew in currency options, Journal of Financial Economics, 86 (2007), 213-247.  doi: 10.1016/j.jfineco.2006.03.010. [11] M. Caporin, A. Kolokolov and R. Renò, Systemic co-jumps, Journal of Financial Economics, 126 (2017), 563-591. [12] S. R. Das and R. Uppal, Systemic risk and international portfolio choice, The Journal of Finance, 59 (2004), 2809-2834. [13] D. Du, General equilibrium pricing of currency and currency options, Journal of Financial Economics, 110 (2013), 730-751.  doi: 10.1016/j.jfineco.2013.08.006. [14] K. Fan, Y. Shen, T. K. Siu and R. Wang, Pricing foreign equity options with regime-switching, Economic Modelling, 37 (2014), 296-305.  doi: 10.1016/j.econmod.2013.11.009. [15] E. Farhi and X. Gabaix, Rare disasters and exchange rates, Nber Working Paper Series, 2008. doi: 10.3386/w13805. [16] J. Fu and H. Yang, Equilibrium approach of asset pricing under Lévy process, European J. Oper. Res., 223 (2012), 701-708.  doi: 10.1016/j.ejor.2012.06.037. [17] M. Garman and S. Kohlhagen, Foreign currency option values, Journal of International Money and Finance, 2 (1983), 231-237.  doi: 10.1016/S0261-5606(83)80001-1. [18] S. Heston, Closed-form solution for options with stochastic volatility with application to bond and currency options, Rev. Financ. Stud., 6 (1993), 327-343.  doi: 10.1093/rfs/6.2.327. [19] Z. Hong, L. Niu and G. Zeng, US and Chinese yield curve responses to RMB exchange rate policy shocks: An analysis with the arbitrage-free Nelson-Siegel term structure model, China Finance Review International, 9 (2019), 360-385.  doi: 10.1108/CFRI-12-2017-0239. [20] S. Hurn, K. A. Lindsay and L. Xu, Revisiting the numerical solution of stochastic differential equations, China Finance Review International, 9 (2019), 312-323.  doi: 10.1108/CFRI-12-2018-0155. [21] O. W. Ibhagui, Monetary model of exchange rate determination under floating and non-floating regimes, China Finance Review International, 9 (2019), 254-283.  doi: 10.1108/CFRI-10-2017-0204. [22] J. Jacod and V. Todorov, Testing for common arrivals of jumps for discretely observed multidimensional processes, Ann. Statist., 37 (2009), 1792-1838.  doi: 10.1214/08-AOS624. [23] P. Jorion, On jump processes in the foreign exchange and stock markets, Review of Financial Studies, 1 (1988), 427-445.  doi: 10.1093/rfs/1.4.427. [24] S. S. Lee and P. A. Mykland, Jumps in financial markets: A new nonparametric test and jump dynamics, Review of Financial Studies, 21 (2008), 2535-2563. [25] J. Liu, Impact of uncertainty on foreign exchange market stability: Based on the LT-TVP-VAR model, China Finance Review International, 11 (2021), 53-72.  doi: 10.1108/CFRI-07-2019-0112. [26] R. E. Lucas, Interest rates and currency prices in a two-country world, Collected Papers on Monetary Theory, (2012).  doi: 10.4159/harvard.9780674067851.c6. [27] Y. Ma, D. Pan, K. Shrestha and W. Xu, Pricing and hedging foreign equity options under Hawkes jump-diffusion processes, Physica A, 537 (2020), 122645, 18 pp. doi: 10.1016/j.physa.2019.122645. [28] Y. Ma, K. Shrestha and W. Xu, Pricing vulnerable options with jump clustering, Journal of Futures Markets, 37 (2017), 1155-1178.  doi: 10.1002/fut.21843. [29] J. Nagayasu, Global and country-specific movements in real effective exchange rates: Implications for external competitiveness, Journal of International Money and Finance, 76 (2017), 88-105.  doi: 10.1016/j.jimonfin.2017.05.005. [30] H. W. Niu and D. C. Wang, Pricing vulnerable options with correlated jump-diffusion processes depending on various states of the economy, Quant. Finance, 16 (2016), 1129-1145.  doi: 10.1080/14697688.2015.1090623. [31] E. O. Ozturk and X. S. Sheng, Measuring global and country-specific uncertainty, Journal of International Money and Finance, 88 (2018), 276-295. [32] X. C. Wang, Pricing power exchange options with correlated jump risk, Finance Research Letters, 19 (2016), 90-97.  doi: 10.1016/j.frl.2016.06.009. [33] Y. Xing and X. P. Yang, Equilibrium valuation of currency options under a jump-diffusion model with stochastic volatility, J. Comput. Appl. Math., 280 (2015), 231-247.  doi: 10.1016/j.cam.2014.12.003. [34] F. Zapatero, Equilibrium asset prices and exchange rates, Journal of Economic Dynamics and Control, 19 (1995), 787-811.

show all references

##### References:
 [1] T. G. Andersen, T. Bollerslev and D. Dobrev, No-arbitrage semi-martingale restrictions for continuous-time volatility models subject to leverage effects, jumps and i.i.d noise: Theory and testable distributional implications, J. Econometrics, 138 (2007), 125-180.  doi: 10.1016/j.jeconom.2006.05.018. [2] G. Bakshi and Z. Chen, Equilibrium valuation of foreign exchange claims, Journal of Finance, 52 (1997), 799-826. [3] G. Bakshi and D. Madan, Spanning and derivative-security valuation, Journal of Financial Economics, 55 (2000), 205-238. [4] O. E. Barndorff-Nielsen and N. Shephard, Econometrics of testing for jumps in financial economics using bipower variation, Journal of Financial Econometrics, 4 (2006), 1-30. [5] J. Barunik and L. Vacha, Do co-jumps impact correlations in currency markets?, Journal of Financial Markets, 37 (2018), 97-119. [6] S. Basak and M. Gallmeyer, Currency prices, the nominal exchange rate, and security prices in a Two country dynamic monetary equilibrium, Math. Finance, 9 (1999), 1-30.  doi: 10.1111/1467-9965.00061. [7] D. Bates, Jumps and stochastic volatility: Exchange rate processes implicit in deutsche mark options, Review of Financial Studies, 9 (1996), 69-107.  doi: 10.1093/rfs/9.1.69. [8] M. Bibinger and L. Winkelmann, Econometrics of co-jumps in high-frequency data with noise, J. Econometrics, 184 (2015), 361-378.  doi: 10.1016/j.jeconom.2014.10.004. [9] M. Cao, Systematic jump risks in a small open economy: Simultaneous equilibrium valuation of options on the market portfolio and the exchange rate, Journal of International Money and Finance, 20 (2001), 191-218.  doi: 10.1016/S0261-5606(00)00053-X. [10] P. Carr and L. Wu, Stochastic skew in currency options, Journal of Financial Economics, 86 (2007), 213-247.  doi: 10.1016/j.jfineco.2006.03.010. [11] M. Caporin, A. Kolokolov and R. Renò, Systemic co-jumps, Journal of Financial Economics, 126 (2017), 563-591. [12] S. R. Das and R. Uppal, Systemic risk and international portfolio choice, The Journal of Finance, 59 (2004), 2809-2834. [13] D. Du, General equilibrium pricing of currency and currency options, Journal of Financial Economics, 110 (2013), 730-751.  doi: 10.1016/j.jfineco.2013.08.006. [14] K. Fan, Y. Shen, T. K. Siu and R. Wang, Pricing foreign equity options with regime-switching, Economic Modelling, 37 (2014), 296-305.  doi: 10.1016/j.econmod.2013.11.009. [15] E. Farhi and X. Gabaix, Rare disasters and exchange rates, Nber Working Paper Series, 2008. doi: 10.3386/w13805. [16] J. Fu and H. Yang, Equilibrium approach of asset pricing under Lévy process, European J. Oper. Res., 223 (2012), 701-708.  doi: 10.1016/j.ejor.2012.06.037. [17] M. Garman and S. Kohlhagen, Foreign currency option values, Journal of International Money and Finance, 2 (1983), 231-237.  doi: 10.1016/S0261-5606(83)80001-1. [18] S. Heston, Closed-form solution for options with stochastic volatility with application to bond and currency options, Rev. Financ. Stud., 6 (1993), 327-343.  doi: 10.1093/rfs/6.2.327. [19] Z. Hong, L. Niu and G. Zeng, US and Chinese yield curve responses to RMB exchange rate policy shocks: An analysis with the arbitrage-free Nelson-Siegel term structure model, China Finance Review International, 9 (2019), 360-385.  doi: 10.1108/CFRI-12-2017-0239. [20] S. Hurn, K. A. Lindsay and L. Xu, Revisiting the numerical solution of stochastic differential equations, China Finance Review International, 9 (2019), 312-323.  doi: 10.1108/CFRI-12-2018-0155. [21] O. W. Ibhagui, Monetary model of exchange rate determination under floating and non-floating regimes, China Finance Review International, 9 (2019), 254-283.  doi: 10.1108/CFRI-10-2017-0204. [22] J. Jacod and V. Todorov, Testing for common arrivals of jumps for discretely observed multidimensional processes, Ann. Statist., 37 (2009), 1792-1838.  doi: 10.1214/08-AOS624. [23] P. Jorion, On jump processes in the foreign exchange and stock markets, Review of Financial Studies, 1 (1988), 427-445.  doi: 10.1093/rfs/1.4.427. [24] S. S. Lee and P. A. Mykland, Jumps in financial markets: A new nonparametric test and jump dynamics, Review of Financial Studies, 21 (2008), 2535-2563. [25] J. Liu, Impact of uncertainty on foreign exchange market stability: Based on the LT-TVP-VAR model, China Finance Review International, 11 (2021), 53-72.  doi: 10.1108/CFRI-07-2019-0112. [26] R. E. Lucas, Interest rates and currency prices in a two-country world, Collected Papers on Monetary Theory, (2012).  doi: 10.4159/harvard.9780674067851.c6. [27] Y. Ma, D. Pan, K. Shrestha and W. Xu, Pricing and hedging foreign equity options under Hawkes jump-diffusion processes, Physica A, 537 (2020), 122645, 18 pp. doi: 10.1016/j.physa.2019.122645. [28] Y. Ma, K. Shrestha and W. Xu, Pricing vulnerable options with jump clustering, Journal of Futures Markets, 37 (2017), 1155-1178.  doi: 10.1002/fut.21843. [29] J. Nagayasu, Global and country-specific movements in real effective exchange rates: Implications for external competitiveness, Journal of International Money and Finance, 76 (2017), 88-105.  doi: 10.1016/j.jimonfin.2017.05.005. [30] H. W. Niu and D. C. Wang, Pricing vulnerable options with correlated jump-diffusion processes depending on various states of the economy, Quant. Finance, 16 (2016), 1129-1145.  doi: 10.1080/14697688.2015.1090623. [31] E. O. Ozturk and X. S. Sheng, Measuring global and country-specific uncertainty, Journal of International Money and Finance, 88 (2018), 276-295. [32] X. C. Wang, Pricing power exchange options with correlated jump risk, Finance Research Letters, 19 (2016), 90-97.  doi: 10.1016/j.frl.2016.06.009. [33] Y. Xing and X. P. Yang, Equilibrium valuation of currency options under a jump-diffusion model with stochastic volatility, J. Comput. Appl. Math., 280 (2015), 231-247.  doi: 10.1016/j.cam.2014.12.003. [34] F. Zapatero, Equilibrium asset prices and exchange rates, Journal of Economic Dynamics and Control, 19 (1995), 787-811.
The time series of the return rate of the daily exchange rate CHF/USD. (Note: Every circle denotes a detected jump.)
The time series of the return rate of the daily exchange rate GBP/USD
The option prices and implied volatilities with different $\lambda^h$ and $\lambda^f$. Parameters values are: $\lambda^g=0.1, K=100$
The option prices and implied volatilities with different $\lambda^g$ and time-to-maturity $\tau$. Parameters values are $\lambda^h=\lambda^f=0.5, K=90$
The option prices and implied volatilities with different $\lambda^g$ and strike prices. Parameters values are $\lambda^h=\lambda^f=0.1$
Implied volatilities under BSGK, SV, SVJD models
Implied volatilities with only single jump process
Implied volatilities with no jumps, no co-jumps and co-jumps
Identification of the jumps
 Jump times detected in the time series of the return rate of CHF/USD 1990-04-25 1994-12-29 1995-03-03 1995-03-06 1996-07-16 1997-10-28 1999-07-20 2002-01-25 2009-03-19 2009-07-31 2011-09-06 2013-03-01 2014-09-04 2015-01-15 2015-01-16 2016-12-15 2017-12-28 2019-08-02 Jump times detected in the time series of the return rate of GBP/USD 1992-08-24 1993-01-05 1994-08-26 1994-09-12 1994-12-22 1994-12-29 1995-03-03 1995-11-13 1996-05-30 1996-12-03 1997-07-28 1998-06-16 1998-08-28 2000-04-28 2002-06-26 2006-06-30 2009-03-19 2016-06-24 2016-12-15 2017-05-26 2017-11-29
 Jump times detected in the time series of the return rate of CHF/USD 1990-04-25 1994-12-29 1995-03-03 1995-03-06 1996-07-16 1997-10-28 1999-07-20 2002-01-25 2009-03-19 2009-07-31 2011-09-06 2013-03-01 2014-09-04 2015-01-15 2015-01-16 2016-12-15 2017-12-28 2019-08-02 Jump times detected in the time series of the return rate of GBP/USD 1992-08-24 1993-01-05 1994-08-26 1994-09-12 1994-12-22 1994-12-29 1995-03-03 1995-11-13 1996-05-30 1996-12-03 1997-07-28 1998-06-16 1998-08-28 2000-04-28 2002-06-26 2006-06-30 2009-03-19 2016-06-24 2016-12-15 2017-05-26 2017-11-29
Identification of the co-jumps
 Detected co-jumps Announcements around the time of jump 1994-12-29 Mexico's financial crisis 1995-03-03 The US dollar depreciated sharply against the Japanese yen 2009-03-19 On March 18, the Federal Reserve announced that it would keep the interest rate of the federal funds unchanged, and it would put 1.05 trillion of money supply into the market at the same time. 2016-12-15 The Federal Reserve announced that it would raise its benchmark interest rate.
 Detected co-jumps Announcements around the time of jump 1994-12-29 Mexico's financial crisis 1995-03-03 The US dollar depreciated sharply against the Japanese yen 2009-03-19 On March 18, the Federal Reserve announced that it would keep the interest rate of the federal funds unchanged, and it would put 1.05 trillion of money supply into the market at the same time. 2016-12-15 The Federal Reserve announced that it would raise its benchmark interest rate.
Compare the call option prices derived from formula (14) and that from Monte Carlo method. Parameters values are $\lambda^h=\lambda^f=0.5, \lambda^g=0.1.$
 Strike price Derived from formula (14) Monte Carlo % difference 70 30.8190 30.8230 0.013% 80 21.9313 21.9430 0.053% 90 14.3142 14.3034 0.075% 100 8.5364 8.5344 0.023% 110 4.7174 4.7180 0.013% 120 2.4964 2.4978 0.056% 130 1.3223 1.3208 0.113%
 Strike price Derived from formula (14) Monte Carlo % difference 70 30.8190 30.8230 0.013% 80 21.9313 21.9430 0.053% 90 14.3142 14.3034 0.075% 100 8.5364 8.5344 0.023% 110 4.7174 4.7180 0.013% 120 2.4964 2.4978 0.056% 130 1.3223 1.3208 0.113%
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