Article Contents
Article Contents

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

• *Corresponding author: Xiaonan Su

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.

Mathematics Subject Classification: Primary: 60G55, 60H15; Secondary: 65C30.

 Citation:

• Figure 1.  The time series of the return rate of the daily exchange rate CHF/USD. (Note: Every circle denotes a detected jump.)

Figure 2.  The time series of the return rate of the daily exchange rate GBP/USD

Figure 3.  The option prices and implied volatilities with different $\lambda^h$ and $\lambda^f$. Parameters values are: $\lambda^g=0.1, K=100$

Figure 4.  The option prices and implied volatilities with different $\lambda^g$ and time-to-maturity $\tau$. Parameters values are $\lambda^h=\lambda^f=0.5, K=90$

Figure 5.  The option prices and implied volatilities with different $\lambda^g$ and strike prices. Parameters values are $\lambda^h=\lambda^f=0.1$

Figure 6.  Implied volatilities under BSGK, SV, SVJD models

Figure 7.  Implied volatilities with only single jump process

Figure 8.  Implied volatilities with no jumps, no co-jumps and co-jumps

Table 1.  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

Table 2.  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.

Table 3.  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%
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Figures(8)

Tables(3)