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

Yu Xing; Wei Wang; Xiaonan Su; Huawei Niu

doi:10.3934/jimo.2022022

Article Contents

2023, Volume 19, Issue 3: 1869-1892. Doi: 10.3934/jimo.2022022

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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
^*Corresponding author: Xiaonan Su

Received: October 2020

Revised: December 2021

Early access: February 2022

Published: March 2023

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)..

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Abstract

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.

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

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

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

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

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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 $

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

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

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Figure 7. Implied volatilities with only single jump process

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

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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

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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.

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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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