
-
Previous Article
On a modified extragradient method for variational inequality problem with application to industrial electricity production
- JIMO Home
- This Issue
-
Next Article
A slacks-based model for dynamic data envelopment analysis
Pricing vulnerable options under a Markov-modulated jump-diffusion model with fire sales
1. | Advanced Modeling and Applied Computing Laboratory, Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong, China |
2. | Department of Actuarial Studies and Business Analytics, Faculty of Business and Economics, Macquarie University, Sydney, Australia |
In this paper, we consider the valuation of vulnerable options under a Markov-modulated jump-diffusion model, where the option writer's asset value is subject to price pressure from other financial institutions due to distressed selling. A change of numéraire technique, proposed by Geman et al. [
References:
[1] |
M. Anton and C. Polk,
Connected stocks, The Journal of Finance, 69 (2014), 1099-1127.
|
[2] |
F. Black and M. Scholes,
The valuation of options and corporate liabilities, Journal of Political Economy, 81 (1973), 637-654.
doi: 10.1086/260062. |
[3] |
G. Cheang and G. Teh,
Change of num$\acute{e}$raire and a jump-diffusion option pricing formula, Springer International Publishing, (2014), 371-389.
|
[4] |
R. Cont and P. Tankov, Financial Modeling with Jump Processes, Chapman & Hall/CRC Financial Mathematics Series, Chapman & Hall/CRC, Boca Raton, FL, 2004. |
[5] |
R. Cont and L. Wagalath,
Fire sales forensics: Measuring endogenous risk, Mathematical Finance, 26 (2016), 835-866.
doi: 10.1111/mafi.12071. |
[6] |
M. Costabile, A. Leccadito, I. Massab |
[7] |
J. Coval and E. Stafford,
Asset fire sales (and purchases) in equity markets, Journal of Financial Economics, 86 (2007), 479-512.
|
[8] |
D. Duffie and K. Singleton,
Modeling Term Structures of Defaultable Bonds, Review of Financial Studies, 12 (1999), 197-226.
|
[9] |
R. Elliott, T. Siu, L. Chan and J. Lau,
Pricing options under a generalized Markov-modulated jump-diffusion model, Stochastic Analysis and Applications, 25 (2007), 821-843.
doi: 10.1080/07362990701420118. |
[10] |
R. Elliott and T. Siu,
Option pricing and filtering with hidden Markov-modulated pure-jump processes, Applied Mathematical Finance, 20 (2013), 1-25.
doi: 10.1080/1350486X.2012.655929. |
[11] |
R. Elliott and T. Siu,
Asset pricing using trading volumes in a hidden regime-switching environment, Asia-Pacific Financial Market, 22 (2015), 133-149.
|
[12] |
F. A. Fard,
Analytical pricing of vulnerable options under a generalized jump-diffusion model, Insurance: Mathematics and Economics, 60 (2015), 19-28.
doi: 10.1016/j.insmatheco.2014.10.007. |
[13] |
I. Florescu, R. Liu and M. Mariani,
Solutions to a partial integro-differential parabolic system arising in the pricing of financial options in regime-switching jump diffusion models, Electronic Journal of Differential Equations, 2012 (2012), 1-12.
|
[14] |
H. Geman, N. Karoui and J. Rochet,
Change of numéaire, changes of probability measure and option pricing, Journal of Applied Probability, 32 (1995), 443-458.
doi: 10.2307/3215299. |
[15] |
R. Greenwood and D. Thesmar,
Stock price fragility, Journal of Financial Economics, 102 (2011), 471-490.
|
[16] |
X. Guo and Q. Zhang,
Closed-form solutions for perpetual American put options with regime switching, SIAM Journal on Applied Mathematics, 64 (2004), 2034-2049.
doi: 10.1137/S0036139903426083. |
[17] |
T. Hida, J. Potthoff and L. Streit,
Dirichlet forms and white noise analysis, Communications in Mathematical Physics, 116 (1988), 235-245.
doi: 10.1007/BF01225257. |
[18] |
M. Hung and Y. Liu,
Pricing vulnerable options in incomplete markets, Journal of Futures Markets, 25 (2005), 135-170.
|
[19] |
R. Jarrow and S. Turnbull,
Credit risk: Drawing the analogy, Risk Magazine, 5 (1992), 63-70.
|
[20] |
R. Jarrow and S. Turnbull,
Pricing derivatives on financial securities subject to credit risk, Journal of Finance, 50 (1995), 53-85.
|
[21] |
H. Johnson and R. Stulz,
The pricing of options with default risk, Journal of Finance, 42 (1987), 267-280.
|
[22] |
P. Klein,
Pricing black-scholes options with correlated credit risk, Journal of Banking and Finance, 20 (1996), 1211-1229.
|
[23] |
S. Kou,
A jump-diffusion model for option pricing, Management Science, 48 (2002), 1086-1101.
|
[24] |
L. Liew and T. Siu,
A hidden markov regime-switching model for option valuation, Insurance: Mathematics and Economics, 47 (2010), 374-384.
doi: 10.1016/j.insmatheco.2010.08.003. |
[25] |
R. Merton,
Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 3 (1976), 125-144.
|
[26] |
V. Naik,
Option valuation and hedging strategies with jumps in the volatility of asset returns, The Journal of Finance, 48 (1993), 1969-1984.
|
[27] |
H. Niu and D. Wang,
Pricing vulnerable options with correlated jump-diffusion processes depending on various states of the economy, Quantitative Finance, 16 (2016), 1129-1145.
doi: 10.1080/14697688.2015.1090623. |
[28] |
A. Pascucci, PDE and Martingale Methods in Option Pricing, Bocconi & Springer Series, Springer-Verlag, New York, 2011. |
[29] |
P. Pedler,
Occupation time for two state markov chains, Journal of Applied Probability, 8 (1971), 381-390.
doi: 10.2307/3211908. |
[30] |
W. J. Runggaldier, Jump diffusion models, In S. T. Rachev (Ed.), Handbook of heavy tailed distributions in finance, (2003), 169-209. |
[31] |
B. Sericola,
Occupation times in Markov processes, Stochastic Models, 16 (2000), 479-510.
doi: 10.1080/15326340008807601. |
[32] |
Y. Shen and T. Siu,
Pricing variance swaps under a stochastic interest rate and volatility model with regime-switching, Operations Research Letters, 41 (2013), 180-187.
doi: 10.1016/j.orl.2012.12.008. |
[33] |
A. Shleifer and R. Vishny,
Liquidation values and debt capacity: A market equilibrium approach, Journal of Finance, 47 (1992), 1343-1366.
|
[34] |
A. Shleifer and R. Vishny,
Fire sales in finance and macroeconomics, Journal of Economic Perspectives, 25 (2011), 29-48.
|
[35] |
S. Shreve,
Stochastic calculus for finance II: Continuous-time models, Springer Finance Series, (2003), 404-459.
|
[36] |
T. Siu, J. Lau and H. Yang, Pricing participating products under a generalized jump-diffusion, Journal of Applied Mathematics and Stochastic Analysis, (2008), Article ID 474623, 30 Pages.
doi: 10.1155/2008/474623. |
[37] |
T. Siu,
A BSDE approach to optimal investment of an insurer with hidden regime switching, Stochastic Analysis and Applications, 31 (2013), 1-18.
doi: 10.1080/07362994.2012.727144. |
[38] |
T. Siu, A stochastic flows approach for asset allocation with hidden economic environment, International Journal of Stochastic Analysis, (2015), Article ID 462524, 11 pages.
doi: 10.1155/2015/462524. |
[39] |
R. Wiggins, T. Piontek and A. Metrick, The Lehman Brothers Bankruptcy A: Overview, Yale Program on Financial Stability Case Study, 2014. |
[40] |
S. Yang, M. Lee and J. Kim,
Pricing Vulnerable Options under a Stochastic Volatility Model, Applied Mathematics Letters, 34 (2014), 7-12.
doi: 10.1016/j.aml.2014.03.007. |
[41] |
Q. Yang, W. Ching, J. Gu and T. Siu, Optimal liquidation strategy across multiple exchanges under a jump-diffusion fast mean-reverting model, (2016), available at arXiv: 1607.04553. |
show all references
References:
[1] |
M. Anton and C. Polk,
Connected stocks, The Journal of Finance, 69 (2014), 1099-1127.
|
[2] |
F. Black and M. Scholes,
The valuation of options and corporate liabilities, Journal of Political Economy, 81 (1973), 637-654.
doi: 10.1086/260062. |
[3] |
G. Cheang and G. Teh,
Change of num$\acute{e}$raire and a jump-diffusion option pricing formula, Springer International Publishing, (2014), 371-389.
|
[4] |
R. Cont and P. Tankov, Financial Modeling with Jump Processes, Chapman & Hall/CRC Financial Mathematics Series, Chapman & Hall/CRC, Boca Raton, FL, 2004. |
[5] |
R. Cont and L. Wagalath,
Fire sales forensics: Measuring endogenous risk, Mathematical Finance, 26 (2016), 835-866.
doi: 10.1111/mafi.12071. |
[6] |
M. Costabile, A. Leccadito, I. Massab |
[7] |
J. Coval and E. Stafford,
Asset fire sales (and purchases) in equity markets, Journal of Financial Economics, 86 (2007), 479-512.
|
[8] |
D. Duffie and K. Singleton,
Modeling Term Structures of Defaultable Bonds, Review of Financial Studies, 12 (1999), 197-226.
|
[9] |
R. Elliott, T. Siu, L. Chan and J. Lau,
Pricing options under a generalized Markov-modulated jump-diffusion model, Stochastic Analysis and Applications, 25 (2007), 821-843.
doi: 10.1080/07362990701420118. |
[10] |
R. Elliott and T. Siu,
Option pricing and filtering with hidden Markov-modulated pure-jump processes, Applied Mathematical Finance, 20 (2013), 1-25.
doi: 10.1080/1350486X.2012.655929. |
[11] |
R. Elliott and T. Siu,
Asset pricing using trading volumes in a hidden regime-switching environment, Asia-Pacific Financial Market, 22 (2015), 133-149.
|
[12] |
F. A. Fard,
Analytical pricing of vulnerable options under a generalized jump-diffusion model, Insurance: Mathematics and Economics, 60 (2015), 19-28.
doi: 10.1016/j.insmatheco.2014.10.007. |
[13] |
I. Florescu, R. Liu and M. Mariani,
Solutions to a partial integro-differential parabolic system arising in the pricing of financial options in regime-switching jump diffusion models, Electronic Journal of Differential Equations, 2012 (2012), 1-12.
|
[14] |
H. Geman, N. Karoui and J. Rochet,
Change of numéaire, changes of probability measure and option pricing, Journal of Applied Probability, 32 (1995), 443-458.
doi: 10.2307/3215299. |
[15] |
R. Greenwood and D. Thesmar,
Stock price fragility, Journal of Financial Economics, 102 (2011), 471-490.
|
[16] |
X. Guo and Q. Zhang,
Closed-form solutions for perpetual American put options with regime switching, SIAM Journal on Applied Mathematics, 64 (2004), 2034-2049.
doi: 10.1137/S0036139903426083. |
[17] |
T. Hida, J. Potthoff and L. Streit,
Dirichlet forms and white noise analysis, Communications in Mathematical Physics, 116 (1988), 235-245.
doi: 10.1007/BF01225257. |
[18] |
M. Hung and Y. Liu,
Pricing vulnerable options in incomplete markets, Journal of Futures Markets, 25 (2005), 135-170.
|
[19] |
R. Jarrow and S. Turnbull,
Credit risk: Drawing the analogy, Risk Magazine, 5 (1992), 63-70.
|
[20] |
R. Jarrow and S. Turnbull,
Pricing derivatives on financial securities subject to credit risk, Journal of Finance, 50 (1995), 53-85.
|
[21] |
H. Johnson and R. Stulz,
The pricing of options with default risk, Journal of Finance, 42 (1987), 267-280.
|
[22] |
P. Klein,
Pricing black-scholes options with correlated credit risk, Journal of Banking and Finance, 20 (1996), 1211-1229.
|
[23] |
S. Kou,
A jump-diffusion model for option pricing, Management Science, 48 (2002), 1086-1101.
|
[24] |
L. Liew and T. Siu,
A hidden markov regime-switching model for option valuation, Insurance: Mathematics and Economics, 47 (2010), 374-384.
doi: 10.1016/j.insmatheco.2010.08.003. |
[25] |
R. Merton,
Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 3 (1976), 125-144.
|
[26] |
V. Naik,
Option valuation and hedging strategies with jumps in the volatility of asset returns, The Journal of Finance, 48 (1993), 1969-1984.
|
[27] |
H. Niu and D. Wang,
Pricing vulnerable options with correlated jump-diffusion processes depending on various states of the economy, Quantitative Finance, 16 (2016), 1129-1145.
doi: 10.1080/14697688.2015.1090623. |
[28] |
A. Pascucci, PDE and Martingale Methods in Option Pricing, Bocconi & Springer Series, Springer-Verlag, New York, 2011. |
[29] |
P. Pedler,
Occupation time for two state markov chains, Journal of Applied Probability, 8 (1971), 381-390.
doi: 10.2307/3211908. |
[30] |
W. J. Runggaldier, Jump diffusion models, In S. T. Rachev (Ed.), Handbook of heavy tailed distributions in finance, (2003), 169-209. |
[31] |
B. Sericola,
Occupation times in Markov processes, Stochastic Models, 16 (2000), 479-510.
doi: 10.1080/15326340008807601. |
[32] |
Y. Shen and T. Siu,
Pricing variance swaps under a stochastic interest rate and volatility model with regime-switching, Operations Research Letters, 41 (2013), 180-187.
doi: 10.1016/j.orl.2012.12.008. |
[33] |
A. Shleifer and R. Vishny,
Liquidation values and debt capacity: A market equilibrium approach, Journal of Finance, 47 (1992), 1343-1366.
|
[34] |
A. Shleifer and R. Vishny,
Fire sales in finance and macroeconomics, Journal of Economic Perspectives, 25 (2011), 29-48.
|
[35] |
S. Shreve,
Stochastic calculus for finance II: Continuous-time models, Springer Finance Series, (2003), 404-459.
|
[36] |
T. Siu, J. Lau and H. Yang, Pricing participating products under a generalized jump-diffusion, Journal of Applied Mathematics and Stochastic Analysis, (2008), Article ID 474623, 30 Pages.
doi: 10.1155/2008/474623. |
[37] |
T. Siu,
A BSDE approach to optimal investment of an insurer with hidden regime switching, Stochastic Analysis and Applications, 31 (2013), 1-18.
doi: 10.1080/07362994.2012.727144. |
[38] |
T. Siu, A stochastic flows approach for asset allocation with hidden economic environment, International Journal of Stochastic Analysis, (2015), Article ID 462524, 11 pages.
doi: 10.1155/2015/462524. |
[39] |
R. Wiggins, T. Piontek and A. Metrick, The Lehman Brothers Bankruptcy A: Overview, Yale Program on Financial Stability Case Study, 2014. |
[40] |
S. Yang, M. Lee and J. Kim,
Pricing Vulnerable Options under a Stochastic Volatility Model, Applied Mathematics Letters, 34 (2014), 7-12.
doi: 10.1016/j.aml.2014.03.007. |
[41] |
Q. Yang, W. Ching, J. Gu and T. Siu, Optimal liquidation strategy across multiple exchanges under a jump-diffusion fast mean-reverting model, (2016), available at arXiv: 1607.04553. |

Parameters | Values | Parameters | Values |
Dimension of |
Tolerance level | ||
Transition rate | Transition rate | ||
Jump in |
Jump in |
||
Jump in |
Jump in |
||
Jump in |
Jump in |
||
Jump in |
Jump in |
||
Probability | Probability | ||
Market depth | MLR | ||
Intensity | Intensity | ||
Default boundary | Outstanding Claims | ||
Deadweight cost | Time to maturity | ||
Initial price | Strike price | ||
Initial price | Initial price | ||
Initial price | Time steps |
Parameters | Values | Parameters | Values |
Dimension of |
Tolerance level | ||
Transition rate | Transition rate | ||
Jump in |
Jump in |
||
Jump in |
Jump in |
||
Jump in |
Jump in |
||
Jump in |
Jump in |
||
Probability | Probability | ||
Market depth | MLR | ||
Intensity | Intensity | ||
Default boundary | Outstanding Claims | ||
Deadweight cost | Time to maturity | ||
Initial price | Strike price | ||
Initial price | Initial price | ||
Initial price | Time steps |
No | Impact | No | Impact | No | Impact | No | Impact | ||||||
0.8 | 1 | 0.5445 | 0.2324 | 0.0469 | 0.0200 | 0.0836 | 0.0357 | 0.3153 | 0.3146 | ||||
2 | 1.7845 | 0.7617 | 0.2812 | 0.1200 | 0.4341 | 0.1853 | 1.1786 | 0.5030 | |||||
1.0 | 1 | 0.5520 | 0.2354 | 0.0475 | 0.0203 | 0.0848 | 0.0362 | 0.3197 | 0.1363 | ||||
2 | 1.8093 | 0.7716 | 0.2851 | 0.1216 | 0.4403 | 0.1877 | 1.1949 | 0.5096 | |||||
1.25 | 1 | 0.5581 | 0.2378 | 0.0480 | 0.0205 | 0.0857 | 0.0365 | 0.3232 | 0.1377 | ||||
2 | 1.8291 | 0.7796 | 0.2882 | 0.1228 | 0.4450 | 0.1897 | 1.2080 | 0.5148 |
No | Impact | No | Impact | No | Impact | No | Impact | ||||||
0.8 | 1 | 0.5445 | 0.2324 | 0.0469 | 0.0200 | 0.0836 | 0.0357 | 0.3153 | 0.3146 | ||||
2 | 1.7845 | 0.7617 | 0.2812 | 0.1200 | 0.4341 | 0.1853 | 1.1786 | 0.5030 | |||||
1.0 | 1 | 0.5520 | 0.2354 | 0.0475 | 0.0203 | 0.0848 | 0.0362 | 0.3197 | 0.1363 | ||||
2 | 1.8093 | 0.7716 | 0.2851 | 0.1216 | 0.4403 | 0.1877 | 1.1949 | 0.5096 | |||||
1.25 | 1 | 0.5581 | 0.2378 | 0.0480 | 0.0205 | 0.0857 | 0.0365 | 0.3232 | 0.1377 | ||||
2 | 1.8291 | 0.7796 | 0.2882 | 0.1228 | 0.4450 | 0.1897 | 1.2080 | 0.5148 |
[1] |
Christoforidou Amalia, Christian-Oliver Ewald. A lattice method for option evaluation with regime-switching asset correlation structure. Journal of Industrial and Management Optimization, 2021, 17 (4) : 1729-1752. doi: 10.3934/jimo.2020042 |
[2] |
Zhuo Jin, Linyi Qian. Lookback option pricing for regime-switching jump diffusion models. Mathematical Control and Related Fields, 2015, 5 (2) : 237-258. doi: 10.3934/mcrf.2015.5.237 |
[3] |
Fuke Wu, George Yin, Zhuo Jin. Kolmogorov-type systems with regime-switching jump diffusion perturbations. Discrete and Continuous Dynamical Systems - B, 2016, 21 (7) : 2293-2319. doi: 10.3934/dcdsb.2016048 |
[4] |
Mourad Bellassoued, Raymond Brummelhuis, Michel Cristofol, Éric Soccorsi. Stable reconstruction of the volatility in a regime-switching local-volatility model. Mathematical Control and Related Fields, 2020, 10 (1) : 189-215. doi: 10.3934/mcrf.2019036 |
[5] |
Jiaqin Wei. Time-inconsistent optimal control problems with regime-switching. Mathematical Control and Related Fields, 2017, 7 (4) : 585-622. doi: 10.3934/mcrf.2017022 |
[6] |
Wensheng Yin, Jinde Cao, Yong Ren. Inverse optimal control of regime-switching jump diffusions. Mathematical Control and Related Fields, 2021 doi: 10.3934/mcrf.2021034 |
[7] |
Engel John C Dela Vega, Robert J Elliott. Conditional coherent risk measures and regime-switching conic pricing. Probability, Uncertainty and Quantitative Risk, 2021, 6 (4) : 267-300. doi: 10.3934/puqr.2021014 |
[8] |
Jun Li, Fubao Xi. Exponential ergodicity for regime-switching diffusion processes in total variation norm. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2021309 |
[9] |
Kun Fan, Yang Shen, Tak Kuen Siu, Rongming Wang. On a Markov chain approximation method for option pricing with regime switching. Journal of Industrial and Management Optimization, 2016, 12 (2) : 529-541. doi: 10.3934/jimo.2016.12.529 |
[10] |
Zhuo Jin, George Yin, Hailiang Yang. Numerical methods for dividend optimization using regime-switching jump-diffusion models. Mathematical Control and Related Fields, 2011, 1 (1) : 21-40. doi: 10.3934/mcrf.2011.1.21 |
[11] |
Ping Chen, Haixiang Yao. Continuous-time mean-variance portfolio selection with no-shorting constraints and regime-switching. Journal of Industrial and Management Optimization, 2020, 16 (2) : 531-551. doi: 10.3934/jimo.2018166 |
[12] |
Yinghui Dong, Kam Chuen Yuen, Guojing Wang. Pricing credit derivatives under a correlated regime-switching hazard processes model. Journal of Industrial and Management Optimization, 2017, 13 (3) : 1395-1415. doi: 10.3934/jimo.2016079 |
[13] |
Chao Xu, Yinghui Dong, Zhaolu Tian, Guojing Wang. Pricing dynamic fund protection under a Regime-switching Jump-diffusion model with stochastic protection level. Journal of Industrial and Management Optimization, 2020, 16 (6) : 2603-2623. doi: 10.3934/jimo.2019072 |
[14] |
Jiaqin Wei, Zhuo Jin, Hailiang Yang. Optimal dividend policy with liability constraint under a hidden Markov regime-switching model. Journal of Industrial and Management Optimization, 2019, 15 (4) : 1965-1993. doi: 10.3934/jimo.2018132 |
[15] |
Jiapeng Liu, Ruihua Liu, Dan Ren. Investment and consumption in regime-switching models with proportional transaction costs and log utility. Mathematical Control and Related Fields, 2017, 7 (3) : 465-491. doi: 10.3934/mcrf.2017017 |
[16] |
Meiqiao Ai, Zhimin Zhang, Wenguang Yu. Valuing equity-linked death benefits with a threshold expense structure under a regime-switching Lévy model. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2022007 |
[17] |
Tak Kuen Siu, Yang Shen. Risk-minimizing pricing and Esscher transform in a general non-Markovian regime-switching jump-diffusion model. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2595-2626. doi: 10.3934/dcdsb.2017100 |
[18] |
Kehan Si, Zhenda Xu, Ka Fai Cedric Yiu, Xun Li. Open-loop solvability for mean-field stochastic linear quadratic optimal control problems of Markov regime-switching system. Journal of Industrial and Management Optimization, 2021 doi: 10.3934/jimo.2021074 |
[19] |
Ishak Alia, Mohamed Sofiane Alia. Open-loop equilibrium strategy for mean-variance Portfolio selection with investment constraints in a non-Markovian regime-switching jump-diffusion model. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2022048 |
[20] |
Caibin Zhang, Zhibin Liang, Kam Chuen Yuen. Portfolio optimization for jump-diffusion risky assets with regime switching: A time-consistent approach. Journal of Industrial and Management Optimization, 2022, 18 (1) : 341-366. doi: 10.3934/jimo.2020156 |
2020 Impact Factor: 1.801
Tools
Metrics
Other articles
by authors
[Back to Top]