American Institute of Mathematical Sciences

Advanced
January  2014, 10(1): 41-55. doi: 10.3934/jimo.2014.10.41

Catastrophe equity put options under stochastic volatility and catastrophe-dependent jumps

Hwa-Sung Kim 1, , Bara Kim 2, and  Jerim Kim 3,

1. 

School of Management, Kyung Hee University, 26 Kyunghee-daero, Dongdaemun-gu, Seoul, 130-701
2. 

Department of Mathematics, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul, 136-713
3. 

Department of Business Administration, Yong In University, 134 Yongindaehak-ro, Cheoin-gu, Yongin-si, Gyeonggi-do, 449-714, South Korea

Received  September 2012 Revised  July 2013 Published  October 2013

This paper develops a catastrophe equity put (CatEPut) option model under realistic assumptions. To reflect the phenomena of real data, we adopt the following assumptions. First, following the reasoning in Lin and Wang [12], we assume that the loss index follows a compound Poisson process with jumps of a mixture of Erlangs. Second, the volatility of stock return is assumed to be stochastic as in Heston [8]. Under the assumptions, we derives a pricing formula for CatEPut options. Numerical examples are given to insist that the pricing formula can be easily implemented numerically. We also confirm the validity and accuracy of implementation of the pricing formula by comparing the numerical results obtained by the pricing formula with those obtained by the Monte Carlo simulation.
Keywords: moment generating transform., CatEPut, option pricing, stochastic volatility, jump-diffusion process.
Mathematics Subject Classification: Primary: 60K10, 90G20; Secondary: 97M3.
Citation: Hwa-Sung Kim, Bara Kim, Jerim Kim. Catastrophe equity put options under stochastic volatility and catastrophe-dependent jumps. Journal of Industrial & Management Optimization, 2014, 10 (1) : 41-55. doi: 10.3934/jimo.2014.10.41
References:
[1]

G. Bakshi, C. Cao and Z. Chen, Empirical performance of alternative option pricing models,, Journal of Finance, 52 (1997), 2003.   Google Scholar

[2]

D. Bates, Jumps and stochastic volatility: Exchange rate processes implicit in Deutschemark options,, Review of Financial Studies, 9 (1996), 69.   Google Scholar

[3]

F. Black and S. Myron, The pricing of options and corporate liabilities,, Journal of Political Economy, 81 (1973), 637.  doi: 10.1086/260062.  Google Scholar

[4]

L.-F. Chang and M.-W. Hung, Analytical valuation of catastrocphe equity options with negative exponential jumps,, Insurance: Mathematics and Economics, 44 (2009), 59.  doi: 10.1016/j.insmatheco.2008.09.009.  Google Scholar

[5]

R. Cont, Empirical properties of asset returns: Stylized facts and statistical issues,, Quantitative Finance, 1 (2001), 223.   Google Scholar

[6]

J. C. Cox and S. A. Ross, The valuation of options for alternative stochastic processes,, Journal of Financial Economics, 3 (1976), 145.  doi: 10.1016/0304-405X(76)90023-4.  Google Scholar

[7]

H. U. Gerber and E. S. W. Shiu, On the time value of ruin,, North American Actuarial Journal, 2 (1998), 48.  doi: 10.1080/10920277.1998.10595671.  Google Scholar

[8]

S. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options,, Review of Financial Studies, 6 (1993), 327.  doi: 10.1093/rfs/6.2.327.  Google Scholar

[9]

J. Hull and A. White, The pricing of options with stochastic volatilities,, Journal of Finance, 42 (1987), 281.   Google Scholar

[10]

S. Jaimungal and T. Wang, Catastrophe options with stochastic interest rates and compound Poisson losses,, Insurance: Mathematics and Economics, 38 (2006), 469.  doi: 10.1016/j.insmatheco.2005.11.008.  Google Scholar

[11]

B. Kim, J. Kim, K.-S. Moon and I.-S. Wee, Valuation of power options under Heston's stochastic volatility model,, Journal of Economic Dynamics and Control, 36 (2012), 1796.  doi: 10.1016/j.jedc.2012.05.005.  Google Scholar

[12]

X. S. Lin and T. Wang, Pricing perpetual American catastrophe put options: A penalty function approach,, Insurance: Mathematics and Economics, 44 (2009), 287.  doi: 10.1016/j.insmatheco.2008.04.002.  Google Scholar

[13]

D. Madan, P. Carr and E. Chang, The variance gamma process and option pricing,, European Finance Review, 2 (1998), 79.  doi: 10.1023/A:1009703431535.  Google Scholar

[14]

R. Merton, Option pricing when the underlying stock returns are discontinuous,, Journal of Financial Economics, 4 (1976), 125.  doi: 10.1016/0304-405X(76)90022-2.  Google Scholar

[15]

L. Scott, Pricing stock options in a jump diffusion model with stochastic volatility and interest rates: Applications of Fourier inversion methods,, Mathematical Finance, 7 (1997), 413.  doi: 10.1111/1467-9965.00039.  Google Scholar

[16]

E. Stein and J. Stein, Stock price distributions with stochastic volatility,, Review of Financial Studies, 4 (1991), 727.  doi: 10.1093/rfs/4.4.727.  Google Scholar

show all references

References:
[1]

G. Bakshi, C. Cao and Z. Chen, Empirical performance of alternative option pricing models,, Journal of Finance, 52 (1997), 2003.   Google Scholar

[2]

D. Bates, Jumps and stochastic volatility: Exchange rate processes implicit in Deutschemark options,, Review of Financial Studies, 9 (1996), 69.   Google Scholar

[3]

F. Black and S. Myron, The pricing of options and corporate liabilities,, Journal of Political Economy, 81 (1973), 637.  doi: 10.1086/260062.  Google Scholar

[4]

L.-F. Chang and M.-W. Hung, Analytical valuation of catastrocphe equity options with negative exponential jumps,, Insurance: Mathematics and Economics, 44 (2009), 59.  doi: 10.1016/j.insmatheco.2008.09.009.  Google Scholar

[5]

R. Cont, Empirical properties of asset returns: Stylized facts and statistical issues,, Quantitative Finance, 1 (2001), 223.   Google Scholar

[6]

J. C. Cox and S. A. Ross, The valuation of options for alternative stochastic processes,, Journal of Financial Economics, 3 (1976), 145.  doi: 10.1016/0304-405X(76)90023-4.  Google Scholar

[7]

H. U. Gerber and E. S. W. Shiu, On the time value of ruin,, North American Actuarial Journal, 2 (1998), 48.  doi: 10.1080/10920277.1998.10595671.  Google Scholar

[8]

S. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options,, Review of Financial Studies, 6 (1993), 327.  doi: 10.1093/rfs/6.2.327.  Google Scholar

[9]

J. Hull and A. White, The pricing of options with stochastic volatilities,, Journal of Finance, 42 (1987), 281.   Google Scholar

[10]

S. Jaimungal and T. Wang, Catastrophe options with stochastic interest rates and compound Poisson losses,, Insurance: Mathematics and Economics, 38 (2006), 469.  doi: 10.1016/j.insmatheco.2005.11.008.  Google Scholar

[11]

B. Kim, J. Kim, K.-S. Moon and I.-S. Wee, Valuation of power options under Heston's stochastic volatility model,, Journal of Economic Dynamics and Control, 36 (2012), 1796.  doi: 10.1016/j.jedc.2012.05.005.  Google Scholar

[12]

X. S. Lin and T. Wang, Pricing perpetual American catastrophe put options: A penalty function approach,, Insurance: Mathematics and Economics, 44 (2009), 287.  doi: 10.1016/j.insmatheco.2008.04.002.  Google Scholar

[13]

D. Madan, P. Carr and E. Chang, The variance gamma process and option pricing,, European Finance Review, 2 (1998), 79.  doi: 10.1023/A:1009703431535.  Google Scholar

[14]

R. Merton, Option pricing when the underlying stock returns are discontinuous,, Journal of Financial Economics, 4 (1976), 125.  doi: 10.1016/0304-405X(76)90022-2.  Google Scholar

[15]

L. Scott, Pricing stock options in a jump diffusion model with stochastic volatility and interest rates: Applications of Fourier inversion methods,, Mathematical Finance, 7 (1997), 413.  doi: 10.1111/1467-9965.00039.  Google Scholar

[16]

E. Stein and J. Stein, Stock price distributions with stochastic volatility,, Review of Financial Studies, 4 (1991), 727.  doi: 10.1093/rfs/4.4.727.  Google Scholar

[1]

Meiqiao Ai, Zhimin Zhang, Wenguang Yu. First passage problems of refracted jump diffusion processes and their applications in valuing equity-linked death benefits. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021039
[2]

Xingchun Wang, Yongjin Wang. Variance-optimal hedging for target volatility options. Journal of Industrial & Management Optimization, 2014, 10 (1) : 207-218. doi: 10.3934/jimo.2014.10.207
[3]

Xiaohong Li, Mingxin Sun, Zhaohua Gong, Enmin Feng. Multistage optimal control for microbial fed-batch fermentation process. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021040
[4]

Teddy Pichard. A moment closure based on a projection on the boundary of the realizability domain: 1D case. Kinetic & Related Models, 2020, 13 (6) : 1243-1280. doi: 10.3934/krm.2020045
[5]

Benrong Zheng, Xianpei Hong. Effects of take-back legislation on pricing and coordination in a closed-loop supply chain. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021035
[6]

Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109
[7]

J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008
[8]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437
[9]

Seung-Yeal Ha, Dongnam Ko, Chanho Min, Xiongtao Zhang. Emergent collective behaviors of stochastic kuramoto oscillators. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1059-1081. doi: 10.3934/dcdsb.2019208
[10]

María J. Garrido-Atienza, Bohdan Maslowski, Jana  Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088
[11]

Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367
[12]

Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189
[13]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258
[14]

Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223
[15]

Dan Wei, Shangjiang Guo. Qualitative analysis of a Lotka-Volterra competition-diffusion-advection system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2599-2623. doi: 10.3934/dcdsb.2020197
[16]

Ardeshir Ahmadi, Hamed Davari-Ardakani. A multistage stochastic programming framework for cardinality constrained portfolio optimization. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 359-377. doi: 10.3934/naco.2017023
[17]

Shangzhi Li, Shangjiang Guo. Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2693-2719. doi: 10.3934/dcdsb.2020201
[18]

Xianming Liu, Guangyue Han. A Wong-Zakai approximation of stochastic differential equations driven by a general semimartingale. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2499-2508. doi: 10.3934/dcdsb.2020192
[19]

Nabahats Dib-Baghdadli, Rabah Labbas, Tewfik Mahdjoub, Ahmed Medeghri. On some reaction-diffusion equations generated by non-domiciliated triatominae, vectors of Chagas disease. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021004
[20]

Vo Anh Khoa, Thi Kim Thoa Thieu, Ekeoma Rowland Ijioma. On a pore-scale stationary diffusion equation: Scaling effects and correctors for the homogenization limit. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2451-2477. doi: 10.3934/dcdsb.2020190

2019 Impact Factor: 1.366

Tools

Metrics

  • PDF downloads (90)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences