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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), 2003-2049. Google Scholar

[2]

D. Bates, Jumps and stochastic volatility: Exchange rate processes implicit in Deutschemark options, Review of Financial Studies, 9 (1996), 69-108. Google Scholar

[3]

F. Black and S. Myron, The pricing of options and corporate liabilities, Journal of Political Economy, 81 (1973), 637-659. 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), 59-69. 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), 223-236. Google Scholar

[6]

J. C. Cox and S. A. Ross, The valuation of options for alternative stochastic processes, Journal of Financial Economics, 3 (1976), 145-166. 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), 48-78. 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), 327-343. 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), 281-300. Google Scholar

[10]

S. Jaimungal and T. Wang, Catastrophe options with stochastic interest rates and compound Poisson losses, Insurance: Mathematics and Economics, 38 (2006), 469-483. 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), 1796-1813. 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), 287-295. 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), 79-105. 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), 125-144. 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), 413-426. 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), 727-752. 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), 2003-2049. Google Scholar

[2]

D. Bates, Jumps and stochastic volatility: Exchange rate processes implicit in Deutschemark options, Review of Financial Studies, 9 (1996), 69-108. Google Scholar

[3]

F. Black and S. Myron, The pricing of options and corporate liabilities, Journal of Political Economy, 81 (1973), 637-659. 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), 59-69. 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), 223-236. Google Scholar

[6]

J. C. Cox and S. A. Ross, The valuation of options for alternative stochastic processes, Journal of Financial Economics, 3 (1976), 145-166. 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), 48-78. 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), 327-343. 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), 281-300. Google Scholar

[10]

S. Jaimungal and T. Wang, Catastrophe options with stochastic interest rates and compound Poisson losses, Insurance: Mathematics and Economics, 38 (2006), 469-483. 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), 1796-1813. 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), 287-295. 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), 79-105. 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), 125-144. 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), 413-426. 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), 727-752. doi: 10.1093/rfs/4.4.727.  Google Scholar

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