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

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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

Abstract
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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 and 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.
[2]	D. Bates, Jumps and stochastic volatility: Exchange rate processes implicit in Deutschemark options, Review of Financial Studies, 9 (1996), 69-108.
[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.
[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.
[5]	R. Cont, Empirical properties of asset returns: Stylized facts and statistical issues, Quantitative Finance, 1 (2001), 223-236.
[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.
[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.
[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.
[9]	J. Hull and A. White, The pricing of options with stochastic volatilities, Journal of Finance, 42 (1987), 281-300.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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.
[2]	D. Bates, Jumps and stochastic volatility: Exchange rate processes implicit in Deutschemark options, Review of Financial Studies, 9 (1996), 69-108.
[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.
[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.
[5]	R. Cont, Empirical properties of asset returns: Stylized facts and statistical issues, Quantitative Finance, 1 (2001), 223-236.
[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.
[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.
[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.
[9]	J. Hull and A. White, The pricing of options with stochastic volatilities, Journal of Finance, 42 (1987), 281-300.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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