# American Institute of Mathematical Sciences

April  2015, 11(2): 493-514. doi: 10.3934/jimo.2015.11.493

## Pricing and hedging catastrophe equity put options under a Markov-modulated jump diffusion model

 1 Department of Financial Engineering, Ningbo University, Ningbo, 315211, China 2 School of Finance and Statistics, East China Normal University, 500 Dongchuan Road, Shanghai, 200241, China 3 School of Science, Nanjing Audit University, Nanjing, 210029, China

Received  January 2014 Revised  April 2014 Published  September 2014

In this paper, we consider pricing and hedging of catastrophe equity put options under a Markov-modulated jump diffusion process with a Markov switching compensator. We assume that the risk free interest rate, the appreciation rate and the volatility of the risky asset depend on a finite-state Markov chain. We investigate the pricing of catastrophe equity put options and obtain the explicit pricing formulas. A numerical analysis is provided to illustrate the effect of regime switching on the price of catastrophe equity put options. In the end, since the market which we consider is not complete, we also provide an optimal hedging strategy by using the local risk minimization method.
Citation: Wei Wang, Linyi Qian, Xiaonan Su. Pricing and hedging catastrophe equity put options under a Markov-modulated jump diffusion model. Journal of Industrial & Management Optimization, 2015, 11 (2) : 493-514. doi: 10.3934/jimo.2015.11.493
##### References:
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Huang, Analytical valuation of catastrophe equity options with negative exponential jumps,, Insurance: Mathematics and Economics, 44 (2009), 59.  doi: 10.1016/j.insmatheco.2008.09.009.  Google Scholar [7] S. H. Cox and H. W. Pedersen, Catastrophe risk bonds,, North American Actuarial Journal, 4 (2000), 56.  doi: 10.1080/10920277.2000.10595938.  Google Scholar [8] S. H. Cox, J. Fairchild and H. W. Pedersen, Valuation of structured risk management products,, Insurance: Mathematics and Economics, 34 (2004), 259.   Google Scholar [9] A. Dassios, J. W. Jang, Pricing of catastrophe reinsurance and derivatives using the Cox process with short noise intensity,, Finance and Stochastics, 7 (2003), 73.  doi: 10.1007/s007800200079.  Google Scholar [10] J. C. Duan, I. Popova and P. Ritchken, Option pricing under regime switching,, Quantitative Finance, 2 (2002), 116.  doi: 10.1088/1469-7688/2/2/303.  Google Scholar [11] R. J. Elliott, L. L. Chan and T. K. Siu, Option pricing and Esscher transform under regime switching,, Annals of Finance, 1 (2005), 423.   Google Scholar [12] R. J. Elliott and C. J. U. Osakwe, Option pricing for pure jump processes with Markov switching compensators,, Finance and Stochastics, 10 (2006), 250.  doi: 10.1007/s00780-006-0004-6.  Google Scholar [13] R. J. Elliott, T. K. Siu, L. L. Chan and J. W. Lau, Pricing options under a generalized Markov-modulated jump-diffusion model,, Stochastic Analysis and Applications, 25 (2007), 821.  doi: 10.1080/07362990701420118.  Google Scholar [14] H. F$\ddoto$llmer and M. Schweizer, Hedging of contingent claims under incomplete information,, In Applied Stochastic Analysis (Eds. M.H.A. Davis and R.J. Elliot)(London, (1991), 389.   Google Scholar [15] M. K. Ghosh, A. Arapostathis and S. I. Marcus, Ergodic control of switching diffusions,, SIAM Journal on Control and Optimization, 35 (1997), 1952.  doi: 10.1137/S0363012996299302.  Google Scholar [16] X. Guo, Information and option pricings,, Quantitative Finance, 1 (2001), 38.  doi: 10.1080/713665550.  Google Scholar [17] H. Gründl and H. Schmeiser, Pricing double-trigger reinsurance contracts: Financial versus actuarial approach,, The Journal of Risk and Insurance, 69 (2002), 449.   Google Scholar [18] 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 [19] K. Lee and S. Song, Insiders' hedging in a jump diffusion model,, Quantitative Finance, 5 (2007), 537.  doi: 10.1080/14697680601043191.  Google Scholar [20] K. Lee and P. Protter, Hedging claims with feedback jumps in the price process,, Communications on Stochastic Analysis, 2 (2008), 125.   Google Scholar [21] J. Lewellen, Predicting returns with financial ratios,, Journal of Financial Economics, 74 (2004), 209.   Google Scholar [22] S. K. Lin, C. C. Chang and M. R. Powers, The valuation of contingent capital with catastrophe risks,, Insurance: Mathematics and Economics, 45 (2009), 65.  doi: 10.1016/j.insmatheco.2009.03.005.  Google Scholar [23] R. C. Merton, Option pricing when underlying stock returns are discontinuous,, Journal of Financial Economics, 3 (1976), 125.   Google Scholar [24] Y. Shen and T. K. Siu, Pricing variance swaps under a stochastic interest rate and volatility model with regime-switching,, Operations Research Letters, 41 (2013), 180.  doi: 10.1016/j.orl.2012.12.008.  Google Scholar [25] T. K. Siu, H. L. Yang and J. W. Lau, Pricing currency options under two-factor Markov-modulated stochastic volatility models,, Insurance: Mathematics and Economics, 43 (2008), 295.  doi: 10.1016/j.insmatheco.2008.05.002.  Google Scholar [26] M. Schweizer, A guided tour through quadratic hedging approaches,, in Option Pricing, (2001), 538.  doi: 10.1017/CBO9780511569708.016.  Google Scholar [27] J. H. Yoon, B. G. Jang and K. H. Roh, An analytic valuation method for multivariate contingent claims with regime-switching volatilities,, Operations Research Letters, 39 (2011), 180.  doi: 10.1016/j.orl.2011.02.010.  Google Scholar

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##### References:
 [1] A. Ang and G. Bekaert, Regime switches in interest rates,, Journal of Business and Economic Statistics, 20 (2002), 163.  doi: 10.1198/073500102317351930.  Google Scholar [2] L. J. Bo, Y. J. Wang and X. W. Yang, Markov-modulated jump-diffusions for currency option pricing,, Insurance: Mathematics and Economics, 46 (2010), 461.  doi: 10.1016/j.insmatheco.2010.01.003.  Google Scholar [3] J. Buffington and R. J. Elliott, American options with regime switching,, International Journal of Theoretical and Applied Finance, 5 (2002), 497.  doi: 10.1142/S0219024902001523.  Google Scholar [4] J. Campbell and L. Hentschel, No news is good news: An asymmetric model of changing volatility in stock returns,, Journal of Financial Economics, 31 (1992), 281.   Google Scholar [5] C. C. Chang, S. K. Lin and M. T. Yu, Valuation of catastrophe equity puts with Markov-modulated Poisson processes,, The Journal of Risk and Insurance, 78 (2011), 447.   Google Scholar [6] L. F. Chang and M. W. Huang, Analytical valuation of catastrophe equity options with negative exponential jumps,, Insurance: Mathematics and Economics, 44 (2009), 59.  doi: 10.1016/j.insmatheco.2008.09.009.  Google Scholar [7] S. H. Cox and H. W. Pedersen, Catastrophe risk bonds,, North American Actuarial Journal, 4 (2000), 56.  doi: 10.1080/10920277.2000.10595938.  Google Scholar [8] S. H. Cox, J. Fairchild and H. W. Pedersen, Valuation of structured risk management products,, Insurance: Mathematics and Economics, 34 (2004), 259.   Google Scholar [9] A. Dassios, J. W. Jang, Pricing of catastrophe reinsurance and derivatives using the Cox process with short noise intensity,, Finance and Stochastics, 7 (2003), 73.  doi: 10.1007/s007800200079.  Google Scholar [10] J. C. Duan, I. Popova and P. Ritchken, Option pricing under regime switching,, Quantitative Finance, 2 (2002), 116.  doi: 10.1088/1469-7688/2/2/303.  Google Scholar [11] R. J. Elliott, L. L. Chan and T. K. Siu, Option pricing and Esscher transform under regime switching,, Annals of Finance, 1 (2005), 423.   Google Scholar [12] R. J. Elliott and C. J. U. Osakwe, Option pricing for pure jump processes with Markov switching compensators,, Finance and Stochastics, 10 (2006), 250.  doi: 10.1007/s00780-006-0004-6.  Google Scholar [13] R. J. Elliott, T. K. Siu, L. L. Chan and J. W. Lau, Pricing options under a generalized Markov-modulated jump-diffusion model,, Stochastic Analysis and Applications, 25 (2007), 821.  doi: 10.1080/07362990701420118.  Google Scholar [14] H. F$\ddoto$llmer and M. Schweizer, Hedging of contingent claims under incomplete information,, In Applied Stochastic Analysis (Eds. M.H.A. Davis and R.J. Elliot)(London, (1991), 389.   Google Scholar [15] M. K. Ghosh, A. Arapostathis and S. I. Marcus, Ergodic control of switching diffusions,, SIAM Journal on Control and Optimization, 35 (1997), 1952.  doi: 10.1137/S0363012996299302.  Google Scholar [16] X. Guo, Information and option pricings,, Quantitative Finance, 1 (2001), 38.  doi: 10.1080/713665550.  Google Scholar [17] H. Gründl and H. Schmeiser, Pricing double-trigger reinsurance contracts: Financial versus actuarial approach,, The Journal of Risk and Insurance, 69 (2002), 449.   Google Scholar [18] 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 [19] K. Lee and S. Song, Insiders' hedging in a jump diffusion model,, Quantitative Finance, 5 (2007), 537.  doi: 10.1080/14697680601043191.  Google Scholar [20] K. Lee and P. Protter, Hedging claims with feedback jumps in the price process,, Communications on Stochastic Analysis, 2 (2008), 125.   Google Scholar [21] J. Lewellen, Predicting returns with financial ratios,, Journal of Financial Economics, 74 (2004), 209.   Google Scholar [22] S. K. Lin, C. C. Chang and M. R. Powers, The valuation of contingent capital with catastrophe risks,, Insurance: Mathematics and Economics, 45 (2009), 65.  doi: 10.1016/j.insmatheco.2009.03.005.  Google Scholar [23] R. C. Merton, Option pricing when underlying stock returns are discontinuous,, Journal of Financial Economics, 3 (1976), 125.   Google Scholar [24] Y. Shen and T. K. Siu, Pricing variance swaps under a stochastic interest rate and volatility model with regime-switching,, Operations Research Letters, 41 (2013), 180.  doi: 10.1016/j.orl.2012.12.008.  Google Scholar [25] T. K. Siu, H. L. Yang and J. W. Lau, Pricing currency options under two-factor Markov-modulated stochastic volatility models,, Insurance: Mathematics and Economics, 43 (2008), 295.  doi: 10.1016/j.insmatheco.2008.05.002.  Google Scholar [26] M. Schweizer, A guided tour through quadratic hedging approaches,, in Option Pricing, (2001), 538.  doi: 10.1017/CBO9780511569708.016.  Google Scholar [27] J. H. Yoon, B. G. Jang and K. H. Roh, An analytic valuation method for multivariate contingent claims with regime-switching volatilities,, Operations Research Letters, 39 (2011), 180.  doi: 10.1016/j.orl.2011.02.010.  Google Scholar
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