# American Institute of Mathematical Sciences

December  2021, 6(4): 343-368. doi: 10.3934/puqr.2021017

## CVaR-hedging and its applications to equity-linked life insurance contracts with transaction costs

 1 Department of Mathematical and Statistical Sciences, University of Alberta, T6G 2G1 Edmonton, Canada

Correspondence: hongxi@ualberta.ca

†Equal contributor

Received  February 17, 2021 Accepted  October 29, 2021 Published  December 2021

Fund Project: The authors are grateful to anonymous reviewers and the editors for fruitful suggestions to improve the paper. This research was supported by Natural Sciences and Engineering Research Council of Canada (Grant No. RES0043487).

This paper analyzes Conditional Value-at-Risk (CVaR) based partial hedging and its applications on equity-linked life insurance contracts in a Jump-Diffusion market model with transaction costs. A nonlinear partial differential equation (PDE) that an option value process inclusive of transaction costs should satisfy is provided. In particular, the closed-form expression of a European call option price is given. Meanwhile, the CVaR-based partial hedging strategy for a call option is derived explicitly. Both the CVaR hedging price and the weights of the hedging portfolio are based on an adjusted volatility. We obtain estimated values of expected total hedging errors and total transaction costs by a simulation method. Furthermore,our results are implemented to derive target clients’ survival probabilities and age of equity-linked life insurance contracts.

Citation: Alexander Melnikov, Hongxi Wan. CVaR-hedging and its applications to equity-linked life insurance contracts with transaction costs. Probability, Uncertainty and Quantitative Risk, 2021, 6 (4) : 343-368. doi: 10.3934/puqr.2021017
##### References:
 [1] Amin, K. I., Jump diffusion option valuation in discrete time, The Journal of Finance, 1993, 48(5): 1833−1863. doi: 10.1111/j.1540-6261.1993.tb05130.x. [2] Black, F. and Scholes, M., The pricing of options and corporate liabilities, Journal of Political Economy, 1973, 81(3): 637−654. doi: 10.1086/260062. [3] Boyle, P. P. and Hardy, M. R., Reserving for maturity guarantees: Two approaches, Insurance: Mathematics and Economics, 1997, 21(2): 113−127. doi: 10.1016/S0167-6687(97)00026-7. [4] Boyle, P. P. and Vorst, T., Option replication in discrete time with transaction costs, The Journal of Finance, 1992, 47(1): 271−293. doi: 10.1111/j.1540-6261.1992.tb03986.x. [5] Bratyk, M. and Mishura, Y., The generalization of the quantile hedging problem for price process model involving finite number of Brownian and fractional Brownian motions, Theory of Stochastic Processes, 2008, 14(3): 27−38. [6] Brennan, M. J. and Schwartz, E. S., The pricing of equity-linked life insurance policies with an asset value guarantee, Journal of Financial Economics, 1976, 3(3): 195−213. doi: 10.1016/0304-405X(76)90003-9. [7] Cox, J. C. and Ross, S. A., The valuation of options for alternative stochastic processes, Journal of Financial Economics, 1976, 3(1): 145−166. [8] Dewynne, J. N., Whalley, A. E. and Wilmott, P., Path-dependent options and transaction costs, Philosophical transactions of the royal society of London, Series A: Physical and Engineering Sciences, 1994, 347(1684): 517−529. doi: 10.1098/rsta.1994.0061. [9] Föllmer, H. and Leukert, P., Quantile hedging, Finance and Stochastics, 1999, 3(3): 251−273. doi: 10.1007/s007800050062. [10] Föllmer, H. and Leukert, P., Efficient hedging: Cost versus shortfall risk, finance and stochastics, 2000, 4(2): 117−146. doi: 10.1007/s007800050008. [11] Hodges, S. D., and Neuberger, A., Optimal replication of contingent claims under transaction costs, Review Futures Market, 1989, 8(2): 222−239. [12] Hoggard, T., Whalley, A. E. and Wilmott, P., Option portfolios in the presence of transaction costs, Advances in Futures and Options Research, 1994, 7(4): 21−35. [13] Kirch, M. and Melnikov, A., Efficient hedging and pricing of life insurance policies in a jump-diffusion model, Stochastic Analysis and Applications, 2005, 23(6): 1213−1233. doi: 10.1080/07362990500292692. [14] Leland, H. E., Option portfolios in the presence of transaction costs, In: Boyle, P. P., Pennacchi G. and Ritchken P. (eds.), Advances in Futures and Options Research, 1985, 7(4): 21–35. [15] Melnikov, A. and Petrachenko, Y. G., On option pricing in binomial market with transaction costs, Finance and Stochastics, 2005, 9(1): 141−149. doi: 10.1007/s00780-004-0134-7. [16] Melnikov, A. and Skornyakova, V., Quantile hedging and its application to life insurance, Statistics & Decisions, 2005, 23(4): 301−316. [17] Merton, R. C., Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 1976, 3(1): 125−144. [18] Merton, R. C., Continuous-time Finance, Basil-Blackwell, Cambridge, 1990. [19] Melnikov, A. and Smirnov, I., Dynamic hedging of conditional value-at-risk, Insurance: Mathematics and Economics, 2012, 51(1): 182−190. doi: 10.1016/j.insmatheco.2012.03.011. [20] Melnikov, A. and Tong, S., Quantile hedging on equity-linked life insurance contracts with transaction costs, Insurance:Mathematics and Economics, 2014, 58: 77−88. doi: 10.1016/j.insmatheco.2014.06.005. [21] Toft, K. B., On the mean-variance tradeoff in option replication with transactions costs, The Journal of Financial and Quantitative Analysis, 1996, 31(2): 233−263. doi: 10.2307/2331181. [22] Melnikov, A. and Nosrati, A., Equity-linked Life Insurance Partial Hedging Methods, Chapman and Hall/CRC, 2017. [23] Mocioalca, O., Jump diffusion options with transaction costs, Rev. Roumaine Math. Pures Appl., 2007, 52(3): 349−366. [24] Zakamulin, V., Option pricing and hedging in the presence of transaction costs and nonlinear partial differential equations, SSRN Electronic Journal, 2008, https://ssrn.com/abstract=938933. [25] Zhou, S., Han, L., Li, W., Zhang, Y. and Han, M., A positivity-preserving numerical scheme for option pricing model with transaction costs under jump-diffusion process, Computational and Applied Mathematics, 2015, 34(3): 881−900. doi: 10.1007/s40314-014-0156-5.

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##### References:
 [1] Amin, K. I., Jump diffusion option valuation in discrete time, The Journal of Finance, 1993, 48(5): 1833−1863. doi: 10.1111/j.1540-6261.1993.tb05130.x. [2] Black, F. and Scholes, M., The pricing of options and corporate liabilities, Journal of Political Economy, 1973, 81(3): 637−654. doi: 10.1086/260062. [3] Boyle, P. P. and Hardy, M. R., Reserving for maturity guarantees: Two approaches, Insurance: Mathematics and Economics, 1997, 21(2): 113−127. doi: 10.1016/S0167-6687(97)00026-7. [4] Boyle, P. P. and Vorst, T., Option replication in discrete time with transaction costs, The Journal of Finance, 1992, 47(1): 271−293. doi: 10.1111/j.1540-6261.1992.tb03986.x. [5] Bratyk, M. and Mishura, Y., The generalization of the quantile hedging problem for price process model involving finite number of Brownian and fractional Brownian motions, Theory of Stochastic Processes, 2008, 14(3): 27−38. [6] Brennan, M. J. and Schwartz, E. S., The pricing of equity-linked life insurance policies with an asset value guarantee, Journal of Financial Economics, 1976, 3(3): 195−213. doi: 10.1016/0304-405X(76)90003-9. [7] Cox, J. C. and Ross, S. A., The valuation of options for alternative stochastic processes, Journal of Financial Economics, 1976, 3(1): 145−166. [8] Dewynne, J. N., Whalley, A. E. and Wilmott, P., Path-dependent options and transaction costs, Philosophical transactions of the royal society of London, Series A: Physical and Engineering Sciences, 1994, 347(1684): 517−529. doi: 10.1098/rsta.1994.0061. [9] Föllmer, H. and Leukert, P., Quantile hedging, Finance and Stochastics, 1999, 3(3): 251−273. doi: 10.1007/s007800050062. [10] Föllmer, H. and Leukert, P., Efficient hedging: Cost versus shortfall risk, finance and stochastics, 2000, 4(2): 117−146. doi: 10.1007/s007800050008. [11] Hodges, S. D., and Neuberger, A., Optimal replication of contingent claims under transaction costs, Review Futures Market, 1989, 8(2): 222−239. [12] Hoggard, T., Whalley, A. E. and Wilmott, P., Option portfolios in the presence of transaction costs, Advances in Futures and Options Research, 1994, 7(4): 21−35. [13] Kirch, M. and Melnikov, A., Efficient hedging and pricing of life insurance policies in a jump-diffusion model, Stochastic Analysis and Applications, 2005, 23(6): 1213−1233. doi: 10.1080/07362990500292692. [14] Leland, H. E., Option portfolios in the presence of transaction costs, In: Boyle, P. P., Pennacchi G. and Ritchken P. (eds.), Advances in Futures and Options Research, 1985, 7(4): 21–35. [15] Melnikov, A. and Petrachenko, Y. G., On option pricing in binomial market with transaction costs, Finance and Stochastics, 2005, 9(1): 141−149. doi: 10.1007/s00780-004-0134-7. [16] Melnikov, A. and Skornyakova, V., Quantile hedging and its application to life insurance, Statistics & Decisions, 2005, 23(4): 301−316. [17] Merton, R. C., Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 1976, 3(1): 125−144. [18] Merton, R. C., Continuous-time Finance, Basil-Blackwell, Cambridge, 1990. [19] Melnikov, A. and Smirnov, I., Dynamic hedging of conditional value-at-risk, Insurance: Mathematics and Economics, 2012, 51(1): 182−190. doi: 10.1016/j.insmatheco.2012.03.011. [20] Melnikov, A. and Tong, S., Quantile hedging on equity-linked life insurance contracts with transaction costs, Insurance:Mathematics and Economics, 2014, 58: 77−88. doi: 10.1016/j.insmatheco.2014.06.005. [21] Toft, K. B., On the mean-variance tradeoff in option replication with transactions costs, The Journal of Financial and Quantitative Analysis, 1996, 31(2): 233−263. doi: 10.2307/2331181. [22] Melnikov, A. and Nosrati, A., Equity-linked Life Insurance Partial Hedging Methods, Chapman and Hall/CRC, 2017. [23] Mocioalca, O., Jump diffusion options with transaction costs, Rev. Roumaine Math. Pures Appl., 2007, 52(3): 349−366. [24] Zakamulin, V., Option pricing and hedging in the presence of transaction costs and nonlinear partial differential equations, SSRN Electronic Journal, 2008, https://ssrn.com/abstract=938933. [25] Zhou, S., Han, L., Li, W., Zhang, Y. and Han, M., A positivity-preserving numerical scheme for option pricing model with transaction costs under jump-diffusion process, Computational and Applied Mathematics, 2015, 34(3): 881−900. doi: 10.1007/s40314-014-0156-5.
Survival probability vs CVaR constraint for life insurance contracts for different revision frequencies, T = 5.
Estimated present values of total hedging errors and total transaction costs with the adjusted volatility $\hat{\sigma}_1$, C = 5
 Maturity T (years) Revision period CVaR price HE TC HE−TC Biweekly 5.46 0.808 0.7688 0.0392 T=1 Weekly 5.7489 1.0306 1.0588 −0.0282 Daily 6.8641 2.2084 2.2208 −0.0124 Biweekly 14.6754 1.4408 1.3929 0.0479 T=3 Weekly 15.1997 1.8731 1.9117 −0.0386 Daily 17.1948 3.9429 3.9721 −0.0292 Biweekly 21.8367 1.6432 1.7092 −0.066 T=5 Weekly 22.488 2.3572 2.3839 −0.0267 Daily 34.958 4.9192 4.9291 −0.0099 Biweekly 35.6998 2.0476 2.083 −0.0354 T=10 Weekly 36.5054 2.9219 2.93 −0.0081 Daily 39.5508 5.9992 5.9938 0.0054 Biweekly 46.291 2.1415 2.1668 −0.0253 T=15 Weekly 47.1447 3.0194 3.0096 0.0098 Daily 50.3664 6.2761 6.2833 −0.0073
 Maturity T (years) Revision period CVaR price HE TC HE−TC Biweekly 5.46 0.808 0.7688 0.0392 T=1 Weekly 5.7489 1.0306 1.0588 −0.0282 Daily 6.8641 2.2084 2.2208 −0.0124 Biweekly 14.6754 1.4408 1.3929 0.0479 T=3 Weekly 15.1997 1.8731 1.9117 −0.0386 Daily 17.1948 3.9429 3.9721 −0.0292 Biweekly 21.8367 1.6432 1.7092 −0.066 T=5 Weekly 22.488 2.3572 2.3839 −0.0267 Daily 34.958 4.9192 4.9291 −0.0099 Biweekly 35.6998 2.0476 2.083 −0.0354 T=10 Weekly 36.5054 2.9219 2.93 −0.0081 Daily 39.5508 5.9992 5.9938 0.0054 Biweekly 46.291 2.1415 2.1668 −0.0253 T=15 Weekly 47.1447 3.0194 3.0096 0.0098 Daily 50.3664 6.2761 6.2833 −0.0073
Estimated present values of total hedging errors and total transaction costs with the original volatility $\sigma_1$, C = 5
 Maturity T (years) Revision period CVaR price HE TC HE−TC Biweekly 4.7298 −0.0298 0.8092 −0.839 T=1 Weekly 4.7298 0.0177 1.104 −1.0863 Daily 4.7298 −0.0041 2.4556 −2.4597 Biweekly 13.3329 0.0305 1.4584 −1.427 T=3 Weekly 13.3329 0.0182 2.0448 −2.0266 Daily 13.3329 −0.0018 4.4939 −4.4957 Biweekly 20.1652 −0.0288 1.7894 −1.8182 T=5 Weekly 20.1652 −0.0108 2.5308 −2.5416 Daily 20.1652 −0.0029 5.5339 −5.5368 Biweekly 33.6293 −0.022 2.1664 −2.1884 T=10 Weekly 33.6293 0.0215 3.0789 −3.0574 Daily 33.6293 0.0053 6.7483 −6.743 Biweekly 44.0962 −0.0259 2.2768 −2.3027 T=15 Weekly 44.0962 0.0081 3.2585 −3.2504 Daily 44.0962 0.0035 6.9676 −6.9641
 Maturity T (years) Revision period CVaR price HE TC HE−TC Biweekly 4.7298 −0.0298 0.8092 −0.839 T=1 Weekly 4.7298 0.0177 1.104 −1.0863 Daily 4.7298 −0.0041 2.4556 −2.4597 Biweekly 13.3329 0.0305 1.4584 −1.427 T=3 Weekly 13.3329 0.0182 2.0448 −2.0266 Daily 13.3329 −0.0018 4.4939 −4.4957 Biweekly 20.1652 −0.0288 1.7894 −1.8182 T=5 Weekly 20.1652 −0.0108 2.5308 −2.5416 Daily 20.1652 −0.0029 5.5339 −5.5368 Biweekly 33.6293 −0.022 2.1664 −2.1884 T=10 Weekly 33.6293 0.0215 3.0789 −3.0574 Daily 33.6293 0.0053 6.7483 −6.743 Biweekly 44.0962 −0.0259 2.2768 −2.3027 T=15 Weekly 44.0962 0.0081 3.2585 −3.2504 Daily 44.0962 0.0035 6.9676 −6.9641
Estimated present values of total hedging errors and total transaction costs with adjusted volatility $\hat{\sigma}_1$ for different levels of CVaR constraint, T = 1
 Revision period $CVaR_{0.95}\leq 5$ $CVaR_{0.95}\leq 7.5$ $CVaR_{0.95}\leq 10$ HE TC HE−TC HE TC HE−TC HE TC HE−TC Biweekly 0.808 0.7688 0.0392 0.7154 0.7505 −0.0351 0.6769 0.7228 −0.0459 Weekly 1.0306 1.0588 −0.0282 1.0291 1.0622 −0.0331 0.9786 1.0173 −0.0387 Daily 2.2084 2.2208 −0.0124 2.13 2.1519 −0.0219 2.0485 2.065 −0.0165
 Revision period $CVaR_{0.95}\leq 5$ $CVaR_{0.95}\leq 7.5$ $CVaR_{0.95}\leq 10$ HE TC HE−TC HE TC HE−TC HE TC HE−TC Biweekly 0.808 0.7688 0.0392 0.7154 0.7505 −0.0351 0.6769 0.7228 −0.0459 Weekly 1.0306 1.0588 −0.0282 1.0291 1.0622 −0.0331 0.9786 1.0173 −0.0387 Daily 2.2084 2.2208 −0.0124 2.13 2.1519 −0.0219 2.0485 2.065 −0.0165
Estimated present values of total hedging errors and total transaction costs with original volatility ${\sigma}_1$ for different levels of CVaR constraint, T = 1
 Revision period $CVaR_{0.95}\leq 5$ $CVaR_{0.95}\leq 7.5$ $CVaR_{0.95}\leq 10$ HE TC HE−TC HE TC HE−TC HE TC HE−TC Biweekly −0.0298 0.8092 −0.839 −0.0064 0.7919 −0.7983 −0.0166 0.7383 −0.7549 Weekly 0.0177 1.104 −1.0863 0.0169 1.0956 −1.0787 0.0281 1.0077 −0.9796 Daily −0.0041 2.4556 −2.4597 0.0042 2.3388 −2.3346 0.021 2.255 −2.234
 Revision period $CVaR_{0.95}\leq 5$ $CVaR_{0.95}\leq 7.5$ $CVaR_{0.95}\leq 10$ HE TC HE−TC HE TC HE−TC HE TC HE−TC Biweekly −0.0298 0.8092 −0.839 −0.0064 0.7919 −0.7983 −0.0166 0.7383 −0.7549 Weekly 0.0177 1.104 −1.0863 0.0169 1.0956 −1.0787 0.0281 1.0077 −0.9796 Daily −0.0041 2.4556 −2.4597 0.0042 2.3388 −2.3346 0.021 2.255 −2.234
Survival probabilities and age of insured in the market with transaction costs
 Maturity T (years) $CVaR_{0.95}\leq 5$ $CVaR_{0.95}\leq 10$ ${}_{T}p_{x}$ age ${}_{T}p_{x}$ age T=3 0.9078 75 0.8237 82 T=5 0.9359 64 0.8762 72 T=10 0.9633 45 0.9284 53 T=15 0.9749 31 0.9507 41
 Maturity T (years) $CVaR_{0.95}\leq 5$ $CVaR_{0.95}\leq 10$ ${}_{T}p_{x}$ age ${}_{T}p_{x}$ age T=3 0.9078 75 0.8237 82 T=5 0.9359 64 0.8762 72 T=10 0.9633 45 0.9284 53 T=15 0.9749 31 0.9507 41
Survival probabilities and age of insured in the complete market
 Maturity T (years) $CVaR_{0.95}\leq 5$ $CVaR_{0.95}\leq 10$ ${}_{T}p_{x}$ age ${}_{T}p_{x}$ age T=3 0.8806 78 0.7741 84 T=5 0.9166 67 0.8398 75 T=10 0.9516 48 0.9058 57 T=15 0.9665 36 0.9343 44
 Maturity T (years) $CVaR_{0.95}\leq 5$ $CVaR_{0.95}\leq 10$ ${}_{T}p_{x}$ age ${}_{T}p_{x}$ age T=3 0.8806 78 0.7741 84 T=5 0.9166 67 0.8398 75 T=10 0.9516 48 0.9058 57 T=15 0.9665 36 0.9343 44
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