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Article Contents

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

†Equal contributor

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.

Mathematics Subject Classification: 91G20, 91G60, 62P05.

 Citation:

• Figure 1.  Survival probability vs CVaR constraint for life insurance contracts for different revision frequencies, T = 5.

Table 1.  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

Table 2.  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

Table 3.  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

Table 4.  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

Table 5.  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

Table 6.  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
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