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

Alexander Melnikov; Hongxi Wan

doi:10.3934/puqr.2021017

Article Contents

2021, Volume 6, Issue 4: 343-368. Doi: 10.3934/puqr.2021017

This issue Previous Article Existence, uniqueness and strict comparison theorems for BSDEs driven by RCLL martingales Next Article Extended conditional G-expectations and related stopping times

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

Alexander Melnikov^†, and
Hongxi Wan^,

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

Correspondence: hongxi@ualberta.ca
Correspondence: hongxi@ualberta.ca

†Equal contributor

Received: February 17, 2021

Accepted: October 29, 2021

Published: December 2021

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).

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Abstract

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.

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

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

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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

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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

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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	$CVaR_{0.95}\leq 5$			$CVaR_{0.95}\leq 7.5$			$CVaR_{0.95}\leq 10$
Revision period	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

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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	$CVaR_{0.95}\leq 5$			$CVaR_{0.95}\leq 7.5$			$CVaR_{0.95}\leq 10$
Revision period	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

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Table 5. Survival probabilities and age of insured in the market with transaction costs

Maturity T (years)	$CVaR_{0.95}\leq 5$		$CVaR_{0.95}\leq 10$
Maturity T (years)	$ {}_{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

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Table 6. Survival probabilities and age of insured in the complete market

Maturity T (years)	$CVaR_{0.95}\leq 5$		$CVaR_{0.95}\leq 10$
Maturity T (years)	${}_{T}p_{x}$	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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