Analysis of VIX-linked fee incentives in variable annuities via continuous-time Markov chain approximation

Abstract

We consider the pricing of variable annuities (VAs) with general fee structures under a class of stochastic volatility models which includes the Heston, Hull-White, Scott, α-Hypergeometric, 3/2, and 4/2 models. In particular, we analyze the impact of different VIX-linked fee structures on the optimal surrender strategy of a VA contract with guaranteed minimum maturity benefit (GMMB). Under the assumption that the VA contract can be surrendered before maturity, the pricing of a VA contract corresponds to an optimal stopping problem with an unbounded, time-dependent, and discontinuous payoff function. We develop efficient algorithms for the pricing of VA contracts using a two-layer continuous-time Markov chain approximation for the fund value process. When the contract is kept until maturity and under a general fee structure, we show that the value of the contract can be approximated by a closed-form matrix expression. We also provide a quick and simple way to determine the value of early surrenders via a recursive algorithm and give an easy procedure to approximate the optimal surrender surface. We show numerically that the optimal surrender strategy is more robust to changes in the volatility of the account value when the fee is linked to the VIX index.

Keywords:

• Variable annuities
• Optimal stopping
• American options
• Stochastic volatility
• Heston model
• Continuous-time Markov chain

JEL Classifications:

• C63
• G12
• G13
• G22

Disclosure statement

No potential conflict of interest was reported by the author(s).

Supplemental data

Supplemental data for this article can be accessed online at http://dx.doi.org/10.1080/14697688.2023.2215278.

Notes

1 There are no consensus among practitioners and scientists for these products' name, and thus, different authors may use different terminologies for the same product.

2 In United-States, the total variable annuity sales were $125 billion in 2021, representing an increase of 25% with respect to the total VA sales in 2020. Source: LIMRA Secure Retirement Institute, U.S. Individual Annuities survey https://www.limra.com/siteassets/newsroom/fact-tank/sales-data/2021/q4/2012-2021-annuity-sales-updated.pdf.

3 See https://www.cboe.com/tradable_products/vix/.

4 See Retirement Income Journal available at https://retirementincomejournal.com/article/sunamerica-links-va-rider-fees-to-volatility-index/.

5 See footnote 6 on p. 9 of the prospectus (the long-form) available at https://aig.onlineprospectus.net/AIG/867018103A/index.php?open=POLARIS!5fPLATINUM!5fO-SERIES!5fISP.pdfhttps://aig.onlineprospectus.net/AIG/867018103A/index.php?open=POLARIS!5fPLATINUM!5fO-SERIES!5fISP.pdf.

6 The integral representation of the value function may be challenging to obtain under general stochastic volatility models unless making some regularity assumptions on the value function as in Ma et al. (2021). Indeed, the smoothness of the value function can be difficult to show under such bi-dimensional models; see Terenzi (2018) and Lamberton and Terenzi (2019).

7 More precisely, the semigroup $\left\{P_t\right\}$ must be standard – that is, $p_{ii}\left(t\right)\to 1$ and $p_{ij}\left(t\right)\to 0$ as $t\downarrow 0$ – and uniform – $\underset{i}{sup}-q_{ii}<\mathrm{\infty }$, see Grimmet and Stirzaker (2001), definitions 6.9.4 and 6.10.3, theorems 6.10.1, 6.10.5 and 6.10.6 for details.

8 An advised reader will notice some differences between the transition rates stated above, and the ones that appear in Lo and Skindilias (2014). However, one can show that the two rate matrices are equivalent with some simple algebra.

9 If $X_0^{(m,N)}$ and $V_0^{\left(m\right)}$ are not part of their respective grids, then the two points can be added to the grids, or the option price must be linearly interpolated between grid points, see remark 5.1 for details.

10 Recall that ${\psi }^{-1}\left(n_y\right)=(x_n,v_l)$ for $n_y\in {\mathcal{S}}_Y^{(m,N)}$, where $n\le N$ is the unique integer such that $n_y=(l-1)N+n$ for some $l\in \{1,2,\dots ,m\}$.

11 Many convergence results, such as the one in Palczewski and Stettner (2010) and Song et al. (2013), require the reward function to be bounded. However, as mentioned in Mijatović and Pistorius (2013), remark 5.4 and Cui et al. (2018), remark 5, the original payoff φ can be replaced by the truncated payoff $\phi \wedge L$ with a constant L sufficiently large without altering the accuracy of the numerical results.

12 A set X is connected if it cannot be divided into two disjoint non-empty open sets.

13 For the 3/2 model, a closed-form expression for the $\mathrm{V}\mathrm{I}\mathrm{X}$ may be found in Carr and Sun (2007, theorem 4). However, as pointed out by Drimus (2012), the integral that appears in the analytical formula is difficult to implement and is not suited for fast and accurate numerical methods. For this reason, the CTMC approximation of the VIX is used in the numerical examples under the 3/2 model.

14 Bloomberg provides historical Heston calibrated parameters to market data on a daily basis via its Option Pricing template (OVME). These parameters are often used in practice for over-the-counter option pricing. Bloomberg's Heston calibrated speed reversion parameter is $\kappa =3.6881$ as of December 31, 2019, $\kappa =5$ as of March 31, 2020 and $\kappa =1.1397$ as of September 30, 2020. The parameter selected for our numerical experiments falls approximately in the middle of those of December 2019 and September 2020. In the financial literature, Aït-Sahalia and Kimmel (2007) obtain $\kappa =5.07$ whereas Garcia et al. (2011) obtain $\kappa =0.173$, and again our values fall between these two values.

15 The function fastExpm is based on Hogben et al. (2011) and Kuprov (2011).

16 See VIX historical data at https://www.cboe.com/tradable_products/vix/vix_historical_data/https://www.cboe.com/tradable_products/vix/vix_historical_data/.

17 Numerical experiments under the Heston model have been performed using equation (22(22) $\begin{array}{rl}& v_e^{(m,N)}(0,F_0,V_0)\\ & :=\mathbb{E}[e^{-rT}max(G,F_T^{(m,N)})|F_0^{(m,N)}=F_0,V_0^{\left(m\right)}=V_0]\\ & =e^{-rT}{\mathbf{e}}_{ik}\exp \left\{T{\mathbf{G}}^{(m,N)}\right\}\mathbf{H},\end{array}$(22) ), and algorithms 2 and 4. Note however that similar results are obtained when using the Fast Algorithms, see appendix B, available online for details.

Additional information

Funding

The first author acknowledges support from the Natural Science and Engineering Research Council of Canada [grant number RGPIN-2016-04869] and from the Fonds de recherche du Québec – Nature et Technologies [grant number 253180]. The second author was supported by Fonds de recherche du Québec – Nature et Technologies [grant number 257320]. The third author acknowledges partial support from National Natural Science Foundation of China [grant number 72271249] and National Science Foundation [grant number CNS-2113906].