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Valuation of variable annuities under the Volterra mortality and rough Heston models

Wenyuan Li, Haoqi Lyu

Abstract

This paper investigates the valuation of variable annuity contracts with an early surrender option under non-Markovian models. Moreover, policyholders are provided with guaranteed minimum maturity and death benefits to protect against the downside risk. Unlike the existing literature, our variable annuity account value is linked to two non-Markovian processes: an equity index modeled by a rough Heston model and a force of mortality following a Volterra-type stochastic model. In this case, the early surrender feature introduces an optimal stopping problem where continuation values depend on the entire path history, rendering traditional numerical methods infeasible. We develop a deep signature Least Squares Monte Carlo approach to learn optimal surrender strategies on a discretized time grid. To mitigate the curse of dimensionality arising from the path-dependent model, we use truncated rough-path signatures to encode the historical paths and approximate the continuation values using a neural network. Numerically, we find that the fair fee increases with the Hurst parameters of both the stock volatility and the force of mortality. Finally, a convergence proof is provided to further support the stability of our method.

Valuation of variable annuities under the Volterra mortality and rough Heston models

Abstract

This paper investigates the valuation of variable annuity contracts with an early surrender option under non-Markovian models. Moreover, policyholders are provided with guaranteed minimum maturity and death benefits to protect against the downside risk. Unlike the existing literature, our variable annuity account value is linked to two non-Markovian processes: an equity index modeled by a rough Heston model and a force of mortality following a Volterra-type stochastic model. In this case, the early surrender feature introduces an optimal stopping problem where continuation values depend on the entire path history, rendering traditional numerical methods infeasible. We develop a deep signature Least Squares Monte Carlo approach to learn optimal surrender strategies on a discretized time grid. To mitigate the curse of dimensionality arising from the path-dependent model, we use truncated rough-path signatures to encode the historical paths and approximate the continuation values using a neural network. Numerically, we find that the fair fee increases with the Hurst parameters of both the stock volatility and the force of mortality. Finally, a convergence proof is provided to further support the stability of our method.

Paper Structure

This paper contains 19 sections, 5 theorems, 92 equations, 7 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Mortality model
  4. Market model and variable annuity account
  5. Guaranteed benefits and early surrender
  6. Fair fee and optimal surrender strategy
  7. Forward simulation of the underlying process
  8. Dynamic programming principle for variable annuity
  9. State variable encoding and neural network structure
  10. Deep signature LSMC method
  11. State Variables.
  12. Deep signature LSMC algorithm.
  13. Fair fee
  14. Convergence analysis of the LSMC algorithm for the optimal surrender strategy
  15. Numerical results
  16. ...and 4 more sections

Key Result

Lemma 4.2

Under Assumption assump: regularity, for each $n\in \{0, \cdots, N-1\}$, there exists $\phi_n^{\widehat{p}, K} \in \Theta_{\widehat{p}}$, the neural network parameters with neural network width $\widehat{p}$ and model input of path signature truncated at level $K$, such that where $C_{\tau_n}$ is defined in (eq: Ctaun). $\blacktriangleleft$$\blacktriangleleft$

Figures (7)

  • Figure 1: Sample Paths of the Stochastic force of Mortality
  • Figure 2: Observed survival probabilities and model survival probabilities
  • Figure 3: Neural network with structure "$41-64-64-64-64-1$".
  • Figure 4: Sample survival probabilities with different $H_m$
  • Figure 5: Kernel function with different Hurst parameter $H_m$
  • ...and 2 more figures

Theorems & Definitions (12)

  • Definition 3.1
  • Definition 3.2
  • Lemma 4.2
  • proof
  • Proposition 4.3
  • proof
  • Lemma 4.4
  • proof
  • Lemma 4.9
  • proof
  • ...and 2 more