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On an Optimal Stopping Problem with a Discontinuous Reward

Anne Mackay, Marie-Claude Vachon

Abstract

We study an optimal stopping problem with an unbounded, time-dependent and discontinuous reward function. This problem is motivated by the pricing of a variable annuity contract with guaranteed minimum maturity benefit, under the assumption that the policyholder's surrender behaviour maximizes the risk-neutral value of the contract. We consider a general fee and surrender charge function, and give a condition under which optimal stopping always occurs at maturity. Using an alternative representation for the value function of the optimization problem, we study its analytical properties and the resulting surrender (or exercise) region. In particular, we show that the non-emptiness and the shape of the surrender region are fully characterized by the fee and the surrender charge functions, which provides a powerful tool to understand their interrelation and how it affects early surrenders and the optimal surrender boundary. Under certain conditions on these two functions, we develop three representations for the value function; two are analogous to their American option counterpart, and one is new to the actuarial and American option pricing literature.

On an Optimal Stopping Problem with a Discontinuous Reward

Abstract

We study an optimal stopping problem with an unbounded, time-dependent and discontinuous reward function. This problem is motivated by the pricing of a variable annuity contract with guaranteed minimum maturity benefit, under the assumption that the policyholder's surrender behaviour maximizes the risk-neutral value of the contract. We consider a general fee and surrender charge function, and give a condition under which optimal stopping always occurs at maturity. Using an alternative representation for the value function of the optimization problem, we study its analytical properties and the resulting surrender (or exercise) region. In particular, we show that the non-emptiness and the shape of the surrender region are fully characterized by the fee and the surrender charge functions, which provides a powerful tool to understand their interrelation and how it affects early surrenders and the optimal surrender boundary. Under certain conditions on these two functions, we develop three representations for the value function; two are analogous to their American option counterpart, and one is new to the actuarial and American option pricing literature.
Paper Structure (17 sections, 26 theorems, 79 equations, 2 figures)

This paper contains 17 sections, 26 theorems, 79 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Financial Setting
  3. Market Model
  4. Variable Annuity Contract
  5. Optimal Stopping Time
  6. Trivial Case
  7. General Case
  8. Analytical study of the value function
  9. Elementary Properties of the Value Function
  10. Free Boundary Value Problem and Variational Inequality
  11. Surrender and Continuation Premium Representation
  12. Characterization of the Surrender Region
  13. Equivalence of the Optimal Stopping Problems
  14. Numerical Examples
  15. Time-dependent fee function
  16. ...and 2 more sections

Key Result

Lemma 3.1

Figures (2)

  • Figure 1: The continuation region (in red) of the optimal stopping problem with the discontinuous reward function \ref{['eqAmOptVA']}. The value of the variable annuity contract is approximated using the continuous-time Markov chain approximation described in MackayVachonCui2022VaCTMC. Market and VA parameters are $r=0.03$, $\sigma=0.2$, $F_0=G=100$ and $T=15$.
  • Figure 2: Continuation region (in red) of the optimal stopping problem \ref{['eqAmOptVA']}. The value of the variable annuity contract is approximated using the continuous-time Markov chain approximation described in MackayVachonCui2022VaCTMC. Market and VA parameters are $r=0.03$, $\sigma=0.165$, $F_0=G=100$ and $T=10$.

Theorems & Definitions (73)

  • Remark 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Proposition 3.1
  • proof
  • Remark 3.1
  • Example 3.1
  • Example 3.2
  • Remark 3.2
  • ...and 63 more