The Existence of Optimal Bang-Bang Controls for GMxB Contracts

Azimzadeh, Parsiad

doi:10.1137/140953885

Cited by 34 publications

(22 citation statements)

References 23 publications

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“…as an impulse control problem in [12]. Optimal controls for GMWBs with annual withdrawals is considered in [1]. A GMWB is composed of investment and guarantee accounts, (S t ) t≥0 and (A t ) t≥0 , respectively.…”

Section: Convergence Testsmentioning

confidence: 99%

Weakly Chained Matrices, Policy Iteration, and Impulse Control

Azimzadeh¹

2016

SIAM J. Numer. Anal.

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100

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This work is motivated by numerical solutions to Hamilton-Jacobi-Bellman quasivariational inequalities (HJBQVIs) associated with combined stochastic and impulse control problems. In particular, we consider (i) direct control, (ii) penalized, and (iii) semi-Lagrangian discretization schemes applied to the HJBQVI problem. Scheme (i) takes the form of a Bellman problem involving an operator which is not necessarily contractive. We consider the well-posedness of the Bellman problem and give sufficient conditions for convergence of the corresponding policy iteration. To do so, we use weakly chained diagonally dominant matrices, which give a graph-theoretic characterization of weakly diagonally dominant M-matrices. We compare schemes (i)-(iii) under the following examples: (a) optimal control of the exchange rate, (b) optimal consumption with fixed and proportional transaction costs, and (c) pricing guaranteed minimum withdrawal benefits in variable annuities. We find that one should abstain from using scheme (i). Definition 2.1 (Monotone matrix). A real square matrix A is monotone (in the sense of Collatz) if for all real vectorsWe use the following assumptions:Proposition 2.2 (Convergence of policy iteration). Suppose (H0), (H1), and that A(P ) is a monotone matrix for all P in P.(v ) ≥1 defined by Policy-Iteration is nondecreasing and converges to the unique solution v of (2.1). Moreover, if P is finite, convergence occurs in at most |P| iterations (i.e. v |P| = v |P|+1 = · · · ).The monotone convergence of (v ) ≥1 to the unique solution of (2.1) can be proven similarly to Theorem A.3 of Appendix A. See [7, Theorem 2.1] for a proof of the finite termination when P is finite. In practice, P is often finite, in which case (H0) and (H1) are trivial.Remark 2.3. Theorem A.3 establishes the existence and uniqueness of solutions to (2.1) independent of (H1.ii). Owing to this, results that rely on Proposition 2.2 can be relaxed to exclude (H1.ii), with the caveat that when P is infinite, Policy-Iteration be replaced by -Policy-Iteration (see Appendix A). In this case, the resulting sequence (v ) ≥1 is not necessarily nondecreasing. Weakly chained diagonally dominant matricesWe say row i of a complex matrix A := (a ij ) is strictly diagonally dominant (SDD) if |a ii | > j =i |a ij |. We say A is SDD if all of its rows are SDD. Weakly diagonally dominant (WDD) is defined with weak inequality instead. Definition 3.1. A complex square matrix A is said to be a weakly chained diagonally dominant (WCDD) if:(i) A is WDD; (ii) for each row r, there exists a path in the graph of A from r to an SDD row p.Recall that the directed graph of an M × M complex matrix A := (a ij ) is given by the vertices {1, . . . , M } and edges defined as follows: there exists an edge from i to j if and only if a ij = 0. Note that if r is itself an SDD row, the trivial path r → r satisfies the requirement of (ii) in the above.The nonsingularity of WCDD matrices is proven in [27]. We provide an elementary proof for the convenience of the reader: Lemma 3.2....

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Section: Convergence Testsmentioning

confidence: 99%

Weakly Chained Matrices, Policy Iteration, and Impulse Control

Azimzadeh¹

2016

SIAM J. Numer. Anal.

Self Cite

100

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“…The authors of [13] prove the existence of an optimal bang-bang control for GLWB contract when thecontract holder can maximize contract writer's losses by only ever performing non-withdrawal, withdrawal at the contract rate or full surrender. However, they also demonstrate that the related GMWB contract does not satisfy the bang-bang principle other than in certain degenerate cases.…”

Section: Introductionmentioning

confidence: 99%

A Unified Pricing of Variable Annuity Guarantees under the Optimal Stochastic Control Framework

Shevchenko

Luo

2016

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Abstract:In this paper, we review pricing of the variable annuity living and death guarantees offered to retail investors in many countries. Investors purchase these products to take advantage of market growth and protect savings. We present pricing of these products via an optimal stochastic control framework and review the existing numerical methods. We also discuss pricing under the complete/incomplete financial market models, stochastic mortality and optimal/sub-optimal policyholder behavior, and in the presence of taxes. For numerical valuation of these contracts in the case of simple risky asset process, we develop a direct integration method based on the Gauss-Hermite quadratures with a one-dimensional cubic spline for calculation of the expected contract value, and a bi-cubic spline interpolation for applying the jump conditions across the contract cashflow event times. This method is easier to implement and faster when compared to the partial differential equation methods if the transition density (or its moments) of the risky asset underlying the contract is known in closed form between the event times. We present accurate numerical results for pricing of a Guaranteed Minimum Accumulation Benefit (GMAB) guarantee available on the market that can serve as a numerical benchmark for practitioners and researchers developing pricing of variable annuity guarantees to assess the accuracy of their numerical implementation.

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“…In particular, both strategies usually make withdrawals only at a few critical levels, including 0, the contracted withdrawal amount, and a higher level. These patterns are reminiscent of the bang-bang control as analyzed in Azimzadeh and Forsyth (2015) for a guaranteed lifetime withdrawal benefit (GLWB) contract. It is unclear to the authors whether the optimal withdrawals are always in the form of bang-bang controls under the current settings.…”

Section: Withdrawal Strategies When Management Fees Are Presentmentioning

confidence: 97%

The Impact of Management Fees on the Pricing of Variable Annuity Guarantees

Sun

Shevchenko

Fung

2018

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Variable annuities, as a class of retirement income products, allow equity market exposure for a policyholder’s retirement fund with optional guarantees to limit the downside risk of the market. Management fees andguarantee insurance fees are charged respectively for the market exposure and for the protection from the downside risk. We investigate the pricing of variable annuity guarantees under optimal withdrawal strategies when management fees are present. We consider from both policyholder’s and insurer’s perspectives optimal withdrawal strategies and calculate the respective fair insurance fees. We reveal a discrepancy where the fees from the insurer’s perspective can be significantly higher due to the management fees serving as a form of market friction. Our results provide a possible explanation of lower guarantee insurance fees observed in the market than those predicted from the insurer’s perspective. Numerical experiments are conducted to illustrate the results.

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The Existence of Optimal Bang-Bang Controls for GMxB Contracts

Cited by 34 publications

References 23 publications

Weakly Chained Matrices, Policy Iteration, and Impulse Control

Weakly Chained Matrices, Policy Iteration, and Impulse Control

A Unified Pricing of Variable Annuity Guarantees under the Optimal Stochastic Control Framework

The Impact of Management Fees on the Pricing of Variable Annuity Guarantees

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