17 research outputs found
Numerical Methods for Pricing a Guaranteed Minimum Withdrawal Benefit (GMWB) as a Singular Control Problem
Guaranteed Minimum Withdrawal Benefits(GMWB) have become popular riders on variable annuities. The pricing of a GMWB contract was originally formulated as a singular stochastic control problem which results in a Hamilton Jacobi Bellman (HJB) Variational Inequality (VI). A penalty method method can then be used to solve the HJB VI. We present a rigorous proof of convergence of the penalty method to the viscosity solution of the HJB VI assuming the underlying asset follows a Geometric Brownian Motion. A direct control method is an alternative formulation for the HJB VI. We also extend the HJB VI to the case of where the underlying asset follows a Poisson jump diffusion.
The HJB VI is normally solved numerically by an implicit method, which gives rise to highly nonlinear discretized algebraic equations. The classic policy iteration approach works well for the Geometric Brownian Motion case. However it is not efficient in some circumstances such as when the underlying asset follows a Poisson jump diffusion process. We develop a combined fixed point policy iteration scheme which significantly increases the efficiency of solving the discretized equations. Sufficient conditions to ensure the convergence of the combined fixed point policy iteration scheme are derived both for the penalty method and direct control method.
The GMWB formulated as a singular control problem has a special structure which results in a block matrix fixed point policy iteration converging about one order of magnitude faster than a full matrix fixed point policy iteration. Sufficient conditions for convergence of the block matrix fixed point policy iteration are derived. Estimates for bounds on the penalty parameter (penalty method) and scaling parameter (direct control method) are obtained so that convergence of the iteration can be expected in the presence of round-off error
Fast Numerical Method for Pricing of Variable Annuities with Guaranteed Minimum Withdrawal Benefit under Optimal Withdrawal Strategy
A variable annuity contract with Guaranteed Minimum Withdrawal Benefit (GMWB)
promises to return the entire initial investment through cash withdrawals
during the policy life plus the remaining account balance at maturity,
regardless of the portfolio performance. Under the optimal withdrawal strategy
of a policyholder, the pricing of variable annuities with GMWB becomes an
optimal stochastic control problem. So far in the literature these contracts
have only been evaluated by solving partial differential equations (PDE) using
the finite difference method. The well-known Least-Squares or similar Monte
Carlo methods cannot be applied to pricing these contracts because the paths of
the underlying wealth process are affected by optimal cash withdrawals (control
variables) and thus cannot be simulated forward in time. In this paper we
present a very efficient new algorithm for pricing these contracts in the case
when transition density of the underlying asset between withdrawal dates or its
moments are known. This algorithm relies on computing the expected contract
value through a high order Gauss-Hermite quadrature applied on a cubic spline
interpolation. Numerical results from the new algorithm for a series of GMWB
contract are then presented, in comparison with results using the finite
difference method solving corresponding PDE. The comparison demonstrates that
the new algorithm produces results in very close agreement with those of the
finite difference method, but at the same time it is significantly faster;
virtually instant results on a standard desktop PC
Valuation of Variable Annuities with Guaranteed Minimum Withdrawal and Death Benefits via Stochastic Control Optimization
In this paper we present a numerical valuation of variable annuities with
combined Guaranteed Minimum Withdrawal Benefit (GMWB) and Guaranteed Minimum
Death Benefit (GMDB) under optimal policyholder behaviour solved as an optimal
stochastic control problem. This product simultaneously deals with financial
risk, mortality risk and human behaviour. We assume that market is complete in
financial risk and mortality risk is completely diversified by selling enough
policies and thus the annuity price can be expressed as appropriate
expectation. The computing engine employed to solve the optimal stochastic
control problem is based on a robust and efficient Gauss-Hermite quadrature
method with cubic spline. We present results for three different types of death
benefit and show that, under the optimal policyholder behaviour, adding the
premium for the death benefit on top of the GMWB can be problematic for
contracts with long maturities if the continuous fee structure is kept, which
is ordinarily assumed for a GMWB contract. In fact for some long maturities it
can be shown that the fee cannot be charged as any proportion of the account
value -- there is no solution to match the initial premium with the fair
annuity price. On the other hand, the extra fee due to adding the death benefit
can be charged upfront or in periodic instalment of fixed amount, and it is
cheaper than buying a separate life insurance.Comment: arXiv admin note: substantial text overlap with arXiv:1410.860
The valuation of GMWB variable annuities under alternative fund distributions and policyholder behaviours
In this paper, we present a dynamic programming algorithm for pricing variable annuities with Guaranteed Minimum Withdrawal Benefits (GMWB) under a general Lévy processes framework. The GMWB gives the policyholder the right to make periodical withdrawals from her policy account even when the value of this account is exhausted. Typically, the total amount guaranteed for withdrawals coincides with her initial investment, providing then a protection against downside market risk. At each withdrawal date, the policyholder has to decide whether, and how much, to withdraw, or to surrender the contract. We show how different policyholder’s withdrawal behaviours can be modelled. We perform a sensitivity analysis comparing the numerical results obtained for different contractual and market parameters, policyholder behaviours and different types of Lévy processes
Optimal initiation of a GLWB in a variable annuity: no arbitrage approach
This paper offers a financial economic perspective on the optimal time (and
age) at which the owner of a Variable Annuity (VA) policy with a Guaranteed
Living Withdrawal Benefit (GLWB) rider should initiate guaranteed lifetime
income payments. We abstract from utility, bequest and consumption preference
issues by treating the VA as liquid and tradable. This allows us to use an
American option pricing framework to derive a so-called optimal initiation
region. Our main practical finding is that given current design parameters in
which volatility (asset allocation) is restricted to less than 20%, while
guaranteed payout rates (GPR) as well as bonus (roll-up) rates are less than
5%, GLWBs that are in-the-money should be turned on by the late 50s and
certainly the early 60s. The exception to the rule is when a non-constant GPR
is about to increase (soon) to a higher age band, in which case the optimal
policy is to wait until the new GPR is hit and then initiate immediately. Also,
to offer a different perspective, we invert the model and solve for the bonus
(roll-up) rate that is required to justify delaying initiation at any age. We
find that the required bonus is quite high and more than what is currently
promised by existing products. Our methodology and results should be of
interest to researchers as well as to the individuals that collectively have
over \$1 USD trillion in aggregate invested in these products. We conclude by
suggesting that much of the non-initiation at older age is irrational (which
obviously benefits the insurance industry.
Impulse Control in Finance: Numerical Methods and Viscosity Solutions
The goal of this thesis is to provide efficient and provably convergent
numerical methods for solving partial differential equations (PDEs) coming from
impulse control problems motivated by finance. Impulses, which are controlled
jumps in a stochastic process, are used to model realistic features in
financial problems which cannot be captured by ordinary stochastic controls.
The dynamic programming equations associated with impulse control problems
are Hamilton-Jacobi-Bellman quasi-variational inequalities (HJBQVIs) Other than
in certain special cases, the numerical schemes that come from the
discretization of HJBQVIs take the form of complicated nonlinear matrix
equations also known as Bellman problems. We prove that a policy iteration
algorithm can be used to compute their solutions. In order to do so, we employ
the theory of weakly chained diagonally dominant (w.c.d.d.) matrices. As a
byproduct of our analysis, we obtain some new results regarding a particular
family of Markov decision processes which can be thought of as impulse control
problems on a discrete state space and the relationship between w.c.d.d.
matrices and M-matrices. Since HJBQVIs are nonlocal PDEs, we are unable to
directly use the seminal result of Barles and Souganidis (concerning the
convergence of monotone, stable, and consistent numerical schemes to the
viscosity solution) to prove the convergence of our schemes. We address this
issue by extending the work of Barles and Souganidis to nonlocal PDEs in a
manner general enough to apply to HJBQVIs. We apply our schemes to compute the
solutions of various classical problems from finance concerning optimal control
of the exchange rate, optimal consumption with fixed and proportional
transaction costs, and guaranteed minimum withdrawal benefits in variable
annuities
Impulse Control in Finance: Numerical Methods and Viscosity Solutions
The goal of this thesis is to provide efficient and provably convergent
numerical methods for solving partial differential equations (PDEs) coming from
impulse control problems motivated by finance. Impulses, which are controlled
jumps in a stochastic process, are used to model realistic features in
financial problems which cannot be captured by ordinary stochastic controls.
The dynamic programming equations associated with impulse control problems
are Hamilton-Jacobi-Bellman quasi-variational inequalities (HJBQVIs) Other than
in certain special cases, the numerical schemes that come from the
discretization of HJBQVIs take the form of complicated nonlinear matrix
equations also known as Bellman problems. We prove that a policy iteration
algorithm can be used to compute their solutions. In order to do so, we employ
the theory of weakly chained diagonally dominant (w.c.d.d.) matrices. As a
byproduct of our analysis, we obtain some new results regarding a particular
family of Markov decision processes which can be thought of as impulse control
problems on a discrete state space and the relationship between w.c.d.d.
matrices and M-matrices. Since HJBQVIs are nonlocal PDEs, we are unable to
directly use the seminal result of Barles and Souganidis (concerning the
convergence of monotone, stable, and consistent numerical schemes to the
viscosity solution) to prove the convergence of our schemes. We address this
issue by extending the work of Barles and Souganidis to nonlocal PDEs in a
manner general enough to apply to HJBQVIs. We apply our schemes to compute the
solutions of various classical problems from finance concerning optimal control
of the exchange rate, optimal consumption with fixed and proportional
transaction costs, and guaranteed minimum withdrawal benefits in variable
annuities
Optimal behavior strategy in the GMIB product
Guaranteed Minimum Income benefit are variable annuities contract, which offer the policyholder the possibility to con- vert the guarantee level into an annuities income for life. This paper focuses on the optimal customer behavior assuming the maximization of the discounted expected future cash flows over the full life of the contract duration. Using convenient scaling properties of the contract value enables to reduce the complexity (dimension) of the problem and to characterize the policyholder’s decision as a function of the contract moneyness across four main choices: zero withdrawals, guaranteed withdrawals, lapse and the income period election. Sensitivities to key drivers such as the market volatility, the interest rate and the roll-up rate illustrate how crucial are not only the environment, but also the product design features, in order to ensure a fair and robust pricing for both customer and life insurer. In particular, the authors find that most empirical contracts are usually underpriced compared to mean optimal behavior pricing, which empirically translated into multiple updates of behavior assumptions and re-reserving by life insurers in the recent years