Fast and simple method for pricing exotic options using Gauss–Hermite quadrature on a cubic spline interpolation

Luo, Xiaolin; Shevchenko, Pavel V.

doi:10.1142/s2345768614500330

Cited by 15 publications

(10 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…is not known in closed form but one can find its moments, then the integration can also be done with similar efficiency and accuracy by the method of matching moments as described in [7,26]. The method also works very well in the two-dimensional case, see, e.g., [12] where it was applied for GMWB pricing in the case of stochastic interest rate.…”

Section: Direct Integration Methodsmentioning

confidence: 99%

“…This method can be applied when transition density of the underlying asset between the contract cashflow event dates or its moments are known in closed form. We have used this for pricing specific financial derivatives and some simple versions of the VA guarantees in [7,26]. Here, we adapt and extend the method to handle pricing VA riders in general.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Shevchenko

Luo

2016

Risks

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Direct Integration Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Shevchenko

Luo

2016

Risks

Self Cite

View full text Add to dashboard Cite

show abstract

“…Applications to pricing Asian, barrier and other financial derivatives with a single underlying risky and stochastic interest rate are straightforward. Also, it should be possible to extend the algorithm to situations when the underlying bivariate transition density is not known in closed-form but its moments are known, similarly as developed in Luo and Shevchenko (2014) for one-dimensional problems; this is a subject of future research.…”

Section: Discussionmentioning

confidence: 99%

“…This allows us to get very fast and accurate results for prices of a typical GMWB contract on the standard desktop computer. Previously, in a similar spirit, we developed algorithm for the case of one underlying stochastic risky asset and non-stochastic interest rate for pricing exotic options in Luo and Shevchenko (2014) and optimal stochastic control problems for pricing GMWB in Luo and Shevchenko (2015a).…”

Section: Introductionmentioning

confidence: 99%

Valuation of Variable Annuities with Guaranteed Minimum Withdrawal Benefit under Stochastic Interest Rate

Shevchenko¹,

Luo²

2016

Preprint

Self Cite

View full text Add to dashboard Cite

This paper develops an efficient direct integration method for pricing of the variable annuity (VA) with guarantees in the case of stochastic interest rate. In particular, we focus on pricing VA with Guaranteed Minimum Withdrawal Benefit (GMWB) that promises to return the entire initial investment through withdrawals and the remaining account balance at maturity. Under the optimal (dynamic) withdrawal strategy of a policyholder, GMWB pricing becomes an optimal stochastic control problem that can be solved using backward recursion Bellman equation. Optimal decision becomes a function of not only the underlying asset but also interest rate. Presently our method is applied to the Vasicek interest rate model, but it is applicable to any model when transition density of the underlying asset and interest rate is known in closed-form or can be evaluated efficiently.Using bond price as a numéraire the required expectations in the backward recursion are reduced to two-dimensional integrals calculated through a high order Gauss-Hermite quadrature applied on a two-dimensional cubic spline interpolation. The quadrature is applied after a rotational transformation to the variables corresponding to the principal axes of the bivariate transition density, which empirically was observed to be more accurate than the use of Cholesky transformation. Numerical comparison demonstrates that the new algorithm is significantly faster than the partial differential equation or Monte Carlo methods. For pricing of GMWB with dynamic withdrawal strategy, we found that for positive correlation between the underlying asset and interest rate, the GMWB price under the stochastic interest rate is significantly higher compared to the case of deterministic interest rate, while for negative correlation the difference is less but still significant. In the case of GMWB with predefined (static) withdrawal strategy, for negative correlation, the difference in prices between stochastic and deterministic interest rate cases is not material while for positive correlation the difference is still significant. The algorithm can be easily adapted to solve similar stochastic control problems with two stochastic variables possibly affected by control. Application to numerical pricing of Asian, barrier and other financial derivatives with a single risky asset under stochastic interest rate is also straightforward.

show abstract

“…For convenience, hereafter we refer this new algorithm as GHQC (Gauss-Hermite quadrature on cubic spline). We adopt the method developed in Luo and Shevchenko (2014) for pricing American options and extend it to solving optimal stochastic control problem for pricing GMWB variable annuity. This allows to get virtually instant results for typical GMWB annuity prices on the standard desktop PC.…”

Section: Introductionmentioning

confidence: 99%

Fast numerical method for pricing of variable annuities with guaranteed minimum withdrawal benefit under optimal withdrawal strategy

Luo

Shevchenko

2015

J. Finan. Eng.

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Fast and simple method for pricing exotic options using Gauss–Hermite quadrature on a cubic spline interpolation

Cited by 15 publications

References 18 publications

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

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

Valuation of Variable Annuities with Guaranteed Minimum Withdrawal Benefit under Stochastic Interest Rate

Fast numerical method for pricing of variable annuities with guaranteed minimum withdrawal benefit under optimal withdrawal strategy

Contact Info

Product

Resources

About