Valuation of capital protection options

TL;DR

This paper introduces an efficient numerical algorithm using Gauss-Hermite quadrature and splines for pricing capital protection options, accounting for optimal policyholder behavior, which significantly impacts the fair fee calculation.

Contribution

It develops a novel Gauss-Hermite quadrature method with spline interpolation for pricing capital protection options under optimal withdrawal policies, improving computational efficiency and accuracy.

Findings

Optimal policyholder behavior increases the fair fee significantly.

Extra fees can reach over 40% at low interest rates and high volatility.

Static withdrawal assumptions underestimate the true cost of capital protection options.

Abstract

This paper presents numerical algorithm and results for pricing a capital protection option offered by many asset managers for investment portfolios to take advantage of market growth and protect savings. Under optimal withdrawal policyholder behaviour the pricing of such a product is an optimal stochastic control problem that cannot be solved using Monte Carlo method. In low dimension case, it can be solved using PDE based methods such as finite difference. In this paper, we develop a much more efficient Gauss-Hermite quadrature method with a one-dimensional cubic spline for calculation of the expectation between withdrawal/reset dates, and a bi-cubic spline interpolation for applying the jump conditions across withdrawal/reset dates. We show results for both static and dynamic withdrawals and for both the asset accumulation and the pension phases (different penalties for any excessive withdrawal) in the retirement investment cycle. To evaluate products with capital protection option, it is common industry practice to assume static withdrawals and use Monte Carlo method. As a result, the fair fee is underpriced if policyholder behaves optimally. We found that extra fee that has to be charged to counter the optimal policyholder behaviour is most significant at smaller interest rate and higher volatility levels, and it is sensitive to the penalty threshold. At low interest rate and a moderate penalty threshold level (15% of the portfolio value per annum) typically set in practice, the extra fee due to optimal withdrawal can be as high as 40% and more on top of the base case of no withdrawal or the case of fixed withdrawals at the penalty threshold.