  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.
