Residual histograms

In the following example we’re going to price a VA contract with GMAB and GMDB riders:

• Expiry T = 5 (years).
• The initial single premium is 100.
• The guaranteed minimum for both GMAB and GMDB is the roll up of premium with rate 2%.
• The base fee is 6% constant. We’re going to compare constant and state-dependent fees.
• The state-dependent barrier is set to be the roll-up of premium at time T= 5 (years).

With regards to the simulations and LSMC algorithm:

• The number of Monte Carlo simulations will be 1000.
• Constant interest rate r = 0.03
• Underlying fund is a GBM with volatility 0.2
• Intensity of mortality is given by the Weibull mortality function with c1=90.43, c2=10.36
• Within the mixed method, the surrender decision is taken every 6 months so 9 times in total. The LSMC works moving backwards from T = 9 to T = 1.

For each decision time, the histogram of the standardized residuals from the state-dependent case will be plotted next to the one from constant fee case. The standard normal curve is also plotted for reference.

## State-dependent

## Static method: 109.0692

## Mixed method:  108.3689

## Constant

## Static method: 101.8561

## Mixed method:  104.5796

## T =  9

## T =  8

## T =  7

## T =  6

## T =  5

## T =  4

## T =  3

## T =  2

## T =  1