Regression of continuation values - Diagnostics plots - CST fee

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 fee is 6% constant.

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 10 times in total. The LSMC works moving backwards from T = 10 to T = 1.

For each decision time, the Standardized Residual Plot will be printed as well as an histogram of the predicted vs ex-post continuation values. The normal density which should be fitting the ex post CV values is also plotted on the latter.

## Static method: 102.1208

## Mixed method:  104.0714

## T =  9

## Ex post CV summary:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   1.089   1.089   1.089   1.168   1.089   3.025 
## Predicted CV summary:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   1.020   1.085   1.095   1.168   1.141   2.782 
## T =  8

## Ex post CV summary:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   1.072   1.072   1.072   1.152   1.072   3.191 
## Predicted CV summary:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   1.065   1.069   1.080   1.152   1.143   2.815 
## T =  7

## Ex post CV summary:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   1.056   1.056   1.056   1.139   1.056   3.269 
## Predicted CV summary:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.9931  1.0590  1.0650  1.1390  1.1280  2.6150 
## T =  6

## Ex post CV summary:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   1.041   1.041   1.041   1.123   1.048   2.817 
## Predicted CV summary:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.9838  1.0460  1.0570  1.1230  1.1130  2.5230 
## T =  5

## Ex post CV summary:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   1.025   1.025   1.025   1.109   1.052   2.775 
## Predicted CV summary:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   1.002   1.032   1.051   1.109   1.108   2.459 
## T =  4

## Ex post CV summary:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   1.010   1.010   1.010   1.098   1.137   2.734 
## Predicted CV summary:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   1.000   1.016   1.043   1.098   1.117   1.947 
## T =  3

## Ex post CV summary:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.9949  0.9949  0.9949  1.0830  1.1370  2.1260 
## Predicted CV summary:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.9773  1.0080  1.0400  1.0830  1.1070  1.7820 
## T =  2

## Ex post CV summary:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.9802  0.9802  0.9802  1.0680  1.1270  2.0180 
## Predicted CV summary:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.9843  1.0010  1.0390  1.0680  1.1030  1.8840 
## T =  1

## Ex post CV summary:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.9655  0.9655  0.9655  1.0550  1.1570  1.9880 
## Predicted CV summary:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.9746  1.0040  1.0390  1.0550  1.0900  1.4950