Regression of continuation values - Diagnostics plots

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 contract fee is state-dependent with base fee 6%.
• 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 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: 109.0692

## Mixed method:  108.3689

## T =  9

## Ex post CV summary:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   1.089   1.089   1.089   1.248   1.181   4.018 
## Predicted CV summary:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   1.070   1.079   1.104   1.248   1.246   3.591 
## T =  8

## Ex post CV summary:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   1.072   1.072   1.072   1.228   1.188   4.111 
## Predicted CV summary:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   1.052   1.061   1.094   1.228   1.239   3.427 
## T =  7

## Ex post CV summary:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   1.056   1.056   1.056   1.205   1.204   4.026 
## Predicted CV summary:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   1.021   1.056   1.081   1.205   1.215   2.937 
## T =  6

## Ex post CV summary:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   1.041   1.041   1.041   1.187   1.197   3.966 
## Predicted CV summary:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.9075  1.0500  1.0690  1.1870  1.1810  2.7820 
## T =  5

## Ex post CV summary:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   1.025   1.025   1.025   1.168   1.184   3.907 
## Predicted CV summary:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   1.017   1.034   1.075   1.168   1.176   3.492 
## T =  4

## Ex post CV summary:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   1.010   1.010   1.010   1.151   1.190   3.849 
## Predicted CV summary:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.9807  1.0190  1.0640  1.1510  1.1800  2.5410 
## T =  3

## Ex post CV summary:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.9949  0.9949  0.9949  1.1340  1.1730  3.7910 
## Predicted CV summary:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.9617  1.0140  1.0640  1.1340  1.1710  2.2810 
## T =  2

## Ex post CV summary:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.9802  0.9802  0.9802  1.1170  1.1560  3.7350 
## Predicted CV summary:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.8904  1.0110  1.0600  1.1170  1.1780  1.6390 
## T =  1

## Ex post CV summary:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.9655  0.9655  0.9655  1.1010  1.1380  3.6790 
## Predicted CV summary:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.9639  1.0270  1.1010  1.1010  1.1680  2.2820