PDE approach

Contract value of a VA (GMAB and GMDB riders) with state-Dependent Fee and Fee Barrier Threshold of 150 I used my implementation of the implicit method to resolve the PDE in MacKay et Al, DOI: 10.1111/jori.120942017

Mortality follows Makeham’s law with parameters A = 0.0001,B = 0.00035, and c = 1.075 The underlying fund is modeled with a GBM with volatility 0.165 and interest rate r = 0.03. The penalty function is exponential = function(t, T) 1 - exp(-0.008 * (10 - min(t, 10))) Initial fund value is F0 = 100 Contract expires in T = 10 years The insured is 50 years old With these assumptions MacKay et Al. calculate the fair fee to be 0.0167

## [1] "Fee:  0.0167  Age:  50  =>  100.013"

Monte Carlo approach

I then calculated the contract value for the same contract and market parameters by means of my implementation of the algorithms proposed in Bacinello et Al. 2011. The number of paths used is 20000.

## [1] "Mixed without adjustment:  99.2231"                       
## [2] "Mixed without adjustment using GLM:  99.2109"             
## [3] "Mixed without adjustment using Ridge regression:  99.2109"
## [4] "Mixed without adjustment using the Lasso:  99.2109"       
## [5] "Mixed with  adjustment using Least Squares:  99.9106"     
## [6] "Static method:  99.577"