# va_pde_pricer: PDE Pricing of Variable Annuity In valuer: Pricing of Variable Annuities

## Description

`va_pde_pricer` returns the price of a VA with GMAB and GMDB guarantees. The underlying fund is a GBM and the intensity of mortality is deterministic. The fee has a state-dependent structure.

## Usage

 ```1 2 3 4``` ```va_pde_pricer(F0 = 100, r = 0.03, sigma = 0.165, x = 50, delta = 0, fee = 0.01, beta = 150, T = 5, dt = 0.1, dF = 0.1, lambda = function(t) makeham(t, x = 50, A = 1e-04, B = 0.00035, c = 1.075), K = function(t, T) { 0.05 * (1 - t/T)^3 }, Fmax = 500) ```

## Arguments

 `F0` numeric scalar with the initial value of the underlying fund `r` numeric scalar with the constant interest rate `sigma` numeric scalar with the constant volatility `x` numeric integer with the age of the insured `delta` numeric scalar with the roll-up rate of the GMAB and GMDB `fee` numeric scalar with the state-dependent base fee `beta` numeric scalar with the state-dependent barrier It should be greater than `F0`. If set to `Inf` the fee structure becomes constant `T` numeric integer with the maturity of the contract `dt` numeric scalar with the discretization step of the time dimension `dF` numeric scalar with the discretization step for the fund dimension `lambda` function with the intensity of mortality. Default is `makeham` with parameters `x = 50, A = 0.0001, B = 0.00035, c = 1.075` `K` function with the surrender penalty. `Fmax` numeric scalar with the maximum fund value

## Details

This function resolves the PDE in [MK2017] by means of the finite difference implicit method. It requires the package limSolve to be installed.

## Value

numeric scalar with the VA price

## References

1. [MK2017] A. MacKay, M. Augustyniak, C. Bernard, and M.R. Hardy. Risk management of policyholder behavior in equity-linked life insurance. The Journal of Risk and Insurance, 84(2):661-690, 2017. DOI: 10.1111/jori.12094

## Examples

 ```1 2 3 4 5 6``` ```lambda <- function(t) mu(t, x = 50, c1 = 90.43 , c2 = 10.36) K <- function(t, T) {0.05 * ( 1 - t / T)^3} va <- va_pde_pricer(lambda = lambda, K = K, Fmax = 200) va ```

valuer documentation built on May 2, 2019, 3:43 p.m.