va_sde_engine: General Variable Annuity pricing engine
In valuer: Pricing of Variable Annuities

Description Usage Format Value Methods References Examples

Class providing a variable annuity pricing engine where the underlying reference fund and the intensity of mortality are specified by an arbitrary system of stochastic differential equations. The simulation is done by means of the yuima package.
The value of the VA contract is estimated by means of the Monte Carlo method if the policyholder cannot surrender (the so called "static" approach), and by means of Least Squares Monte Carlo in case the policyholder can surrender the contract (the "mixed" approach).
See References -[BMOP2011] for a description of the mixed and static approaches and the algorithm implemented by this class, [LS2001] for Least Squares Monte Carlo and [YUIMA2014] for yuima.

1	va_sde_engine

R6Class object.

Object of R6Class

new

Constructor method with arguments:

product: A va_product object with the VA product.
financial_parms: A list of parameters specifying the financial processes. See financials_BMOP2011 for an example.
mortality_parms: A list of parameters specifying the demographic processes. See mortality_BMOP2011 for an example.

death_time

Returns the time of death index. If the death doesn't occur during the product time-line it returns the last index of the product time-line plus one.

simulate_financial_paths

Simulates npaths paths of the underlying fund of the VA contract and the discount factors (interest rate) and saves them into private fields for later use.

simulate_mortality_paths

Simulates npaths paths of the intensity of mortality and saves them into private fields for later use.

get_fund

Gets the i-th path of the underlying fund where i goes from 1 to npaths.

do_static

Estimates the VA contract value by means of the static approach (Monte Carlo), see References. It takes as arguments:

the_gatherer: gatherer object to hold the point estimates
npaths: positive integer with the number of paths to simulate
simulate: boolean to specify if the paths should be simulated from scratch, default is TRUE.

do_mixed

Estimates the VA contract by means of the mixed approach (Least Squares Monte Carlo), see References. It takes as arguments:

the_gatherer: gatherer object to hold the point estimates
npaths: positive integer with the number of paths to simulate
degree: positive integer with the maximum degree of the weighted Laguerre polynomials used in the least squares by LSMC
freq: string which contains the frequency of the surrender decision. The default is "3m" which corresponds to deciding every three months if surrendering the contract or not.
simulate: boolean to specify if the paths should be simulated from scratch, default is TRUE.

get_discount

Arguments are i,j. Gets the j-th discount factor corresponding to the i-th simulated path of the discount factors.

fair_fee

Calculates the fair fee for a contract using the bisection method. Arguments are:

fee_gatherer: data_gatherer object to hold the point estimates
npaths: numeric scalar with the number of MC simulations to run
lower: numeric scalar with the lower fee corresponding to positive end of the bisection interval
upper: numeric scalar with the upper fee corresponding to the negative end of the bisection interval
mixed: boolean specifying if the mixed method has to be used. The default is FALSE
tol: numeric scalar with the tolerance of the bisection algorithm. Default is 1e-4
nmax: positive integer with the maximum number of iterations of the bisection algorithm
simulate: boolean specifying if financial and mortality paths should be simulated.

[BMOP2011] Bacinello A.R., Millossovich P., Olivieri A. ,Pitacco E., "Variable annuities: a unifying valuation approach." In: Insurance: Mathematics and Economics 49 (2011), pp. 285-297.
[LS2001] Longstaff F.A. e Schwartz E.S. Valuing american options by simulation: a simple least-squares approach. In: Review of Financial studies 14 (2001), pp. 113-147
[YUIMA2014] Alexandre Brouste, Masaaki Fukasawa, Hideitsu Hino, Stefano M. Iacus, Kengo Kamatani, Yuta Koike, Hiroki Masuda, Ryosuke Nomura, Teppei Ogihara, Yasutaka Shimuzu, Masayuki Uchida, Nakahiro Yoshida (2014). The YUIMA Project: A Computational Framework for Simulation and Inference of Stochastic Differential Equations. Journal of Statistical Software, 57(4), 1-51. URL http://www.jstatsoft.org/v57/i04/.

#Sets up the payoff as a roll-up of premiums with roll-up rate 2%

rate <- constant_parameters$new(0.02)

premium <- 100
rollup <- payoff_rollup$new(premium, rate)

#Five years time-line
begin <- timeDate::timeDate("2016-01-01")
end <- timeDate::timeDate("2020-12-31")

#Age of the policyholder.
age <- 60
# A constant fee of 2% per year (365 days)
fee <- constant_parameters$new(0.02)

#Barrier for a state-dependent fee. The fee will be applied only if
#the value of the account is below the barrier
barrier <- 200
#Withdrawal penalty applied in case the insured surrenders the contract
#It is a constant penalty in this case
penalty <- penalty_class$new(type = 1, 0.02)
#Sets up the contract with GMAB guarantee
contract <- GMAB$new(rollup, t0 = begin, t = end, age = age, fee = fee,
barrier = barrier, penalty = penalty)

#Sets up a gatherer of the MC point estimates
the_gatherer  <- mc_gatherer$new()
no_of_paths <- 10

#Sets up the pricing engine
engine <- va_sde_engine$new(contract, financials_BMOP2011,
mortality_BMOP2011)

#Estimates the contract value by means of the static approach

engine$do_static(the_gatherer, no_of_paths)
the_gatherer$get_results()


#Estimates the contract value by means of the mixed approach
#To compare with the static approach we don't simulate the underlying
#fund paths again.

the_gatherer_2 <- mc_gatherer$new()

engine$do_mixed(the_gatherer_2, no_of_paths, degree = 3, freq = "3m",
simulate = FALSE)
the_gatherer_2$get_results()