Standard mixed approach

In the following example we’re going to price VA contracts with GMAB and GMDB riders with the following features:

• 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 fees are state-dependent.
• The state-dependent barrier is set to be the roll-up of premium at time T= 5 (years).

Contract values corresponding to a base fee ranging from 6% to 12% will be estimated by means of both the static and mixed approaches. The number of Monte Carlo simulations will be 20000.

## Fee:  0.06 
## Static: 107.4425 
## Mixed:  105.9045 
## Fee:  0.07 
## Static: 106.284 
## Mixed:  104.6254 
## Fee:  0.08 
## Static: 105.1618 
## Mixed:  103.3781 
## Fee:  0.09 
## Static: 104.0878 
## Mixed:  102.2999 
## Fee:  0.1 
## Static: 103.0511 
## Mixed:  101.3848 
## Fee:  0.11 
## Static: 102.0823 
## Mixed:  100.655 
## Fee:  0.12 
## Static: 101.287 
## Mixed:  100.1511 
## Fee:  0.13 
## Static: 100.4824 
## Mixed:  99.77228 
## Fee:  0.14 
## Static: 99.79763 
## Mixed:  99.45206 
## Fee:  0.15 
## Static: 99.20684 
## Mixed:  99.2208 
## Fee:  0.16 
## Static: 98.74772 
## Mixed:  98.99112 
## Fee:  0.17 
## Static: 98.29769 
## Mixed:  98.77757 
## Fee:  0.18 
## Static: 97.9244 
## Mixed:  98.59506

Adjusted mixed approach

The contract values are now estimated by means of a mixed approach with a tweak. In this case, at each step the linear regression in LSMC will be done only for paths where the GMAB guarantee is out of the money.

## Fee:  0.06 
## Mixed:  106.4116 
## Fee:  0.07 
## Mixed:  105.1883 
## Fee:  0.08 
## Mixed:  104.0934 
## Fee:  0.09 
## Mixed:  103.0915 
## Fee:  0.1 
## Mixed:  102.1724 
## Fee:  0.11 
## Mixed:  101.3135 
## Fee:  0.12 
## Mixed:  100.5289 
## Fee:  0.13 
## Mixed:  99.83394 
## Fee:  0.14 
## Mixed:  99.20233 
## Fee:  0.15 
## Mixed:  98.70446 
## Fee:  0.16 
## Mixed:  98.36157 
## Fee:  0.17 
## Mixed:  98.02577 
## Fee:  0.18 
## Mixed:  97.68262

Conclusions

With the adjusted mixed approach the contract values get closer to the static ones but remain below. In addition, they stay below even for high fee values while with the original method they eventually become greater as they should.