Pricing variable annuity with surrender guarantee
A variable annuity with guaranteed sum \(G_t = G e^{g t}\) at time \(t\) is invested in a fund modelled as a Black-Scholes market \[d S_t = r S_t \; d t + \sigma S_t \;d W_t\;,\] where all processes are considered under the pricing measure. A fixed part of the wealth is taken for the management of the fund, so that the fund follows \[d F_t = (r-c) F_t \; d t + \sigma F_t \;d W_t\;.\] At maturity, the policy holder will get \(\max\{F_T,G_T\}\). Moreover, the policy holder has the possibility to surrender. Three possible surrender values are considered: (1) \(e^{-\kappa(T-t)} \max\{F_t, G_t\}\), (2) \(e^{-\kappa(T-t)} F_t\), (3) \(e^{-\kappa(T-t)} G_t\). The corresponding optimal stopping theorems are formulated, the variational inequalities are given and the continuation and surrender regions are defined. The main theorem shows that the value function can be split into an European part and an early surrender premium part. The European part is given explicitly, the early surrender part in terms of the boundaries between the continuation and surrender regions. Several figures illustrate the regions.
scientific article; zbMATH DE number 7337484
scientific article