We consider the problem of evaluating variable annuities with a guaranteed minimum withdrawal benefit under a regime-switching model. We propose a trinomial lattice model to approximate the evolution of the investment fund value and the policy value at inception is computed through a backward induction scheme. Finally, the insurance fee is computed as the solution of the equation that makes the contract actuarially fair. Numerical results are reported to illustrate the consistency of the proposed model.

This paper provides an econometric analysis aiming at evidencing the dynamics showed by the S&P 500 market index during the period of 4 January 2001–28 April 2020, in which the subprime crisis has taken place and the COVID-19 crisis has begun. In particular, we fit a three-regime switching model that allows market parameters to behave differently during economic downturns, with the regimes representative of the tranquil, volatile, and turbulent states. We document that the tranquil regime is the most frequent for the whole period, while the dominant regime is the volatile one for the crisis of 2008 and the turbulent one for the first four months of 2020. We fit the same model to the returns of the Dow Jones Industrial Average index and find that during the same period of investigation, the most frequent regime has been the tranquil one, while the volatile and turbulent regimes share the same frequencies. Additionally, we use a multinomial logit model to describe the probabilities of volatile or turbulent regimes. We show that, in the case of the S&P 500 index, the returns from the Volatility Index (VIX) index are significant for both the volatile and the turbulent regimes, while the gold, WTI oil, and the dollar indices have some explanatory power only for the turbulent regime.

One of the most common sources of path dependency in derivatives arises when the volatility is stochastic. This is apparent in the basic binomial model, where time-varying volatility causes the lattice to splinter rather than recombine, leading to n² different nodes at the nth time step instead of n + 1 nodes in a tree that recombines. Various methods have been developed to deal with that problem within a lattice framework, by constructing three-dimensional lattices with both the price and the volatility as state variables. An alternative technique is the forward shooting grid that builds a lattice for the stock price and carries along a set of possible values for the volatility at each price node as auxiliary variables. But both of those approaches can run into problems with negative transition probabilities and difficulty in achieving the right correlation between returns and volatility changes. In this article, Costabile, Massabó, and Russo develop a different forward shooting grid approach, in which the squared volatility is the primary path-dependent variable and stock prices are carried along as the auxiliary variables. Negative transition probabilities are avoided, and the procedure produces highly accurate results very efficiently in a compact tree.