Pricing Equity Index Annuities with Surrender Options in Four Models

Equity-indexed annuities (EIAs) have generated a great deal of interest since Keyport Life first launched "Key Index" in February 1995. They are considered to be the most innovative products to appear on the market in years. EIAs are, essentially equity-linked deferred annuities which provide the policyholder with a guaranteed accumulation rate on their premium, and also at maturity, benefits from an additional return based on the performance of an equity mutual fund or a family of mutual funds or a stock index, typically of the Standard and Poor's (S&P500) index, so the customer can profit from the growth of the stock market. Product designs of EIAs can vary, depending on the companies that sell them. In this article, we focus on the pricing of one of the product designs in the market which gives the possibility to surrender their policy before maturity. Such options can be valued using Monte Carlo simulation method proposed for pricing American options but, here, we use the least squares Monte Carlo suggested by Longstaff and Schwartz added to the control variate tool in order to construct efficient estimators. We analyze these equity-linked life insurance contracts under four different models. The frameworks differ from the way we model the price of the fund associated with the contract. The first setting is the usual Black and Scholes model, the second is the environment of jump diffusions, especially a Kou process, the third is the regime switching log normal model (RSLN) developed by Hamilton and the fourth is a mixed of a regime switching and a jump diffusion. The surrender option is priced using the Longstaff and Schwartz methodology.