Model risks in the valuation of equity indexed annuities

Abstract

Over last several decades, a lot of work has been done to incorporate various features of asset price movements into stochastic models, including (but not limited to) stochastic volatility of stock price, randomness of interest rate and the correlation between stock return and return volatility (i.e. the leverage effect). This thesis presents an empirical study in the pricing errors for equity indexed annuities (EIAs) arising from the use of different interest rate models and volatility models. For the interest rate risk, I will inspect the pricing differences between the use of the stochastic volatility model of Heston (1993) and the use a combination of the Heston model with the stochastic interest rate model of Cox, Ingersoll and Ross (1985). The classical Black-Scholes (1973) model will also be compared against its combination with the extended Vasicek model due to Hull and White (1990). Unlike most equity options in the literature, EIAs typically have moderate to long maturities. While in the valuing the former one may consider the interest rate as deterministic, this may not be the case for pricing EIAs. This part of the thesis is a partial attempt to answer this question. For volatility risks, although the number of existing volatility models is vast, most of them are impractical for EIA valuation. This is because the EIAs are Bermudan options (owing to the presence of surrender terms) but most popular volatility models simply do not admit closed form or semi-closed form formulas for the densities or characteristic functions of the stock return conditional on the initial and final volatilities. To solve this problem, I adopt the finite-state, continuous-time regime switching Levy stock return model proposed by Chourdakis (2004). This model also has the merit that the leverage effect can be built in easily. However, the model was initially intended to be used as an approximation to Heston's (1993) model. As such, it is not risk-neutral. In addition, for general time-changed Levy processes, that how to find an equivalent martingale measure is currently a very confusing-issue in the literature. In this thesis I modify Chourdakis' model so that it is used directly as the model of, but not an approximation to, a physical process. A way to obtain a structure-preserving equivalent martingale measure is also proposed. Given the conditional density or characteristic function for a regime switching model, one can compute Bermudan option prices by using the sequential quadrature method developed by Sullivan (2000a,b). We will explain the idea of this method and give a convergence proof to it in this thesis.