Valuing American Derivatives by Least Squares Methods

Least Squares estimators are notoriously known to generate sub-optimal exercise decisions when determining the optimal stopping time. The consequence is that the price of the option is underestimated. We show how variance reduction methods can be implemented to obtain more accurate option prices. We also extend the Longsta¤ and Schwartz (2001) method to price American options under stochastic volatility. These are two important contributions that are particularly relevant for practitioners. Finally, we extend the Glasserman and Yu (2004b) methodology to price Asian options and basket options.

Optimal Martingales and American Option Pricing

Pricing American options is an interesting research topic since there is no analytical solution to value these derivatives. Different numerical methods have been proposed in the literature with some, if not all, either limited to a specific payoff or not applicable to multidimensional cases. Applications of Monte Carlo methods to price American options is a relatively new area that started with Longstaff and Schwartz (2001). Since then, few variations of that methodology have been proposed. The general conclusion is that Monte Carlo estimators tend to underestimate the true option price. The present paper follows Glasserman and Yu (2004b) and proposes a novel Monte Carlo approach, based on designing "optimal martingales" to determine stopping times. We show that our martingale approach can also be used to compute the dual as described in Rogers (2002).

The Greying Church: The Impact of Life Expectancy on Religiosity

Many churches are concerned about older and dwindling congregations. We develop a theoretical framework to explain not only the downward trend in church attendance, but also the increase in the proportion of older people in the congregations. Religiosity depends positively on the expected social and spiritual benefits attached to religious adherence, as well as the probability of entering heaven in the afterlife. While otherworldly compensation in terms of salvation and spiritual benefits motivates religiosity, the costs of formal religion in terms of time allocated to communal activities and foregone income work in the opposite direction. We show that higher life expectancy discounts expected benefits in the afterlife and is hence likely to lead to postponement of religiosity. For this reason, religious organizations should be prepared to attract older members to their congregations, while emphasizing contemporaneous religious benefits to increase overall church attendance.