This paper develops a new pricing model for American-style indexed executive stock options. We rely on a basic model framework and an indexation scheme first proposed by Johnson and Tian (2000a) in their analysis of European-style indexed options. Our derivation of the valuation formula represents an instructive example of the usefulness of the change-of-numeraire technique. In the paper's numerical section we implement the valuation formula and demonstrate that not only may the early exercise premium be significant but also that the delta of the American-style option is typically much larger than the delta of the otherwise identical (value-matched) European-style option. Vega is higher for indexed options than for conventional options but largely independent of whether the options are European- or American-style. This has important implications for the design of executive compensation contracts. We finally extend the analysis to cover the case where the option contracts are subject to delayed vesting . We show that for realistic parameter values, delayed vesting leads only to a moderate reduction in the value of the American-style indexed executive stock option.

In the European Union financial regulation requires that life and pension (L&P) companies use the Smith and Wilson (2000) model for the term structure of risk-free interest rates when valuing their liabilities and long term guarantees. Some key features of this model are that it allows for a perfect fit to market observed bond prices, and that its extrapolated long rates converge towards a constant level, the Ultimate Forward Rate (UFR). Both this level and the rate at which convergence towards it takes place are directly specified via parameters of the model. Since the Smith–Wilson model is not one of finance theory's standard term structure models, we introduce the model and summarize its most important mathematical properties. We also describe how the European Solvency II regulation came to embrace this particular model. The paper moves on to document how the regulation also imposes quite detailed and tight restrictions on how the Smith–Wilson model should be parameterized and applied. We argue that many of these implementation instructions – one of which is the regulator's specification of a very high UFR – seem biased in the same direction and that this could indicate a systematic attempt to “lift” the term structure curve up and away from its true location whereby artificially high discount rates are induced. The result of the bias is not only significant undervaluation of L&P liabilities but also a peculiar contradiction of the Solvency II overall objective of enhancing financial stability and of protecting policyholders via the promotion of economic valuation in accordance with market consistent principles. The paper's analysis is accompanied by valuation illustrations based on data on the liability composition of an actual medium-sized Danish pension fund. The results suggest that the undervaluation of liabilities resulting from use of the regulation compliant Smith–Wilson model can be massive compared to results obtained from some alternative and more freely calibrated models.

This paper introduces, prices, and analyzes traffic light options. The traffic light option is an innovative structured OTC derivative developed independently by several London-based investment banks to suit the needs of Danish life and pension (L&P) companies, which must comply with the traffic light solvency stress test system introduced by the Danish Financial Supervisory Authority (DFSA) in June 2001. This monitoring system requires L&P companies to submit regular reports documenting the sensitivity of the companies’ base capital to certain pre-defined market shocks – the red and yellow light scenarios. These stress scenarios entail drops in interest rates as well as in stock prices, and traffic light options are thus designed to pay off and preserve sufficient capital when interest rates and stock prices fall simultaneously. Sweden’s FSA implemented a traffic light system in January 2006, and supervisory authorities in many other European countries have implemented similar regulation. Traffic light options are therefore likely to attract the attention of a wider audience of pension fund managers in the future. Focusing on the valuation of the traffic light option we set up a Black–Scholes/Hull–White model to describe stock market and interest rate dynamics, and analyze the traffic light option in this framework.