Narrow Results

In this paper, we analyze the exercise behavior of warrant holders and its impact on warrant values. For this purpose, we propose a parametric model to describing the exercise volume of warrants and calibrate it to exercise data of 40 warrants from the German market. We find that few too-early exercises but also a significant number of too-late exercises occur. This observed exercise behavior results in warrant values that are more than 3% below those under the optimal exercise strategy for at-the-money warrants and the differences are even much higher for in- and out-of-the-money warrants.

There has been considerable interest in developing stochastic volatility and jump-diffusion option pricing models, e.g. Hull and White (1987, Journal of Finance, 42, 281–300) and Merton (1976, Journal of Financial Economics, 3, 125–144). These models, however, have some undesirable aspects that arise from introducing some non-traded sources of risks to the models. Furthermore, the models require much analytical complications; thus, if they are applied to American options then it is not easy to acquire practical implications for hedging and optimal exercise strategies. This paper examines the American option prices and optimal exercise strategies where the volatility of the underlying asset changes over time in a deterministic way. The paper considers two simple cases: monotonically increasing and decreasing volatilities. The discussion of these two simple cases gives useful implications for the possibility of early-exercise and optimal exercise strategies.

This paper examines how forfeiture, vesting, and early exercise affect the value of employee stock options. The forfeiture and exogenous exercise of the options are modeled as two Poisson processes with constant intensity. Rational exercise by the employee due to the option's American feature is modeled as endogenous exercise. The Crank–Nicholson numerical method is used to solve for the resulting value of employee stock options. Results are consistent with the findings that the employee stock options are worth substantially less than the Black–Scholes formula value. The model demonstrates that the option holder would rationally exercise the option early even when the underlying stock pays no dividend. The model also provides results on the expected time to exercise of employee stock options and the expected exercise price relative to strike price. The method used in this paper is proposed as an alternative to the Black–Scholes formula for measuring fair market value of employee stock options for accounting and regulation purposes.

We identify a problem in the widely used binomial option pricing model when it is used to value options on an asset paying continuous dividends. It does not value pairs of European spot and futures options consistently even though they are theoretically equivalent. The inconsistency arises from the way dividend yield is incorporated into the jumps and probabilities. In addition, the model also has the tendency to undervalue American options due to suboptimal early exercise decisions. While the lingering effect of this problem diminishes asymptotically, it is nonetheless a concern for someone just beginning to learn the model or in applications where the use of a sufficiently large binomial tree is not practical or economical. We propose a simple modification to solve the problem and demonstrate the effectiveness of the solution.