[While American calls on non-dividend-paying stocks may be valued as European, there is no completely explicit exact solution for the values of American puts. We use a technique called randomization to value American puts and calls on dividend-paying stocks. This technique yields a new semiexplicit approximation for American option values in the Black-Scholes model. Numerical results indicate that the approximation is both accurate and computationally efficient.]

While American calls on non-dividend-paying stocks may be valued as European, there is no completely explicit exact solution for the values of American puts. We use a technique called randomization to value American puts and calls on dividend-paying stocks. This technique yields a new semiexplicit approximation for American option values in the Black-Scholes model. Numerical results indicate that the approximation is both accurate and computationally efficient.

This paper develops static hedges for several exotic options using standard options. The method relies on a relationship between European puts and calls with different strike prices. The analysis allows for constant volatility or for volatility smiles or frowns.

A three parameter stochastic process, termed the variance gamma process, that generalizes Brownian motion is developed as a model for the dynamics of log stock prices. Theprocess is obtained by evaluating Brownian motion with drift at a random time given by a gamma process. The two additional parameters are the drift of the Brownian motion and the volatility of the time change. These additional parameters provide control over the skewness and kurtosis of the return distribution. Closed forms are obtained for the return density and the prices of European options.The statistical and risk neutral densities are estimated for data on the S&P500 Index and the prices of options on this Index. It is observed that the statistical density is symmetric with some kurtosis, while the risk neutral density is negatively skewed with a larger kurtosis. The additional parameters also correct for pricing biases of the Black Scholes model that is a parametric special case of the option pricing model developed here.

ABSTRACT This paper develops static hedges for several exotic options using standard options. The method relies on a relationship between European puts and calls with different strike prices. The analysis allows for constant volatility or for volatility smiles or frowns.