Displaced Jump-Diffusion Option Valuation

Abstract

The lognormal diffusion process was the most convenient assumption Black and Scholes could make to capture the general features of stock price movements; it allows stochastic evolution of returns in continuous-time, and stock prices are bounded below by zero. But once we gained more empirical knowledge about returns distributions and observed the persistent volatility skew in real world option prices, alternative processes such as jump-diffusions were introduced. In this article, Câmara, Krehbiel, and Li note that, unlike an individual stock, a stock index will have a minimum value that is strictly positive, because a component stock whose price is going towards zero will be replaced in the index by a different stock. Thus a stock index should follow a displaced jump-diffusion. With this assumption, the authors derive an option pricing formula and test it on 10 years of S&P 500 Index and index option data. The model’s behavior with regard to implied jump intensity and frequency, the shape of the volatility skew, and the existence of a positive lower bound on the index are quite plausible and the goodness of fit is better than either Rubinstein’s displaced diffusion model (without jumps) or Merton’s jump-diffusion model (with lower bound of zero).