Counting jumps: does the counting process count?

Abstract

In this paper, we develop a ‘jump diffusion type’ financial model based on renewal processes for the discontinuous part of the risk driver, and study its ability to price options and accurately reproduce the corresponding implied volatility surfaces. In this model, the log-returns process displays finite mean and variance, and non-vanishing skewness and excess kurtosis over a long period. The proposed construction is parsimonious, and it allows for a simple and intuitive pricing formula for European vanilla options; furthermore, it offers an efficient Monte Carlo sample path generator for the pricing of exotic options. We illustrate the performance of the proposed framework using observed market data, and we study the features of the best fitting model specifications.