Time changes for Lévy processes (Q2707163)

Such assumptions as continuity of price processes have long served finance as a convenient and powerful approach, delivering market completeness and unique pricing of derivatives by arbitrage. They are crucial to the validity of some option pricing theories and the associated dynamic hedging strategy. The goal of this paper is to consider instead pure jump Lévy processes of finite variation with an infinite arrival rate of jumps as models for the logarithm of asset prices. These processes replace the role of Brownian motion in the classical models and the resulting price process can be expressed as the difference between two increasing random processes that account for the upward and downward moves of the market. Thus, the proposed processes are both purely discontinuous and, unlike Brownian motion, of finite variation.NEWLINENEWLINENEWLINEThe authors argue that price processes, being semimartingales are time-changed Brownian motion. They observe that as time changes are increasing random processes they are for practical purposes purely discontinuous if they are not locally deterministic. This leads them to consider purely discontinuous models for the prices of financial assets. By various examples they show that one may generally relate this time change to a measure of price moves. The specific time changes are Poisson processes, gamma processes, general subordinators, and the inverse local time of Brownian motion at zero. In each case the authors exhibit the price process as the above-mentioned difference between two increasing processes and show how the price process may be viewed as Brownian motion evaluated at a random time that is related to the price moves. They also provide numerous examples of potentially empirically relevant Lévy processes with closed-form expressions for their Lévy densities and characteristic functions. The resulting models are tractable for both option pricing and statistical estimation.