Jump diffusion models and the evolution of financial prices

Abstract

We analyze a stochastic model to describe the evolution of financial prices. We consider the stochastic term as a sum of the Wiener noise and a jump process. We point to the effects of the jumps on the return time evolution, a central concern of the econophysics literature. The presence of jumps suggests that the process can be described by an infinitely divisible characteristic function belonging to the De Finetti class. We then extend the De Finetti functions to a generalized nonlinear model and show the model to be capable of explaining return behavior.

On the time-homogeneous Ornstein–Uhlenbeck process in the foreign exchange rates

“…Thus, Eq. (3) belongs to the De Finetti class of infinitely divisible characteristic functions [14].…”

“…17, we may infer that the rand-dollar rate returns are not linear or time-homogeneous because its cumulants do not reach an equilibrium state as t increases. If linearity assumption does not hold for these data, the type of diffusion equation may not be appropriate [15,16].…”

“…Continuous-time stochastic models have been shown to be useful in modeling the behavior of financial time series, such as exchange rates and stock prices [15,18]. As an instance, volatility can be modeled at different observational frequencies to derive an option pricing formula.…”