Hi, there:
I have been studying a little bit of stochastic calculus and a little bit of time series analysis for a while… and I wondered:
What is the equivalent equation in Stochastic calculus for the Arima Model??
I wondered this, since in the literaure, stock prices are modelled using Geometric Brownian Motion (langevin's equation), but when I reviewed time series analysis, ARIMA model is used to model Stock prices instead (even thought, I have found some stocks that don't seem to follow an arima model). So I wonder…
Is Geometric Brownian Motion the equivalent of ARIMA model in continous time??
Thanks
Stochastic equations model unknown generating processes in continuous time -e.g. evolution of stock prices or a pendulum under noise. By definition, if we observe such systems over a defined time interval in N different experiments - we will record N different sequences of observation. A time series is a particular realization of such a model - one of the N sequences. So essentially, time series is discrete and is usually indexed by integer (discrete time).
When we need to estimate the parameters of a stochastic equation, we need to discretize the differential equation and use an observed time-series for parameter estimations. For simple case like GBM or OU process, this discretization results in familiar models. GBM results in a model with just mean and noise. OU results in an AR(1) process. I am sure one can come up with a stochasic equation whose discrete version will approximate an AR(p,q) model.
When we are interested in forecasting, we need to estimate the parameters and hence need a discrete model - i.e. a time-series model. If we are only interested in forecasting, in fact, we can even stop worrying about the underlying stochastic model and start with the time series model specification itself. So if we are only interested in forecasting, we do not usually use much stochastic equations and calculus. Instead we talk about time-series models like ARMA, GARCH or VECM.
On the other hand, in finance, we heavily rely on stochastic models for derivatives pricing. Here the objective is not forecasting at all. We start with a stochastic model of the underlying, and then for given derivatives payoff (mathematically the boundary conditions), we solve the equation. And then calibrate the model from observed prices to get a working pricing model. This works since we price the derivatives in terms of underlying, so we need not forecast the underlying price (for e.g. the underlying price is an input to the option price itself).