Time series modeling in finance

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See also: Time series introduction, Geometric Brownian Motion models, Autoregressive models, Insurance and finance risk analysis modeling introduction

There are a couple of commonly used time series for financial models of variables like stock prices, exchange rates, interest rates and economic indicators like producers' price index (PPI) and gross domestic product (GDP). Although these were originally developed for financial markets they have much wider applications.

The Time Series Fit window

Financial time series are considered to vary continuously even if perhaps we only observe them at certain moments in time. They are based on stochastic differential equations (SDEs), which are the most general descriptions of the random system describing a continuous evolution of the variable. The problem with SDEs from a simulation perspective is that they are not always amenable to being exactly converted to algorithms that will generate random possible observations at specific moments in time, and there are often no exact methods for estimating their parameters from data. On the other hand, the advantage is that we have a consistent framework for comparing the time series and there are sometimes analytical solution available to us for determining, say, the probability that the variable exceeds some value at a certain point in time - answers that are useful for pricing derivatives and other financial instruments, for example.

Financial time series model a variable in one of two forms: the actual price of the stock (or the valuable of the variable like exchange rate, interest rate, etc if it is not a stock) at some time t, S_t, or its return (aka its relative change if it is not an investment) r_t over a period Dt: DS/S_t. It might seem that modelling S_t would be more natural but in fact modelling the return of the variable is often more helpful: apart from making the mathematics simpler, it is usually the more fundamental variable. In this section, we will refer to S_t when talking specifically about a price, r_t when talking specifically about a return and x_t when it could be either.

• Gometric Brownian Motion (GBM) models are the most simple and common financial time series. They are the basis of the Black-Scholes model, and also the launching pad for a number of more advanced models.

• Autoregressive models are another important category in finance - some examples are the ARCH, GARCH and EGARCH.

• Markov Chain models

• Birth and death models

• Wilkie models

ModelRisk provides facilities both to fit and to model all of the time series described.

Read on: Geometric Brownian Motion models