Uwe Hassler - Stochastic Processes and Calculus

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Springer International Publishing Switzerland 2016

Uwe Hassler Stochastic Processes and Calculus Springer Texts in Business and Economics 10.1007/978-3-319-23428-1_1

1. Introduction

Uwe Hassler 1

(1)

Faculty of Economics and Business Administration, Goethe University Frankfurt, Frankfurt, Germany

1.1 Summary

Stochastic calculus is used in finance and econom(etr)ics for instance for solving stochastic differential equations and handling stochastic integrals. This requires stochastic processes. Although stemming from a rather recent area of mathematics, the methods of stochastic calculus have shortly come to be widely spread not only in finance and economics. Moreover, these techniques along with methods of time series modeling are central in the contemporary econometric tool box. In this introductory chapter some motivating questions are brought up being answered in the course of the book, thus providing a brief survey of the topics treated.

1.2 Finance

The names of two Nobel prize winners dealing with finance are closely connected to one field of applications treated in the textbook at hand. The analysis and the modeling of stock prices and returns is central to this work.

Stock Prices

Let S ( t ), t 0, be the continuous stock price of a stock with return expressed as growth rate. We assume constant returns,

This differential equation for the stock price is usually also written as follows:

(1.1)

The corresponding solution is (see Problem )

(1.2)

i.e. if c >0 the exponential process is explosive. The assumption of a deterministic stock price movement is of course unrealistic which is why a stochastic differential equation consistent with ( ),

(1.3)

where dW ( t ) are the increments of a so-called Wiener process W ( t ) (also referred to as Brownian motion, cf. Chap. on stochastic differential equations.

Interest Rates

Next, r ( t ) denotes an interest rate for t 0. Assume it is given by the differential equation

(1.4)

with or equivalently by

Expression () can alternatively be written as the following integral equation:

(1.5)

The solution to this reads (see Problem )

(1.6)

For c <0 therefore it holds that the interest rate converges to as time goes by. Again, a deterministic movement is not realistic. This is why Vasicek ():

(1.7)

As aforementioned, dW ( t ) denotes the increments of a Wiener process. How is the interest rate movement (on average) affected by the parameter c ? Which kind of stochastic process is described by ( on interest rate models.

Empirical Returns

Looking at return time series one can observe that the variance (or volatility) fluctuates a lot as time passes by. Long quiet market phases characterized by only mild variation are followed by short periods characterized by extreme observations where extreme amplitudes again tend to entail extreme observations. Such a behavior is in conflict with the assumption of normally distributed data. It is an empirically well confirmed law (stylized fact) that financial market data in general and returns in particular produce outliers with larger probability than it would be expected under normality.

It is crucial, however, that extreme observations occur in clusters (volatility clusters). Even though returns are not correlated over time in efficient markets, they are not independent as there exists a systematic time dependence of volatility. Engle (

1.3 Econometrics

Clive Granger (19342009) was a British econometrician who created the concept of cointegration (Granger, ). He shared the Nobel prize for methods of analyzing economic time series with common trends (cointegration) (official statement of the Nobel Committee) with R.F. Engle. The leading example of trending time series he considered is the random walk.

Random Walks

In econometrics, we are often concerned with time series not fluctuating with constant variance around a fixed level. A widely-used model for accounting for this nonstationarity are so-called integrated processes. They form the basis for the cointegration approach that has become an integral part of common econometric methodology since Engle and Granger (). Lets consider a special case the random walk as a preliminary model,

(1.8)

where is a random process, i.e. and , t s , are uncorrelated or even independent with zero expected value and constant variance 2. For a random walk with zero starting value x 0=0 it holds by definition that:

(1.9)

The increments can also be written using the difference operator ,

Regressing two stochastically independent random walks on each other, a statistically significant relationship is identified which is a statistical artefact and therefore nonsense (see Chap.).

Dickey-Fuller Distribution

If one wants to test whether a given time series indeed follows a random walk, then equation () suggests to estimate the regression

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