Stochastic Processes in Discrete Time
A stochastic process or random process consists of chronologically ordered random variables \(\{X_{t};\,\,t \geq 0\}.\) For simplicity we assume that the process starts at time t = 0 in \(X_{0}\,=\,0.\)
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Stochastic Processes in Discrete Time
Processus artis coniectandi, qui spatio temporis discreto fiunt Vitam regit fortuna, non sapientia. Fortune, not wisdom, rules lives. Marcus Tullius Cicero, Tusculanarum Disputationum LIX
A stochastic process or random process consists of chronologically ordered random variables fXt I t 0g: For simplicity we assume that the process starts at time t D 0 in X0 D 0: This means that even if the starting point is known, there are many possible routes the process might take, some of them with a higher probability. In this section, we exclusively consider processes in discrete time, i.e. processes which are observed at equally spaced points of time t D 0; 1; 2; : : : : In other words, a discrete process is considered to be an approximation of the continuous counterpart. Hence, it is important to start with discrete processes in order to understand sophisticated continuous processes. In particular, a Brownian motion is a limit of random walks and a stochastic differential equation is a limit of stochastic difference equations. A random walk is a stochastic process with independent, identically distributed binomial random variables which can serve as the basis for many stochastic processes. Typical examples are daily, monthly or yearly observed economic data as stock prices, rates of unemployment or sales figures. In order to get an impression of stochastic processes in discrete time, we plot the time series for the Coca-Cola stock price. The results are displayed in Fig. 4.1. If prices do not vary continuously, at least they vary frequently, and the stochastic process has thus proved its usefulness as an approximation of reality. Exercise 4.1 (Geometric Brownian Motion). Construct a simulation for a random stock price movement in discrete time with the characteristics given in Table 4.1 from a geometric Brownian motion.
S. Borak et al., Statistics of Financial Markets, Universitext, DOI 10.1007/978-3-642-33929-5 4, © Springer-Verlag Berlin Heidelberg 2013
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4 Stochastic Processes in Discrete Time COCA COLA series 56 Price (USD)
54 52 50 48 46 44 42 40 38 Jan-02
Jan-03 Time (days)
Jan-04
Fig. 4.1 Stock price of Coca-Cola Table 4.1 Characteristics to simulate a random stock price movement in discrete time
Default values Initial stock price S0 Initial time Time to maturity T Time interval t Volatility p.a. Expected return p.a.
49 0 20 weeks 1 week 0.20 0.13
The numerical procedure to simulate the stock price movement in discrete time with characteristics described in Table 4.1 is given by defining the process Si D p Si 1 expfXi T =n C . 2 =2/T =ng, with i D 0; : : : ; n where n denotes the number of time intervals, t D T =n and X N.0; 1/ denotes a standard normal r.v. Fig. 4.2 displays the simulation of a random stock price movement in discrete time with t D 1 week and 1 day respectively. P Exercise 4.2 (Random Walk). Consider an ordinary random walk Xt D tkD1 Zk for t D 1; 2; : : :, X0 D 0, where Z1 ; Z2 ; : : : are i.i.d. with P.Zk D 1/ D p and P.Zk
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