Random Walk Hypothesis

Random Walk Model

Example 1: Random Walk with Fixed Moves

  • Suppose price of a stock move up by 10 paisa with probability 0.5 or move down by 10 paisa with probability 0.5 every seconds.
  • If the price of the stock is Re 1/-; then what will be the price of the stock after 21600 seconds
  • The model \(P_t=P_{t-1}\pm M_t\), where \(M_t=5~a.s.\)
set.seed(321)
n<-21600
P<-rep(NA,n)
P[1]<-100 ## Current price 100 paisa or Re 1/-
for(sec in 2:n){
  toss<-sample(c("H","T"),1,replace = TRUE,prob = c(0.5,0.5))
  if(toss=="H")P[sec]<-P[sec-1]+5
  if(toss=="T")P[sec]<-P[sec-1]-5
}
plot(ts(P))
abline(h=0,lwd=2,col="red")

  • Notice: Price of the stock is negative
  • This model is good candidate to model the stock price movement.
  • However, it cannot take care of the limited liability feature of the stock market.
  • Check what happens price of the stock move up by 10 paisa with probability 0.51 or move down by 10 paisa with probability 0.49 on every seconds!

Example 2: Random Walk with Random Moves

  • Suppose price of a stock move up or down with probability 0.5 or
  • Size of the movement follow \(Poisson(\lambda=5)\)
  • If the price of the stock is Re 1/-; then what will be the price of the stock after 21600 seconds
  • The model: \[ P_t=P_{t-1}+\pm M_t, \] where \(M_t \sim Poisson(\lambda=5)\)
set.seed(321)
n<-21600
P<-M<-rep(NA,n)
P[1]<-100 ## Current price 100 paisa or Re 1/-
for(sec in 2:n){
  toss<-sample(c("H","T"),1,replace = TRUE,prob = c(0.5,0.5))
  M[sec]<-rpois(1,lambda = 5)
  if(toss=="H")P[sec]<-P[sec-1]+ M[sec]
  if(toss=="T")P[sec]<-P[sec-1]- M[sec]
}
par(mfrow=c(1,2))
plot(ts(P))
abline(h=0,lwd=2,col="red")
plot(ts(M))
abline(h=0,lwd=2,col="red")

Example 3: Random Walk with Random Return

  • Simple return of an asset is nothing but movement of the price with respect to previous price. \[ \begin{eqnarray} R_t&=&\frac{P_t-P_{t-1}}{P_{t-1}}\\ R_t.P_{t-1}&=&P_t-P_{t-1}\\ P_t&=&P_{t-1}(1+R_t) \end{eqnarray} \]
  • Suppose \(R_t\sim N(\mu=0,\sigma=0.01)\) on every seconds.
  • If the price of the stock is Re 1/-; then what will be the price of the stock after 21600 seconds
set.seed(321)
n<-21600
P<-rep(NA,n)
P[1]<-100 ## Current price 100 paisa or Re 1/-

rt<-rnorm(n,mean=0,sd=0.01)
for(sec in 2:n) P[sec]<-P[sec-1]*(1+rt[sec])

par(mfrow=c(1,2))
plot(ts(P))
abline(h=0,lwd=2,col="red")
plot(ts(rt))
abline(h=0,lwd=2,col="red")

  • Check what happend if you choose mean= 0.01 !

Example 4: Random Walk with Random log-Return

  • The log-return of an asset indicates continuous compounding effect of the proce movement; \[ \begin{eqnarray*} r_t &=& \log(P_t) - \log(P_{t-1})\\ &=& \log\Big(\frac{P_t}{P_{t-1}}\Big)\\ P_t &=& P_{t-1}e^{r_t} \end{eqnarray*} \]
  • Suppose \(r_t\sim N(\mu=0,\sigma=0.01)\) on every seconds.
  • If the price of the stock is Re 1/-; then what will be the price of the stock after 21600 seconds
set.seed(321)
n<-21600
P<-rep(NA,n)
P[1]<-100 ## Current price 100 paisa or Re 1/-

rt<-rnorm(n,mean=0,sd=0.01)
for(sec in 2:n) P[sec]<-P[sec-1]*exp(rt[sec])

par(mfrow=c(1,2))
plot(ts(P))
abline(h=0,lwd=2,col="red")
plot(ts(rt))
abline(h=0,lwd=2,col="red")

Arbitrage

  • In example 1, if fixed move takes place with unequal probability it means it has an arbitrage opportunity for either party.
  • Similarly, in example 2,3 and 4 if mean: \(\mu \neq 0\) then it means it has an arbitrage opportunity.

Random Walk Model

  • Suppose \(r_1,r_2,...\) be i.i.d with mean \(\mu\) and standard deviation \(\sigma\) .

  • Let \(P_0\) be an arbitrary starting point and \[ P_t=P_0+r_1+r_2+...+r_t,~~t\geq 1. \]

  • The process \(P_0 , P_1,P_2,...\) is known as random walk and \(r_1,r_2,...\) are corresponding steps of that random walk.

  • The conditional expectation and variance of \(P_t\) given \(P_0\) is \(E(P_t|P_0)=P_0+\mu t\) and \(Var(P_t|P_0)=\sigma^2 t\) .

  • The parameter \(\mu\) is the drift and set an overall trend of the random walk.

  • The parameter \(\sigma\) is the volatility and controls how much it fluctuates around \(P_0+\mu t\).

  • Since the standard deviation of \(P_t\) given \(P_0\) is \(\sigma\sqrt{t}\) , as \(t\) increases the range of variability in the process increases.

  • This means at the \(t = 0\) you know very little about where the random walk will be in the remote future compared to its current spot value.

Modeling with log-return

  • Suppose \(R_1,R_2,...\) are simple return from single investment.

  • You can make a common working assumption as the returns are mutually independent and identically distributed (\(i.i.d\)) random variables with mean \(\mu\) and variance \(\sigma^2\).

  • However, this is not a good model from a theoretical perspective.

  • The model assumes \(P_t\) is distributed over the real line, which means the model implies unlimited loss.

  • But typically in the world of limited liability structure of the companies, the loss is usually limited.

  • That is \(R_t \geq -1\) means the maximum loss that an investor can experience is the total investment.

  • You see, for the log return, \[ 1+R_t=\exp(r_t)\geq 0, \] which implies \(R_t\geq -1\).

  • This satisfies the condition of limited liability, i.e., possible maximum loss is the total investment. In addition, \[ \begin{eqnarray*} 1+R_t(k)&=&(1+R_t)(1+R_{t-1})...(1+R_{t-k+1}),\\ &=&\exp(r_t)\exp(r_{t-1})...\exp(r_{t-k+1}),\\ &=&\exp(r_t+r_{t-1}+...+r_{t-k+1}). \end{eqnarray*} \]

  • So to sum of \(k\) period log-returns yield \(k\)-period gross return. Now note that \[ \frac{P_t}{P_{t-k}}=1+R_t(k)=\exp(r_t+r_{t-1}+...+r_{t-k+1}). \] can be expressed as for \(k=t\), \[ P_t=P_0\exp(r_t+...+r_1). \]

  • Therefore, if the log returns are assumed to be i.i.d. random variables, then the price of the stock or market index, denoted by the process \(P=\{P_t:t \geq 0\}\), is the exponential of random walk or also known as geometric random walk.

Stationary Process

  • In probability, a stochastic process is known as stationary if the joint probability distribution is independent of time.
  • Parameters such as mean, volatility, correlation etc., do not change over time.
  • The process \(P=\{P_t: t\geq 0\}\) is strongly stationary if \(\forall\) \(h\) and \(n\) \[ F_P(p_{t_1+h},p_{t_2+h},...,p_{t_n+h})=F_P(p_{t_1},p_{t_2},...,p_{t_n}), \] since \(h\) doe not affect \(F_P\), it is not function of time.
  • Note that the families \(\{P(t_1),...,P(t_n)\}\) and \(\{P(t_1+h),...,P(t_n+h)\}\) have the same joint distribution for all \(t_1,t_2,...,t_n\) and \(h>0\).
  • A weak form of stationarity is defined based on the parameters of the process.
  • The (WSS) process only require that mean and autocovariance do not vary with respect to time.
  • A continuous time random process \(P_t\) is , if it has the following restrictions on its mean function, \[ \begin{eqnarray} \mathbb{E}(P_t)=\mu(t)=\mu(t+h)=\mu~~\forall t,h \in \mathbb{R},\label{RFR_eq_wss_mean_fun} \end{eqnarray} \] and autocovariance function \[ \begin{eqnarray} \mathbb{C}ov_P(t_1,t_2)=\mathbb{E}[\{P(t_1)-\mu(t_1)\}\{P(t_2)-\mu(t_2)\}]=\mathbb{C}ov_P(t_1-t_2).\label{RFR_eq_wss_cov_fun} \end{eqnarray} \]
  • The mean function \(\mathbb{E}(P_t)\) must be constant over time
  • The covariance function depends only on the difference between \(t_{1}\) and \(t_{2}\)
  • Distribution of log-return must be stationary

Simulate from stationary process

mu<-0
s<-1
n<-100
r<-rnorm(n,mu,s)
plot(ts(r))

P<-r
P[1]<-1
P<-cumsum(P)
plot(ts(P))

  • Price may be non-stationary. However, its steps could be a stationary process.

Unit Root of Random Walk

  • The random walk is said to have unit root. To understand what this means, you should consider the AR(1) model (i.e., Auto-Regressive model with lag 1), \[ p_t=\phi p_{t-1}+r_t \] where \(\phi=1\) and \(p_t=\log(P_t)\) is price of the asset in log-scale.

  • The generic AR(1) model can be presented as \[ \begin{eqnarray*} p_t&=&\phi p_{t-1}+r_t\\ &=& \phi (\phi p_{t-2} + r_{t-1}) + r_t\\ &=& \phi^2 p_{t-2} + \phi r_{t-1} + r_t \\ &=& \phi^2 (\phi p_{t-3} + r_{t-2}) + \phi r_{t-1} + r_t\\ &=& \phi^3 p_{t-3} + \phi^2 r_{t-2} + \phi r_{t-1} + r_t\\ &\vdots&\\ &=& \phi^k p_{t-k} + \phi^{k-1} r_{t-(k-1)}+... + \phi r_{t-1} + r_t\\ &=& \phi^k p_{t-k} + \sum_{i=0}^{k-1}\phi^{i-1} r_{t-(i-1)}. \end{eqnarray*} \]

  • If \(\phi = 1\) then the process is non-stationary. Because \(\sum_{i=0}^{k-1}\phi^{i-1} r_{t-(i-1)}\) accumulates the information over time. Hence a random walk is a non-stationary process.
  • However \(|\phi|<1\), i.e., \(-1<\phi<1\) implies, the process is stationary.

  • If \(\phi=0\) that means the process is stationary and \(p_t\) and \(p_{t-1}\) are independent \(\forall t\).

  • As the series \(\{p_t: t \geq 0\}\) is a random walk (i.e., \(\phi=1\)) the incremental steps (i.e., log-returns) are independent and stationary process, you can write it as \[ r_t=\phi_1 r_{t-1}+\epsilon_t \] where \(\phi_1=0\) and \(\epsilon_t\) is white noise with mean \(\mu\) and variance \(\sigma^2\).

  • In order to check if the price of a stock follows the geometric random walk, you have to check following three things.
    1. First, you should check if \(\{p_t\}\) is a non-stationary process, i.e., \[ p_t=\phi p_{t-1}+r_t; \] check if \(\phi=1\) or \(\phi<1\).
    1. Second, you check if the log-returns are stationary process, i.e., \[ r_t=\phi_1 r_{t-1}+\epsilon_t; \] check if \(\phi_1=1\) or \(\phi_1<1\).
    1. Second check only tells you if a log-returns are stationary, but it does not check if \(\phi_1=0\) or not. In addition \(\phi_1=0\) only implies pairwise independence. It does not check the mutual independence of \(r_t\). So you should check if the serial correlations of \(r_t\) are 0 or not. That is check if \(\rho_1=\rho_2=...=\rho_H=0\), where \(\rho_h=corr(r_t,r_{t+h})\) is the lag \(h\) auto-correlation.

Dickey-Fuller test for Stationarity in a Time Series

  • The Dickey-Fuller test (1979) compares the null hypothesis \[ H_0: p_t=p_{t-1}+r_t \] i.e., that the series is a random walk without drift, where \(r_t\) is a white noise with mean 0 and variance \(\sigma^2\).

  • The alternative hypothesis is \[ H_1: p_t=\mu + \phi p_{t-1} + r_t \] where \(\mu\) and \(\phi\) are constant with \(|\phi|<1\).

  • According to \(H_1\), the process is stationary AR(1) with mean \(\frac{\mu}{1-\phi}\).

  • You can implement the Dickey Fuller test using adf.test function in tseries package.

library(tseries)
head(EuStockMarkets)
##          DAX    SMI    CAC   FTSE
## [1,] 1628.75 1678.1 1772.8 2443.6
## [2,] 1613.63 1688.5 1750.5 2460.2
## [3,] 1606.51 1678.6 1718.0 2448.2
## [4,] 1621.04 1684.1 1708.1 2470.4
## [5,] 1618.16 1686.6 1723.1 2484.7
## [6,] 1610.61 1671.6 1714.3 2466.8
FTSE<-EuStockMarkets[,"FTSE"]
## Plot FTSE
n<-length(FTSE)
plot(ts(FTSE))
grid(col = "red")

## Compute the log-return of the FTSE
log_return<-diff(log(FTSE))*100
n<-length(log_return)
## The time-series plot of the log-return 
plot(ts(log_return))
grid(col="red")

## Dickey-Fuller test for unit-root

## Step 1: Check if log-Price is non-stationary
adf.test(log(FTSE))
## 
##  Augmented Dickey-Fuller Test
## 
## data:  log(FTSE)
## Dickey-Fuller = -2.4976, Lag order = 12, p-value = 0.3677
## alternative hypothesis: stationary
## Step 2: Check if log-return is non-stationary
adf.test(log_return)
## Warning in adf.test(log_return): p-value smaller than printed p-value
## 
##  Augmented Dickey-Fuller Test
## 
## data:  log_return
## Dickey-Fuller = -11.162, Lag order = 12, p-value = 0.01
## alternative hypothesis: stationary

Ljung-Box test for independence in a Time Series

  • The null hypothesis is \(\rho_h = 0\) for. That is, \[ H_0: \rho_1=\rho_2=...=\rho_H=0 ~\forall~ t, \]

    \[ H_1: \text{At least one inequality.} \]

  • The test statistic for the Ljung-Box test is \[ Q=n(n+2)\sum_{h=1}^H\frac{\hat{\rho}_h^2}{n-h}, \] where \(n\) is the sample size, \(\hat{\rho}_h\) is the sample autocorrelation of lag \(h\).

  • You can show under \(H_0\), \(Q\) follows a chi-square distribution, \(\chi^2_{(h)}\).

  • The Ljung-Box test can be done in R using Box.test function available in stats package.

## Step 3: Check if the log-returns are uncorrelated

Box.test(log_return,lag=10,type = "Ljung-Box")
## 
##  Box-Ljung test
## 
## data:  log_return
## X-squared = 29.815, df = 10, p-value = 0.0009183

Is Market really efficient?