Geometric Brownian Motion
The evolution of stock prices over a time period [\(0,T\)], from initial value \(P_0\) to final value \(P_T\).
The model is probabilistic: it treats \(P_T\) as a random variable.
For the risk-free assets the price grows at risk-free rate: \[ P_T=P_0e^{rT}. \]
Consider the following model: \[ P_T=P_0e^{\mu T+\lambda_T Z_1}, \] where \(Z_1\sim N(0,1)\), and \(\mu\) and \(\lambda_T\) are some trend and volatility parameters.
The \(\mu\) is the trend parameter of the model, to which \(\lambda_T Z\) term combines random fluctuations.
If \(Z\sim N(0,1)\) then you can show \(\mathbb{E}(e^{cZ})=e^{c^2/2}\).
Hence the random term causes a steady increase in the expected return, since \[ \mathbb{E}(e^{\lambda_T Z_1})=e^{\lambda_T^2/2}>1. \]
Add a damping effect to the model to nullify the effect of random term of growth and the adjusted model is: \[ P_T=P_0e^{\mu T}e^{\lambda_T Z_1 - \lambda_T^2/2}. \]
This additional term is the remainder term of Ito formula.
Suppose \(Z_1\) is the random movement over \([0,T]\) and \(Z_2\) is the random movement over \([T,2T]\).
The spot price at \(2T\) is: \[ \begin{eqnarray*} P_{2T}&=&P_Te^{\mu(T)+\lambda_T Z_2-\lambda_T^2/2},\\ &=&P_0e^{\mu(2T)+\lambda_T (Z_1+Z_2)-\lambda_T^2}. \end{eqnarray*} \]
Assumption : random returns over non-overlapping time intervals are independent random normal variables.
That is
\[ Z_1+Z_2\sim N(0,2). \]
- Therefore,
\[ P_{2T}=P_0 e^{\mu(2T)+\sqrt{2}\lambda_T Z - \lambda_T^2}, \]
where \(Z\sim N(0,1)\).
- On the other hand, if you treat [\(0,2T\)] as a single interval, you get
\[ P_{2T}=P_0e^{\mu(2T)+\lambda_{2T}Z-\lambda_{2T}^2/2}. \]
You must have \(\lambda_{2T}=\sqrt{2}\lambda_{T}\).
You can have it by setting
\[ \lambda_T^2=\sigma^2T, \]
where \(\sigma^2\) is a positive constant.
- Thus the final model for spot price at time \(T\):
\[ P_{T}=P_0e^{(\mu-\sigma^2/2)T+\sigma W_T}, \]
where \(W_T\sim N(0,T)\), where \(\mu\) and \(\sigma\) are known as drift and volatility parameters respectively.
If return follow GBM
\(P_i (i=0,1,...,n)\) : spot prices observed over a consecutive time period of \(\Delta t\) each.
Then the log-return is
\[ r_i=\log(P_{i+1})-\log(P_i). \]
- According to Geometric Brownian Motion,
\[ r_i=(\mu-\sigma^2/2)\Delta t + \sigma W_{\Delta t}, \] where \(W_{\Delta t} \sim N(0,\Delta t)\).
- Therefore,
\[ r_i\sim N((\mu-\sigma^2/2)\Delta t,\sigma \sqrt{\Delta t}). \]
\(\bar{r}=\frac{1}{n}\sum_{i=1}^nr_i\) is the sample mean and variance \(s^2=\frac{1}{n-1}\sum_{i=1}^n(r_i-\bar{r})^2\) of returns.
Estimate \(\mu\) and \(\sigma\) as
\[ \begin{eqnarray*} (\mu-\sigma^2/2)\Delta t &\approx& \bar{r}, \\ \sigma^2 \Delta t &\approx& s^2. \end{eqnarray*} \]
- This results
\[ \begin{eqnarray*} \hat{\mu} &\approx& \frac{\bar{r}+s^2/2}{\Delta t}, \\ \hat{\sigma} &\approx& \frac{s}{\Delta t}. \end{eqnarray*} \]
library(tseries)
library(zoo)##
## Attaching package: 'zoo'## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numericFTSE<-EuStockMarkets[,"FTSE"]
Asset<-FTSE
## Simulation size
sim.size<-500
n<-length(Asset)
## log-return
rt<-diff(log(Asset))
rbar<-mean(rt)
s<-sd(rt)
delta_t<-1
mu_hat<-rbar+s^2/2
set.seed(321)
## Simulate log-return from Normal distribution
rt.sim<-rnorm(sim.size,mean=(mu_hat-s^2/2),sd=s)
Asset.sim<-rep(NA,sim.size)
Asset.sim[1]<-Asset[n]*exp(rt.sim[1])
for(i in 2:sim.size)Asset.sim[i]<-Asset.sim[i-1]*exp(rt.sim[i])
yl<-min(Asset)*0.85
yu<-max(Asset)*1.9
plot(ts(Asset),xlim=c(0,(n+sim.size)),ylim=c(yl,yu))
lines((n+1):(n+sim.size),Asset.sim,col="red",lwd=2)
grid(col="black",lwd=2)
- Change the value of the seed
set.seed(_ _ _)and see how the plot changes !!