Geometric Brownian Motion

\[ Z_1+Z_2\sim N(0,2). \]

\[ P_{2T}=P_0 e^{\mu(2T)+\sqrt{2}\lambda_T Z - \lambda_T^2}, \]

where \(Z\sim N(0,1)\).

\[ P_{2T}=P_0e^{\mu(2T)+\lambda_{2T}Z-\lambda_{2T}^2/2}. \]

\[ \lambda_T^2=\sigma^2T, \]

where \(\sigma^2\) is a positive constant.

\[ P_{T}=P_0e^{(\mu-\sigma^2/2)T+\sigma W_T}, \]

where \(W_T\sim N(0,T)\), where \(\mu\) and \(\sigma\) are known as drift and volatility parameters respectively.

If return follow GBM

\[ r_i=\log(P_{i+1})-\log(P_i). \]

\[ r_i=(\mu-\sigma^2/2)\Delta t + \sigma W_{\Delta t}, \] where \(W_{\Delta t} \sim N(0,\Delta t)\).

\[ r_i\sim N((\mu-\sigma^2/2)\Delta t,\sigma \sqrt{\Delta t}). \]

\[ \begin{eqnarray*} (\mu-\sigma^2/2)\Delta t &\approx& \bar{r}, \\ \sigma^2 \Delta t &\approx& s^2. \end{eqnarray*} \]

\[ \begin{eqnarray*} \hat{\mu} &\approx& \frac{\bar{r}+s^2/2}{\Delta t}, \\ \hat{\sigma} &\approx& \frac{s}{\Delta t}. \end{eqnarray*} \]

library(tseries)
library(zoo)
## 
## Attaching package: 'zoo'
## The following objects are masked from 'package:base':
## 
##     as.Date, as.Date.numeric
FTSE<-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)

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