Simulating Stock Prices from Historical Returns with Time-Series Models
Summary
The document considers how to simulate future stock prices after backtesting a strategy on historical prices. It critiques a simple return simulation based only on the historical mean and standard deviation, noting that returns form a time series and that this approach imposes a basic model. It suggests fitting time-series models such as ARMA or ARCH-GARCH to log returns, then using the fitted model’s simulation features to generate future returns.
To turn simulated returns into prices, the answer distinguishes ordinary returns from log returns. With ordinary returns, each simulated period’s return compounds against the prior price, so a multi-period path is built by multiplying successive gross returns by the starting price. The example’s random draw is uniform, while the stated simulation equation typically calls for an appropriate random innovation; the answer does not clarify this modeling choice or assess the resulting distribution. It also gives no validation procedure, so model choice and forecast quality remain dependent on the return series and diagnostics.
Key ideas
- Historical mean and volatility alone give a simple model that may not capture return time-series behavior.
- ARMA and ARCH-GARCH are presented as possible models for fitting and simulating returns.
- Ordinary returns produce prices by compounding each period’s gross return from the starting price.
- Log returns require a corresponding conversion when translating simulated returns into prices.
- The document does not establish that a particular time-series model will forecast well for every stock.
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Full text
# How to generate simulated stock price from historical data using R?
# How to generate simulated stock price from historical data using R?
I have created a strategy specifically for a particular stock which I backtested with its historical data. Now I want to forward test it with simulated stock price generated using Monte Carlo. I have used this websites formula for generating simulated return.
> $$\operatorname{Return} = \mu\Delta t + \sigma r\sqrt{Δt}$$
This is my code
```
library(quantmod)
set.seed(100)
aapl=getSymbols("AAPL",from="2014-01-01",auto.assign=F)
N=1000 #number of iterations
ret=ROC(Cl(aapl))
plot(ret)
t=1
mu=mean(na.omit(ret))
sigma=sd(na.omit(ret))
new_ret=NULL
for( i in 1:N){
phi=runif(1, min=0, max=1)
new_ret[i]=mu*t+sigma*phi*sqrt(t)
}
plot(new_ret)
```
The simulated return(new_ret) looks some what odd and not proper? How to generate simulated data using historical data of a stock price?
Return is calculated using `ROC()` function which in turn uses `diff(log())` function. How can I generate price from simulated return vales?
## Answer by user24039 (score 5)
https://quant.stackexchange.com/a/29728
This approach is rather crude. It only takes the mean and volatility of the historical returns and assumes a very simple model. I'm not sure if you have much experience with Time Series, but your returns series is a Time series.
You can now perform tests on these log returns to ensure you can continue with Time series models. One very simple model is ARMA. You can extend it to ARCH-GARCH. R has these function built in and it also has features which will simulate future values using the model it builds.
Check out he `arima()` function, or uGARCH. Sorry if you have not come across this yet. It is a fairly simple model but very good in my experience for returns. Id expect it to be better than your model above. It is a common approach in modelling financial time series.
- Which econometric models can be used to forecast security returns + ARIMA/GARCH questions.
- Autoregressive–moving-average model.
- Statistical modelling of financial time series: An introduction
Hope this is useful.
For your final question. Your code looks fine, although inefficient if you are simulating a lot of data. To forecast prices, you have some return $r$. IF you take today's stock price $S_0$, then tomorrows price will be $S_0(1+r)$ if you use normal returns. Similar formulae hold if you use log prices as I mentioned before. IF your using standard returns, then predicting $n$ days ahead requires $$S_0(1+r_1)(1+r_2)\cdots(1+r_n).$$ This can be applied in a crude for loop or more elegant methods in R.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.