Modeling Future Prices with Lognormal Distributions
Summary
The document explains how volatility can be used to model a future stock price distribution. It distinguishes the normal distribution of the log price from the resulting lognormal distribution of the price itself. Under a simplified assumption of no dividends and a zero risk-free rate, it describes the log price after a time interval as having a mean based on the current log price and a standard deviation that scales with volatility and the square root of elapsed time. This log-price uncertainty does not depend on the current price level.
To simulate prices, the document suggests drawing a standard normal value and exponentiating a drift and volatility term. It also gives the variance formula for an exponentiated normal variable. The simulation expression includes interest, dividends, and a volatility adjustment to drift, while the simpler description omits those effects. These are model-based relationships, not evidence that actual returns follow a lognormal distribution; day-count conventions and assumptions must match the intended application.
Key ideas
- A normally distributed log price corresponds to a lognormally distributed price level.
- Under the stated simplified assumptions, log-price standard deviation scales with volatility and the square root of time.
- Simulated lognormal prices can be generated by exponentiating a normally distributed shock with drift and volatility terms.
- The variance of an exponentiated normal variable depends on both its normal mean and variance.
- Interest rates, dividends, and day-count assumptions affect how the future-price model is specified.
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Full text
# How to calculate future distribution of price using volatility?
# How to calculate future distribution of price using volatility?
I want to create a lognormal distribution of future stock prices. Using a monte carlo simulation I came up with the standard deviation as being $\sqrt{(days/252)}$ $*volatility*mean*$ $\log(mean)$. Is this correct?
## Answer by shabbychef (score 6, accepted)
https://quant.stackexchange.com/a/49
I'm not sure I understand, but if you want to compute the variance of $exp(X)$, where $X$ is normally distributed with mean $\mu$ and variance $\sigma^2$, that variance is (from Wikipedia): $$\left(\exp{(\sigma^2)} - 1\right) \exp{(2\mu + \sigma^2)}$$
## Answer by user59 (score 3)
https://quant.stackexchange.com/a/55
The distribution of the log of a stock price in n days is a normal distribution with mean of $\log(current_price)$ and standard deviation of $volatility*\sqrt(n/365.2425)$ if you're using calendar days, and assuming no dividends and 0% risk-free interest rate.
Note that the standard deviation is independent of the current_price: if $\log(current_price)$ increases by 0.3 (for example), the stock has increased by 35%, regardless of its current_price.
To include dividends and the risk-free interest rate, see:
http://en.wikipedia.org/wiki/Black-Scholes
which models future stock prices w/ an eye towards pricing options.
## Answer by Brian B (score 0)
https://quant.stackexchange.com/a/2078
To create a lognormal distribution (that is, to generate values from it), you need to start with normally distributed numbers and then exponentiate them.
That is to say, take a sample $z$ from the standard normal distribution, and form the lognormally distributed underlying value
$$ U_T = U_0 \exp\left( (r-q-\sigma^2/2)T + \sigma \sqrt{T} z \right) $$
The probability density function of $U_T$ is formed from solving this for $z$ and then applying the normal PDF.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.