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?
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1$\begingroup$ Thanks for the help, I figured it was a simple mistake. Actual formula for stdev is sqrt(days/252)*volatility*mean. The mistake I was making is days doesn't include first day. So if you have 5 days of random walk, days=4 in above formula. $\endgroup$– CptanPanicFeb 3, 2011 at 1:34
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3 Answers
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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): $$\left(\exp{(\sigma^2)} - 1\right) \exp{(2\mu + \sigma^2)}$$
answered Feb 1, 2011 at 0:37
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$\begingroup$ I don't think I asked my question correctly. Assuming a random walk, starting at $\mu$ and yearly volatility of $\sigma$ . What is $\sigma$ for the distribution (expected prices) in like 1 month? I must be overthinking this, but it doesn't seem like $\sigma / 12 works either. $\endgroup$ Feb 1, 2011 at 1:47
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1$\begingroup$ The volatility scales as the square root of time. So in one month, you would have $\sigma / \sqrt{12}$, not $\sigma / 12$. This has nothing to do with being log normal, though. $\endgroup$ Feb 1, 2011 at 3:29
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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.
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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.
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$\begingroup$ I get that $r-\sigma^2/2$ comes from ito calculus, but I don't get the extra variables $z$ and $q$, can you explain what that means or from where I can get that information? $\endgroup$– silgonJul 27, 2017 at 13:13