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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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): $$\left(\exp{(\sigma^2)} - 1\right) \exp{(2\mu + \sigma^2)}$$ |
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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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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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