Selecting "Random Walk" for the History Type field of a History Generator element allows you to define a history that randomly walks through time.
In its simplest form, this type of history requires two inputs (Annual Volatility, and Initial Value). The other two inputs (Target and Annual Reversion Rate) are only required if you wish the history to track an underlying target or trend (and this is discussed in the next topic).
Annual Volatility. This is the standard deviation of the annual values (σ):
where μ is the mean of the annual values:
If the volatility is measured over a different time period (e.g., monthly), this can be scaled by direct multiplication. For example a mean monthly volatility can be converted to a mean annual volatility by multiplying it by √12.
It has the same dimensions as the output, and must be non-negative.
Initial Value. This is the initial value of the time history (the value at time zero). It has the same dimensions as the output and can be positive, negative or zero.
You can control whether your history can take on negative values using the Allow Negative Values checkbox (which defaults on). If this box is cleared, negative values are "truncated". Note that this has the effect of artificially reducing the standard deviation of the values.
If the Annual Reversion Rate is set to zero (the default), GoldSim generates successive values as follows:
where Vnew is the new value, Vold is the previous value, Δt is the time (in years) between the two values, ε is a random standard normal value (sampled from a distribution with mean 0 and standard deviation 1), and σ is the volatility.
For constant values of the Annual Volatility, this results in a history whose standard deviation increases as the square root of time.
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