Monday, March 16, 2015

from BM's drift rate to GBM's drift rate - eroded by .5 sigma^2

Let's start with a regular BM with a known drift rate denoted "m", and known variance parameter value, denoted "s":

   dX = m dt + s dBt

In other words, 

  Xt – X0 = m*t + s*Bt 

Here, "... + t" has a non-trivial meaning. It is not same as adding two numbers or adding two variables, but rather a signal-noise formula.. . It describes a Process, with a non-random, deterministic part and a random part whose variance at time t is equal to s2 t

Next, we construct or encounter a random process related but not derived from this BM:

  dG/G = m dt + s dBt

It turns out this process can be exactly described by

  G = G0 exp[ (m- ½ s2)t  + s Bt ]

Again, the simple-looking "... + s Bt" expression has a non-trivial meaning. It describes a Process, whose log value has a deterministic component, and a random component whose variance is s2 t.

Note in the formula above (m- ½ s2) isn't  the drift of GBM process, because left hand size is "dG / G" rather than dG itself.

In contrast, (m- ½ s2) is a drift rate in the "log" process, i.e. the log of the GBM. This log process is a BM. We can think of the original drift eroded by (½ s2)

In summary
* Starting from a BM with drift = (u) dt
** the exponential process Y_t derived from the BM has drift = ([u + ½ s2 ]Y_t) dt

* Starting from a GBM (Not something derived from BM) process with drift = (m*X_t) dt
** the log process L_t, derived from the GBM process is a BM with drift = (m - ½ s2) dt, not "...L_t dt"

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http://www.linkedin.com/in/tanbin