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## Friday, January 31, 2014

### quasi constant parameters in BS

dS/S = a dt + b dW [1]

[[Hull]] says this is the most widely used model of stock price behavior. I guess this is the basic GBM dynamic. Many "treasures" hidden in this simple equation. Here are some of them.

I now realize a and b (usually denoted σ) are "quasi-constant parameters". The initial model basically assumes constant [2] a and b. In a small adaptation, a and b are modeled as time-varying parameters. In a sense, 'a' can be seen as a Process too, as it changes over time unpredictably. However, few researchers regard a as a Process. I feel a is a long-term/steady-state drift. In contrast, many treat b as a Process -- the so-called stochastic vol.

Nevertheless in equation [1], a and b are assumed to be fairly slow-changing, more stable than S. These 2 parameters are still, strictly speaking, random and unpredictable. On a trading desk, the value of b is typically calibrated at least once a day (OCBC), and up to 3 times an hour (Lehman). How about on a volatile day? Do we calibrate b more frequently? I doubt it. Instead, implied vol would be high, and market maker may jack up the bid/ask spread even wider.

As an analogy, the number of bubbles in a large boiling kettle is random and fast-changing (changing by the second). It is affected by temperature and pressure. These parameters change too, but much slower than the "main variable". For a short period, we can safely assume these parameters constant.

Q: where is √ t
A: I feel equation [1] doesn't have it. In this differential equation about the instantaneous change in S, dt is assumed infinitesimal. However, for a given "distant future" from now, t is given and not infinitesimal. Then the lognormal distribution has a dispersion proportional to √ t

[2] The adjective "constant" is defined along time axis. Remember we are talking about Processes where the Future is unknown and uncertain.