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.
Friday, January 31, 2014
quasi constant parameters in BS
Labels: fMathOption, original_content
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