Wednesday, November 3, 2010

About Ito's Process

Besides Black & Scholes, Kiyoshi Ito is one of the really popular guys in mathematical finance. Everybody in the scene knows his diffusion process, and I have some problems with it!

Ito's Diffusion Process (Simplest form):

Given that X(t) is a random variable, representing price of a financial security at time t, the change in X can be represented by,

dX(t)= aX(t)dt + sdW(t)


a, s: constants with respect to the model applied
W(t): Brownian motion

Stylized Facts
Empirical findings have however shown that the markets are NOT truly random as suggested by the Brownian motion. D. Whitcomb has pointed out a negative serial correlation off Index returns since 1979! Here's an article about his findings. This means the drift parameter a above does not always make sense. Yeah, there's always the drift due to inflation on indexes, but not all securities can even keep up with it, especially those with high credit risk.

The conditional variance contradicts with a lot of existing academic assumptions around Heston's stochastic volatility paper. Well it isn't so stochastic if you've actually observed volatility time series critically. I can't get into the volatility thing until a later time.

That's it for now!

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