Saturday, November 1, 2014

ret rate vs log ret - numerically close but LN vs N

Given IBM price is known now, the price at a future time is a N@T random var. The "return rate" over the same period is another N@T random var. BS and many models assume --

* Price ~ logNormal
* return ~ Normal i.e. a random var following a Normal distro

The "return" is actually the log return. In contrast,

* return rate ~ a LogNormal random variable shifted down by 1.0
* price relative := (return rate +1) ~ LogNormal

N@T means Noisegen Output at a future Time, a useful concept illustrated in other posts

Q (Paradox): As pointed out on P29 [[basic black scholes]], for small returns, return rate and log return are numerically very close, so why only log return (not return rate) can be assumed Normal?

A: "for small returns"... But for large (esp. neg) returns, the 2 return calculations are not close at all. One is like -inf, the other is like -100%
A: log return can range from -inf to +inf. In contrast, return rate can only range from -100% to +inf => can't have a Normal distro as a N@T.

Basic assumption so far -- daily returns are iid. Well, if we look at historical daily returns and compare adjacent values, they are uncorrelated but not independent. One simple set-up is, construct 2 series – odd days and even days. Uncorrelated, but not independent. The observed volatility of returns is very much related from day to day.

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