See also P402 [[CFA textbook on stats]]
http://www.hoadley.net/options/bs.htm says Black-Scholes “model is based on a normal distribution of underlying asset returns which is the same thing as saying that the underlying asset prices themselves are log-normally distributed.”. Actually, many non-BS models also assume the same, but my focus today is the 2nd part of the sentence.
At expiration, the asset has exactly one price as reported on WSJ. However, if we simulate 1000 experiments, we get 1000 (non-unique) expiration prices. If we plot them in a __histogram__, we get a kind of bell curve. But in Black-Schole’s (and other people’s) simulations, the curve will resemble a log-normal bell. Reason? …..
Well, they tweak their simulator according to their model. They assume underlying price is a random walker taking many small steps, whose probability of reaching 125% equals probability of dropping to 80% at each step. (But remember the walks are tiny steps, so 80% is huge;) Now the reason behind the paradoxical numbers —
log(new_px/old_px) is normally distributed, so log(1.25)=0.97 and log (0.8)= – 0.97 are equally likely.
Now if we do 1000 experiments and compute the log(price_relative), we get another histogram – a normal (NOT log-normal) curve. Note Price-relative is the ratio of new_Price / old_Price over a holding period.
Here’s Another experiment to illustrate log-normal. Imagine a volatile stock (say SUN) price is now $64. How about after a year ? Black-Scholes basically says it’s
equally likely to double or half.
Double to $128 or half to $32. log2(new_Price / old_Price) would be 1 or -1 with equal likelihood. Intuitively,
log (new_Price / old_Price) is normally distributed.
Now consider prices after Year1, Year2, Year3… log2(S2/currentPx) = log2(S2/S1 * S1/currentPx) = log2(S2/S1) + log2(S1/currentPx). In English this says base-2 log of overall price-relative is sum of the log of annual price-relatives. Among the 3 possible outcomes below, the $256 likelihood equals the $16 likelihood, and is 50% the $64 likelihood.
double-double -> $256
double-half -> $64 unchanged
half-double -> $64 unchanged
half-half -> $16
This stock can also appreciate/drop to other values beside $256,$64,$16, but IF the $256 likelihood is 1.71%, then so is the $16 likelihood, and the $64 likelihood would be 3.42%. We assume no other price “path” will end up at $64 — an unsound assumption but ok for now.
Since log(S2/S1) is normally distributed, so is the sum-of-log. Therefore log(S2/currentPx) is normally distributed.
log(price-relative) is normal.
log(cumulative price-relative) is normal for any number of intervals. For example,
Price_After_2years/current_Price is equally likely to double or half.
Price_After_2years/current_Price is equally likely to grow to 125% or drop to 80%.
More realistic numbers — when we shrink the interval to 1 day, the expected price relative looks more like
“equally likely to hit 101.0101% or drop to 99%”