i****k
发帖数: 39 | 1 Some naive questions come to my mind when I try to understand the derivation
of Black-Scholes equation.
1). Why we assume stock price S follows GBM? I know data suggest S~GBM, but
is there any fundamental reason?
2). Is it a MUST to assume S ~ GBM to derive B-S? If I assume stock price
follows BM (not GBM), what equation will European call option satisfy? Will
it still be BS or not? Why? How to prove?
3). A more general question is for any asset S, assume we don't know what
process it follows. Will option price satisfy B-S equation? Why or why not?
4). Which assumption(s) can be violated such that we still get the B-S
equation? Which assumption(s), when violated, lead to non B-S description?
Thank you! | l***m
发帖数: 920 | 2 BS is only a model. all models are wrong, some are useful. before inventing
a better pricing theory, markets find BS useful to get agreed-on prices.
you can of course assume stock follow other dynamics such as BM, but it is a
different model. A major drawback is following BM stock price can go
negative.
If asset is not stock, evidently it is not necessary to assume BS, thus
options are not governed by BS PDE, since SDE has changed.
BS has a bunch of assumptions over vol, rate among other factors. if you
want to prioritize them, i think determining one is a perfectly liquid and
efficient market where stock is lognormal, that gives your BS equation.
changing other assumptions on rate, vol or dividends will just have some
terms play in your equation. | i****k
发帖数: 39 | 3 thank you, lainm!
Why stock price follows GBM? Any economic reason? | a*********r
发帖数: 139 | 4 The motivation from the finance point of view is documented in Hull (7th
edition). Look at chapter 12 and problem 12.8. You should also read the
paper " How to Use the Holes in Black-Scholes" by Black, F. These two
resources should answer all your questions. Good luck!
【在 i****k 的大作中提到】
: thank you, lainm!
: Why stock price follows GBM? Any economic reason?
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