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I know this is a naïve question but how does the BS formula have a closed form solution? It seems from what I am reading Price impacts delta, price influences volatility which in turn influeces delta and gamma. BS seems to be an endogenous models. Is price the only exogenous variable in the model? How can find a closed form solution if every all the variables and are impacting one another? I appreciate any advice on the question.

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doesn't this model assume constant interest rate and volatility? –  bits_international Mar 20 '13 at 0:09
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up vote 1 down vote accepted

Just as an aside: The BS formula is not so closed after all (it of course all depends on your definition of closed form):

$$c(S,K,t,r,\sigma)=$$ $$S\frac{1}{\sqrt{2\pi{}}}\int_{-\infty{}}^{\frac{\ln{\left(\frac{S}{K}\right)}+\left(r+\frac{{\sigma{}}^2}{2}\right)t}{\sigma{}\sqrt{t}}}e^{-\frac{x^2}{2}}dx-e^{-rt}K\frac{1}{\sqrt{2\pi{}}}\int_{-\infty{}}^{\frac{\ln{\left(\frac{S}{K}\right)}+\left(r-\frac{{\sigma{}}^2}{2}\right)t}{\sigma{}\sqrt{t}}}e^{-\frac{x^2}{2}}dx$$

So lots of integrals, logarithms, square roots, fractions, $e$’s, $π$’s and infinity here and everything has to be evaluated. The only thing that makes life easier is that these integrals (normal distribution) turn up so often in maths that they get their own abbreviation, where everything is hidden behind and this is then called closed form. This is why BS always looks so nice and simple. Just saying...

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BS does not make any assumptions about the existence or lack thereof of endogenous variables. According to BS, an options price (equity option) can be determined from 5(6) variables: Price of the underlying, volatility, strike, time to expiration, interest rate, dividends (if applicable). None of those directly are derived from each other, however, a case can be made that there are correlation patterns that can be observed especially between asset prices (and their returns) and volatility. Also obviously interests are a function of time (some models even state that rates are composed of a deterministic function (sole function of time)).

However none of those possible causality patterns negate the validity of the BS model. Nobody argues that the BS model is perfect, it suffers from many simplifying assumptions but then please keep in mind that BS is merely a translation tool between prices and volatility and vice versa. Nothing more, nothing less. An options trader cannot make money simply by knowledge and application of the BS model. The model itself does not make any predictions, it does not spot any arbitrate opportunities (in fact one faulty assumption is that arbitrage does not exist).

In summary, so yes there are inputs to the BS model that may exhibit causality relationships (endogeneity), however, that does not mathematically contradict the correct derivation of BS, given all the assumptions made.

By the way, the WIKI article regarding BS wrongly makes the assumption that dividends do not exist. The extension is so trivial that I would still consider it part of BS. Black's model is also by most considered a trivial extension of BS and hence part of the family.

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