# Endogeniety of Black-Scholes

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? –  private data public channel 2 Mar 20 '13 at 0:09

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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