Why is it that when we differentiate Black-Scholes option pricing formulae for European call and put options with respect to St, but take d1 and d2 as constants instead of functions of St, we still get the correct formula? Is this purely coincidence or is there some underlying mathematical reason?
See the 2nd sticky post above titled "The Greeks" Page 4 of the pdf shows the derivation. The terms cancel out.
If you step through the whole differentiation, you will find the algebra falls out and d1 and d2 magically "reappear" as if they were constants. I think this was asked as an exam question once, but not sure.
It's because: (1) the partial derivatives of d1 and d2 with respect to St are both the same. This is because St appears identically in the formulae for d1 and d2, as: d2 = d1 - sigma*root(T-t) (2) St*exp[-q(T-t)]*littlephi(d1) = K*exp[-r(T-t)]*littlephi(d2) So, when you differentiate the B-S formulae for a call partially with respect to St and get: delta = exp[-q(T-t)]*bigphi(d1) + St*exp[-q(T-t)]*littlephi(d1)*dd1/dSt - K*exp[-r(T-t)]*littlephi(d2)*dd2/dSt the second and third terms sum to zero leaving only the first term.
