gargi.adhikari
Active Member
Those of you wondering where the delta came from in the above derivation,
call price,
c=Sexp(-qT)*N(d1)-X*exp(-rT)*N(d2)
c=Sexp(-qT)*N(d1)-X* exp(-rT)*N(d2)
dc/dS= d/dS[Sexp(-qT)*N(d1)-X exp(-rT)*N(d2)]
dc/dS= d/dS[Sexp(-qT)N(d1)]- exp(-rT)*d/dS[XN(d2)]
dc/dS= exp(-qT) [N(d1)+S dN(d1)/dS]-X exp(-rT)d/dS[N(d2)]
dc/dS= exp(-qT) [N(d1)+S dN(d1)/dd1*dd1/dS]-X exp(-rT) [dN(d2)/d2*dd2/dS)]
dc/dS= exp(-qT) [N(d1)+S *(N’(d1))*(1/S*sigma*root(T))]-X exp(-rT) [(1/S*sigma*root(T))*N’(d2))]….1
Assuming delta to be constant w.r.t S for very small changes in S=>
N(d1)=const
Differentiating w.r.t S both sides,
N’(d1)=0……2
Also d2= d1-sigma*root(T)=>(1/root(2pi))*exp(-.5*d2^2)=(1/root(2pi))*exp(-.5*d1^2)-sigma*root(T)
integrating both sides w.r.t x,
N(d1)=N(d2)-sigma*root(T)*x
Differentiating w.r.t S both sides,
N’(d2)= N’(d1)…..3
From 1,2 and 3,
dc/dS= exp(-qT) *[N(d1)+S *(0)*(1/S*sigma*root(T))]-X[(1/S*sigma*root(T))*0))]
dc/dS= exp(-qT) *N(d1)
or delta of option is exp(-qT) *N(d1)
thanks
Hi @ShaktiRathore Have a quick question here ...in this derivation of the Delta, you started with the Black Scholes formula for the Option Price:-
c=Sexp(-qT)*N(d1)-X*exp(-rT)*N(d2)
Isn't the formula c=S*N(d1)-X*exp(-rT)*N(d2) without the extra factor of exp(-qT) ......? We are not supposed to 'Discount' the Current Stock Price because it's just current- right..?....What am I missing something here....