Greeks

[gargi.adhikari]

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....

 

[David Harper CFA FRM]

Hi @gargi.adhikari He is using the general form of the BSM where the underlying is a stock that pays a continuous dividend at a rate of (q), such that when the underlying is a non-dividend-paying stock q = 0 and exp(-qT) = 1.0. You are correct that S(0) is already a discounted (ie, current) stock price. The reason for the reduction relates to the fact that the option holder forgoes (does not receive) dividends, and dividends of course are a component of the stock's total return. Thanks!

 

[gargi.adhikari]

@David Harper CFA FRM Perfect !! Thanks so much David !

 

[FM22]

can we further elaborete on Rho??

if Interest rates increase and Decrease what will happen to Amerivan/European Put and Calls and why pleaase.

thanks

 

[HarshilDave]

Hi @ShaktiRathore , is there any chance I can find a grid anywhere that explains the movement/direction of option greeks in a grid with call/put, long/short and at-the-money, in-the-money or out-of-money? I'm struggling to have a single view where I can understand how the option greeks move depending on combination of 'Option type' 'Position Type' and the In/out of money.

 

[David Harper CFA FRM]

Hi @FM22 Rhos for European options are given by:

• rho(call) = K*T*e^(-rT)*N(d2); such that call rho is always positive. N(d2) is a cumulative probability so it is always positive between 0 and 1.
  
• rho(put) = -K*T*e^(-rT)*N(-d2); such that put rho is always negative

For me, the easiest way to remember is with minimum values:

• Euro call min value --> c ≥ S - K*e^(-rT); i.e., higher discount rate, r, implies lower subtracted term implies higher min bound, so call value increases with riskfree rate
  
• Euro put min value --> p ≥ K*e^(-rT) - S; i.e., higher discount rate, r, implies lower initial term implies lower min bound, so put value decreases with riskfree rate. I hope that's helpful!

 

[FM22]

Thank you very much David.

Indeed Helpful