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#### Nicole Seaman

##### Director of FRM Operations
Staff member
Subscriber
Option delta is sensitivity of the option's value to a change in the underlying stock price. As illustrated, a call option delta of 0.61 implies that we expect the option's value to increase by $0.61 if the stock price increases by$1.00. Call option delta is bounded by (0, 1.0) and put option delta is bounded by (-1.0, 0)

David's XLS here: https://trtl.bz/2SBoNP6

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#### Eustice_Langham

##### Active Member
David, I found this YT presentation to be extremely useful...thankyou. I do however have a question concerning the shape of the "s" curve when looking at the Call Option Delta vs Stock Price graph. Will this shape always be the case and given its shape what determines the "s" shape.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @Eustice_Langham Thank you for liking the video! The call option delta is given by exp(-qT)*N(d1) where q is the dividend yield such that, in the case of a non-dividend-paying stock, the Δ reduces to N(d1). Even when there is a dividend yield, exp(-qT) is almost 1.0 so that N(d1) approximates regardless.

N(.) is the standard normal cumulative distribution function. In my XLS, like others, I am simply using =NORM.S.DIST(z,TRUE) = =NORM.S.DIST(d1,TRUE). So the shape of call option delta is literally the shape of a normal CDF; see right-hand panel https://en.wikipedia.org/wiki/Normal_distribution

Notice that put Δ = N(d1) - 1, so it's shape is also the shape of the normal CDF but just happens to be shifted down by -1.0, below the X-axis.

BTW, if you take the derivative of a CDF, get the corresponding density PDF; in the case of the normal, that's the familiar bell-shaped distribution. The option's greek vega is given by v = S(0)*sqrt(T)*N'(d1) where N'(.) is the probability density function for a standard normal; ie, the first derivative of N(.). For this reason, vega has a shape that resembles the normal PDF, although not exactly due to the S(0)*sqrt(T) multiplier. I hope that's interesting, thanks,

#### Eustice_Langham

##### Active Member
Thanks David, this make sense and being perfectly honest I should have realised this myself.

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