I came across two formulas to compute the Delta of European Call Options.

The First:

$\frac{\partial C}{\partial S} = e^{(b - r)T} N(d_{1})$

The Second:

$\frac{\partial C}{\partial S} = e^{-qr}N(d_{1})$

Here, q is the Annual dividend yield. r, the risk-free interest rate. b, the cost of carry, T the time to maturity (entered as a decimal). And N is the Normal CDF.

Assuming zero dividends, why are these formulas equivalent?

Basically, why is $(b - r)T = -qr$

  • 3
    $\begingroup$ The second formula should have $e^{-qT}$. Cost of carry is $b=r-q$ , essentially you fund the asset at r but then the asset holding produces q, so cost of carry is lower if q is positive. Now you can substitute this for b to get the result with the typo corrected to make it $-qT$. $\endgroup$ – Magic is in the chain Jul 12 '19 at 7:03

The cost of carry $b$ is, as the name says, the cost which rises when you hold this asset. For instance, if you buy a share, you cannot invest your money in a risk-free bond. Thus, the risk-free rate $r$ is part of your cost of carry. It is the cost you give up in order to be able to own this particular share.

On the other hand, owning this share may give you some dividends with yield $q$. These are benefits of carry and thus, reduce your overall cost of carry. Consequently, $b=r-q$. If $q>r$, i.e. if you have a stock which pays a lot of dividends (relative to its share price) or if the interest rate is very low, you can have a positive cost of carry.


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