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Cost of Carry Links Two European Call Delta Formulas

Article Quant Q&A · Author: Mutating Algorithm
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Summary

The note explains why two expressions for a European call’s delta agree when the cost of carry is defined as the risk-free rate minus the dividend yield. The risk-free rate is a cost of holding the asset because that capital could earn interest elsewhere, while dividends are a benefit that offsets part of this cost. Substituting this relationship into the carry-based delta expression gives the dividend-yield form.

The explanation is conceptual and uses no empirical evidence or derivation beyond the relationship between carry, interest, and dividends. Its equivalence depends on consistent definitions and units for the rates and maturity; the displayed dividend expression in the question appears to have a possible notation error, so the intended formula should be checked before applying it.

Key ideas

  • Cost of carry includes the financing opportunity cost of holding an asset.
  • Dividend yield reduces the net cost of carry.
  • When cost of carry equals the risk-free rate minus dividend yield, the delta formulas are equivalent after substitution.
  • The formulas require consistent rate and maturity conventions.

Tags

OptionsDerivatives pricingCarry
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# Equivalence of formulas for pricing the Delta of a European Call Option?


# Equivalence of formulas for pricing the Delta of a European Call Option?












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 $(b - r)T = -qr$

## Answer by Kevin (score 1, accepted)

https://quant.stackexchange.com/a/46622

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.