Why a Call Option’s Delta Can Exceed One
Summary
The document explains why a call option’s delta can exceed one when the underlying’s financing cost is higher than the rate used to discount future cash flows. The familiar Black–Scholes result, in which call delta is bounded by one, applies under a simplified zero-cost-of-carry assumption. With positive excess carry, the delta includes a multiplier reflecting that financing exposure.
A one-year, zero-strike call illustrates the effect: with no discounting and a 10% annual financing cost tied to a spot price of $100, the seller’s financing expense rises if spot rises. The answer says hedging this exposure can require an initial delta of 110%. The example demonstrates the mechanism, while the specific figure depends on its assumed rates, maturity, strike, and carry. A second answer gives the bounded-delta formula but does not account for the more general carry setting, so it is incomplete outside its stated traditional-option assumptions.
Key ideas
- Call delta is bounded by one in the simplified zero-cost-of-carry setting.
- When the underlying’s financing cost exceeds its discount rate, a call’s delta can exceed one.
- The carry-adjusted call delta includes an excess-carry multiplier.
- Higher spot can increase the option seller’s financing exposure and hedge requirement.
Tags
Full text
# Possibility of delta greater than 1
# Possibility of delta greater than 1
Can delta of an option be greater than 1? Please illustrate it with an example.
## Answer by Ivan (score 8, accepted)
https://quant.stackexchange.com/a/49237
Only constrained to be <1 in the simplified Black-Scholes setting with zero cost of carry on the underlying. In the more realistic and common setting where the cost of carry of the underlying is higher than the discounting rate, then it is entirely possible for a call to have a delta > 1.
This is the case because your future costs are proportional to the spot value and hence require an additional hedge in terms of spot.
Consider for example a one-year zero-strike call in a situation where rates are 0 and the financing cost for the underlying is an annual rate of 10% of spot (USD 100) paid continuously. Then clearly as the option seller this is initially expected to cost you $10 in financing over the year but will actually cost you more in USD terms if the spot doubles today. This risk needs to be hedged, and calls for you to buy 110% delta in total.
The delta of a call is not $N(d_1)$ but $e^{qT}N(d_1)$ where q is the excess of the cost of carry over the risk-free rate, i.e. financing spread minus dividend yield.
## Answer by Hydraxize (score 3)
https://quant.stackexchange.com/a/49235
The delta of a European call is: $$ \Delta(call)=N(d_1) $$ where $N$ is the cumulative probability function which return value between 0 and 1. Therefore, for a traditional option, your $\Delta$ cannot be greater than 1.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.