How Underlying Price Changes Option Delta in Black–Scholes
Summary
The document explains whether an option’s delta remains fixed when the underlying asset price changes while other Black–Scholes inputs are held constant. It gives the standard call and put delta relationships through the normal cumulative distribution of d1, and shows that d1 itself depends on the ratio of underlying price to strike. As the underlying price rises, call delta rises while put delta falls under the stated setup.
It also connects this sensitivity to gamma, the rate at which delta changes with the underlying price. The answer describes gamma as small for options far in or out of the money and more material near the money, where a hedge may need adjustment as the underlying moves. These are model-based explanations under fixed remaining assumptions; the document offers no empirical comparison and does not discuss changes in volatility, time, rates, dividends, or model limitations.
Key ideas
- In Black–Scholes, delta depends on the underlying price through d1.
- With other inputs fixed, a rising underlying price increases call delta and decreases put delta.
- Gamma measures how delta changes as the underlying price moves.
- Near-the-money options can require more frequent delta hedge adjustments than options far from the money.
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Full text
# Will the Delta of an Option always be the same irrespective of the underlying stock price?
# Will the Delta of an Option always be the same irrespective of the underlying stock price?
Suppose, under the Black Scholes model we keep all the parameters the same except that we vary the asset price. Will the Delta of the option always remain the same?
## Answer by 10uss (score 1, accepted)
https://quant.stackexchange.com/a/46598
You can just check this yourself with the BSM formulas. You fill out the regular $d_{1}$ formula in Excel and hold all variables fixed except for the $S_{t}$. Excel example is provided below the formulas.
$$\Delta_{Call}=N(d_{1})$$ $$\Delta_{Put}=-N(-d_{1})$$
$$d_{1}={ln(S_{t}/K)+(r+\sigma^2/2)t\over\sigma\sqrt{t}}$$
As you can see in the image when $S_{t}$ increases (holding other factors fixed), the delta of the call option increases and the delta of the put option decreases.
## Answer by Dimitri Vulis (score 0)
https://quant.stackexchange.com/a/46601
You can think of the gamma (second derivative, convexity) as the change in delta cause by the change in the price of the underlying. In other words, the gamma tells you the adjustment in the delta hedge necessitated by the change in the pricer of the underlying. So if the option is either far in the money or far out of the money then the gamma is zero and a small change in the price of the underlying does not change the delta and does not necessitate an adjustment of your selta hedge. But if the option is close to being at the money, then you have a gamma, and have to adjust the delta hedge in order to flatten the exposure to the underlying.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.