Distinguishing the Option Volatility Smile from Delta Changes
Summary
The document clarifies why implied volatility and delta can appear to move in opposite directions without contradicting each other. A volatility smile compares different options at one point in time: with the underlying price held fixed, a lower-strike call can have both higher delta and higher implied volatility than a higher-strike call. This is a cross-sectional comparison across strikes.
A separate effect occurs when the underlying price changes for one option with a fixed strike. As the stock price rises, that call generally becomes more in the money and its delta increases, while implied volatility often falls. The response identifies this dynamic relationship as the leverage effect, distinct from the smile. It recommends describing smile patterns by strike and discussing dynamic volatility changes by underlying price. The explanation is qualitative; it notes that the relationship is usual rather than guaranteed and provides no model or empirical test.
Key ideas
- A volatility smile compares options with different strikes at the same time and underlying price.
- A lower-strike call may have both higher delta and higher implied volatility than a higher-strike call.
- For one fixed-strike call, a rising underlying can increase delta while implied volatility usually declines.
- The cross-sectional smile and the dynamic leverage effect describe different comparisons.
- Describe smile patterns by strike and dynamic changes by underlying price.
Tags
Full text
# Option Volatility Smile vs Delta
# Option Volatility Smile vs Delta
I am new to options trading and have been trying to better understand the relationship between implied volatility, delta, and moneyness. I was wondering how a call option's implied volatility can go up the further in the money it gets (volatility smile) and at the same time fall, since delta rises, approaching 1 as you go further into the money. Is there any mistake in my reasoning? Any answers are very much appreciated!
## Answer by nbbo2 (score 3)
https://quant.stackexchange.com/a/51623
It is not a contradiction, we are looking at two different phenomena:
The Vol Smile is about a comparison on two call options $C_1$ and $C_2$ at a point in time:
S is the same for both options (and does not change!), but $C_1$ has strike $K_1$ and $C_2$ has strike $K_2$. To fix ideas let's say $K_2 > K_1$. Then:
$$\Delta_2 < \Delta_1$$
and
$$IV_2<IV_1$$
In words the low strike Call option (or in you terminology the high moneyness Call option) has a higher Delta and a higher IV.
Nothing is "changing", it is just two static comparisons of 2 pairs of constants at a point in time.
On the other hand if $S$ rises from $S^{OLD}$ to $S^{NEW}$ (so that $S^{NEW}>S^{OLD}$) then for any one call option C: the option has more moneyness, $\Delta^{NEW} > \Delta^{OLD}$ but usually $IV^{NEW} < IV^{OLD}$. That's a dynamic effect on a single option. K is fixed and S changes. That's the "leverage effect on vol" (or negative relation between stock price and vol), distinct from the "vol smile".
My suggestion: don't talk about moneyness in an IV context. For the Smile talk about low strike versus high strike (and it is true for Puts as well as Calls) or for the dynamic vol effect talk about higher stock price vs lower stock price (again true for Puts and Calls).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.