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!
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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).
