Moneyness, implied volatility and option greeks

Question

I know that the more an option is ITM, the more is the implied volatility. I would like to deep dive into the concept, what is the logic that drives this statement? Also comparing an option with a higher delta to one with a lower delta, why the one with higher delta will have a lower vega? The explanation that I found is that options that are deeper ITM have a lower vega exposure, because with higher volatility, a large stock price move in both directions is more likely. Since a deep ITM option is likely to be exercised, the payoff may increase or decrease. In contrast, an ATM option (lower delta) needs volatility to have any chance of ending in the money. Due to the asymmetric payoff of the option, an increase in volatility is very beneficial for such options. Hence, they have higher vega. What I do not get is "hence they have a higher vega", why is this 100% true?

Answer 1

For the first question:

The differences in IV across strikes is largely due to the market having a different view on potential movements than a constant-volatility model (like Black-Scholes) would imply. For example, the market may think that large downward movements are more likely that large upward movements, so they may be willing to pay more (relative to intrinsic value) for lower strikes than higher strikes, which would mean a higher implied volatility for lower-strike options. Or, they may be willing to pay more for ITM options as a way to create leverage - they pay less than the spot price for an option that has a very small chance of not paying off. There are just theories; since the differences in IV are market-driven, they depend on motivations that cannot be definitively quantified.

For the second question:

Vega is the sensitivity to volatility, meaning how much does an extra 1% of volatility affect my option price.

Option values have two components. The intrinsic value (how much the option would be worth if exercised now) and time value (how much extra you pay for the benefit of optionality between now and expiry).

With deep TIM (high delta) options, the time value of the option is very small, because the probability of the spot crossing the strike and making the option worthless is much smaller. Therefore, the volatility has a much smaller impact on the value. For example, if a stock price must double to cross the strike, it doesn't matter much if the volatility is 20% or 30%, the probability of that big of a move is very small. On the other hand, an ATM option (delta=50) with a higher volatility has roughly the same probability of crossing the strike (50/50). but the possible positive outcomes are better, meaning that there the probability of a larger increase is higher with higher volatility.

Put short, with deep ITM options, the fate of the option is largely decided, so volatility is less of a factor. For ATM options, the probability of better results is higher as volatility increases (the "bell curve" of future prices flattens).