If it were not related, you would get lots of arbitrage opportunities. This answer shows how to use SVI to fit a vol surface from market quotes.
If you look at these two fictional examples, you will realize that one has several mispriced options, while the other has all options more or less aligned with the entire vol surface.
Now in reality, you face issues like stale prices, unreasonable small or large bid/ask spreads, you need figure out a way to handle that the underlying asset and options on the asset can trade during different times on different exchanges, that there exist erratic prices, especially close to the opening and closing times of the trading session...
That is why actual shapes of IV surfaces are quite fascinating at times. Voladynamics has some great examples on their webpage.
The underlying message is the same though: IVs must be related to each other, otherwise there will be arbitrage opportunities. Since there are many firms constantly monitoring this, any such opportunity will be very short lived.
In my opinion the cleanest way to show why this makes sense is to look at FX OTC options. It is a very liquid market with lots of market makers constantly quoting implied vol for At-the-Money Delta Neutral Straddles, Risk Reversals and Butterflies for several deltas. Combined, they give you the entire Vol surface. In a nutshell,
- ATM determines the level (you can think of it as the Black Scholes IVOL for a specific tenor),
- RR the skew (how its tilted, towards OTM puts for RUB and GBP in the examples below) and
- BF the kurtosis (how pronounced the general wings are).
Getting call and put IV from these quotes requires a combination of these using the following relations:
- RR = C - P (for deltas)
- BF =(C + P)/2 - ATM
- C = ATM + BF + RR/2
- P = ATM + BF - RR/2
Plotting this you can see how they depend on each other:
It is changes in ATM vol that move the entire surface up and down. The vol surface (smile, smirk, skew,...) exists mainly because there are fat tails, skewness, heteroscedasticity, jumps (crashes), and so forth. None of these real-world phenomena are featured in the Black Scholes formula. The market just developed ways to account for many of the shortcomings of Black Scholes.
Now, equity is not vol quoted (at least not exchange traded / listed options). However, the general idea is the same. In fact, some market makers even quote IV surfaces for equities alongside variance swap rates. If you have access to Bloomberg (many universities and libraries like the New York Public Library have Windows PCs running Bloomberg Terminal software), you can look at
OVDV to see FX quotes and the surface I displayed above. If you have access at work, you will likely also have access to some OTC IV quotes for equities on OVDV (just select the dropdown where BVOL stands to switch quotes).