A delta profile is a graphical representation showing the delta (directional exposure) of an option for all different levels of the underlying security’s price. If depicted horizontally, it takes the shape of an S-curve which is centered at the strike price, which is generally aligned with about 50 delta. Without exception, deltas become lower when a strike is further out-of-the-money and higher when a strike is further in-the-money. 

Intermediate: Identifying Delta on an Option Chain

On any option chain at a reputable broker, we can see delta across the whole option chain for each strike (the target price an option is based on). The main inefficiency to look out for is how close 50 delta is to the inflection point at-the-money, which is usually separated graphically by differently-colored shaded regions. 

<SPX option chain (20 DTE) retrieved from thinkorswim mobile>

Advanced: Discounted Gamma

In extreme examples such as fringe conditions for VIX options, call deltas can be very high near the money. This robs call holders of part of their convexity ramp, which is how much higher (on the way to 100 delta) that an option can increase and accelerate toward as it continues to be right about direction. The magic of options which allows those deltas to change is gamma, which is the acceleration of directional exposure.

Expert: Lognormal Analysis

Due to lognormal distortions, which pertain to how the options market prices-in how stocks can only drop to zero but increase indefinitely, 50 delta would often be below the money for calls and above it for puts. Sometimes this characteristic gets inverted, mostly because of volatility skew (differences in implied volatility for different strikes on the same date), and so this can be important information to organize graphically. 

However, one can get a good idea of it simply by looking at the option chain. For example, here is a 20-day view of VIX options (expiring on April 19), where this lognormal penalty is actually lower than it generally has been over the past year, which for a while was closer to about 65 delta for the first ITM collar strike. 

< VIX option chain (20 DTE) retrieved from thinkorswim mobile>

This temporarily softer lognormal penalty might actually be part of a bull case for VIX calls as a partial aspect because it suggests more value from a gamma perspective (it has more deltas to ride up a convexity ramp on for favorable and accelerating delta exposure). Either way, that penalty is still somewhat present with the first strike ITM truncating to 57 delta and the first strike OTM truncating to 50 delta. 

Ironically, we say penalty but the original design of options was for calls to be weighted this way as a principal lognormal assumption. What this means is that calls should have slightly higher IV% (implied volatility) because they can theoretically increase forever as compared to puts–which can only benefit [intrinsically] from an underlying until it drops down to zero. 

The reality however is that the original lognormal assumptions are reliably inverted for index products as a result of heavy call writing (selling to open) and put buying in order to protect long-only portfolios. This protection is done with covered calls (only the short calls for mild protection) and option collars (both the short calls and long puts for strong protection). However, since the VIX is usually inverse to SPX, the normal inversion actually produces the intended effect of the original lognormal design of the theoretical pricing for options.