When you sell options, you’re making a very specific bet:
You collect a premium today in exchange for the chance the option finishes in-the-money.
Delta gives us a convenient shortcut to think about that bet.
Under Black-Scholes:
- Delta (Δ) is roughly the probability the option expires ITM
- 1 − Δ is roughly the probability it expires OTM — meaning you keep the full premium
So a natural question arises:
How much premium am I getting per unit probability that the option expires worthless?
A simple way to express that is:
Efficiency (Selling an Option) = Option Price / (1 – abs(Delta))
This isn’t a standard options metric — but it turns out to be a very revealing one.
The Trade-Off Every Option Seller Faces
As you move the strike price, two things change at the same time:
- Option price
- Delta (probability of exercise)
This creates a classic trade-off:
Deep OTM options
- Very high probability of expiring worthless
- Very small premium
At-the-money options
- Very large premium
- Only ~50% chance of expiring worthless
- High gamma risk
In-the-money options
- Huge premium
- Near-certain exercise (not meaningful for sellers)
The question is:
Where does the premium fall slowest relative to the probability of losing?
What Black-Scholes Tells Us
Under pure Black-Scholes assumptions (lognormal prices, constant volatility, no skew):
The ratio Price ÷ (1 − Delta Abstract Value) is maximised for options that are slightly out-of-the-money — not deep OTM and not ATM.
More precisely:
- Calls: Δ ≈ 0.25–0.40
- Puts: Δ ≈ −0.25 to −0.40
This is the sweet spot where:
- The option still carries meaningful time value
- The probability of expiring worthless remains high
- The premium hasn’t yet collapsed into tail-risk pricing
Why Deep OTM Looks Attractive — but Isn’t
Deep OTM options feel safe:
- Low delta
- High probability of success
But under Black-Scholes, option prices decay exponentially in the tails.
That means:
- Price drops faster than (1 − Δ) increases
- Each additional unit of “safety” costs you disproportionate premium
In short:
You gain probability faster than you gain dollars — and that’s a bad trade.
Why ATM Isn’t Optimal Either
ATM options maximise:
- Vega
- Absolute premium
But they also:
- Cut your probability of full profit roughly in half
- Expose you to maximum gamma risk
So while ATM options are expensive, they are not efficient when measured against the probability of keeping the full premium.
The Seller’s Sweet Spot
Under Black-Scholes, the most efficient options to sell cluster around:
- 25–40 delta
- Moderately OTM
- Time value still dominant
- Tail risk not yet dominant
This is not an accident.
It explains why:
- 30-delta shorts are a default in many systematic strategies
- Iron condors and strangles gravitate toward this region
- “Far OTM lottery selling” consistently underperforms
One Important Caveat
This result holds only under Black-Scholes.
In real markets:
- Volatility skew
- Jumps
- Fat tails
…all change the calculus — especially for puts.
That’s a topic for another post.
Final Takeaway
If you think like an option seller, the key question isn’t:
What has the highest probability of success?
It’s:
What pays me the most per unit of probability I’m taking?
Under Black-Scholes, the answer is clear:
Sell moderately OTM options — roughly 25–40 delta — where premium efficiency is highest.