According to most books the ATM option is the option with a delta of 0.50. However, this is only the case when the distribution is normal. The more positively skewed the distribution, the further the 0.50 delta option is out-of-the-money (for calls).
The formula to calculate the 0.50 delta option strike is equal to: S . e^(σ^2/2)
Interesting Observations About Options
- Even in a continuous distribution, the higher the volatility, the more positively skewed the distribution, the further OTM the 50d call strike lives.
- The cheapest straddle will occur at the median outcome or the ATM strike.
- The most expensive butterfly will have its “body” near the theoretical mode. This makes sense since a butterfly which is just a spread of 2 vertical spreads is a pure bet on the distribution. If you chart the price of all the butterflies equidistantly across strikes you will have drawn the probability density function implied by the options market!
In Black Scholes:- The term for delta is N(d1).
- The term for the probability of finishing in the money is N(d2).
What’s the relationship between d2 and d1?
- d2 = d1 – σ√t
The math defines the relationship we figured out intuitively:
The higher the volatility the more delta and probability will diverge!
Delta and probability are only similar when an option is near expiration or when it’s vol is “low”.
For more read:
https://moontowermeta.com/lessons-from-the-50-delta-option/
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