I understand that delta can be seen as a probability proxy for an option expiring in the money, as well as deltas for call options ranging from 0 to 1 and deltas for put options ranging from 0 to -1.
How/can you use delta to calculate a probability of profit proxy for spreads or positions with multiple legs? For example, a position that consists of both long and short calls and puts, as well as a common stock position. How would one go about calculating the overall probability of profit proxy of that portfolio using delta?
Consider the following portfolio scenario:
Current price of AAPL: 150.00 AAPL Sep 17 2021 157.5 Put (Delta: -0.889) AAPL Sep 17 2021 149 Put (Delta: -0.427) AAPL Sep 17 2021 148 Call (Delta: 0.651) AAPL Sep 17 2021 146 Call (Delta: 0.778) AAPL Common Stock; 5 shares purchased at 150.00
Furthermore, if using delta is not a suggested way to go about calculating the overall probability of profit proxy of a portfolio, how can we use N(d1) from the Black-Scholes formula to do the same?