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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?

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  • $\begingroup$ The delta of a call is the risk-neutral probability of that call being in the money. The risk neutral probability is not the same as a 'real' probability - if you examine the empirical probability of a certain moneyness ending up in the money there is no reason that you would get the options delta. One way to think about risk neutral probabilities (in black scholes framework) is that they give the distribution of stock prices if you are allowed to hedge the first moment (drift) risk away. What you are left with is exposure to some random dispersion due to the second and higher moments. $\endgroup$ Sep 14, 2021 at 2:34
  • $\begingroup$ Where should I look to calculate the empirical probability of the scenario I've included in my question if delta is not my answer? $\endgroup$ Sep 14, 2021 at 22:02
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    $\begingroup$ The empirical probability is just the frequency of seeing a certain return. Why not start with a histogram of stock returns over the lifetime of the option? $\endgroup$ Sep 15, 2021 at 3:42

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