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I'm attempting to write a tool that will automatically calculate the expected value of arbitrary options positions, and I need to clarify my understanding. I am neither a statistician nor a professional quant so please permit me a bit of (perhaps unnecessary) exposition:

Scenario

Suppose I am interested in ticker $\\\$XYZ$ and I open a standard put credit spread by shorting a put at strike $K_{short}$ with delta $\Delta_{short}$ and buying a protective put with strike $K_{long}$ and delta $\Delta_{long}$ for a net credit/premium of $C$. Obviously, I can divide my calculation into 3 cases for ease of computation:

  1. At expiration the short leg expires out of the money with $\\\$XYZ$ closing above $K_{short}$.

  2. At expiration the short leg expires in the money with $\\\$XYZ$ closing between $K_{short}$ and $K_{long}$.

  3. At expiration the short and long legs expire in the money with $\\\$XYZ$ closing below $K_{long}$.

The question in 2 parts

Among retail traders it is commonly accepted that an option's Delta approximates the chance of that option expiring in the money. My questions are:

a. Is it valid to use delta in this case? I.e. for case 1 above, is it reasonable to calculate EV by:

$EV = \Delta_{short} \cdot (K_{short}-C) + (1-\Delta_{short}) \cdot K_{short}$

?

Or, given that Black Scholes is primarily applicable to European options, is delta not appropriate here? Would it be better to use a stochastic model and derive probabilities from something like monte carlo trials? Is there some other limitation of which I'm not aware?

b. For case 2 above, some of the sources I've read online suggest "using the lognormal distribution" but don't go into any further detail. I'm familiar with the log-normal distribution in the sense that I understand its basic properties, but it's not clear to me specifically how to calculate/estimate the probability that the price falls between the strikes. If the log-normal CDF was $F(x)$, is it as simple as $F(\Delta_{short}) - F(\Delta_{long})$ ?

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