Most Black-Scholes tutorials skip the parts where things break
You copy-paste the formula from Wikipedia, wrap it in a Python function, and suddenly your call options are worth negative money or your Greeks explode near expiry. The Black-Scholes equation itself is elegant — five parameters, one closed-form solution. But between the math and production-ready code lies a minefield of numerical instability, parameter validation, and domain constraints that most tutorials quietly ignore.
I'm going to build a Black-Scholes pricer twice: first the naive version that replicates what you'd write after reading the paper, then a hardened version that survives edge cases I've actually hit. The goal isn't just to compute option prices — it's to show you where the formula betrays you and how to defend against it.
The formula everyone starts with
The Black-Scholes price for a European call option is:
$$C = S_0 N(d_1) - K e^{-rT} N(d_2)$$
where:
$$d_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}$$
$$d_2 = d_1 - \sigma\sqrt{T}$$
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