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The Black Scholes Model and Implied Volatility

C=S Φ(d1)−Ke−rTΦ(d2)C = S\,\Phi(d_1) - Ke^{-rT}\Phi(d_2)d1=ln⁡(S/K)+(r+12σ2)TσT,d2=d1−σTd_1 = \frac{\ln(S/K) + (r + \tfrac{1}{2}\sigma^2)T}{\sigma\sqrt{T}}, \qquad d_2 = d_1 - \sigma\sqrt{T}

Reading it

The formula is less opaque than it looks once you see the two pieces.

Ke−rTΦ(d2)Ke^{-rT}\Phi(d_2) is the discounted strike times Φ(d2)\Phi(d_2), and Φ(d2)\Phi(d_2) is the risk-neutral probability the option finishes in the money. So this term is the expected cost of exercising.

SΦ(d1)S\Phi(d_1) is the expected value of receiving the stock, conditional on exercise.

The call price is the difference: what you expect to get, minus what you expect to pay.

Key takeaway

Φ(d2)\Phi(d_2) is the probability of finishing in the money under the risk-neutral measure. Knowing that turns the formula from a black box into two readable terms.

Running it backwards

Every input except σ\sigma is observable. So in practice nobody uses the formula to produce a price. They observe the market price and solve for the volatility that reproduces it:

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