Option pricing models

Black–Scholes

The continuous-hedging limit of the binomial argument, and the industry’s quoting convention.

C=S0Φ(d1)−Ke−rTΦ(d2),d1=ln⁡(S0/K)+(r+σ2/2)TσTC = S_0\Phi(d_1) - Ke^{-rT}\Phi(d_2), \qquad d_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}

Where

Φ(d2)\Phi(d_2)
Risk-neutral probability of finishing in the money.
Φ(d1)\Phi(d_1)
The delta. Not the same number as Φ(d2)\Phi(d_2).
d2=d1−σTd_2 = d_1 - \sigma\sqrt{T}
The two arguments differ by one unit of total volatility.

Assumptions

  • Continuous costless hedging — broken by spreads and discrete rebalancing.
  • Constant volatility — contradicted by the smile.
  • Lognormal returns with no jumps — contradicted by every crash.

Sanity check. The expected return of the stock does not appear anywhere. If it does, check your working.

Where this is taught