Section 26.2

Black-Scholes Rules of Thumb

The at-the-money approximation, implied volatility from a price, limiting cases (infinite maturity or volatility), the Black-Scholes PDE.

Toolkit

For $S_0=Ke^{-rT}$ (forward-ATM) and $r=0$ , Black-Scholes reduces to a near-linear formula in volatility and maturity:

C \approx S\,\sigma\sqrt{T/2\pi}\approx 0.4\,\sigma S\sqrt{T},

useful for mental pricing and for backing out implied volatility, $\sigma \approx \dfrac{C}{0.4S\sqrt{T}}$ .

Limiting cases follow from no-arbitrage bounds $S_0-Ke^{-rT}\le C\le S_0$ : as $T\to\infty$ (with $r>0$ ) or $\sigma\to\infty$ , $C\to S_0$ (and similarly for puts, bounds collapse appropriately).

Every option value, regardless of payoff, satisfies the Black-Scholes PDE

\Theta+\tfrac12\sigma^2S^2\Gamma+rS\Delta=rV,

so knowing any four of $\Theta,\Gamma,\Delta,V,r,S,\sigma$ pins down the fifth.

Problems

26.2 At-the-Money Call Approximation+2 variations Easy Free
26.3 Call with Infinite Maturity Medium
26.4 Option Price from the Greeks Easy