Implied volatility

Every input to Black-Scholes is observable except volatility. So traders run the formula backwards: given a market price, find the \(\sigma\) that reproduces it. That number, the implied volatility, is how options are actually quoted. It is the market's forecast of future volatility, plus a premium for risk.

There is no closed form, but the price is increasing in \(\sigma\) so a root exists and is unique whenever the price lies between the no-arbitrage bounds. Newton's method uses Vega as the slope:

It converges in a handful of steps near the money. Far from the money Vega is tiny and Newton can overshoot, so a careful solver keeps a bracket and falls back to bisection.

The smile

If Black-Scholes were right, every strike would imply the same volatility. In practice out-of-the-money puts trade at higher implied volatility than at-the-money options, and often calls do too: a smile or a skew when plotted against strike. The shape encodes fat tails and the market's fear of crashes. It is the single clearest piece of evidence that the model is a quoting convention rather than a description of reality.

American options without a tree

Two classic approximations give near-closed-form American prices.

• Barone-Adesi and Whaley (1987). Write the American price as the European price plus an early-exercise premium. The premium satisfies the same PDE, and dropping a small term turns it into an ordinary differential equation with a power-law solution \(A (S/S^*)^{q}\). One root-find locates the critical spot \(S^*\) at which exercise becomes optimal.
• Bjerksund and Stensland (1993). Assume the exercise boundary is flat over the option's life and integrate the payoff of exercising when the boundary is first hit. Everything is closed form, so it is the fastest method there is. Because the true boundary is not flat, the price is biased slightly low.

Both are accurate to a few cents for typical maturities and are widely used where speed matters more than the last basis point. On the comparison page they appear beside the tree and grid prices for any American contract.