Module baw 

Barone-Adesi & Whaley (1987) quadratic approximation for American options.

A fast, closed-form-ish alternative to a tree or PDE for American vanillas: the early-exercise premium is approximated by the dominant term of the quadratic (MacMillan) approximation to the American PDE, so pricing is a European Black-Scholes evaluation plus a short Newton solve for the critical exercise price. Accuracy is a few cents versus a fine binomial tree for typical parameters — use it when speed matters more than the last basis point (e.g. a large book revalued many times), and a tree/FD solve when precision is paramount.

Everything is expressed with a continuous cost of carry b = r - q, where q is the total carry (dividend yield plus borrow). With b = r - q the generalized Black-Scholes formula is exactly bs_price, so the European leg is shared with the rest of the library.

Key properties the implementation preserves:

• an American call on a non-dividend payer (b >= r, i.e. q <= 0) is never exercised early, so it equals the European call;
• the price is bounded below by intrinsic value and by the European price (the early-exercise premium is non-negative).

Functions

critical_spot

Critical early-exercise spot for this option (the BAW boundary S*).

early_exercise_premium

Early-exercise premium: the BAW American price minus the European price. Non-negative by construction.

npv

price

American vanilla price via the Barone-Adesi-Whaley approximation.

price_with

Reprice under a market move for portfolio PnL attribution: spot + d_spot, a parallel vol shift + d_vol, rate + d_rate, and d_time years of elapsed calendar time (which shortens maturity).