Module perpetual 

Perpetual (infinite-maturity) American options — Merton (1973) closed forms. Unlike the finite-maturity approximations (baw, bjerksund_stensland) these are exact: with no expiry the American value solves the stationary ODE 1/2 sigma^2 S^2 V'' + b S V' - r V = 0 with value matching and smooth pasting at a constant exercise boundary.

The exponents y1 > 1 > 0 > y2 are the roots of the quadratic 1/2 sigma^2 y (y - 1) + b y - r = 0 — the same beta that drives the Barone-Adesi-Whaley and Bjerksund-Stensland boundaries, whose infinite-maturity limit these formulas are. Finite-maturity American prices increase in maturity toward the perpetual value (tested).

Conventions match the rest of the library: q is the total carry (dividend yield + borrow), b = r - q.

Functions

exercise_boundary

The constant early-exercise boundary: exercise the call once the spot rises to y1/(y1-1) K, the put once it falls to y2/(y2-1) K.

perpetual_call

Perpetual American call.

perpetual_put

Perpetual American put. Requires r > 0 (with no discounting the optimal-stopping problem degenerates).