Module lookback 

Lookback options with closed forms (continuous monitoring, GBM).

• Floating strike (Goldman-Sosin-Gatto 1979): the call pays S_T - min S, the put max S - S_T — “buy at the low, sell at the high”, never out of the money;
• Fixed strike (Conze-Viswanathan 1991): the call pays (max S - K)+, the put (K - min S)+.

Both support seasoned contracts through the running extremum argument (pass the spot for a fresh option). The formulas take the usual carry b = r - q; the b = 0 singularity (the sigma^2/2b factor) is handled by nudging b by 1e-7, accurate to ~1e-6 in price and validated against Monte Carlo in the tests.

Monte Carlo pricing of the same payoffs monitors discretely on the simulation grid, so it sits below these continuous forms for max-based payoffs (above for min-based) by the O(sigma sqrt(dt)) extremum gap — the tests assert direction and convergence rather than pretending the two conventions coincide.

Functions

fixed_strike_lookback_price

Fixed-strike lookback (Conze-Viswanathan). extremum is the running maximum for a call, the running minimum for a put; pass the spot for a freshly issued option.

floating_strike_lookback_price

Floating-strike lookback (Goldman-Sosin-Gatto). extremum is the running minimum for a call, the running maximum for a put; pass the spot for a freshly issued option.