How does a lookback option guarantee the best possible payoff over the option's life, and why is it so expensive?

Lookback options grant the holder the benefit of hindsight — the payoff is determined using the most favorable price observed during the option's life. This path-dependent feature makes lookbacks the most expensive standard exotic option.\n\nTwo Types of Lookback Options:\n\n`mermaid\ngraph TD\n A[\"Lookback Options\"] --> B[\"Fixed Strike Lookback\"]\n A --> C[\"Floating Strike Lookback\"]\n B --> D[\"Call: max(S_max - K, 0)
Put: max(K - S_min, 0)\"]\n C --> E[\"Call: S_T - S_min
Put: S_max - S_T\"]\n D --> F[\"Uses extreme price
against fixed K\"]\n E --> G[\"Strike is set retroactively
to optimal level\"]\n G --> H[\"Always in-the-money
(if any price movement)\"]\n`\n\nWhy Floating Strike Lookbacks Always Pay Off:\n\nA floating strike lookback call has payoff S_T - S_min. Since S_min is by definition less than or equal to S_T (the minimum price observed must be at or below the terminal price), the option always finishes in the money if there has been any price variation at all.\n\nWorked Example:\nGranville Asset Management buys a 3-month floating strike lookback call on Stellarion Corp stock. Initial price is $65.\n\nPrice path observed (weekly): $65, $68, $63, $61, $67, $72, $69, $74, $70, $66, $71, $73\n\n- S_min = $61 (lowest observed)\n- S_T = $73 (terminal price)\n- Payoff = $73 - $61 = $12.00\n\nA vanilla ATM call with strike $65 would have paid only $73 - $65 = $8.00. The lookback captures an additional $4.00 by retroactively setting the strike at the path minimum.\n\nPricing Considerations:\n\nThe Goldman-Sosin-Gatto (1979) formula provides closed-form pricing for continuously monitored lookback options. Key inputs:\n\n- The lookback premium increases rapidly with volatility (more volatile paths produce more extreme observations)\n- Longer maturities substantially increase the cost (more time = more opportunities for extreme prices)\n- Discrete monitoring reduces the premium because the true extreme may occur between observation points\n\nCost Comparison:\n\nLookback options typically cost 1.5x to 3x a comparable vanilla option. The multiplier rises with volatility and maturity. For a 6-month ATM option with 30% volatility, a lookback might cost approximately 2.2x the vanilla premium.\n\nPractical Uses:\n- Timing-sensitive hedges where entry or exit timing is uncertain\n- Executive compensation structures (guaranteeing the best exercise price)\n- Reducing regret in strategic hedging programs\n\nPractice exotic option pricing in our FRM question bank.