How do Bermuda option exercise windows work, and where do they fall in the American-European pricing spectrum?

Bermuda options occupy the middle ground between European and American options by allowing exercise only on specific predetermined dates during the option's life. This intermediate flexibility results in a price between the two extremes.\n\nExercise Structure:\n\nA typical Bermuda option might allow exercise on monthly or quarterly dates:\n\n| Feature | European | Bermuda | American |\n|---|---|---|---|\n| Exercise dates | Expiry only | Specified dates | Any business day |\n| Flexibility | Lowest | Moderate | Highest |\n| Premium | Lowest | Middle | Highest |\n| Pricing method | Closed-form (BSM) | Lattice / Monte Carlo | Lattice (binomial) |\n\nPricing Approach:\n\nBermuda options are typically priced using backward induction on a binomial or trinomial tree, but evaluation of early exercise is only performed at the permitted exercise nodes.\n\nAt each permitted exercise date t_i, the holder compares:\n- Immediate exercise value: max(S - K, 0) for calls\n- Continuation value: risk-neutral expected value of holding to the next exercise date\n\nOptimal exercise occurs when immediate value exceeds continuation value.\n\nWorked Example:\nFalconridge Insurance purchases a 1-year Bermuda put on Dalton Power Corp bonds (current price $102, strike $100) with quarterly exercise dates at months 3, 6, 9, and 12.\n\nUsing a 4-period binomial tree with u = 1.08, d = 0.926, r = 1% per quarter:\n\nPeriod 0: S = $102\nPeriod 1: S_u = $110.16, S_d = $94.45\nPeriod 2: S_uu = $118.97, S_ud = $102.00, S_dd = $87.47\n\nAt month 6 (Period 2), evaluate the down-down node:\n- Exercise value: max($100 - $87.47, 0) = $12.53\n- Continuation value (expected value of period 3-4 paths): $11.80\n- Decision: Exercise early (since $12.53 > $11.80)\n\nThe Bermuda put value at inception comes out to approximately $4.85, compared to:\n- European put: $4.20\n- American put: $5.10\n\nCommon Markets for Bermuda Options:\n- Interest rate swaptions (Bermuda swaptions are the most liquid exotic in fixed income)\n- Callable bonds (the embedded option is Bermuda-style with quarterly or semi-annual call dates)\n- Convertible bonds with discrete conversion windows\n- Energy markets with seasonal exercise dates\n\nWhy Not Just Buy American?\n- The American premium over Bermuda is often small (the early exercise premium concentrates at discrete points)\n- Bermuda options are easier to hedge because exercise decisions occur at known dates\n- Dealers can more efficiently manage inventory risk with discrete exercise events\n\nExplore option exercise strategies in our FRM course materials.