How does a range accrual note work, and what type of exotic option is embedded in its coupon structure?

A range accrual note pays a coupon that accumulates day by day, but only on days when a reference rate (or index, or FX rate) falls within a predetermined range. Each daily observation is equivalent to a digital (binary) option, making the range accrual a portfolio of hundreds of daily digitals.\n\nAccrual Mechanism:\n\nThe coupon for each period is:\n\nCoupon = C x (n / N)\n\nWhere C is the stated annual coupon, n is the number of business days the reference rate was inside the range, and N is the total business days in the period.\n\nWorked Example -- Thornfield Structured Products:\n\nThornfield issues a 1-year range accrual linked to 3-month SOFR:\n\n| Parameter | Value |\n|---|---|\n| Notional | $10,000,000 |\n| Stated coupon | 7.60% p.a. |\n| Lower bound | 3.75% |\n| Upper bound | 5.25% |\n| Payment frequency | Quarterly |\n| Day count | Act/360 |\n\nQ1 results (63 business days): SOFR inside range for 58 days.\nQ1 coupon = $10,000,000 x 7.60% x (90/360) x (58/63) = $190,000 x 0.9206 = $174,914\n\nQ2 results (65 business days): SOFR rises above 5.25% for 20 days.\nQ2 coupon = $190,000 x (45/65) = $131,538\n\n`mermaid\ngraph TD\n A[\"Range Accrual
= Bond + Strip of Daily Digitals\"] --> B[\"Day 1: Is rate in [3.75%, 5.25%]?\"]\n A --> C[\"Day 2: Is rate in [3.75%, 5.25%]?\"]\n A --> D[\"...
Day N\"]\n B -->|\"Yes\"| E[\"Accrues 1/N of coupon\"]\n B -->|\"No\"| F[\"Zero accrual for that day\"]\n C -->|\"Yes\"| E\n C -->|\"No\"| F\n D --> E\n D --> F\n E --> G[\"End of period:
Sum accrued days / Total days\"]\n F --> G\n`\n\nOption Decomposition:\n\nEach daily observation embeds a long digital call at the lower bound (pays if rate >= 3.75%) and a short digital call at the upper bound (cancels payoff if rate >= 5.25%). Across 252 business days, the note contains 504 individual digital options.\n\nHedging Challenge:\n\nDigital options have theoretically infinite gamma at the strike. Near the boundaries (e.g., SOFR at 5.24%), the payoff switches discontinuously from full accrual to zero. Dealers hedge using call spreads (a tight bull spread that approximates the digital), creating a 'ramp' payoff near each boundary. This ramp width determines the dealer's hedging cost and affects the coupon offered.\n\nSensitivity Analysis:\n- Implied volatility: Higher vol increases the probability of range excursion, reducing expected accrual. Paradoxically, higher vol means the investor should demand a higher stated coupon.\n- Rate level: If the current rate is centered in the range, more days accrue. Near boundaries, accrual risk spikes.\n- Correlation (multi-asset ranges): Some range accruals reference two variables (e.g., SOFR stays in range AND EUR/USD stays in range). Lower correlation between references dramatically reduces expected accrual.\n\nPractice digital option decomposition problems in our FRM question bank.