Gap Options: Trigger Rules, Signed Payoffs, and Digital Replication

Audit terminal gap options through exact trigger and payment strikes, signed payoff regions, digital replication, model values, executable quotes, and settlement terms.

For educational purposes only; not investment advice. Investing may result in loss.

Direct answer

A terminal gap option uses one level to decide whether payment activates and another to calculate the contractual amount. For multiplier M, a strict European gap call can pay M x (S_T - K_pay) x 1{S_T > K_trig}; a strict gap put can pay M x (K_pay - S_T) x 1{S_T < K_trig}. The confirmation must define S_T, both strikes, the comparator, observation and rounding order, payment direction, floor, cap, rebate, currency, and settlement.

The algebraic amount can be negative. Adding max(..., 0) changes the claim and can make the trigger economically redundant. A terminal trigger is also not a path-monitored barrier, and the word “gap” does not mean overnight-gap insurance.

Reconstruct the legal claim before valuing it

Identify the legal wrapper, issuer or counterparty, clearing status, collateral, closeout terms, underlying, currency, signed quantity, multiplier, exercise style, and cash or physical settlement.
Copy K_trig, K_pay, call or put, strict or inclusive comparator, floor, cap, rebate, negative-payment provision, and which party owes each signed amount.
Lock the final reference source, fixing time and timezone, averaging, rounding sequence, correction, disruption, fallback, holiday, corporate-action, and calculation-agent rules.
Draw every payoff region and the exact boundary value; calculate left and right limits, jump direction, zero, negative-payment domain, and cash or share units. Keep payoff separate from premium, fees, funding, tax, and P/L.
For an unfloored terminal claim, verify gap call = call(K_trig) + (K_trig - K_pay) x cash digital call and gap put = put(K_trig) + (K_pay - K_trig) x cash digital put, using identical observation and settlement terms.
Benchmark a controlled model, then stress volatility skew, jumps, rates, dividends, borrow, credit, collateral, liquidity, discontinuous hedging, and executable entry and unwind quotes. A model or issuer mark is not a fill.
At maturity, apply the contractual evaluation order to the official reference, trigger, gross payment, corrections, cash or delivery ledger, fees, tax, counterparty recovery, and final reconciliation.

Four worked examples

Strict versus inclusive jump. A cash gap call has K_trig = $105, K_pay = $100, M = 100, and strict S_T > $105. At S_T = $104.99, payment is $0; at $105.00, it is $0, while an inclusive term would pay $500; at $105.01, it is $501; and at $120, it is $2,000. A $0.02 reference change produces a $501 payment change.
Negative-payment domain. An unfloored signed gap call has K_trig = $95, K_pay = $105, M = 100, and strict >. At $94.99 and $95.00, payment is $0; at $95.01, it is -$999; at $105, it is $0; and at $110, it is $500. Flooring the amount at zero would instead produce the payoff of a vanilla $105 call, apart from any boundary convention.
Black-Scholes replication benchmark. Let S_0 = $100, K_trig = $105, K_pay = $100, T = 0.5, r = 4%, q = 1%, and sigma = 25%. Then d_1 = -0.102757686125 and d_2 = -0.279534381421. The model gap-call value is $7.459148829072: a $5.548166344537 vanilla call at $105 plus a $1.910982484535 digital adjustment. This is a theoretical value before credit, liquidity, funding, and fees.
Rounding order controls settlement. Using the first contract, suppose terms round the raw official value to cents before applying the strict trigger. Raw 105.004 becomes $105.00 and pays $0; raw 105.006 becomes $105.01 and pays $501. Testing the unrounded values first would instead calculate $500.40 and $500.60, so evaluation order is contractual rather than cosmetic.

Contract, model, and settlement risks

Reversing the trigger and payment strikes.
Applying the wrong call, put, long, short, or payment-direction sign.
Confusing strict and inclusive comparators at the boundary.
Missing a negative-payment region, floor, cap, or rebate.
Using the wrong quantity, multiplier, deliverable, point value, currency, or FX rule.
Confusing terminal observation with path or barrier monitoring.
Using the wrong official reference source, fixing time, or timezone.
Applying averaging, rounding, correction, or fallback steps in the wrong order.
Ignoring market disruption, holiday, calculation-agent discretion, or later correction.
Missing a corporate action or adjusted contract term.
Treating contractual payoff as net P/L before premium, fees, funding, and tax.
Confusing cash settlement with a physical strike-cash and delivery ledger.
Putting K_pay into the probability boundary or K_trig into the cash term in a model.
Ignoring volatility skew, jumps, rates, dividends, borrow, and forward inputs.
Understating digital-like hedge instability, gaps, halts, and model error near the trigger.
Substituting midpoint, model, or issuer marks for executable price, size, and unwind liquidity.
Ignoring transfer limits, early termination, distribution costs, and secondary-market absence.
Assuming issuer, bilateral counterparty, collateral, netting, CCP, and closeout risks are interchangeable.
Assuming FLEX eligibility, listing, margin treatment, or OCC clearing without the exact venue and series record.
Missing last trade, exercise, final fixing, payment timing, legal, accounting, tax, recovery, or reconciliation details.

Common misconceptions

“A gap option protects against overnight gaps.” Its name refers to two contractual strike levels.
“The two strikes are interchangeable.” One controls activation and the other controls payment size.
“The amount is always floored at zero.” An unfloored algebraic contract can require a negative payment.
“Touching the level activates the option.” Terminal observation and the exact comparator control.
“An option or FLEX label guarantees listing, liquidity, or OCC clearing.” The actual wrapper, rules, and series govern.