{ "id": "961d13ee-bdad-556b-8daa-66e1457c0735", "slug": "binary-option", "term": "Binary Option", "aliases": [], "category": "Derivatives & Options", "category_slug": "derivatives-options", "difficulty": "intermediate", "definition": "A binary option (also called a digital option) is a type of option contract with a fixed, all-or-nothing payoff: the holder receives either a predetermined cash amount if the option expires in-the-money, or nothing if it expires out-of-the-money. There is no continuous payoff profile — the payout is binary.", "key_takeaways": [ "The two primary structures are the cash-or-nothing option (pays a fixed cash amount Q if in-the-money) and the asset-or-nothing option (pays the value of the underlying asset if in-the-money).", "Binary options are priced as the risk-neutral probability that the option expires in-the-money, discounted at the risk-free rate: C = e^(-rT) × Q × N(d2) in the Black-Scholes framework.", "Retail-marketed binary options have been widely associated with fraud and have been banned for retail investors in the EU, UK, Canada, and many other jurisdictions.", "Institutional uses include hedging event-driven binary outcomes (regulatory approvals, M&A closings), structured products, and as building blocks for other exotic payoffs.", "Delta of a binary option at-the-money is extremely high relative to its premium, creating complex hedging dynamics as the underlying approaches the strike near expiry." ], "detailed_explanation": "Binary options have a payoff structure that is discontinuous at the strike: the function jumps from zero to Q (or from zero to S_T for asset-or-nothing) as the underlying price crosses the strike. This discontinuity creates unique pricing and hedging challenges relative to vanilla options, which have a smooth, linear payoff profile above the strike.\n\nIn the Black-Scholes framework, the value of a cash-or-nothing binary call with payoff Q is: C_binary = Q × e^(-rT) × N(d2), where d2 = [ln(S/K) + (r − σ²/2)T] / (σ√T), and N(·) is the cumulative standard normal distribution. This formula is intuitive: N(d2) is the risk-neutral probability that the option expires in-the-money, and the discounting converts the expected future payoff to present value. The vanilla call option's N(d2) term has exactly the same interpretation — a vanilla call can be decomposed as a combination of asset-or-nothing and cash-or-nothing binary options.\n\nThe hedging of binary options is notoriously difficult near expiry and near the strike. The delta of a binary option spikes sharply as the underlying approaches the strike close to expiration, creating a near-vertical profile. A market maker holding a short binary call position faces exponentially growing delta exposure as the underlying oscillates near the strike in the final hours — a phenomenon known as 'gamma risk at the boundary.' To manage this, dealers typically add a small buffer spread around the strike or use replication strategies involving vanilla options.\n\nInstitutional-grade binary options are used in specific legitimate contexts. A pharmaceutical company might use a binary option to hedge the payoff profile of an FDA drug approval decision — if the approval occurs, the company's stock jumps; if it is rejected, the stock falls sharply. A binary call option on the company's stock with a strike at the pre-announcement price provides a clean hedge against the binary outcome. Similarly, merger arbitrageurs use binary options on acquirer stock to hedge against deal-break scenarios in pending M&A transactions.\n\nThe retail binary options market that grew on unregulated offshore platforms in the 2010s was largely fraudulent — platforms manipulated payouts, refused to honor withdrawals, and misrepresented win rates. Regulatory crackdowns by the FCA, ESMA, SEC, and CFTC effectively shut down the retail binary options industry in regulated markets by 2019.", "example": "An institutional trader believes that the ECB will announce a 25bps rate cut at its Thursday meeting (probability assessed at 65%). The trader buys a one-week binary call option on EUR/USD with a strike at the current spot (1.0850) and a fixed payout of $1 million if EUR/USD is above 1.0850 at Friday's close. The option is priced at approximately $490,000 — reflecting the risk-neutral probability of the ECB cutting and EUR/USD rallying (roughly 49% at market pricing). If the ECB cuts 25bps and EUR/USD rallies to 1.0950, the trader receives $1 million — a gain of $510,000. If EUR/USD finishes below 1.0850 for any reason (ECB holds, or risk-off sentiment overrides the cut), the trader loses the $490,000 premium. The fixed, known risk and reward makes this instrument useful for expressing a precisely-defined binary macro view.", "formula": "Cash-or-Nothing Binary Call: C = Q × e^(-rT) × N(d2)\nd2 = [ln(S/K) + (r - σ²/2) × T] / (σ × √T)\nCash-or-Nothing Binary Put: P = Q × e^(-rT) × N(-d2)", "formula_latex": null, "interactive_type": "calculator", "calculator_id": null, "related_terms": [ "asian-option", "call-option", "delta", "digital-option", "equity-swap", "esma", "expiration-date", "fungibility", "gamma", "hedging", "in-the-money", "market-maker", "normal-distribution", "option", "out-of-the-money" ], "backlinks": [ "class-of-options", "digital-option", "distant-months", "gamma", "hybrid-security", "straddle" ], "cross_references": [ "call-option", "delta", "digital-option", "esma", "gamma", "hedging", "in-the-money", "market-maker", "normal-distribution", "option", "out-of-the-money", "premium", "present-value", "stock" ], "tags": [ "level:intermediate", "cat:derivatives-options" ], "asset_classes": [ "derivatives" ], "regulators": [], "see_also": [], "sources": [], "wordcount": 745, "checksum": "b30b9b39c7e1c6c9", "version": "2026.05.03", "license": "CC-BY-4.0", "updated_at": "2026-09-07T02:15:24+00:00", "_links": { "self": "https://hedgefund.wiki/api/v1/terms/binary-option", "jsonld": "https://hedgefund.wiki/api/v1/terms/binary-option?format=jsonld", "markdown": "https://hedgefund.wiki/api/v1/terms/binary-option?format=md", "graph": "https://hedgefund.wiki/api/v1/graph/binary-option", "category": "https://hedgefund.wiki/api/v1/categories/derivatives-options", "schema": "https://hedgefund.wiki/schema/term.schema.json", "html": "https://hedgefund.wiki/#/terms/binary-option" } }