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Barrier Option Calculator

Standard single-barrier options: knock-in and knock-out calls and puts with rebate. Closed-form Haug (1997) pricing with full Greeks and barrier profile diagram.

Knock-In — option activates if barrier is touched
Knock-Out — option extinguishes if barrier is touched

Option Inputs

$
$
$
H must be < S for Down barriers, > S for Up barriers.
Warning: barrier direction may be inconsistent with current spot.
$
Amount paid if knocked out (KO) or if never touched (KI). Set to 0 for standard barriers.
%
%
%
For stocks: b = r. For FX: b = rd − rf. For futures: b = 0.
Down-Out Call
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USD
Barrier Price
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Vanilla Price
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Ratio
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Delta
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Gamma
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Vega (per 1%)
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Theta (per day)
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What Are Barrier Options?

Barrier options are path-dependent derivatives whose existence depends on whether the underlying asset price reaches a pre-specified barrier level H during the option's life. They are among the most traded exotic derivatives in FX, equity, and commodity markets — primarily because they are cheaper than vanilla options (the barrier restricts the payoff, reducing the option's value).

The Eight Standard Types

There are 8 standard single-barrier option types, organized by call/put × knock-in/knock-out × up/down:

  • Down-In Call (CDI) — becomes a call only if spot falls to H. Useful for bullish views with a "only hedge if it gets dangerous" trigger.
  • Up-In Call (CUI) — activates if spot rises to H. Exotic: the option only becomes active if the market has already moved strongly up.
  • Down-In Put (PDI) — put activates if spot falls to H. Common in structured products as a conditional floor.
  • Up-In Put (PUI) — put activates if spot rises to H. Rarely natural, used in reverse-convertible structures.
  • Down-Out Call (CDO) — standard call that dies if spot falls to H. Cheaper than vanilla; protects buyer if spot stays above H.
  • Up-Out Call (CUO) — call knocked out if spot rises above H. The buyer effectively has a capped upside — explains the lower premium.
  • Down-Out Put (PDO) — put knocked out if spot falls to H. Counterintuitive: the protection dies exactly when the market crashes. Popular in yield enhancement products.
  • Up-Out Put (PUO) — put knocked out if spot rises to H. Sometimes used in covered call / put spread structures.

Pricing Formula (Haug 1997)

This calculator uses the analytical closed-form for standard European barriers under Generalized Black-Scholes (GBM, constant vol, continuous monitoring). The formula, derived by Haug (1997) building on Rubinstein & Reiner (1991), constructs each barrier type from 6 component terms (f₁…f₆), selecting the appropriate combination based on whether X > H or X ≤ H.

Key parameters:

μ = (b − σ²/2) / σ², λ = √(μ² + 2r/σ²)

The component terms use both N(·) and (H/S)^(2μ) reflection terms that capture the probability that the barrier has been touched under a log-normal diffusion.

Barrier Parity

A fundamental identity: Knock-In + Knock-Out = Vanilla. For any consistent parameters:

CDI + CDO = Vanilla Call  ·  PDI + PDO = Vanilla Put

This can be verified using this calculator: sum CDI and CDO with the same inputs — they must equal the Black-Scholes call price (shown in the "Vanilla Price" comparison).

Greeks and Risk Management

Delta near the barrier is the biggest risk in barrier options. A knock-out call near H has large negative delta (every $1 move toward H causes major value loss). A knock-in option near H has large positive delta. The delta can be discontinuous at H, making exact hedging impossible — this is the "pin risk" dealers face near expiry.

Gamma can be negative for knock-out options near the barrier. This reverses the normal gamma convention and requires careful risk monitoring. A dealer who sells a down-out call and tries to delta-hedge is short gamma near H — bad news in volatile markets.

Vega for knock-out options is typically lower than vanilla; for knock-in options near the barrier, vega can be large because higher vol increases the probability of touching H.

Rebate

Many traded barrier options include a rebate k — a consolation payment that mitigates the harsh all-or-nothing nature of the knock-out. For knock-out options, the rebate is paid at expiry if the option was knocked out. Setting k = 0 gives standard barrier behavior. The rebate component uses the formula:

f₆ = k × [(H/S)^(μ+λ) × N(η·z) + (H/S)^(μ−λ) × N(η·z − 2ηλσ√T)]

Worked Example — Down-Out Call (CDO)

S = $100, X = $100, H = $90, k = 0, T = 90 days, σ = 20%, r = 4.5%, b = 4.5%.

  • Vanilla call ≈ $4.00
  • CDO ≈ $3.26 (82% of vanilla — you save 18% in premium, but lose everything if spot drops to $90)
  • CDI ≈ $0.74 (the "missing" 18% — only pays off if you first trade through $90)
  • Delta CDO ≈ +0.51 (similar to vanilla when well away from barrier)
  • If spot moves from $100 to $91, CDO delta spikes dramatically — gap risk kicks in

This is why traders say barrier options are "cheap for a reason" — the premium savings come with path-dependency risk that is hard to hedge near expiry.

Applications

  • FX structured products — range binary notes, barrier reverse convertibles, accumulator contracts
  • Equity structured products — auto-callable notes (knock-out + coupon), principal protected notes with KI puts
  • Commodities — barrier options on oil, gold reduce hedging cost for producers / consumers
  • Credit-linked notes — knock-in puts tied to credit events

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