Exotic Derivatives Lab

Barrier Option Pricing & Greeks Lab

Price European knock-in and knock-out barrier options using closed-form analytical formulas. Model in-out parity, barrier pin risk, hitting probabilities, and discrete monitoring shifts.

Exotic Archetypes:

Contract & Barrier Parameters

Down-Barrier (H < S)

Exotic Valuation Summary & Hitting Metrics

Knock-Out Price
$0.00
Down-and-Out Call
Vanilla Option Price
$0.00
Black-Scholes
Barrier Discount
$0.00
0.0% Savings
Knockout Probability
0.0%
P(Touch Barrier)
Knock-Out Price Knock-In Price Knock-Out + Knock-In Vanilla Price In-Out Parity Check
$0.00 $0.00 $0.00 $0.00 Matched (±0.0000)

Barrier Greeks (Finite Difference Engine)

Delta (Δ) 0.0000
Gamma (Γ) 0.0000
Vega (ν) 0.0000
Theta (Θ / day) 0.0000
Rho (ρ) 0.0000
Barrier Pin Risk: As spot approaches barrier H, delta exhibits sudden reversals and gamma can become extremely negative for knock-out options.

Visual Analytics & Pin Risk Profile

Blue: Barrier Option Value | Red Dash: Vanilla European Value | Gold Dash: Barrier Level (H)

Matrix 1: Spot Price (S) vs. Barrier (H) on Option Price

Evaluates barrier option valuation across spot price shifts (±20%) and barrier levels.

Matrix 2: Volatility (σ) vs. Expiry (T) on Knockout Probability

Analyzes first-passage barrier hitting probability P(τ ≤ T) across market volatility and maturities.

Mathematical Foundation: Closed-Form Barrier Formulas

1. Analytical Reiner-Rubinstein Architecture

Robert Merton (1973) and Reiner-Rubinstein (1991) solved European single barrier options by decomposing the reflection principle of geometric Brownian motion into standard building blocks A, B, C, D, E, F:

μ = (b - 0.5σ²) / σ²
A = φ S e^{(b-r)T} N(φ x_1) - φ K e^{-rT} N(φ(x_1 - σ√T))
B = φ S e^{(b-r)T} N(φ x_2) - φ K e^{-rT} N(φ(x_2 - σ√T))
C = φ S e^{(b-r)T} (H/S)^{2(μ+1)} N(η y_1) - φ K e^{-rT} (H/S)^{2μ} N(η(y_1 - σ√T))
D = φ S e^{(b-r)T} (H/S)^{2(μ+1)} N(η y_2) - φ K e^{-rT} (H/S)^{2μ} N(η(y_2 - σ√T))

For instance, a Down-and-Out Call with strike K ≥ H is evaluated simply as C_{do} = A - C + E.

2. In-Out Parity & Discrete Monitoring Shift

Because a barrier option must either touch the barrier or not, holding both a knock-in and a knock-out option replicates a standard European vanilla contract with mathematical certainty:

In-Out Parity: V_{in} + V_{out} = V_{vanilla}
Hitting Probability: P(τ ≤ T) = N(-d_{hit1}) + (H/S)^{2μ} N(-d_{hit2})
Discrete Shift: H_{eff} = H · exp(±0.5826 · σ√Δt)

Broadie, Glasserman, and Kou (1997) proved that applying the 0.5826 continuity correction adjusts continuous closed-form formulas to match discrete daily monitoring with sub-basis-point accuracy.

Barrier Option Mastery Quiz

Frequently Asked Questions

A barrier option is a path-dependent exotic financial derivative whose payoff depends not only on the underlying asset's price at expiration, but also on whether the underlying asset reaches a predetermined price barrier H during the option's lifetime. Options can either knock in (activate) or knock out (terminate) upon touching the barrier.

In-out parity states that the sum of a knock-in option and a knock-out option with the same strike, underlying, expiration, and barrier level exactly equals the price of an otherwise identical standard European vanilla option: V_in + V_out = V_vanilla (assuming zero rebate). This fundamental no-arbitrage law enables rapid cross-validation of exotic pricing engines.

Barrier pin risk occurs when the underlying asset approaches the barrier level H near expiration. For a knock-out option, the payoff jumps abruptly from a substantial intrinsic value to zero if the barrier is crossed. This causes delta to change signs rapidly and gamma to become extremely large or negative, making dynamic delta hedging challenging for market makers.

Standard closed-form formulas assume the barrier is monitored continuously (24/7). In practical exchange contracts, barriers are often monitored discretely (e.g., daily closing prices). Because discrete monitoring offers fewer opportunities to touch the barrier, knock-out options are worth more and knock-in options are worth less. Broadie, Glasserman, and Kou (1997) showed that shifting the continuous barrier by H * exp(+-0.5826 * sigma * sqrt(dt)) accurately prices discretely monitored contracts.

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