Barrier Options: Pricing, Risk Management, and Desk Charging

Abstract

These notes develop a unified framework for the pricing, risk management, and desk charging of barrier options. The starting point is the windowed double barrier option — a contract whose knockout or knockin condition is active only during a specified time window [T₁,T₂] within the option’s life — from which all standard barrier structures emerge as special cases by varying the window bounds, the barrier levels, or both. Single barriers, European barriers, one-touch and no-touch options are each identified as particular limits of this general case. Pricing is treated in two ways. Analytic results are derived under continuous monitoring: for single barriers, the reflection principle delivers closed forms for the American window T₁ = 0, T₂ = T and for each of the three non-degenerate sub-window configurations (early window, late start, full window), with prices expressed as bivariate and trivariate Normal CDFs; for double barriers, a Kunitomo–Ikeda image series extends these to every windowed configuration, converging geometrically in the number of image reflections. The Broadie–Glasserman–Kou continuity correction connects these results to the practically important discrete-monitoring case. Richer model dynamics (local-vol, stochastic-vol, jumps) are treated numerically via a three-stage backward PDE or a Brownian-bridge-corrected Monte Carlo. The model risk problem is identified as the central challenge for barrier options: multiple models calibrated to the same vanilla surface can give materially different barrier prices, in contrast to vanilla options where the implied volatility is the sufficient statistic. A survey of the main model families — Black–Scholes, local volatility, Heston, local stochastic volatility, and jump-diffusion — assesses the strengths and limitations of each for barrier option applications. A detailed treatment of the Greeks — delta discontinuity at the barrier, gamma blow-up near the barrier approaching expiry, and the sign reversal of vega for knock-out options — motivates the hedging section, which evaluates dynamic delta hedging, static replication with vanilla options, and the hybrid approach appropriate when the vanilla market has finite depth. The final sections address the desk charging problem. A three-component decomposition of the charge is proposed: a vanilla bid-offer component, a barrier survival premium linked to the conditional knockout probability, and an interaction adjustment that accounts for the correlation between terminal payoff and the barrier survival event. This decomposition yields a principled client-facing upfront fee and a dynamic internal reserve. The arbitrage constraint imposed by in-out parity on the combined charge is made explicit. Asset class specifics for foreign exchange and commodities are discussed throughout, with the role of surface liquidity and gap risk examined in each.