Designed Dependence for Tractable Two-Asset First-Passage Pricing

Closed-form multi-asset first-passage prices become rare once dependence is allowed to vary with the state. We build a two-asset local-volatility/local-correlation family in which relative-performance passage problems stay one-dimensional by construction. Under a common instantaneous volatility, the log-ratio and log-product are orthogonal, and their variance rates form a simple budget. Holding the variance rate of the log-ratio fixed while leaving the orthogonal index variance free produces the Brownian-dispersion model: the log-ratio is an exact Brownian motion under the pricing measure, whereas the geometric-mean index carries the state-dependent volatility and correlation. Cash-settled and one-homogeneous claims whose path dependence is carried by the ratio are then invariant to the local-correlation specification; the second property holds more generally under constant relative variance, while the first is specific to the orthogonal design. We derive double-knock-out exchange-option, cash-rebate and relative-performance lookback formulas and identify an index-dependent subclass in which ratio and index are independent. There, a two-dimensional exit law factorises and an exchange option knocked out on a curvilinear region is priced as a ratio double-knock-out value times a one-dimensional index survival probability. Simulations under sharply different local-correlation functions confirm the invariant prices while generating very different fixed-strike basket prices.