Designed Dependence for Tractable Two-Asset First-Passage Pricing
- 1Department of Statistical and Actuarial Sciences, Western University, London, ON N6A 5B7, Canada
Abstract
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.
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BibTeX
@article{escobaranel2026designed,
title = {{Designed Dependence for Tractable Two-Asset First-Passage Pricing}},
author = {Marcos Escobar-Anel},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14213895},
url = {https://www.mdpi.com/2227-7390/14/21/3895},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Designed Dependence for Tractable Two-Asset First-Passage Pricing
AU - Marcos Escobar-Anel
PY - 2026
DA - 2026/10/25
JO - Mathematics
DO - 10.3390/math14213895
UR - https://www.mdpi.com/2227-7390/14/21/3895
AB - 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.
SN - 2227-7390
ER -