Short-term barrier option price expansion
Abstract
We derive a short-maturity expansion for up-and-out put barrier option prices under continuous stochastic volatility when the strike and the barrier approach the spot at the diffusive scale. Assuming joint weak convergence of the normalized terminal return, the relative volatility fluctuation, and the running maximum, together with uniform integrability, we show that the leading term is the time-inhomogeneous Black-Scholes barrier price fitted to the forward variance curve. The first model-dependent correction is of order θH+1/2, where θH, H ∈ (0,1/2], is the order of the relative volatility fluctuation, and is represented explicitly through killed Brownian transition densities. For regular volatility models, where H= 1/2, the coefficient is determined by the short-maturity at-the-money implied-volatility skew. For rough volatility models with H < 1/2, the coefficient reduces to a one-dimensional integral. Numerical experiments show that the correction materially improves the Black-Scholes approximation.
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BibTeX
@misc{fukasawa2026shortterm,
title = {{Short-term barrier option price expansion}},
author = {Masaaki Fukasawa},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00340},
eprint = {2610.00340},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00340},
}
RIS
TY - UNPB
TI - Short-term barrier option price expansion
AU - Masaaki Fukasawa
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00340
UR - https://arxiv.org/abs/2610.00340
AB - We derive a short-maturity expansion for up-and-out put barrier option prices under continuous stochastic volatility when the strike and the barrier approach the spot at the diffusive scale. Assuming joint weak convergence of the normalized terminal return, the relative volatility fluctuation, and the running maximum, together with uniform integrability, we show that the leading term is the time-inhomogeneous Black-Scholes barrier price fitted to the forward variance curve. The first model-dependent correction is of order θ^H+1/2, where θ^H, H ∈ (0,1/2], is the order of the relative volatility fluctuation, and is represented explicitly through killed Brownian transition densities. For regular volatility models, where H= 1/2, the coefficient is determined by the short-maturity at-the-money implied-volatility skew. For rough volatility models with H < 1/2, the coefficient reduces to a one-dimensional integral. Numerical experiments show that the correction materially improves the Black-Scholes approximation.
ER -