Extrema, Barrier Options, and Semi-Analytic Leverage Corrections in Stochastic-Clock Volatility Models
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| Format: | Preprint |
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2026
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| _version_ | 1866914540394905600 |
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| author | Guillaume, Tristan |
| author_facet | Guillaume, Tristan |
| contents | Barrier derivatives depend on extrema and first-passage events and are therefore highly sensitive to volatility dynamics -- especially to the instantaneous return-volatility correlation $ρ$, often called ``leverage''. This sensitivity makes accurate and fast pricing under realistic stochastic-volatility specifications difficult: two-dimensional PDE solvers are expensive inside calibration loops, while Monte Carlo methods converge slowly when barrier hits are rare and discretely monitored. In equity markets in particular, the pronounced implied-volatility skew motivates factoring in a negative return-volatility correlation. We study a class of continuous-path stochastic-clock volatility models in which the log-price is represented as a Brownian motion run on a random increasing clock. In the baseline independent-clock case (ρ=0), a broad family of barrier-relevant objects-maximum distributions, survival probabilities, and killed joint laws-reduces to one-dimensional quantities determined by the Laplace transform of the terminal clock. This yields transform-only pricing formulas for single- and double-barrier contracts that are fast and numerically stable once the clock transform is available, notably for affine and quadratic clocks. To incorporate leverage without forfeiting tractability, we develop a systematic small-ρexpansion around the ρ=0 backbone. The expansion produces a hierarchy of forced problems whose forcing terms are semi-analytic and computable from baseline barrier objects. We provide two implementable leverage-correction routes\,: forced PDEs and a Duhamel-type Monte Carlo representation, and we show how Pad{é} acceleration can extend practical accuracy to equity-like correlations. Calibration then proceeds by\,: (i) fitting clock parameters from vanillas using only one-dimensional transforms, (ii) precomputing the ρ=0 barrier backbone once, and (iii) iterating on ρ(and any remaining parameters) using the fast semi-analytic corrections-optionally Pad{é}-accelerated-inside a standard least-squares loop. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_06677 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Extrema, Barrier Options, and Semi-Analytic Leverage Corrections in Stochastic-Clock Volatility Models Guillaume, Tristan Computational Finance Probability Pricing of Securities Barrier derivatives depend on extrema and first-passage events and are therefore highly sensitive to volatility dynamics -- especially to the instantaneous return-volatility correlation $ρ$, often called ``leverage''. This sensitivity makes accurate and fast pricing under realistic stochastic-volatility specifications difficult: two-dimensional PDE solvers are expensive inside calibration loops, while Monte Carlo methods converge slowly when barrier hits are rare and discretely monitored. In equity markets in particular, the pronounced implied-volatility skew motivates factoring in a negative return-volatility correlation. We study a class of continuous-path stochastic-clock volatility models in which the log-price is represented as a Brownian motion run on a random increasing clock. In the baseline independent-clock case (ρ=0), a broad family of barrier-relevant objects-maximum distributions, survival probabilities, and killed joint laws-reduces to one-dimensional quantities determined by the Laplace transform of the terminal clock. This yields transform-only pricing formulas for single- and double-barrier contracts that are fast and numerically stable once the clock transform is available, notably for affine and quadratic clocks. To incorporate leverage without forfeiting tractability, we develop a systematic small-ρexpansion around the ρ=0 backbone. The expansion produces a hierarchy of forced problems whose forcing terms are semi-analytic and computable from baseline barrier objects. We provide two implementable leverage-correction routes\,: forced PDEs and a Duhamel-type Monte Carlo representation, and we show how Pad{é} acceleration can extend practical accuracy to equity-like correlations. Calibration then proceeds by\,: (i) fitting clock parameters from vanillas using only one-dimensional transforms, (ii) precomputing the ρ=0 barrier backbone once, and (iii) iterating on ρ(and any remaining parameters) using the fast semi-analytic corrections-optionally Pad{é}-accelerated-inside a standard least-squares loop. |
| title | Extrema, Barrier Options, and Semi-Analytic Leverage Corrections in Stochastic-Clock Volatility Models |
| topic | Computational Finance Probability Pricing of Securities |
| url | https://arxiv.org/abs/2605.06677 |