Extrema, Barrier Options, and Semi-Analytic Leverage Corrections in Stochastic-Clock Volatility Models

Tristan Guillaume ()
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Tristan Guillaume: CYU - CY Cergy Paris Université

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Abstract: Barrier derivatives depend on extrema and first-passage events and are therefore highly sensitive to volatility dynamics—especially to the instantaneous return–volatility correlation \rho, 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 (\rho=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-\rho expansion around the \rho=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 Padé 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 \rho=0 barrier backbone once, and (iii) iterating on \rho (and any remaining parameters) using the fast semi-analytic corrections—optionally Padé-accelerated—inside a standard least-squares loop.

Keywords: stochastic clock; time change; barrier options; maximum distribution; killed diffusion; Laplace transform; leverage effect; implied-volatility skew; small-correlation expansion; Padé approximation; calibration (search for similar items in EconPapers)
Date: 2026-02
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