{
  "abstract": "We compute $\\E[(S_T-K)^+]$ by Monte Carlo for a scalar stochastic-volatility model with a fast mean-reverting factor of time scale $\\eps$, for $\\eps$ ranging from $1$ down to $10^{-3}$. A conditional (mixing) estimator gives finite variance, whereas the direct estimator has infinite variance for this model. The volatility factor is simulated with its exact Ornstein--Uhlenbeck transition. As $\\eps\\to0$ the price converges, at rate $O(\\eps)$, to the Black--Scholes price with the averaged volatility $\\barσ$, and the implied-volatility smile flattens to $\\barσ$. Finally, we test a martingale control variate built on the Black--Scholes delta with volatility $\\barσ$. If the martingale is driven by the true volatility $σ(Y_t)$, the variance is reduced by a factor that grows like $1/\\eps$, about $160$ at $\\eps=10^{-3}$. If it is driven by the constant $\\barσ$, the variance is essentially not reduced. Last, we calibrate the model with $ρ\\neq0$ to S\\&P~500 implied volatilities by full simulation of $S$. At the same number of paths the control variate reduces the variance by a factor $1.7$--$16$ (median $5.6$) and gives more accurate calibrated parameters; at equal CPU time it pays off only if the delta is rebalanced on a coarser grid than the time step. For a basket of four indices (state dimension $8$) the gain is larger (median $8$) and the relative cost smaller, so that the control variate is about $4$--$5$ times more efficient than plain Monte Carlo at equal CPU time.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32765",
  "author_total": 2,
  "authors": [
    {
      "name": "Laurent Mertz"
    },
    {
      "name": "Olivier Pironneau"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bn9esv1ch9sgx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32765",
  "fulltext_url": null,
  "id": "bn9esv1ch9sgx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32765",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.32765",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bn9esv1ch9sgx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.32765",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32765"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Monte Carlo pricing under fast mean-reverting stochastic volatility: the multi-scale limit ${ε\\to 0}$",
  "title_text": "Monte Carlo pricing under fast mean-reverting stochastic volatility: the multi-scale limit ε→ 0",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/bn9esv1ch9sgx",
  "venue": "arXiv",
  "versions": []
}