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Calibrating Local Volatility Models with Stochastic Drift and Diffusion

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  • Orcan Ogetbil
  • Narayan Ganesan
  • Bernhard Hientzsch

Abstract

We propose Monte Carlo calibration algorithms for three models: local volatility with stochastic interest rates, stochastic local volatility with deterministic interest rates, and finally stochastic local volatility with stochastic interest rates. For each model, we include detailed derivations of the corresponding SDE systems, and list the required input data and steps for calibration. We give conditions under which a local volatility can exist given European option prices, stochastic interest rate model parameters, and correlations. The models are posed in a foreign exchange setting. The drift term for the exchange rate is given as a difference of two stochastic short rates, domestic and foreign, each modeled by a G1++ process. For stochastic volatility, we model the variance for the exchange rate by a CIR process. We include tests to show the convergence and the accuracy of the proposed algorithms.

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  • Orcan Ogetbil & Narayan Ganesan & Bernhard Hientzsch, 2020. "Calibrating Local Volatility Models with Stochastic Drift and Diffusion," Papers 2009.14764, arXiv.org, revised May 2023.
    • Handle: RePEc:arx:papers:2009.14764
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    References listed on IDEAS

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    1. Griselda Deelstra & Gr�gory Ray�e, 2013. "Local Volatility Pricing Models for Long-Dated FX Derivatives," Applied Mathematical Finance, Taylor & Francis Journals, vol. 20(4), pages 380-402, September.
    2. Stein, Elias M & Stein, Jeremy C, 1991. "Stock Price Distributions with Stochastic Volatility: An Analytic Approach," The Review of Financial Studies, Society for Financial Studies, vol. 4(4), pages 727-752.
    3. Orcan Ogetbil & Bernhard Hientzsch, 2020. "Extensions of Dupire Formula: Stochastic Interest Rates and Stochastic Local Volatility," Papers 2005.05530, arXiv.org, revised Feb 2023.
    4. Rainer Schöbel & Jianwei Zhu, 1999. "Stochastic Volatility With an Ornstein–Uhlenbeck Process: An Extension," Review of Finance, European Finance Association, vol. 3(1), pages 23-46.
    5. John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 5, pages 129-164, World Scientific Publishing Co. Pte. Ltd..
    6. Julien Hok & Shih-Hau Tan, 2018. "Calibration of Local Volatility Model with Stochastic Interest Rates by Efficient Numerical PDE Method," Papers 1803.03941, arXiv.org.
    7. Marc Atlan, 2006. "Localizing Volatilities," Papers math/0604316, arXiv.org.
    8. Yuri F. Saporito & Xu Yang & Jorge P. Zubelli, 2017. "The Calibration of Stochastic-Local Volatility Models - An Inverse Problem Perspective," Papers 1711.03023, arXiv.org.
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    Cited by:

    1. Arun Kumar Polala & Bernhard Hientzsch, 2023. "Parametric Differential Machine Learning for Pricing and Calibration," Papers 2302.06682, arXiv.org, revised Feb 2023.
    2. Kim, Sangkwon & Kim, Junseok, 2021. "Robust and accurate construction of the local volatility surface using the Black–Scholes equation," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).

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