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A Stochastic Approach to the Valuation of Barrier Options in Heston's Stochastic Volatility Model

Author

Listed:
  • Susanne Griebsch
  • Kay Pilz

Abstract

In the valuation of continuous barrier options the distribution of the first hitting time plays a substantial role. In general, the derivation of a hitting time distribution poses a mathematically challenging problem for continuous but otherwise arbitrary boundary curves. When considering barrier options in the Heston model the non-linearity of the variance process leads to the problem of a non-linear hitting boundary. Here, we choose a stochastic approach to solve this problem in the reduced Heston framework, when the correlation is zero and foreign and domestic interest rates are equal. In this context one of our main findings involves the proof of the reection principle for a driftless Itô process with a time-dependent variance. Combining the two results, we derive a closed-form formula for the value of continuous barrier options. Compared to an existing pricing formula, our solution provides further insight into how the barrier option value in the Heston model is constructed. Extending the results to the general Heston framework with arbitrary correlation and drift, we obtain approximations for the joint random variables of the Itô process and its maximum in a weak sense. As a consequence, an approximate formula for pricing barrier options is established. A numerical case study is also performed which illustrates the agreement in results of our developed formulas with standard finite difference methods

Suggested Citation

  • Susanne Griebsch & Kay Pilz, 2012. "A Stochastic Approach to the Valuation of Barrier Options in Heston's Stochastic Volatility Model," Research Paper Series 309, Quantitative Finance Research Centre, University of Technology, Sydney.
  • Handle: RePEc:uts:rpaper:309
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    File URL: https://www.uts.edu.au/sites/default/files/qfr-archive-03/QFR-rp309.pdf
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    References listed on IDEAS

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    1. Andrew Ziogas & Carl Chiarella, 2005. "Pricing American Options under Stochastic Volatility," Computing in Economics and Finance 2005 77, Society for Computational Economics.
    2. Susanne Griebsch & Uwe Wystup, 2011. "On the valuation of fader and discrete barrier options in Heston's stochastic volatility model," Quantitative Finance, Taylor & Francis Journals, vol. 11(5), pages 693-709.
    3. Carl Chiarella & Boda Kang & Gunter H. Meyer, 2010. "The Evaluation Of Barrier Option Prices Under Stochastic Volatility," Research Paper Series 266, Quantitative Finance Research Centre, University of Technology, Sydney.
    Full references (including those not matched with items on IDEAS)

    More about this item

    Keywords

    Heston model; barrier options; reflection principle;

    JEL classification:

    • C15 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Statistical Simulation Methods: General
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates

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