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Generic pricing of FX, inflation and stock options under stochastic interest rates and stochastic volatility

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  • Alexander van Haastrecht
  • Antoon Pelsser

Abstract

We consider the pricing of FX, inflation and stock options under stochastic interest rates and stochastic volatility, for which we use a generic multi-currency framework. We allow for a general correlation structure between the drivers of the volatility, the inflation index, the domestic (nominal) and the foreign (real) rates. Having the flexibility to correlate the underlying FX/inflation/stock index with both stochastic volatility and stochastic interest rates yields a realistic model that is of practical importance for the pricing and hedging of options with a long-term exposure. We derive explicit valuation formulas for various securities, such as vanilla call/put options, forward starting options, inflation-indexed swaps and inflation caps/floors. These vanilla derivatives can be valued in closed form under Schobel and Zhu [Eur. Finance Rev., 1999, 4, 23-46] stochastic volatility, whereas we devise an (Monte Carlo) approximation in the form of a very effective control variate for the general Heston [Rev. Financial Stud., 1993, 6, 327-343] model. Finally, we investigate the quality of this approximation numerically and consider a calibration example to FX and inflation market data.

Suggested Citation

  • Alexander van Haastrecht & Antoon Pelsser, 2011. "Generic pricing of FX, inflation and stock options under stochastic interest rates and stochastic volatility," Quantitative Finance, Taylor & Francis Journals, vol. 11(5), pages 665-691.
  • Handle: RePEc:taf:quantf:v:11:y:2011:i:5:p:665-691
    DOI: 10.1080/14697688.2010.504734
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    References listed on IDEAS

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    1. Alan L. Lewis, 2001. "A Simple Option Formula for General Jump-Diffusion and other Exponential Levy Processes," Related articles explevy, Finance Press.
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    1. repec:wsi:ijfexx:v:04:y:2017:i:01:n:s242478631750013x is not listed on IDEAS
    2. Mordecai Avriel & Jens Hilscher & Alon Raviv, 2013. "Inflation Derivatives Under Inflation Target Regimes," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 33(10), pages 911-938, October.
    3. Andrei Cozma & Christoph Reisinger, 2015. "A mixed Monte Carlo and PDE variance reduction method for foreign exchange options under the Heston-CIR model," Papers 1509.01479, arXiv.org, revised Apr 2016.
    4. Andrei Cozma & Matthieu Mariapragassam & Christoph Reisinger, 2015. "Convergence of an Euler scheme for a hybrid stochastic-local volatility model with stochastic rates in foreign exchange markets," Papers 1501.06084, arXiv.org, revised Oct 2016.
    5. Griselda Deelstra & Grégory Rayée, 2013. "Local Volatility Pricing Models for Long-Dated FX Derivatives," Applied Mathematical Finance, Taylor & Francis Journals, vol. 20(4), pages 380-402, September.
    6. Lech A. Grzelak & Cornelis W. Oosterlee, 2012. "On Cross-Currency Models with Stochastic Volatility and Correlated Interest Rates," Applied Mathematical Finance, Taylor & Francis Journals, vol. 19(1), pages 1-35, February.
    7. Singor, Stefan N. & Grzelak, Lech A. & van Bragt, David D.B. & Oosterlee, Cornelis W., 2013. "Pricing inflation products with stochastic volatility and stochastic interest rates," Insurance: Mathematics and Economics, Elsevier, vol. 52(2), pages 286-299.
    8. Alessandro Gnoatto & Martino Grasselli, 2013. "An analytic multi-currency model with stochastic volatility and stochastic interest rates," Papers 1302.7246, arXiv.org, revised Mar 2013.

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