Accelerating Closed-Form Heston Pricers for Calibration

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Abstract

Calibrating models against the markets is a crucial step to obtain meaningful results in the subsequent pricing processes. In general, calibration can be seen as a minimization problem that tries to fit modeled product prices to the observed ones on the market (compare Chap.2 by Sayer and Wenzel). This means that during the calibration process the modeled prices need to be calculated many times, and therefore the run time of the product pricers have the highest impact on the overall calibration run time. Therefore, in general, only products are used for calibration for which closed-form mathematical pricing formulas are known. While for the Heston model (semi) closed-form solutions exist for simple products, their evaluation involves complex functions and infinite integrals. So far these integrals can only be solved with time-consuming numerical methods. However, over the time, more and more theoretical and practical subtleties have been revealed for doing this and today a large number of possible approaches are known. Examples are different formulations of closed-formulas and various integration algorithms like quadrature or Fourier methods. Nevertheless, all options only work under specific conditions and depend on the Heston model parameters and the input setting. In this chapter we present a methodology how to determine the most appropriate calibration method at run time. For a practical setup we study the available popular closed-form solutions and integration algorithms from literature. In total we compare 14 pricing methods, including adaptive quadrature and Fourier methods. For a target accuracy of 10-3 we show that static Gauss-Legendre are best on Central Processing Units (CPUs) for the unrestricted parameter set. Further we show that for restricted Carr-Madan formulation the methods are 3.6× faster. We also show that Fourier methods are even better when pricing at least 10 options with the same maturity but different strikes.

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APA

Liu, G., Brugger, C., De Schryver, C., & Wehn, N. (2015). Accelerating Closed-Form Heston Pricers for Calibration. In FPGA Based Accelerators for Financial Applications (pp. 221–242). Springer International Publishing. https://doi.org/10.1007/978-3-319-15407-7_10

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