Quanto Option Pricing with Lévy Models

Abstract

We develop a multivariate Lévy model and apply the bivariate model for the pricing of quanto options that captures three characteristics observed in real-world markets for stock prices and currencies: jumps, heavy tails and skewness. The model is developed by using a bottom-up approach from a subordinator. We do so by replacing the time of a Brownian motion with a Lévy process, exponential tilting subordinator. We refer to this model as a multivariate exponential tilting process. We then compare using a time series of daily log-returns and market prices of European-style quanto options the relative performance of the exponential tilting process to that of the Black–Scholes and the normal tempered stable process. We find that, due to more flexibility on capturing the information of tails and skewness, the proposed modeling process is superior to the other two processes for fitting market distribution and pricing quanto options.

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Fig. 1

Notes

  1. 1.

    Detailed information about the Chicago Mercantile Exchange (CME) and this contract can be found in http://www.cmegroup.com. Figure C.1 of the Online Appendix shows the daily exchange volume of the CME Nikkei 225 from January 11 to March 13, 2017.

  2. 2.

    See, among others, Das and Uppal (2004) and Carr et al. (2003).

  3. 3.

    Detailed information about the subordinator and time-changed Lévy process can be found in Swishchuk (2016) and Barndorff-Nielsen and Shiryaev (2010). There are other terms for the subordinator used in the literature. For example, Barndorff-Nielsen and Shephard (2001a) use chronometer rather than subordinator.

  4. 4.

    See, among others, Rachev and Mittnik (2000) and Kim et al. (2008).

  5. 5.

    One possible explanation for realizing different performance for the multivariate rapidly decreasing process for different maturities is the numerical error of implementing hypergeometric function for small maturities. Detailed information about this hypergeometric function can be found in Section A of the Online Appendix.

  6. 6.

    Note that in general, the moment-generating function of a Lévy process may not exist for some u. Detailed information can be found in Cont and Tankov (2004).

  7. 7.

    To be consistent with literature of the stable and tempered stable process, we replace \( \alpha \) with \( \frac{\alpha }{2} \) in empirical study and allow the range of \( \alpha \) be (0, 2).

  8. 8.

    The results discussed in this section can be proved by Tassinari and Bianchi (2014) and Bianchi et al. (2016).

  9. 9.

    Detailed information about calculating this pdf and cdf, and the linear transformation property can be found in Fallahgoul et al. (2016), Rachev et al. (2011), and the references therein.

  10. 10.

    See, Fallahgoul et al. (2016), among others.

  11. 11.

    It should be noted that the way we construct the ETP is similar to the way Barndorff-Nielsen and Shephard (2001b) use the classical tempered stable subordinator to construct the NTS process, see, Barndorff-Nielsen and Shephard (2001b), Cont and Tankov (2004), and Kim et al. (2012), among others.

  12. 12.

    Note that this \( Y_t \) is different from \( Y_t \) on page 13.

  13. 13.

    See Baxter and Rennie (1996).

  14. 14.

    Detailed information about the sufficient conditions of a characteristic function being well defined can be found in Lewis (2001).

  15. 15.

    Graphs of the scaled price process as well as exchange rates are provided in the Online Appendix.

  16. 16.

    In this paper, we deal with problems up to two dimensions. Implementing the expectation-maximization approach for higher dimensions is more effective than MLE, see Bianchi et al. (2016).

  17. 17.

    The hypergeometric function we used in the paper is implemented based on the code provided in Jin and Jjie (1996). The numerical method used in the function is basically a series representation and for the large input number, the function uses an asymptotic approximation.

  18. 18.

    All estimates are expressed in annualized form. In order to construct 95% confidence intervals, we use the same algorithm as discussed in Kim et al. (2015).

  19. 19.

    Information about these test statistics as well as calculating the related p values may be found in Anderson and Darling (1954) and Marsaglia and Marsaglia (2004).

  20. 20.

    In both cases, model 1 is MNTS or METP and model 2 is BS.

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Correspondence to Young S. Kim.

Additional information

The authors thank Julien Hugonnier, Loriano Mancini, and David Veredas for helpful comments. Financial support from the Swiss National Science Foundation Sinergia Grant “Empirics of Financial Stability” [154445] is gratefully acknowledged. The Centre for Quantitative Finance and Investment Strategies has been supported by BNP Paribas.

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Fallahgoul, H.A., Kim, Y.S., Fabozzi, F.J. et al. Quanto Option Pricing with Lévy Models. Comput Econ 53, 1279–1308 (2019). https://doi.org/10.1007/s10614-018-9807-8

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Keywords

  • Quanto option pricing
  • Lévy process
  • Stable and tempered stable process
  • Subordinator

JEL Classification

  • C0
  • C02
  • C1