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Computational Economics

, Volume 53, Issue 2, pp 507–532 | Cite as

Pricing Swaps on Discrete Realized Higher Moments Under the Lévy Process

  • Wenli Zhu
  • Xinfeng RuanEmail author
Article
  • 132 Downloads

Abstract

This paper designs and prices the swaps on discrete realized higher moments under the Lévy process in order to hedge the higher-moment risks, e.g., skewness and kurtosis risks. A comparison with Monte-Carlo simulations provides a verification of the correctness of our pricing formula. This paper is a further extension of Zhu and Lian’s (Math Finance 21:233–256, 2011; Appl Math Comput 219:1654–1669, 2012), which are under the Heston model and only price the variance swaps.

Keywords

Lévy process Stochastic volatility Skewness swaps Kurtosis swaps 

JEL Classification

G12 G13 

References

  1. Amaya, D., Christoffersen, P., Jacobs, K., & Vasquez, A. (2015). Does realized skewness predict the cross-section of equity returns? Journal of Financial Economics, 118, 135–67.CrossRefGoogle Scholar
  2. Bakshi, G., Cao, C., & Chen, Z. (1997). Empirical performance of alternative option pricing models. The Journal of Finance, 52, 2003–49.CrossRefGoogle Scholar
  3. Bakshi, G., Kapadia, N., & Madan, D. (2003). Stock return characteristics, skew laws, and the differential pricing of individual equity options. Review of Financial Studies, 16, 101–43.CrossRefGoogle Scholar
  4. Ballestra, L. V., & Cecere, L. (2013). A numerical method to compute the volatility of the fractional brownian motion implied by american options. International Journal of Applied Mathematics, 26, 203–20.CrossRefGoogle Scholar
  5. Ballestra, L. V., & Cecere, L. (2016). A fast numerical method to price american options under the bates model. Computers & Mathematics with Applications, 72, 1305–19.CrossRefGoogle Scholar
  6. Barndorff-Nielsen, O. E. (1997). Processes of normal inverse gaussian type. Finance and Stochastics, 2, 41–68.CrossRefGoogle Scholar
  7. Bates, D. S. (1996). Jumps and stochastic volatility: Exchange rate processes implicit in deutsche mark options. Review of Financial Studies, 9, 69–107.CrossRefGoogle Scholar
  8. Bates, D. S. (2012). Us stock market crash risk, 1926–2010. Journal of Financial Economics, 105, 229–59.CrossRefGoogle Scholar
  9. Carr, P., Geman, H., Madan, D. B., & Yor, M. (2002). The fine structure of asset returns: An empirical investigation*. The Journal of Business, 75, 305–33.CrossRefGoogle Scholar
  10. Carr, P., & Wu, L. (2003). What type of process underlies options? A simple robust test. The Journal of Finance, 58, 2581–610.CrossRefGoogle Scholar
  11. Carr, P., & Wu, L. (2004). Time-changed lévy processes and option pricing. Journal of Financial economics, 71, 113–41.CrossRefGoogle Scholar
  12. Cont, R., Tankov, P., & Tankov, P. (2004). Financial modelling with jump processes (Vol. 133). Boca Raton: Chapman & Hall/CRC.Google Scholar
  13. Cox, J. C., Ingersoll, J. E., & Ross, S. A. (1985). An intertemporal general equilibrium model of asset prices. Econometrica, 53, 363–84.CrossRefGoogle Scholar
  14. Deville, D. (2007). On lévy processes for option pricing: Numerical methods and calibration to index options. PhD Thesis (Università Politecnica delle Marche).Google Scholar
  15. Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies, 6, 327–43.CrossRefGoogle Scholar
  16. Hull, J., & White, A. (1987). The pricing of options on assets with stochastic volatilities. The Journal of Finance, 42, 281–300.CrossRefGoogle Scholar
  17. Karatzas, I., Lehoczky, J. P., Shreve, S. E., & Xu, G.-L. (1991). Martingale and duality methods for utility maximization in an incomplete market. SIAM Journal on Control and Optimization, 29, 702–30.CrossRefGoogle Scholar
  18. Korn, R., Korn, E., & Kroisandt, G. (2010). Monte carlo methods and models in finance and insurance. Boca Raton: CRC Press.CrossRefGoogle Scholar
  19. Kozhan, R., Neuberger, A., & Schneider, P. (2013). The skew risk premium in the equity index market. Review of Financial Studies, 26, 2174–203.CrossRefGoogle Scholar
  20. Madan, D. B., Carr, P. P., & Chang, E. C. (1998). The variance gamma process and option pricing. European Finance Review, 2, 79–105.CrossRefGoogle Scholar
  21. Merton, R. C. (1976). Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics, 3, 125–44.CrossRefGoogle Scholar
  22. Neuberger, A. (2012). Realized skewness. Review of Financial Studies, 25, 3423–55.CrossRefGoogle Scholar
  23. Ornthanalai, C. (2014). Levy jump risk: Evidence from options and returns. Journal of Financial Economics, 112, 69–90.CrossRefGoogle Scholar
  24. Poularikas, A. D. (2009). Transforms and applications handbook. Boca Raton: CRC Press.Google Scholar
  25. Rompolis, L., & Tzavalis, E. (2013). Pricing and hedging contingent claims using variance and higher-order moment swaps. Quantitative Finance, 17, 531–550.CrossRefGoogle Scholar
  26. Ruan, X., Zhu, W., Huang, J., & Zhang, J. E. (2016). Equilibrium asset pricing under the lévy process with stochastic volatility and moment risk premiums. Economic Modelling, 54, 326–38.CrossRefGoogle Scholar
  27. Rujivan, S., & Zhu, S.-P. (2012). A simplified analytical approach for pricing discretely-sampled variance swaps with stochastic volatility. Applied Mathematics Letters, 25, 1644–50.CrossRefGoogle Scholar
  28. Rydberg, T. H. (1997). The normal inverse gaussian lévy process: Simulation and approximation. Communications in Statistics. Stochastic Models, 13, 887–910.CrossRefGoogle Scholar
  29. Salmi, S., Toivanen, J., & von Sydow, L. (2013). Iterative methods for pricing american options under the bates model. Procedia Computer Science, 18, 1136–44.CrossRefGoogle Scholar
  30. Schoutens, W. (2005). Moment swaps. Quantitative Finance, 5, 525–30.CrossRefGoogle Scholar
  31. Stein, E. M., & Stein, J. C. (1991). Stock price distributions with stochastic volatility: An analytic approach. Review of Financial Studies, 4, 727–52.CrossRefGoogle Scholar
  32. Zhang, J. E., Zhen, F., Sun, X., & Zhao, H. (2017). The skewness implied in the Heston model and its application. Journal of Futures Markets, 37(3), 211–37.CrossRefGoogle Scholar
  33. Zhao, H., Zhang, J. E., & Chang, E. C. (2013). The relation between physical and risk-neutral cumulants. International Review of Finance, 13, 345–81.CrossRefGoogle Scholar
  34. Zheng, W., & Kwok, Y. K. (2014). Closed form pricing formulas for discretely sampled generalized variance swaps. Mathematical Finance, 24, 855–81.CrossRefGoogle Scholar
  35. Zhu, S.-P., & Lian, G.-H. (2011). A closed-form exact solution for pricing variance swaps with stochastic volatility. Mathematical Finance, 21, 233–56.Google Scholar
  36. Zhu, S.-P., & Lian, G.-H. (2012). On the valuation of variance swaps with stochastic volatility. Applied Mathematics and Computation, 219, 1654–69.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.School of Economic MathematicsSouthwestern University of Finance and EconomicsChengduPeople’s Republic of China
  2. 2.Department of Accountancy and Finance, Otago Business SchoolUniversity of OtagoDunedinNew Zealand

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