DOI QR코드

DOI QR Code

ACCURATE AND EFFICIENT COMPUTATIONS FOR THE GREEKS OF EUROPEAN MULTI-ASSET OPTIONS

  • Lee, Seunggyu (DEPARTMENT OF MATHEMATICS, KOREA UNIVERSITY) ;
  • Li, Yibao (DEPARTMENT OF COMPUTATIONAL SCIENCE AND ENGINEERING, YONSEI UNIVERSITY) ;
  • Choi, Yongho (DEPARTMENT OF MATHEMATICS, KOREA UNIVERSITY) ;
  • Hwang, Hyoungseok (DEPARTMENT OF FINANCIAL ENGINEERING, KOREA UNIVERSITY) ;
  • Kim, Junseok (DEPARTMENT OF MATHEMATICS, KOREA UNIVERSITY)
  • Received : 2014.02.06
  • Accepted : 2014.02.25
  • Published : 2014.03.25

Abstract

This paper presents accurate and efficient numerical methods for calculating the sensitivities of two-asset European options, the Greeks. The Greeks are important financial instruments in management of economic value at risk due to changing market conditions. The option pricing model is based on the Black-Scholes partial differential equation. The model is discretized by using a finite difference method and resulting discrete equations are solved by means of an operator splitting method. For Delta, Gamma, and Theta, we investigate the effect of high-order discretizations. For Rho and Vega, we develop an accurate and robust automatic algorithm for finding an optimal value. A cash-or-nothing option is taken to demonstrate the performance of the proposed algorithm for calculating the Greeks. The results show that the new treatment gives automatic and robust calculations for the Greeks.

Acknowledgement

Supported by : National Institute for Mathematical Sciences(NIMS)

References

  1. M. Chesney, R.J. Elliott, D. Madan, and H. Yang, Diffusion coefficient estimation and asset pricing when risk premia and sensitivities are time varying, Math. Financ. 313 (1993), pp. 85-99.
  2. A.J. McNeil and R. Frey, Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach, J. Empirical Financ. 7 13 (2000), pp. 271-300. https://doi.org/10.1016/S0927-5398(00)00012-8
  3. J. Huang, M.G. Subrahmanyam, and G.G. Yu, Pricing and hedging American options: a recursive investigation method, Rev. Financ. Stud. 9 13 (1996), pp. 277-300. https://doi.org/10.1093/rfs/9.1.277
  4. P. Wilmott, Paul Wilmott on quantitative finance. Wiley, New York, 2000.
  5. Y. El-Khatib and N. Privault, Computations of Greeks in a market with jumps via the Mulliavin calculus, Financ. Stoch. 8 13 (2004), pp. 161-179. https://doi.org/10.1007/s00780-003-0111-6
  6. M. Namihira and D.A. Kopriva, Computation of the effects of uncertainty in volatility on option pricing and hedging, Int. J. Comput. Math. 13 89 (2012), pp. 1281-1302. https://doi.org/10.1080/00207160.2012.688819
  7. F. Black and M. Sholes, The pricing of options and corporate liabilities. J. Polit. Econ. 81 13 (1973), pp. 637-59. https://doi.org/10.1086/260062
  8. D.J. Duffy, Finite difference methods in financial engineering : a partial differential equation approach, John Wiley and Sons, New York, 2006.
  9. S. Ikonen and J. Toivanen, Operator splitting methods for American option pricing, Appl. Math. Lett. 17 13 (2004), pp. 809-814. https://doi.org/10.1016/j.aml.2004.06.010
  10. D. Jeong and J. Kim, A comparison study of ADI and operator splitting methods on option pricing models, J. Comput. Appl. Math. 247 13 (2013), pp. 162-171. https://doi.org/10.1016/j.cam.2013.01.008
  11. E.G. Haug, The complete guide to option pricing formulas, McGraw-Hill, New York, 1997.
  12. D. Jeong, J. Kim, and I.S. Wee, An accurate and efficient numerical method for black-scholes equations, Commun. Korean Math. Soc. 24 13 (2009), pp. 617-628. https://doi.org/10.4134/CKMS.2009.24.4.617