The Korean Journal of Applied Statistics (응용통계연구)

The Korean Statistical Society (한국통계학회)
권호 표지
DOI QR코드

DOI QR Code

Comparison of methods of approximating option prices with Variance gamma processes

Variance gamma 확률과정에서 근사적 옵션가격 결정방법의 비교

  • Received : 2015.12.15
  • Accepted : 2016.01.05
  • Published : 2016.02.29
Download PDF
⟨ Previous Next ⟩

Abstract

We consider several methods to approximate option prices with correction terms to the Black-Scholes option price. These methods are able to compute option prices from various risk-neutral distributions using relatively small data and simple computation. In this paper, we compare the performance of Edgeworth expansion, A-type and C-type Gram-Charlier expansions, a method of using Normal inverse gaussian distribution, and an asymptotic method of using nonlinear regression through simulation experiments and real KOSPI200 option data. We assume the variance gamma model in the simulation experiment, which has a closed-form solution for the option price among the pure jump $L{\acute{e}}vy$ processes. As a result, we found that methods to approximate an option price directly from the approximate price formula are better than methods to approximate option prices through the approximate risk-neutral density function. The method to approximate option prices by nonlinear regression showed relatively better performance among those compared.

옵션의 가격을 결정하는 문제에서 블랙-숄즈 모형이 가지는 단점을 보완하기 위해 블랙-숄즈 가격을 선도항으로 하여 보정항을 구하는 근사적 옵션가격의 결정방법을 고려하였다. 이러한 근사적 가격결정 방법들은 비교적 적은 자료를 가지고 간단한 계산으로 다양한 형태의 위험중립 확률분포에 의한 옵션가격을 계산할 수 있다. 이 논문에서는 일반적으로 관찰되는 시장상황을 모사한 모의실험과 실제 시장에서 관측되는 KOSPI200 옵션가격 자료를 통해 몇 가지 근사방법들의 적합성과를 비교, 평가하였다. 헤르미트 다항식 계열의 Edgeworth 확장과 A-type Gram-Charlier, C-type Gram-Charlier 방법, NIG 분포를 이용하는 방법, 비선형 회귀를 이용한 점근적 근사방법이 고려되었다. 모의실험에서는 순수 점프 레비 확률과정 가운데 옵션가격이 닫힌 해의 형태로 존재하는 Variance gamma 과정을 가정하여 자료를 생성하였다. 모의실험과 실제 자료분석의 결과, 분포함수를 먼저 근사하여 가격을 계산하는 것보다 근사적 가격식을 유도하여 직접 가격을 근사하는 방법들의 성능이 좀 더 좋았으며, 그 가운데 비선형 회귀를 이용한 점근적 근사방법이 상대적으로 좋은 성능을 보였다.

Keywords

References

  1. Ait-Sahalia, Y. (1999). Transition densities for interest rate and other nonlinear diffusions, Journal of Finance, 54, 1361-1395. https://doi.org/10.1111/0022-1082.00149
  2. Ait-Sahalia, Y. (2002). Maximum-likelihood estimation of discretely-sampled diffusions: a closed-form approximation approach, Econometrica, 70, 223-262. https://doi.org/10.1111/1468-0262.00274
  3. Bakshi, G., Kapadia, N., and Madan, D. (2003). Stock return characteristics, skew laws, and the differential pricing of individual equity options, The Review of Financial Studies, 16, 101-143. https://doi.org/10.1093/rfs/16.1.0101
  4. Barndorff-Nielsen, O. E. (1998). Processes of normal inverse gaussian type, Finance and Stochastics, 2, 41-68.
  5. Barndorff-Nielsen, O. E. and Cox, D. R. (1989). Asymptotic Techniques for Use in Statistics, Chapman and Hall.
  6. Carr, P., Geman, H., Madan, D., and Yor, M. (2002). The fine structure of asset returns: An empirical investigation, Journal of Business, 75, 305-333. https://doi.org/10.1086/338705
  7. Charlier, C. (1928). A new form of the frequency function, Maddalende fran Lunds Astronomiska Observatorium, II, 51.
  8. Cont, R. and Tankov, P. (2004). Financial Modelling with Jump Processes, Chapman & Hall/CRC.
  9. Corrado, C. and Su, T. (1997). Implied volatility skews and stock index skewness and kurtosis implied by S&P500 index option prices, Journal of Derivatives, 4, 8-19. https://doi.org/10.3905/jod.1997.407978
  10. Eberlein, E. and Keller, U. (1995). Hyperbolic distributions in finance, Bernoulli, 1, 281-299. https://doi.org/10.2307/3318481
  11. Eriksson, A., Ghysels, E., and Wang, F. (2009). The normal inverse Gaussian distribution and the pricing of derivatives, The Journal of Derivatives, 16, 23-37. https://doi.org/10.3905/JOD.2009.16.3.023
  12. Geman, H. (2002). Pure jump Levy processes for asset price modeling, Journal of Banking and Finance, 26, 1297-1316. https://doi.org/10.1016/S0378-4266(02)00264-9
  13. Heston, S. (1993). A closed-form solution for options with stochastic volatility with application to bond and currency options, Review of Financial Studies, 6, 327-343. https://doi.org/10.1093/rfs/6.2.327
  14. Jarrow, R. and Rudd, A. (1982). Approximate option valuation for arbitrary stochastic processes, Journal of Financial Economics, 10, 347-369. https://doi.org/10.1016/0304-405X(82)90007-1
  15. Madan, D., Carr, P., and Chang, E. (1998). The variance gamma process and option pricing, European Finance Review, 2, 79-105. https://doi.org/10.1023/A:1009703431535
  16. Madan, D. and Milne, F. (1994). Contingent claims valued and hedged by pricing and investing in a basis, Mathematical Finance, 4, 223-245. https://doi.org/10.1111/j.1467-9965.1994.tb00093.x
  17. Madan, D. and Seneta, E. (1990). The VG model for share market returns, Journal of Business, 63, 511-524. https://doi.org/10.1086/296519
  18. Rompolis, L. S. and Tzavalis, E. (2007). Retrieving risk neutral densities based on risk neutral moments through a Gram-Charlier series expansion, Mathematical and Computer Modelling, 46, 225-234. https://doi.org/10.1016/j.mcm.2006.12.021
  19. Sato, K. (1999). Levy Processes and Infinitely Divisible Distributions, Cambridge University Press.
  20. Schoutens, W. (2003). Levy Processes in Finance: Pricing Financial Derivatives, John Wiley & Sons, New York.
  21. Song, S., Jeong, J., and Song, J. (2011). Asymptotic option pricing under pure jump Levy processes via nonlinear regression, Journal of the Korean Statistical Society, 40, 227-238. https://doi.org/10.1016/j.jkss.2010.10.001
  22. Xiu, D. (2014). Hermite polynomial based expansion of European option prices, Journal of Econometrics, 179, 158-177. https://doi.org/10.1016/j.jeconom.2014.01.003