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2024 Vol.12, Issue 2

Research Article

Black­Scholes Partial Differential Equation with the Sensitivities(Greeks) as Solution

민감도(Greeks)를 해(Solution)로 갖는 블랙숄즈 편미분방정식(PDE)

Young Sung Kim1, Ki­Soo Suh2, and Hee­Geon Kang3

김영성1, 서기수2, 강희건3

1Adjunct Professor, Department of Business Administration, Soongsil University
2Assistant Professor, Department of Financial Information Engineering, Seokyeong University
3Ph.D Candidate, Department of Statistics, University of Seoul
1숭실대학교 경영학부
2서경대학교 금융정보공학과
3서울시립대학교 통계학과
30 June 2024. pp. 1-10
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Abstract
파생상품과 같은 조건부청구권(Contingent Claim)의 민감도(Greeks)를 산출하는 것은 위험관리에서 매우 중요하고, 헤지 운용을 위한 수익창출에도 매우 중요하다. 본 연구에서는 가격이 아닌, 민감도가 해(solution)인 편미분방정식(partial differential equation)을 이용하여 민감도를 계산하는 방법을 유한차분법(finite difference method)을 적용하여 제안한다. 또한, 실제 폐쇄형해(closed form solution)와 비교하였다. 분석결과, 민감도를 해로 설정하여 편미분방정식으로 계산한 값(numerical solution)과 실제 폐쇄형해에서 산출된 값이 수치적으로 동일하게 나타났다. 또한, 1일을 격자 1개로 설정하여 계산한 경우보다 3개의 격자로 세분하여 계산한 값이 더 정확하게 나타났다. 그리고 변동성이 높을수록 격자의 범위를 높여야 하며 500%까지 설정한 부분에서 오차가 현저하게 떨어졌다. 따라서 하루의 격자를 적절하게 설정하고 변동성에 따라 기초자산 가격의 범위만 잘 정의할 수 있다면, 본 연구에서 제시한 민감도 자체의 편미분방정식(PDE)의 해는 실제 폐쇄형해와 일치한다는 것을 모델링하여 검증하였다. 이 방법을 이용하면 향후 시간대비 효율적인 민감도를 산출 할 수 있을 것이라 기대한다.
Calculating the sensitivities(Greeks) of contingent claims such as derivatives is very important in risk management and also in generating profits for hedging operations. In this study, we propose a method of calculating sensitivity using the finite difference method using a partial differential equation in which sensitivity itself, not price, is the solution. Additionally, it was compared with an actual closed form solution. As a result of the analysis, by setting the sensitivities to the solution, the value calculated from the partial differential equation(numerical solution) and the value calculated from the actual closed form solution were numerically identical. In addition, the value calculated by dividing one day into three grids appears more accurately than when calculating by setting one day to one grid. Also, the higher the volatility, the higher the grid range should be, and the error dropped significantly when set to 500%. Therefore, if the daily grid can be properly set and the range of the underlying asset price can be well defined according to volatility, the solution of the partial differential equation(PDE) of the sensitivities presented in this study is actually a closed form solution. It was verified by modeling that it appeared consistent with the solution. Using this method, it is expected that efficient sensitivities compared to time will be calculated in the future.
Keywords
Partial Differential Equation(PDE)
Finite Difference Method(FDM)
Sensitivities(Greeks)
Black-scholes
Call option
References

  1. M. Kac, "On distributions of certain Wiener functionals", Transactions of the American Mathematical Society, Vol. 6,5 No. 1, 1-13, 1949.

     10.1090/S0002-9947-1949-0027960-X

  2. F. Black and M. Scholes, "The Pricing of Options and Corporate Liabilities", Journal of Political Economy, Vol. 81, No. 3, pp. 637-654, 1973.

     10.1086/260062

  3. R. Merton, "The theory of rational option pricing", The Bell Journal of Economics and Management Science, Vol. 4, pp. 141-183, 1973.

     10.2307/3003143

  4. J. M. Harrison and D. M. Kreps, "Martingales and arbitrage in multiperiod securities markets", Journal of Economic Theory, Vol. 20, pp. 381-408, 1979.

     10.1016/0022-0531(79)90043-7

  5. J. M. Harrison and S. R. Pliska, "Martingales and stochastic integrals in the theory of continuous trading", Stochastic Processes and their Applications, Vol.11, pp. 215-260, 1981.

     10.1016/0304-4149(81)90026-0

  6. J. M. Harrison and S. R. Pliska, "A stochastic calculus model of continuous trading: Complete markets", Stochastic Processes and their Applications, Vol. 15, pp. 313-316, 1983.

     10.1016/0304-4149(83)90038-8

  7. C. Jaksa and Z. Fernando, "Introduction to the Economics and Mathematics of Financial Markets", The MIT Press, 2004.

  8. B. Øksendal, "Stochastic Differential Equations", Universitext. Springer, Berlin, Heidelberg, 2003.

     10.1007/978-3-642-14394-6

  9. J. C. Hull, "Options, Futures and Other Derivatives", Pearson, 2022.

  10. P. Wilmott, "Paul Wilmott on Quantitative Finance", 2nd Edition, John Wiley and Sons, 2006.

     10.1002/wilm.1

  11. J. W. Thomas, "Numerical partial differential equations: finite difference methods", New York, 1995.

     10.1007/978-1-4899-7278-1

  12. B. Flannery., W. Press., S. Teukolsky and W. Vetterling, "Numerical recipes in C: The Art of Scientific Computing", Cambridge University Press. 1992.

  13. 김동석, 변석준 "옵션에 대한 수치해법상의 초기값 불연속성 문제에 관한 연구", 선물연구, 제7권 제1호, pp. 21-39, 2000.

  14. 김다애, 추정호 "편미분 방정식을 이용한 분산 스와프의 평가 방법론", 금융공학연구, 제22권 제3호, pp. 31-48, 2023.

  15. J. Crank and P. Nicolson, "A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type", Advances in Computational Mathematics, Vol. 6, No. 1, pp. 207-226, 1996.

     10.1007/BF02127704

  16. D. Duffy, "A critique of the Crank-Nicolson scheme., strengths and weakness for financial instrument pricing", Wilmott Magazine, Vol. 4, pp. 68-76, 2004.

     10.1002/wilm.42820040417

  17. 김솔, 홍충완, "한국의 HSCEI지수 ELS발행이 홍콩 옵션 시장의 변동성 스큐에 미치는 영향", 재무연구, 제34권 제4호, pp. 125-148, 2021.

     10.37197/ARFR.2021.34.4.4
Information
  • Publisher :The Society of Convergence Knowledge
  • Publisher(Ko) :융복합지식학회
  • Journal Title :The Society of Convergence Knowledge Transactions
  • Journal Title(Ko) :융복합지식학회논문지
  • Volume : 12
  • No :2
  • Pages :1-10
  • DOI :https://doi.org/10.22716/sckt.2024.12.2.001
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