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PRICING AMERICAN LOOKBACK OPTIONS UNDER A STOCHASTIC VOLATILITY MODEL

윤지훈|김동현|우준희
대한수학회 (The Korean Mathematical Society)|2023.03|대한수학회보 (Bulletin of the KMS)|제60권 제2호|pp. 361-388|DOI 10.4134/BKMS.b220134

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키워드

Monte-Carlo simulationstochastic volatilityAmerican lookback optionsLaplace-Carson transform
주제분류수학

초록

In this study, we deal with American lookback option prices on dividend-paying assets under a stochastic volatility (SV) model. By using the asymptotic analysis introduced by Fouque et al.~\cite{fouque2011} and the Laplace-Carson transform (LCT), we derive the explicit formula for the option prices and the free boundary values with a finite expiration whose volatility is driven by a fast mean-reverting Ornstein-Uhlenbeck process. In addition, we examine the numerical implications of the SV on the American lookback option with respect to the model parameters and verify that the obtained explicit analytical option price has been obtained accurately and efficiently in comparison with the price obtained from the Monte-Carlo simulation.

Abstract

In this study, we deal with American lookback option prices on dividend-paying assets under a stochastic volatility (SV) model. By using the asymptotic analysis introduced by Fouque et al.~\cite{fouque2011} and the Laplace-Carson transform (LCT), we derive the explicit formula for the option prices and the free boundary values with a finite expiration whose volatility is driven by a fast mean-reverting Ornstein-Uhlenbeck process. In addition, we examine the numerical implications of the SV on the American lookback option with respect to the model parameters and verify that the obtained explicit analytical option price has been obtained accurately and efficiently in comparison with the price obtained from the Monte-Carlo simulation.

참고문헌 (37)

  1. Joseph Abate, "A Unified Framework for Numerically Inverting Laplace Transforms", INFORMS Journal on Computing, 18(4), pp.408~421 (2006), Institute for Operations Research and the Management Sciences (INFORMS)[DOI]
  2. Ankush Agarwal, "American options under stochastic volatility: control variates, maturity randomization & multiscale asymptotics", Quantitative Finance, 16(1), pp.17~30 (2015), Informa UK Limited[DOI]
  3. Fischer Black, "The Pricing of Options and Corporate Liabilities", Journal of Political Economy, 81(3), pp.637~654 (1973), University of Chicago Press[DOI]
  4. P. Carr, "Randomization and the American put", Rev. Financ. Stud, 11(3), pp.597~626 (1998)
  5. SUN-YONG CHOI, "Option pricing under hybrid stochastic and local volatility", Quantitative Finance, 13(8), pp.1157~1165 (2013), Informa UK Limited[DOI]
  6. S.-Y. Choi, "A Mellin transform approach to the pricing of options with default risk, Comput" (2021), Economics
  7. Sun-Yong Choi, "Pricing of Fixed-Strike Lookback Options on Assets with Default Risk", Mathematical Problems in Engineering, 2019, pp.1~10 (2019), Hindawi Limited[DOI]
  8. A. Conze, "Path dependent options : The case of lookback options", J. Finance, 46(5), pp.1893~1907 (1991)
  9. M. Dai, "A closed form solution to perpetual American floating strike lookback option", J. Comput. Finance, 4(2), pp.63~68 (2000)
  10. Min Dai, "American Options with Lookback Payoff", SIAM Journal on Applied Mathematics, 66(1), pp.206~227 (2005), Society for Industrial & Applied Mathematics (SIAM)[DOI]
  11. Min Dai, "CHARACTERIZATION OF OPTIMAL STOPPING REGIONS OF AMERICAN ASIAN AND LOOKBACK OPTIONS", Mathematical Finance, 16(1), pp.63~82 (2006), Wiley[DOI]
  12. Min Dai, "QUANTO LOOKBACK OPTIONS", Mathematical Finance, 14(3), pp.445~467 (2004), Wiley[DOI]
  13. J.-P. Fouque, "Financial modelling in a fast meanreverting stochastic volatility environment", Asia-Pacific Financial Markets, 6(1), pp.37~49 (1999)
  14. J.-P. Fouque, "Mean-reverting stochastic volatility", Intern. J. Theor. Appl. Finance, 3(1), pp.101~142 (2000)
  15. JEAN-PIERRE FOUQUE, "FROM THE IMPLIED VOLATILITY SKEW TO A ROBUST CORRECTION TO BLACK-SCHOLES AMERICAN OPTION PRICES", International Journal of Theoretical and Applied Finance, 04(04), pp.651~675 (2001), World Scientific Pub Co Pte Lt[DOI]
  16. J. P. Fouque, "Singular Perturbations in Option Pricing", SIAM Journal on Applied Mathematics, 63(5), pp.1648~1665 (2003), Society for Industrial & Applied Mathematics (SIAM)[DOI]
  17. J.-P. Fouque, "Multiscale stochastic volatility for equity, interest rate, and credit derivatives" (2011), Cambridge University Press
  18. D. P. Gaver, "Observing Stochastic Processes, and Approximate Transform Inversion", Operations Research, 14(3), pp.444~459 (1966), Institute for Operations Research and the Management Sciences (INFORMS)[DOI]
  19. M. B. Goldman, "Path dependent options: Buy at the low, sell at the high", J. Finance, 34(5), pp.1111~1127 (1979)
  20. Steven L. Heston, "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options", Review of Financial Studies, 6(2), pp.327~343 (1993), Oxford University Press (OUP)[DOI]
  21. Myungjoo Kang, "Analytic solution for American strangle options using Laplace–Carson transforms", Communications in Nonlinear Science and Numerical Simulation, 47, pp.292~307 (2017), Elsevier BV[DOI]
  22. I. Karatzas, "Graduate Texts in Mathematics", pp.113~ (1991), Springer-Verlag
  23. Toshikazu Kimura, "Valuing finite-lived Russian options", European Journal of Operational Research, 189(2), pp.363~374 (2008), Elsevier BV[DOI]
  24. Toshikazu Kimura, "American Fractional Lookback Options: Valuation and Premium Decomposition", SIAM Journal on Applied Mathematics, 71(2), pp.517~539 (2011), Society for Industrial & Applied Mathematics (SIAM)[DOI]
  25. T. Kimura, "Applications of the Laplace-Carson transform to option pricing : A tutorial", Kyoto University Research Information Repository, 1983, pp.68~89 (2016)
  26. Toshikazu Kimura, "A REFINED LAPLACE-CARSON TRANSFORM APPROACH TO VALUING CONVERTIBLE BONDS", Journal of the Operations Research Society of Japan, 60(2), pp.50~65 (2017), The Operations Research Society of Japan[DOI]
  27. A. Kuznetsov, "On the Convergence of the Gaver--Stehfest Algorithm", SIAM Journal on Numerical Analysis, 51(6), pp.2984~2998 (2013), Society for Industrial & Applied Mathematics (SIAM)[DOI]
  28. M. -K. Lee, "Pricing perpetual American lookback options under stochastic volatility", Comput. Econ, 53, pp.1265~1277 (2019)
  29. Kwai Sun Leung, "An analytic pricing formula for lookback options under stochastic volatility", Applied Mathematics Letters, 26(1), pp.145~149 (2013), Elsevier BV[DOI]
  30. A. G. Ramm, "A Simple Proof of the Fredholm Alternative and a Characterization of the Fredholm Operators", The American Mathematical Monthly, 108(9), pp.855~ (2001), Informa UK Limited[DOI]
  31. H. Stehfest, "Algorithm 368 : Numerical inversion of Laplace transforms", Communications of the ACM, 13, pp.47~49 (1970)
  32. L. N. TAO, "THE ANALYTICITY AND GENERAL SOLUTION OF THE CAUCHY-STEFAN PROBLEM", The Quarterly Journal of Mechanics and Applied Mathematics, 36(4), pp.487~504 (1983), Oxford University Press (OUP)[DOI]
  33. Hoi Ying Wong, "Lookback options and dynamic fund protection under multiscale stochastic volatility", Insurance: Mathematics and Economics, 40(3), pp.357~385 (2007), Elsevier BV[DOI]
  34. H. Y. Wong, "Sub-replication and replenishing premium : Efficient pricing of multi-state lookbacks", Rev. Deriv. Res, 6(2), pp.83~106 (2003)
  35. Hoi Ying Wong, "Valuing American options under the CEV model by Laplace–Carson transforms", Operations Research Letters, 38(5), pp.474~481 (2010), Elsevier BV[DOI]
  36. Zhiqiang Zhou, "Laplace Transform Method for Pricing American CEV Strangles Option with Two Free Boundaries", Discrete Dynamics in Nature and Society, 2018, pp.1~12 (2018), Hindawi Limited[DOI]
  37. Song-Ping Zhu, "Pricing perpetual American options under a stochastic-volatility model with fast mean reversion", Applied Mathematics Letters, 24(10), pp.1663~1669 (2011), Elsevier BV[DOI]

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