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A finite element approach to the pricing of discrete lookbacks with stochastic volatility

Author

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  • P. A. Forsyth
  • K. R. Vetzal
  • R. Zvan

Abstract

Finite element methods are described for valuing lookback options under stochastic volatility. Particular attention is paid to the method for handling the boundary equations. For some boundaries, the equations reduce to first-order hyperbolic equations which must be discretized to ensure that outgoing waves are correctly modelled. Some example computations show that for certain choices of parameters, the option price computed for a lookback under stochastic volatility can differ from the price under the usual constant volatility assumption by as much as 35% (i.e. $7.30 compared with $5.45 for an at-the-money put), even though the models are calibrated so as to produce exactly the same price for an at-the-money vanilla European option with the same time remaining until expiry.

Suggested Citation

  • P. A. Forsyth & K. R. Vetzal & R. Zvan, 1999. "A finite element approach to the pricing of discrete lookbacks with stochastic volatility," Applied Mathematical Finance, Taylor & Francis Journals, vol. 6(2), pages 87-106.
    • Handle: RePEc:taf:apmtfi:v:6:y:1999:i:2:p:87-106
    DOI: 10.1080/135048699334564
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    2. Garnadi, Agah D., 2017. "Valuasi Opsi Beli ({\it Call Options}) Eropa bervolatilitas Stokastik dengan menggunakan Modifikasi Metode Karakteristik dan Metode Elemen Hingga," INA-Rxiv fhbsx, Center for Open Science.
    3. Windcliff, H. & Vetzal, K. R. & Forsyth, P. A. & Verma, A. & Coleman, T. F., 2003. "An object-oriented framework for valuing shout options on high-performance computer architectures," Journal of Economic Dynamics and Control, Elsevier, vol. 27(6), pages 1133-1161, April.
    4. P. Forsyth & K. Vetzal & R. Zvan, 2002. "Convergence of numerical methods for valuing path-dependent options using interpolation," Review of Derivatives Research, Springer, vol. 5(3), pages 273-314, October.
    5. Simona Sanfelici, 2004. "Galerkin infinite element approximation for pricing barrier options and options with discontinuous payoff," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 27(2), pages 125-151, December.
    6. Gongqiu Zhang & Lingfei Li, 2021. "A General Approach for Lookback Option Pricing under Markov Models," Papers 2112.00439, arXiv.org.
    7. Raahauge, Peter, 2004. "Higher-Order Finite Element Solutions of Option Prices," Working Papers 2004-5, Copenhagen Business School, Department of Finance.
    8. Mark Broadie & Jerome B. Detemple, 2004. "ANNIVERSARY ARTICLE: Option Pricing: Valuation Models and Applications," Management Science, INFORMS, vol. 50(9), pages 1145-1177, September.
    9. Fard, Farzad Alavi & Siu, Tak Kuen, 2013. "Pricing participating products with Markov-modulated jump–diffusion process: An efficient numerical PIDE approach," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 712-721.
    10. Carole Bernard & Junsen Tang, 2016. "Simplified Hedge For Path-Dependent Derivatives," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(07), pages 1-32, November.
    11. Bertram Düring & Michel Fournié & Ansgar Jüngel, 2003. "High Order Compact Finite Difference Schemes for a Nonlinear Black-Scholes Equation," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 6(07), pages 767-789.
    12. Farzad Alavi Fard, 2014. "Optimal Bid-Ask Spread in Limit-Order Books under Regime Switching Framework," Review of Economics & Finance, Better Advances Press, Canada, vol. 4, pages 33-48, November.
    13. Li, Hongshan & Huang, Zhongyi, 2020. "An iterative splitting method for pricing European options under the Heston model☆," Applied Mathematics and Computation, Elsevier, vol. 387(C).
    14. Hongshan Li & Zhongyi Huang, 2020. "An iterative splitting method for pricing European options under the Heston model," Papers 2003.12934, arXiv.org.
    15. Rakhymzhan Kazbek & Yogi Erlangga & Yerlan Amanbek & Dongming Wei, 2023. "Valuation of the Convertible Bonds under Penalty TF model using Finite Element Method," Papers 2301.10734, arXiv.org.

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