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A Path Integral Approach to Derivative Security Pricing: I. Formalism and Analytical Results

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  • Marco Rosa-Clot
  • Stefano Taddei

Abstract

We use a path integral approach for solving the stochastic equations underlying the financial markets, and we show the equivalence between the path integral and the usual SDE and PDE methods. We analyze both the one-dimensional and the multi-dimensional cases, with point dependent drift and volatility, and describe a covariant formulation which allows general changes of variables. Finally we apply the method to some economic models with analytical solutions. In particular, we evaluate the expectation value of functionals which correspond to quantities of financial interest.

Suggested Citation

  • Marco Rosa-Clot & Stefano Taddei, 1999. "A Path Integral Approach to Derivative Security Pricing: I. Formalism and Analytical Results," Papers cond-mat/9901277, arXiv.org.
    • Handle: RePEc:arx:papers:cond-mat/9901277
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    Citations

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    Cited by:

    1. Paolinelli, Giovanni & Arioli, Gianni, 2019. "A model for stocks dynamics based on a non-Gaussian path integral," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 517(C), pages 499-514.
    2. Paolinelli, Giovanni & Arioli, Gianni, 2018. "A path integral based model for stocks and order dynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 510(C), pages 387-399.
    3. Bustamante, M. & Contreras, M., 2016. "Multi-asset Black–Scholes model as a variable second class constrained dynamical system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 457(C), pages 540-572.
    4. Zura Kakushadze, 2014. "Path Integral and Asset Pricing," Papers 1410.1611, arXiv.org, revised Aug 2016.
    5. Marco Rosa-Clot & Stefano Taddei, 2002. "A Path Integral Approach To Derivative Security Pricing Ii: Numerical Methods," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 5(02), pages 123-146.
    6. Ingber, Lester, 2000. "High-resolution path-integral development of financial options," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 283(3), pages 529-558.
    7. Andrew Matacz, 2000. "Path dependent option pricing: the path integral partial averaging method," Science & Finance (CFM) working paper archive 500034, Science & Finance, Capital Fund Management.
    8. Decamps, Marc & De Schepper, Ann & Goovaerts, Marc, 2006. "A path integral approach to asset-liability management," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 363(2), pages 404-416.
    9. Contreras, Mauricio & Pellicer, Rely & Villena, Marcelo, 2017. "Dynamic optimization and its relation to classical and quantum constrained systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 479(C), pages 12-25.
    10. Montagna, Guido & Nicrosini, Oreste & Moreni, Nicola, 2002. "A path integral way to option pricing," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 310(3), pages 450-466.
    11. G., Mauricio Contreras & Peña, Juan Pablo, 2019. "The quantum dark side of the optimal control theory," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 515(C), pages 450-473.
    12. L. Ingber & C. Chen & R.P. Mondescu & D. Muzzall & M. Renedo, 2001. "Probability tree algorithm for general diffusion processes," Lester Ingber Papers 01pt, Lester Ingber.
    13. Cassagnes, Aurelien & Chen, Yu & Ohashi, Hirotada, 2014. "Path integral pricing of outside barrier Asian options," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 394(C), pages 266-276.
    14. Andrew Matacz, 2000. "Path Dependent Option Pricing: the path integral partial averaging method," Papers cond-mat/0005319, arXiv.org.
    15. Giovanni Paolinelli & Gianni Arioli, 2018. "A model for stocks dynamics based on a non-Gaussian path integral," Papers 1809.01342, arXiv.org, revised Oct 2018.
    16. Moore, Ryleigh A. & Narayan, Akil, 2022. "Adaptive density tracking by quadrature for stochastic differential equations," Applied Mathematics and Computation, Elsevier, vol. 431(C).
    17. Yu. A. Kuperin & P. A. Poloskov, 2010. "Analytical and Numerical Approaches to Pricing the Path-Dependent Options with Stochastic Volatility," Papers 1009.4587, arXiv.org.
    18. Igor Halperin, 2021. "Distributional Offline Continuous-Time Reinforcement Learning with Neural Physics-Informed PDEs (SciPhy RL for DOCTR-L)," Papers 2104.01040, arXiv.org.
    19. Yu. A. Kuperin & P. A. Poloskov, 2010. "American Options Pricing under Stochastic Volatility: Approximation of the Early Exercise Surface and Monte Carlo Simulations," Papers 1009.5495, arXiv.org.

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