Convergence Speed of Bermudan, Randomized Bermudan, and Canadian Options
American options have long received considerable attention in the literature, with numerous publications dedicated to their pricing. Bermudan and randomized Bermudan options are broadly used to estimate their prices efficiently. Notably, the penalty method yields option prices that coincide with tho...
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2025-01-01
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| author | Guillaume Leduc |
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| description | American options have long received considerable attention in the literature, with numerous publications dedicated to their pricing. Bermudan and randomized Bermudan options are broadly used to estimate their prices efficiently. Notably, the penalty method yields option prices that coincide with those of randomized Bermudan options. However, theoretical results regarding the speed of convergence of these approximations to the American option price remain scarce. In this paper, we address this gap by establishing a general result on the convergence speed of Bermudan and randomized Bermudan option prices to their American limits. We prove that for convex payoff functions, the convergence speed is linear; that is, of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><mi>n</mi></mrow></semantics></math></inline-formula>, where <i>n</i> denotes the number of exercisable opportunities in the Bermudan case and serves as the intensity parameter of the underlying Poisson process in the randomized Bermudan case. Our framework is quite general, encompassing Lévy models, stochastic volatility models, and nearly any risk-neutral model that can be incorporated within a strong Markov framework. We extend our analysis to Canadian options, showing under mild conditions a convergence rate of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><msqrt><mi>n</mi></msqrt></mrow></semantics></math></inline-formula> to their American limits. To our knowledge, this is the first study addressing the speed of convergence in Canadian option pricing. |
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| spelling | doaj-art-4e343b9b9b2a4c54ba89fbfc7e55bea82025-01-24T13:39:46ZengMDPI AGMathematics2227-73902025-01-0113221310.3390/math13020213Convergence Speed of Bermudan, Randomized Bermudan, and Canadian OptionsGuillaume Leduc0Department of Mathematics and Statistics, American University of Sharjah, Sharjah P.O. Box 26666, United Arab EmiratesAmerican options have long received considerable attention in the literature, with numerous publications dedicated to their pricing. Bermudan and randomized Bermudan options are broadly used to estimate their prices efficiently. Notably, the penalty method yields option prices that coincide with those of randomized Bermudan options. However, theoretical results regarding the speed of convergence of these approximations to the American option price remain scarce. In this paper, we address this gap by establishing a general result on the convergence speed of Bermudan and randomized Bermudan option prices to their American limits. We prove that for convex payoff functions, the convergence speed is linear; that is, of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><mi>n</mi></mrow></semantics></math></inline-formula>, where <i>n</i> denotes the number of exercisable opportunities in the Bermudan case and serves as the intensity parameter of the underlying Poisson process in the randomized Bermudan case. Our framework is quite general, encompassing Lévy models, stochastic volatility models, and nearly any risk-neutral model that can be incorporated within a strong Markov framework. We extend our analysis to Canadian options, showing under mild conditions a convergence rate of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><msqrt><mi>n</mi></msqrt></mrow></semantics></math></inline-formula> to their American limits. To our knowledge, this is the first study addressing the speed of convergence in Canadian option pricing.https://www.mdpi.com/2227-7390/13/2/213Bermudan optionAmerican optionrandomizationconvergence speed |
| spellingShingle | Guillaume Leduc Convergence Speed of Bermudan, Randomized Bermudan, and Canadian Options Mathematics Bermudan option American option randomization convergence speed |
| title | Convergence Speed of Bermudan, Randomized Bermudan, and Canadian Options |
| title_full | Convergence Speed of Bermudan, Randomized Bermudan, and Canadian Options |
| title_fullStr | Convergence Speed of Bermudan, Randomized Bermudan, and Canadian Options |
| title_full_unstemmed | Convergence Speed of Bermudan, Randomized Bermudan, and Canadian Options |
| title_short | Convergence Speed of Bermudan, Randomized Bermudan, and Canadian Options |
| title_sort | convergence speed of bermudan randomized bermudan and canadian options |
| topic | Bermudan option American option randomization convergence speed |
| url | https://www.mdpi.com/2227-7390/13/2/213 |
| work_keys_str_mv | AT guillaumeleduc convergencespeedofbermudanrandomizedbermudanandcanadianoptions |