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Recursive marginal quantization: extensions and applications in finance
Quantization techniques have been used in many challenging finance applications, including pricing claims with path dependence and early exercise features, stochastic optimal control, filtering problems and the efficient calibration of large derivative books. Recursive marginal quantization of an Eu...
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| Format: | Thesis |
| Language: | English |
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African Inst. of Fin. Markets and Risk Mngnt
2018
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| _version_ | 1867613142159196160 |
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| access_status_str | Open Access |
| author | Rudd, Ralph |
| author2 | Kienitz, Jorg |
| author_browse | Kienitz, Jorg Rudd, Ralph |
| author_facet | Kienitz, Jorg Rudd, Ralph |
| author_sort | Rudd, Ralph |
| collection | Thesis |
| description | Quantization techniques have been used in many challenging finance applications, including pricing claims with path dependence and early exercise features, stochastic optimal control, filtering problems and the efficient calibration of large derivative books. Recursive marginal quantization of an Euler scheme has recently been proposed as an efficient numerical method for evaluating functionals of solutions of stochastic differential equations. This algorithm is generalized and it is shown that it is possible to perform recursive marginal quantization for two higher-order schemes: the Milstein scheme and a simplified weak-order 2.0 scheme. Furthermore, the recursive marginal quantization algorithm is extended by showing how absorption and reflection at the zero boundary may be incorporated. Numerical evidence is provided of the improved weak-order convergence and computational efficiency for the geometric Brownian motion and constant elasticity of variance models by pricing European, Bermudan and barrier options. The current theoretical error bound is extended to apply to the proposed higher-order methods. When applied to two-factor models, recursive marginal quantization becomes computationally inefficient as the optimization problem usually requires stochastic methods, for example, the randomized Lloyd’s algorithm or Competitive Learning Vector Quantization. To address this, a new algorithm is proposed that allows recursive marginal quantization to be applied to two-factor stochastic volatility models while retaining the efficiency of the original Newton-Raphson gradientdescent technique. The proposed method is illustrated for European options on the Heston and Stein-Stein models and for various exotic options on the popular SABR model. Finally, the recursive marginal quantization algorithm, and improvements, are applied outside the traditional risk-neutral pricing framework by pricing long-dated contracts using the benchmark approach. The growth-optimal portfolio, the central object of the benchmark approach, is modelled using the time-dependent constant elasticity of variance model. Analytic European option prices are derived that generalize the current formulae in the literature. The time-dependent constant elasticity of variance model is then combined with a 3/2 stochastic short rate model to price zerocoupon bonds and zero-coupon bond options, thereby showing the departure from risk-neutral pricing. |
| format | Thesis |
| id | oai:open.uct.ac.za:11427/28378 |
| institution | University of Cape Town (South Africa) |
| language | eng |
| last_indexed | 2026-06-10T12:31:26.417Z |
| license_str | Not specified — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UCTD — University of Cape Town Open Access Repository |
| publishDate | 2018 |
| publishDateRange | 2018 |
| publishDateSort | 2018 |
| publisher | African Inst. of Fin. Markets and Risk Mngnt |
| publisherStr | African Inst. of Fin. Markets and Risk Mngnt |
| record_format | dspace |
| source_str | UCTD — University of Cape Town Open Access Repository |
| spelling | oai:open.uct.ac.za:11427/28378 Recursive marginal quantization: extensions and applications in finance Rudd, Ralph Kienitz, Jorg Platen, Eckhard finance commerce Quantization techniques have been used in many challenging finance applications, including pricing claims with path dependence and early exercise features, stochastic optimal control, filtering problems and the efficient calibration of large derivative books. Recursive marginal quantization of an Euler scheme has recently been proposed as an efficient numerical method for evaluating functionals of solutions of stochastic differential equations. This algorithm is generalized and it is shown that it is possible to perform recursive marginal quantization for two higher-order schemes: the Milstein scheme and a simplified weak-order 2.0 scheme. Furthermore, the recursive marginal quantization algorithm is extended by showing how absorption and reflection at the zero boundary may be incorporated. Numerical evidence is provided of the improved weak-order convergence and computational efficiency for the geometric Brownian motion and constant elasticity of variance models by pricing European, Bermudan and barrier options. The current theoretical error bound is extended to apply to the proposed higher-order methods. When applied to two-factor models, recursive marginal quantization becomes computationally inefficient as the optimization problem usually requires stochastic methods, for example, the randomized Lloyd’s algorithm or Competitive Learning Vector Quantization. To address this, a new algorithm is proposed that allows recursive marginal quantization to be applied to two-factor stochastic volatility models while retaining the efficiency of the original Newton-Raphson gradientdescent technique. The proposed method is illustrated for European options on the Heston and Stein-Stein models and for various exotic options on the popular SABR model. Finally, the recursive marginal quantization algorithm, and improvements, are applied outside the traditional risk-neutral pricing framework by pricing long-dated contracts using the benchmark approach. The growth-optimal portfolio, the central object of the benchmark approach, is modelled using the time-dependent constant elasticity of variance model. Analytic European option prices are derived that generalize the current formulae in the literature. The time-dependent constant elasticity of variance model is then combined with a 3/2 stochastic short rate model to price zerocoupon bonds and zero-coupon bond options, thereby showing the departure from risk-neutral pricing. 2018-09-04T10:42:25Z 2018-09-04T10:42:25Z 2018 2018-09-03T06:34:01Z Thesis http://hdl.handle.net/11427/28378 eng application/pdf African Inst. of Fin. Markets and Risk Mngnt Faculty of Commerce University of Cape Town |
| spellingShingle | finance commerce Rudd, Ralph Recursive marginal quantization: extensions and applications in finance |
| title | Recursive marginal quantization: extensions and applications in finance |
| title_full | Recursive marginal quantization: extensions and applications in finance |
| title_fullStr | Recursive marginal quantization: extensions and applications in finance |
| title_full_unstemmed | Recursive marginal quantization: extensions and applications in finance |
| title_short | Recursive marginal quantization: extensions and applications in finance |
| title_sort | recursive marginal quantization extensions and applications in finance |
| topic | finance commerce |
| url | http://hdl.handle.net/11427/28378 |
| work_keys_str_mv | AT ruddralph recursivemarginalquantizationextensionsandapplicationsinfinance |