Recursive marginal quantization: extensions and applications in finance
| dc.contributor.advisor | Kienitz, Jorg | |
| dc.contributor.advisor | Platen, Eckhard | |
| dc.contributor.author | Rudd, Ralph | |
| dc.date.accessioned | 2018-09-04T10:42:25Z | |
| dc.date.available | 2018-09-04T10:42:25Z | |
| dc.date.issued | 2018 | |
| dc.date.updated | 2018-09-03T06:34:01Z | |
| dc.description.abstract | 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. | |
| dc.identifier.apacitation | Rudd, R. (2018). <i>Recursive marginal quantization: extensions and applications in finance</i>. (). University of Cape Town ,Faculty of Commerce ,African Inst. of Fin. Markets and Risk Mngnt. Retrieved from http://hdl.handle.net/11427/28378 | en_ZA |
| dc.identifier.chicagocitation | Rudd, Ralph. <i>"Recursive marginal quantization: extensions and applications in finance."</i> ., University of Cape Town ,Faculty of Commerce ,African Inst. of Fin. Markets and Risk Mngnt, 2018. http://hdl.handle.net/11427/28378 | en_ZA |
| dc.identifier.citation | Rudd, R. 2018. Recursive marginal quantization: extensions and applications in finance. University of Cape Town. | en_ZA |
| dc.identifier.ris | TY - Thesis / Dissertation AU - Rudd, Ralph AB - 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. DA - 2018 DB - OpenUCT DP - University of Cape Town LK - https://open.uct.ac.za PB - University of Cape Town PY - 2018 T1 - Recursive marginal quantization: extensions and applications in finance TI - Recursive marginal quantization: extensions and applications in finance UR - http://hdl.handle.net/11427/28378 ER - | en_ZA |
| dc.identifier.uri | http://hdl.handle.net/11427/28378 | |
| dc.identifier.vancouvercitation | Rudd R. Recursive marginal quantization: extensions and applications in finance. []. University of Cape Town ,Faculty of Commerce ,African Inst. of Fin. Markets and Risk Mngnt, 2018 [cited yyyy month dd]. Available from: http://hdl.handle.net/11427/28378 | en_ZA |
| dc.language.iso | eng | |
| dc.publisher.department | African Inst. of Fin. Markets and Risk Mngnt | en_ZA |
| dc.publisher.faculty | Faculty of Commerce | en_ZA |
| dc.publisher.institution | University of Cape Town | |
| dc.subject.other | finance | |
| dc.subject.other | commerce | |
| dc.title | Recursive marginal quantization: extensions and applications in finance | |
| dc.type | Thesis | |
| uct.type.filetype | Text | |
| uct.type.filetype | Image |