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Mixing Monte Carlo and Partial Diffe...

Farahany, David.

Mixing Monte Carlo and Partial Differential Equation Methods for Multi-Dimensional Optimal Stopping Problems under Stochastic Volatility.

Record Type:	Electronic resources : Monograph/item
Title/Author:	Mixing Monte Carlo and Partial Differential Equation Methods for Multi-Dimensional Optimal Stopping Problems under Stochastic Volatility.
Author:	Farahany, David.
Published:	Ann Arbor : ProQuest Dissertations & Theses, 2019
Description:	125 p.
Notes:	Source: Dissertations Abstracts International, Volume: 81-04, Section: A.
Notes:	Advisor: Jaimungal, Sebastian;Jackson, Kenneth.
Contained By:	Dissertations Abstracts International81-04A.
Subject:	Statistics.
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=10979340
ISBN:	9781085755528

Mixing Monte Carlo and Partial Differential Equation Methods for Multi-Dimensional Optimal Stopping Problems under Stochastic Volatility.
Farahany, David.

Mixing Monte Carlo and Partial Differential Equation Methods for Multi-Dimensional Optimal Stopping Problems under Stochastic Volatility. - Ann Arbor : ProQuest Dissertations & Theses, 2019 - 125 p.

Source: Dissertations Abstracts International, Volume: 81-04, Section: A.

Thesis (Ph.D.)--University of Toronto (Canada), 2019.

This item must not be sold to any third party vendors.

In this thesis, we develop a numerical approach for solving multi-dimensional optimal stopping problems (OSPs) under stochastic volatility (SV) that combines least squares Monte Carlo (LSMC) with partial differential equation (PDE) techniques. The algorithm provides dimensional reduction from the PDE and regression perspective along with variance and dimensional reduction from the MC perspective. In Chapter 2, we begin by laying the mathematical foundation of mixed MC-PDE techniques for OSPs. Next, we show the basic mechanics of the algorithm and, under certain mild assumptions, prove it converges almost surely. We apply the algorithm to the one dimensional Heston model and demonstrate that the hybrid algorithm outperforms traditional LSMC techniques in terms of both estimating prices and optimal exercise boundaries (OEBs).In Chapter 3 we describe methods for reducing the complexity and run time of the algorithm along with techniques for computing sensitivities. To reduce the complexity, we apply two methods: clustering via sufficient statistics and multi-level Monte Carlo (mlMC)/multi-grids. While the clustering method allows us to reduce computational run times by a third for high dimensional problems, mlMC provides an order of magnitude reduction in complexity. To compute sensitivities, we employ a grid based method for derivatives with respect to the asset, S, and an MC method that uses initial dispersions for sensitivities with respect to variance, v. To test our approximations and computation of sensitivities, we revisit the one-dimensional Heston model and find our approximations introduce little-to-no error and that our computation of sensitivities is highly accurate in comparison to standard LSMC. To demonstrate the utility of our new computational techniques, we apply the hybrid algorithm to the multi-dimensional Heston model and show that the algorithm is highly accurate in terms of estimating prices, OEBs, and sensitivities, especially in comparison to standard LSMC.In Chapter 4 we highlight the importance of multi-factor SV models and apply our hybrid algorithm to two specific examples: the Double Heston model and a mean-reverting commodity model with jumps. Again, we were able to obtain low variance estimates of the prices, OEBs, and sensitivities.

ISBN: 9781085755528Subjects--Topical Terms:

182057
Statistics.

Mixing Monte Carlo and Partial Differential Equation Methods for Multi-Dimensional Optimal Stopping Problems under Stochastic Volatility.
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520 $a In this thesis, we develop a numerical approach for solving multi-dimensional optimal stopping problems (OSPs) under stochastic volatility (SV) that combines least squares Monte Carlo (LSMC) with partial differential equation (PDE) techniques. The algorithm provides dimensional reduction from the PDE and regression perspective along with variance and dimensional reduction from the MC perspective. In Chapter 2, we begin by laying the mathematical foundation of mixed MC-PDE techniques for OSPs. Next, we show the basic mechanics of the algorithm and, under certain mild assumptions, prove it converges almost surely. We apply the algorithm to the one dimensional Heston model and demonstrate that the hybrid algorithm outperforms traditional LSMC techniques in terms of both estimating prices and optimal exercise boundaries (OEBs).In Chapter 3 we describe methods for reducing the complexity and run time of the algorithm along with techniques for computing sensitivities. To reduce the complexity, we apply two methods: clustering via sufficient statistics and multi-level Monte Carlo (mlMC)/multi-grids. While the clustering method allows us to reduce computational run times by a third for high dimensional problems, mlMC provides an order of magnitude reduction in complexity. To compute sensitivities, we employ a grid based method for derivatives with respect to the asset, S, and an MC method that uses initial dispersions for sensitivities with respect to variance, v. To test our approximations and computation of sensitivities, we revisit the one-dimensional Heston model and find our approximations introduce little-to-no error and that our computation of sensitivities is highly accurate in comparison to standard LSMC. To demonstrate the utility of our new computational techniques, we apply the hybrid algorithm to the multi-dimensional Heston model and show that the algorithm is highly accurate in terms of estimating prices, OEBs, and sensitivities, especially in comparison to standard LSMC.In Chapter 4 we highlight the importance of multi-factor SV models and apply our hybrid algorithm to two specific examples: the Double Heston model and a mean-reverting commodity model with jumps. Again, we were able to obtain low variance estimates of the prices, OEBs, and sensitivities.
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