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Some aspects of optimal switching and pricing Bermudan options
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
2013 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of four papers that are all related to the Snell envelope. In the first paper, the Snell envelope is used as a formulation of a two-modes optimal switching problem. The obstacles are interconnected, take both profit and cost yields into account, and switching is based on both sides of the balance sheet. The main result is a proof of existence of a continuous minimal solution to a system of Snell envelopes, which fully characterizes the optimal switching strategy. A counter-example is provided to show that uniqueness does not hold.

The second paper considers the problem of having a large number of production lines with two modes of production, high-production and low-production. As in the first paper, we consider both expected profit and cost yields and switching based on both sides of the balance sheet. The production lines are assumed to be interconnected through a coupling term, which is the average optimal expected yields. The corresponding system of Snell envelopes is highly complex, so we consider the aggregated yields where a mean-field approximation is used for the coupling term. The main result is a proof of existence of a continuous minimal solution to a system of Snell envelopes, which fully characterizes the optimal switching strategy. Furthermore, existence and uniqueness is proven for the mean-field reflected backward stochastic differential equations (MF-RBSDEs) we consider, a comparison theorem and a uniform bound for the MF-RBSDEs is provided.

The third paper concerns pricing of Bermudan type options. The Snell envelope is used as a representation of the price, which is determined using Monte Carlo simulation combined with the dynamic programming principle. For this approach, it is necessary to estimate the conditional expectation of the future optimally exercised payoff. We formulate a projection on a grid which is ill-posed due to overfitting, and regularize with the PDE which characterizes the underlying process. The method is illustrated with numerical examples, where accurate results are demonstrated in one dimension.

In the fourth paper, the idea of the third paper is extended to the multi-dimensional setting. This is necessary because in one dimension it is more efficient to solve the PDE than to use Monte Carlo simulation. We relax the use of a grid in the projection, and add local weights for stability. Using the multi-dimensional Black-Scholes model, the method is illustrated in settings ranging from one to 30 dimensions. The method is shown to produce accurate results in all examples, given a good choice of the regularization parameter.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2013. , v, 20 p.
Series
Trita-MAT, ISSN 1401-2286 ; 13:02
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:kth:diva-120478ISBN: 978-91-7501-707-5 (print)OAI: oai:DiVA.org:kth-120478DiVA: diva2:615257
Public defence
2013-05-17, F3, Lindstedtsvägen 26, KTH, Stockholm, 10:00 (English)
Opponent
Supervisors
Note

QC 20130416

Available from: 2013-04-16 Created: 2013-04-09 Last updated: 2013-04-16Bibliographically approved
List of papers
1. A Full Balance Sheet Two-modes Optimal Switching problem
Open this publication in new window or tab >>A Full Balance Sheet Two-modes Optimal Switching problem
(English)In: Mathematical Methods of Operations Research, ISSN 1432-2994, E-ISSN 1432-5217Article in journal (Other academic) Submitted
Abstract [en]

We formulate and solve a finite horizon full balance sheet two-modes optimal switching problem related to trade-off strategies between expected profit and cost yields. The optimal switching problem is formulated in terms of a system of Snell envelopes for the profit and cost yields which act as obstacles to each other. We prove existence of a continuous minimal solution of this system using an approximation scheme and fully characterize the optimal switching strategy.

Keyword
real options, backward SDEs, Snell envelope, stopping time, optimal switching, impulse control, balance sheet, merger and acquisition
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:kth:diva-42154 (URN)
Note
QS 2011 QS 20120326Available from: 2011-10-05 Created: 2011-10-05 Last updated: 2017-12-08Bibliographically approved
2. A Two-modes Mean-field Optimal Switching Problem for The Full Balance Sheet
Open this publication in new window or tab >>A Two-modes Mean-field Optimal Switching Problem for The Full Balance Sheet
2014 (English)In: International Journal of Stochastic Analysis, ISSN 2090-3332, E-ISSN 2090-3340, 159519Article in journal (Refereed) Published
Abstract [en]

We consider the problem of switching a large number of production lines between two modes, high-production and low-production. The switching is based on the optimal expected profit and cost yields of the respective production lines, and considers both sides of the balance sheet. Furthermore, the production lines are all assumed to be interconnected through a coupling term, which is the average of all optimal expected yields. Intuitively, this means that each individual production line is compared to the average of all its peers which acts as a benchmark.

Due to the complexity of the problem, we consider the aggregated optimal expected yields, where the coupling term is approximated with the mean of the optimal expected yields. This turns the problem into a two-modes optimal switching problem of mean-field type, which can be described by a system of Snell envelopes where the obstacles are interconnected and nonlinear.

The main result of the paper is a proof of a continuous minimal solution to the system of Snell envelopes, as well as the full characterization of the optimal switching strategy.

Keyword
Real options, backward SDEs, Snell envelope, stopping time, optimal switching, impulse control, balance sheet, merger and acquisition, mean-field.
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:kth:diva-120476 (URN)10.1155/2014/159519 (DOI)2-s2.0-84902171942 (Scopus ID)
Note

Updated from "Manuscript" to "Article". QC 20150918

Available from: 2013-04-09 Created: 2013-04-09 Last updated: 2017-12-06Bibliographically approved
3. Pricing Bermudan options: A nonparametric estimation approach
Open this publication in new window or tab >>Pricing Bermudan options: A nonparametric estimation approach
(English)Manuscript (preprint) (Other academic)
Abstract [en]

A nonparametric alternative to the Longstaff-Schwartz estimation of conditional expectations is suggested for pricing of Bermudan options. The method is based on regularization of a least-squares minimization, with a Tikhonov-type smoothing put on the partial differential equation which characterizes the underlying price processes. This approach can hence be viewed as a combination of the Monte Carlo method and the PDE method for the estimation of conditional expectations. The estimation method turns out to be robust with regard to the size of the smoothing parameter.

Keyword
Optimal stopping, regularization, nonparametric estimation, conditional expectations, Bermudan options, Snell envelope.
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:kth:diva-42155 (URN)
Note

QS 2011105

Available from: 2011-10-05 Created: 2011-10-05 Last updated: 2013-04-16Bibliographically approved
4. PDE-regularization for pricing multi-dimensional Bermudan options with Monte Carlo simulation
Open this publication in new window or tab >>PDE-regularization for pricing multi-dimensional Bermudan options with Monte Carlo simulation
(English)Manuscript (preprint) (Other academic)
Abstract [en]

This paper considers the problem of pricing multi-dimensional Bermudan derivatives using Monte Carlo simulation. A new method for computing conditional expectations is proposed, which combined with the dynamic programming principle provides a way of pricing the derivatives.

The method is a non-parametric projection with regularization. The regularization penalizes deviations from the PDE that the true conditional expectation satisfies. The point being that it is less costly to compute the norm of the PDE than it is to solve it, thus avoiding the curse of dimensionality.

The method is shown to produce accurate numerical results in multi-dimensional settings, given a good choice of the regularization parameter. It is illustrated with the multi-dimensional Black-Scholes model and compared to the Longstaff-Schwartz approach.

Keyword
Optimal stopping, regularization, non-parametric estimation, conditional expectations, Bermudan options, Snell envelope.
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:kth:diva-120477 (URN)
Note

QS 2013

Available from: 2013-04-09 Created: 2013-04-09 Last updated: 2013-04-16Bibliographically approved

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