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Pricing Bermudan options: A nonparametric estimation approach
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
(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 [en]
Optimal stopping, regularization, nonparametric estimation, conditional expectations, Bermudan options, Snell envelope.
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:kth:diva-42155OAI: oai:DiVA.org:kth-42155DiVA: diva2:445877
Note

QS 2011105

Available from: 2011-10-05 Created: 2011-10-05 Last updated: 2013-04-16Bibliographically approved
In thesis
1. On the Snell envelope approach to optimal switching and pricing Bermudan options
Open this publication in new window or tab >>On the Snell envelope approach to optimal switching and pricing Bermudan options
2011 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of two papers related to systems of Snell envelopes. The first paper uses a system of Snell envelopes to formulate the problem of two-modes optimal switching for the full balance sheet in finite horizon. This means that the switching problem is formulated in terms of trade-off strategies between expected profit and cost yields, which act as obstacles to each other. Existence of a minimal solution of this system is obtained by using an approximation scheme. Furthermore, the optimal switching strategies are fully characterized.

The second paper uses the Snell envelope to formulate the fair price of Bermudan options. To evaluate this formulation of the price, the optimal stopping strategy for such a contract must be estimated. This may be done recursively if some method of estimating conditional expectations is available. The paper focuses on nonparametric estimation of such expectations, by using regularization of a least-squares minimization, with a Tikhonov-type smoothing put on the partial diferential 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 tothe size of the smoothing parameter.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2011. viii p.
Series
Trita-MAT, ISSN 1401-2286 ; 11:01
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:kth:diva-42274 (URN)978-91-7501-108-0 (ISBN)
Presentation
2011-10-20, 3721, Lindstedtsvägen 25, KTH, Stockholm, 15:00 (English)
Opponent
Supervisors
Note
QC 20111013Available from: 2011-10-13 Created: 2011-10-06 Last updated: 2011-10-13Bibliographically approved
2. Some aspects of optimal switching and pricing Bermudan options
Open this publication in new window or tab >>Some aspects of optimal switching and pricing Bermudan options
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:nbn:se:kth:diva-120478 (URN)978-91-7501-707-5 (ISBN)
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

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