Implied Stopping Rules for American Basket Options from Markovian Projection

Handle URI:
http://hdl.handle.net/10754/626502
Title:
Implied Stopping Rules for American Basket Options from Markovian Projection
Authors:
Bayer, Christian; Häppölä, Juho; Tempone, Raul ( 0000-0003-1967-4446 )
Abstract:
This work addresses the problem of pricing American basket options in a multivariate setting, which includes among others, the Bachelier and the Black-Scholes models. In high dimensions, nonlinear partial differential equation methods for solving the problem become prohibitively costly due to the curse of dimensionality. Instead, this work proposes to use a stopping rule that depends on the dynamics of a low-dimensional Markovian projection of the given basket of assets. It is shown that the ability to approximate the original value function by a lower-dimensional approximation is a feature of the dynamics of the system and is unaffected by the path-dependent nature of the American basket option. Assuming that we know the density of the forward process and using the Laplace approximation, we first efficiently evaluate the diffusion coefficient corresponding to the low-dimensional Markovian projection of the basket. Then, we approximate the optimal early-exercise boundary of the option by solving a Hamilton-Jacobi-Bellman partial differential equation in the projected, low-dimensional space. The resulting near-optimal early-exercise boundary is used to produce an exercise strategy for the high-dimensional option, thereby providing a lower bound for the price of the American basket option. A corresponding upper bound is also provided. These bounds allow to assess the accuracy of the proposed pricing method. Indeed, our approximate early-exercise strategy provides a straightforward lower bound for the American basket option price. Following a duality argument due to Rogers, we derive a corresponding upper bound solving only the low-dimensional optimal control problem. Numerically, we show the feasibility of the method using baskets with dimensions up to fifty. In these examples, the resulting option price relative errors are only of the order of few percent.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Publisher:
arXiv
Issue Date:
1-May-2017
ARXIV:
arXiv:1705.00558
Type:
Preprint
Additional Links:
http://arxiv.org/abs/1705.00558v4; http://arxiv.org/pdf/1705.00558v4
Appears in Collections:
Other/General Submission; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorBayer, Christianen
dc.contributor.authorHäppölä, Juhoen
dc.contributor.authorTempone, Raulen
dc.date.accessioned2017-12-28T07:32:13Z-
dc.date.available2017-12-28T07:32:13Z-
dc.date.issued2017-05-01en
dc.identifier.urihttp://hdl.handle.net/10754/626502-
dc.description.abstractThis work addresses the problem of pricing American basket options in a multivariate setting, which includes among others, the Bachelier and the Black-Scholes models. In high dimensions, nonlinear partial differential equation methods for solving the problem become prohibitively costly due to the curse of dimensionality. Instead, this work proposes to use a stopping rule that depends on the dynamics of a low-dimensional Markovian projection of the given basket of assets. It is shown that the ability to approximate the original value function by a lower-dimensional approximation is a feature of the dynamics of the system and is unaffected by the path-dependent nature of the American basket option. Assuming that we know the density of the forward process and using the Laplace approximation, we first efficiently evaluate the diffusion coefficient corresponding to the low-dimensional Markovian projection of the basket. Then, we approximate the optimal early-exercise boundary of the option by solving a Hamilton-Jacobi-Bellman partial differential equation in the projected, low-dimensional space. The resulting near-optimal early-exercise boundary is used to produce an exercise strategy for the high-dimensional option, thereby providing a lower bound for the price of the American basket option. A corresponding upper bound is also provided. These bounds allow to assess the accuracy of the proposed pricing method. Indeed, our approximate early-exercise strategy provides a straightforward lower bound for the American basket option price. Following a duality argument due to Rogers, we derive a corresponding upper bound solving only the low-dimensional optimal control problem. Numerically, we show the feasibility of the method using baskets with dimensions up to fifty. In these examples, the resulting option price relative errors are only of the order of few percent.en
dc.publisherarXiven
dc.relation.urlhttp://arxiv.org/abs/1705.00558v4en
dc.relation.urlhttp://arxiv.org/pdf/1705.00558v4en
dc.rightsArchived with thanks to arXiven
dc.titleImplied Stopping Rules for American Basket Options from Markovian Projectionen
dc.typePreprinten
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.eprint.versionPre-printen
dc.contributor.institutionWeierstrass Institute, Mohrenstrasse 39, 10117 Berlin, Germanyen
dc.identifier.arxividarXiv:1705.00558en
kaust.authorHäppölä, Juhoen
kaust.authorTempone, Raulen
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