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Sparse Modeling Approach to the Arbitrage-Free Interpolation of Plain-Vanilla Option Prices and Implied Volatilities

  • We present a method for the arbitrage-free interpolation of plain-vanilla option prices and implied volatilities, which is based on a system of integral equations that relates terminal density and option prices. Using a discretization of the terminal density, we write these integral equations as a system of linear equations. We show that the kernel matrix of this system is, in general, ill-conditioned, so that it cannot be solved for the discretized density using a naive approach. Instead, weWe present a method for the arbitrage-free interpolation of plain-vanilla option prices and implied volatilities, which is based on a system of integral equations that relates terminal density and option prices. Using a discretization of the terminal density, we write these integral equations as a system of linear equations. We show that the kernel matrix of this system is, in general, ill-conditioned, so that it cannot be solved for the discretized density using a naive approach. Instead, we construct a sparse model for the kernel matrix using singular value decomposition (SVD), which allows us not only to systematically improve the condition number of the kernel matrix, but also determines the computational effort and accuracy of our method. In order to allow for the treatment of realistic inputs that may contain arbitrage, we reformulate the system of linear equations as an optimization problem, in which the SVD-transformed density minimizes the error between the input prices and the arbitrage-free prices generated by our method. To further stabilize the method in the presence of noisy input prices or arbitrage, we apply an 𝐿1-regularization to the SVD-transformed density. Our approach, which is inspired by recent progress in theoretical physics, offers a flexible and efficient framework for the arbitrage-free interpolation of plain-vanilla option prices and implied volatilities, without the need to explicitly specify a stochastic process, expansion basis functions or any other kind of model. We demonstrate the capabilities of our method in a number of artificial and realistic test cases.show moreshow less

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Metadaten
Author:Daniel Guterding
URL:https://www.mdpi.com/2227-9091/11/5/83
DOI:https://doi.org/https://doi.org/10.3390/risks11050083
Parent Title (English):Risks
Document Type:Article
Language:English
Year of Publishing:2023
Date of Publication (online):2024/02/06
Publishing Institution:Technische Hochschule Brandenburg
Release Date:2024/02/07
Issue:11 (5)
Page Number:24
Institutes:Fachbereich Wirtschaft
University Bibliography:Hochschulbibliografie
Licence (German):Creative Commons - CC BY - Namensnennung 4.0
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