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Local Volatility and Hopfield Neural Network



Author(s)

MarioDell′Era

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DOI:10.17265/1537-1514/2022.03.003

London,United Kingdom

In quantitative finance, numerical methods are commonly used for the valuation of financial quantities. Derivative price models are often multi-dimensional and moreover, closed-form solutions are not available. Over the time different numerical methods have been developed and introduced in literature, to solve partial differential  equations (PDEs) or integral equations. To this list of numerical approaches, in the recent literature, appears more often the use of Neural Networks. The mathematical foundation that allows the Neural Networks to approximate financial quantities and not only, is the “Universal Approximation Theorem”. In this paper we are going to introduce the Stochastic Hopfield Neural Network (or Boltzmann machine) to estimate the Local Volatility σ(S_t, t), in order to speed up the calibration process, which often is time consuming and becomes a key problem to overcome.

 Quantitative Finance, Option Pricing, Local Volatility, Numerical Methods, Neural Networks