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Deep Learning and American Options via Free Boundary Framework

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Abstract

We propose a deep learning method for solving the American options model with a free boundary feature. To extract the free boundary known as the early exercise boundary from our proposed method, we introduce the Landau transformation. For efficient implementation of our proposed method, we further construct an implicit dual solution framework consisting of a novel auxiliary function and free boundary equations. The auxiliary function is formulated to include the feed-forward deep neural network (DNN) output and further mimic the far boundary behaviour, smooth pasting condition, and remaining boundary conditions due to the second-order space derivative and first-order time derivative. Because the early exercise boundary and its derivative are not a priori known, the boundary values mimicked by the auxiliary function are in approximate form. Concurrently, we then establish equations that approximate the early exercise boundary and its derivative directly from the DNN output based on some linear relationships at the left boundary. Furthermore, the option Greeks are obtained from the derivatives of this auxiliary function. We test our implementation with several examples and compare them with the existing numerical methods. All indicators show that our proposed deep learning method presents an efficient and alternative way of pricing options with early exercise features.

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Code and Data Availability

The code and data could be shared at a reasonable request.

Notes

  1. Adam is an optimization algorithm based on adaptive estimation of momentums of the first order and second order (Kingma Diederik & Adam, 2014) It has the reputation of converging rapidly and being computationally efficient.

  2. It is essential to mention that all the plot profiles in Sect. 4.2 are in log scale. It enables better visualization of the LTM profile for the final loss function value and the computational time.

  3. After performing several experiments, we obtained the average of the optimal exercise boundary (denoted by Mean \([\hat{s}_f(T)]\)), SD, final loss value, and training time. The averaged values over the number of sample data are displayed in Table 14.

  4. Based on symmetry properties, the American call options can be obtain from our DNN approximation of the American put options. Similarly, the early exercise boundary for the American call options can also be obtained from the DNN approximation of the early exercise boundary for the American put options. We refer the reader to the work of Detemple (2001).

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Funding

The first author is funded in part by an NSERC Discovery Grant. The second author was funded in part by PIMS and the School of Mathematics, Cardiff University, during her research visit to the University of Calgary.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Calgary, Calgary, T2N 1N4, Canada

    Chinonso Nwankwo & Tony Ware

  2. School of Mathematics, Cardiff University, Cardiff, CF24 4AG, UK

    Nneka Umeorah

  3. Mathematics and Statistics, Louisiana Tech University, Ruston, LA, 71272, USA

    Weizhong Dai

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  1. Chinonso Nwankwo
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Correspondence to Chinonso Nwankwo.

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Nwankwo, C., Umeorah, N., Ware, T. et al. Deep Learning and American Options via Free Boundary Framework. Comput Econ (2023). https://doi.org/10.1007/s10614-023-10459-3

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