Deep learning for quadratic hedging in incomplete jump market

Agram, Nacira (author), Øksendal, Bernt Karsten (author), Rems, Jan (author)

We propose a deep learning approach to study the minimal variance pricing and hedging problem in an incomplete jump diffusion market. It is based on a rigorous stochastic calculus derivation of the optimal hedging portfolio, optimal option price, and the corresponding equivalent martingale measure through the means of the Stackelberg game approach. A deep learning algorithm based on the combination of the feed-forward and LSTM neural networks is tested on three different market models, two of which are incomplete. In contrast, the complete market Black–Scholes model serves as a benchmark for the algorithm’s performance. The results that indicate the algorithm’s good performance are presented and discussed. In particular, we apply our results to the special incomplete market model studied by Merton and give a detailed comparison between our results based on the minimal variance principle and the results obtained by Merton based on a different pricing principle. Using deep learning, we find that the minimal variance principle leads to typically higher option prices than those deduced from the Merton principle. On the other hand, the minimal variance principle leads to lower losses than the Merton principle.

• Language: English
• Keywords: option pricing, incomplete market, equivalent martingale measure, Merton model, deep learning, LSTM
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Year: 2024
• Number of pages: Str. 463-499
• Numbering: Vol. 6, iss. 3
• PID: 20.500.12556/RUL-162188
• UDC: 519.2:004.8
• ISSN on article: 2524-6984
• DOI: 10.1007/s42521-024-00112-5
• COBISS.SI-ID: 208122627
• Publication date in RUL: 19.09.2024

Record is a part of a journal

• Title: Digital finance
• Shortened title: Digit. finance
• Publisher: Springer Nature
• ISSN: 2524-6984
• COBISS.SI-ID: 114564611

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Projects

• Funder: Other - Other funder or multiple funders
• Funding programme: Swedish Research Council
• Project number: 2020-04697

• Funder: ARIS - Slovenian Research and Innovation Agency
• Project number: P1-0448
• Name: Stohastične metode in njihova uporaba