Deep learning for quadratic hedging in incomplete jump market

Nacira Agram (), Bernt Øksendal () and Jan Rems ()
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Nacira Agram: KTH Royal Institute of Technology
Bernt Øksendal: University of Oslo
Jan Rems: University of Ljubljana

Digital Finance, 2024, vol. 6, issue 3, No 5, 463-499

Abstract: Abstract 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.

Keywords: Option pricing; Incomplete market; Equivalent martingale measure; Merton model; Deep learning; LSTM (search for similar items in EconPapers)
JEL-codes: C22 C45 C61 G11 G12 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s42521-024-00112-5

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