Deep Learning for Quadratic Hedging in Incomplete Jump Market

32 Pages Posted: 3 Mar 2023

See all articles by Nacira Agram

Nacira Agram

Royal Institute of Technology (KTH)

Bernt Oksendal

University of Oslo - Department of Mathematics

Jan Rems

University of Ljubljana

Date Written: February 26, 2023

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 upon 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 feedforward 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

Suggested Citation

Agram, Nacira and Oksendal, Bernt and Rems, Jan, Deep Learning for Quadratic Hedging in Incomplete Jump Market (February 26, 2023). Available at SSRN: https://ssrn.com/abstract=4371396 or http://dx.doi.org/10.2139/ssrn.4371396

Nacira Agram

Royal Institute of Technology (KTH) ( email )

Stockholm

Bernt Oksendal

University of Oslo - Department of Mathematics ( email )

P.O. Box 1053
Blindern, N-0162, Os
Norway
+47-2285 5913 (Phone)
+47-2285 4349 (Fax)

Jan Rems (Contact Author)

University of Ljubljana ( email )

Dunajska 104
Ljubljana, 1000
Slovenia

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