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Robust utility maximisation with intractable claims

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Abstract

We study a continuous-time expected utility maximisation problem where the investor at maturity receives the value of a contingent claim in addition to the investment payoff from the financial market. The investor knows nothing about the claim other than its probability distribution; hence the name “intractable claim”. In view of the lack of necessary information about the claim, we consider a robust formulation to maximise her utility in the worst scenario. We apply the quantile formulation to solve the problem, express the quantile function of the optimal terminal investment income as the solution of certain variational inequalities of ordinary differential equations, and obtain the resulting optimal trading strategy. In the case of exponential utility, the problem reduces to a (non-robust) rank-dependent utility maximisation with probability distortion whose solution is available in the literature. The results can also be used to determine the utility indifference price of the intractable claim.

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Acknowledgements

Li and Xu are supported by the NSFC (No.11971409), The Hong Kong RGC (GRF No.15202421 and No.15204622), the PolyU-SDU Joint Research Center on Financial Mathematics, the CAS AMSS-POLYU Joint Laboratory of Applied Mathematics, Research Centre for Quantitative Finance (1-CE03), and internal research grants from the Hong Kong Polytechnic University. Zhou is supported by a start-up grant and the Nie Center for Intelligent Asset Management at Columbia University. His work is also part of a Columbia–CityU/HK collaborative project that is supported by the InnoHK Initiative, the Government of the HKSAR and the AIFT Lab. The authors thank the Editor and two anonymous referees for their constructive comments that have led to an improved version of the paper.

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Authors and Affiliations

  1. Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China

    Yunhong Li & Zuo Quan Xu

  2. Department of Industrial Engineering and Operations Research & The Data Science Institute, Columbia University, New York, NY, 10027, USA

    Xun Yu Zhou

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  1. Yunhong Li
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  2. Zuo Quan Xu
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Correspondence to Xun Yu Zhou.

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Dedicated to Tomas Björk for his friendship and inspiration

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Li, Y., Xu, Z.Q. & Zhou, X.Y. Robust utility maximisation with intractable claims. Finance Stoch 27, 985–1015 (2023). https://doi.org/10.1007/s00780-023-00512-2

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