Abstract
In this paper the connection between the utility pricing methodology and risk sensitive control is explored for stochastic volatility models. It is proved that the utility based price of a European option can be written as the difference of the value functions of two different stochastic optimal control problems. The smoothness of those value functions and gradient estimates are proved, to give a complete solution to these problems. As a consequence of these results, the relation with quadratic BSDEs, as well as the description of a risk neutral measure associated with this pricing approach are formalized.
The research od D. Hernández was partially supported by Conacyt, through the Laboratory LEMME. The research of S.-J. Sheu was supported by the grants from the Ministry of Science and Technology (No. NSC 102-2115-M-008 -002 -MY3), NCTS (MOST), NCTS (NCU), and the Ministry of Education (No. 103G906-9).
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Acknowledgements
The authors thank the anonymous referee for his/her insightful comments that help to improve the presentation of the paper. This work was partially developed during the visit of the first author to Academia Sinica, which support and hospitality is greatly appreciated.
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Hernández–Hernández, D., Sheu, SJ. (2015). Solution of the HJB Equations Involved in Utility-Based Pricing. In: Mena, R., Pardo, J., Rivero, V., Uribe Bravo, G. (eds) XI Symposium on Probability and Stochastic Processes. Progress in Probability, vol 69. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-13984-5_9
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