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Hedging and utility valuation of a defaultable claim driven by Hawkes processes

Hedging and utility valuation of a defaultable claim driven by Hawkes processes This article studies the problem of hedging a defaultable claim via the maximization of the mean value of exponential utility, over a set of admissible strategies. The dynamics of the underlying asset is assumed to be governed by mutually exciting Hawkes processes, which captures the jumps clustering phenomenon observed in the market. The resulting market is incomplete and does not allow perfect replication. Hence, a dynamic programming approach is adopted to characterize the value function as the largest solution to a suitable backward stochastic differential equation (BSDE) with a non‐Lipschitz generator. The value function of the optimal investment problem and of the indifference prices are represented in terms of limits of the sequence of value functions of suitable Lipschitz BSDEs and, further, a result of uniqueness is achieved. Finally, numerical experiments are performed to demonstrate the applicability of the proposed framework and to understand the impact of the jump‐clustering on the values of the claims. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Stochastic Models in Business and Industry Wiley

Hedging and utility valuation of a defaultable claim driven by Hawkes processes

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Publisher
Wiley
Copyright
© 2022 John Wiley & Sons, Ltd.
ISSN
1524-1904
eISSN
1526-4025
DOI
10.1002/asmb.2663
Publisher site
See Article on Publisher Site

Abstract

This article studies the problem of hedging a defaultable claim via the maximization of the mean value of exponential utility, over a set of admissible strategies. The dynamics of the underlying asset is assumed to be governed by mutually exciting Hawkes processes, which captures the jumps clustering phenomenon observed in the market. The resulting market is incomplete and does not allow perfect replication. Hence, a dynamic programming approach is adopted to characterize the value function as the largest solution to a suitable backward stochastic differential equation (BSDE) with a non‐Lipschitz generator. The value function of the optimal investment problem and of the indifference prices are represented in terms of limits of the sequence of value functions of suitable Lipschitz BSDEs and, further, a result of uniqueness is achieved. Finally, numerical experiments are performed to demonstrate the applicability of the proposed framework and to understand the impact of the jump‐clustering on the values of the claims.

Journal

Applied Stochastic Models in Business and Industry – Wiley

Published: Mar 1, 2022

Keywords: backward stochastic differential equations; default time; exponential utility; Hawkes processes; optimal investment

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