# Utility maximisation and utility indifference price for exponential semi-martingale models with random factor

## Author Info

• Anastasia Ellanskaya
• Lioudmila Vostrikova
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## Abstract

We consider utility maximization problem for semi-martingale models depending on a random factor $\xi$. We reduce initial maximization problem to the conditional one, given $\xi=u$, which we solve using dual approach. For HARA utilities we consider information quantities like Kullback-Leibler information and Hellinger integrals, and corresponding information processes. As a particular case we study exponential Levy models depending on random factor. In that case the information processes are deterministic and this fact simplify very much indifference price calculus. Then we give the equations for indifference prices. We show that indifference price for seller and minus indifference price for buyer are risk measures. Finally, we apply the results to Geometric Brownian motion case. Using identity in law technique we give the explicit expression for information quantities. Then, the previous formulas for indifference price can be applied.

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File URL: http://arxiv.org/pdf/1303.1134

## Bibliographic Info

Paper provided by arXiv.org in its series Papers with number 1303.1134.

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Handle: RePEc:arx:papers:1303.1134

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## References

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1. Kramkov, D. & Sîrbu, M., 2007. "Asymptotic analysis of utility-based hedging strategies for small number of contingent claims," Stochastic Processes and their Applications, Elsevier, vol. 117(11), pages 1606-1620, November.
2. Thomas Goll & Ludger Rüschendorf, 2001. "Minimax and minimal distance martingale measures and their relationship to portfolio optimization," Finance and Stochastics, Springer, vol. 5(4), pages 557-581.
3. Marco Frittelli, 2000. "The Minimal Entropy Martingale Measure and the Valuation Problem in Incomplete Markets," Mathematical Finance, Wiley Blackwell, vol. 10(1), pages 39-52.
4. Albert N. Shiryaev & Jan Kallsen, 2002. "The cumulant process and Esscher's change of measure," Finance and Stochastics, Springer, vol. 6(4), pages 397-428.
5. Esche, Felix & Schweizer, Martin, 2005. "Minimal entropy preserves the Lévy property: how and why," Stochastic Processes and their Applications, Elsevier, vol. 115(2), pages 299-327, February.
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