Utility maximisation and utility indifference price for exponential semi-martingale models and HARA utilities

详细信息    查看全文

• 作者：A. Ellanskaya (1)
  L. Vostrikova (1)
  
  1. LAREMA ; D茅partement de Math茅matiques ; Universit茅 d鈥橝ngers ; 2 Bd. Lavoisier ; 49045 ; Angers cedex 01 ; France
  
• 刊名：Proceedings of the Steklov Institute of Mathematics
• 出版年：2014
• 出版时间：December 2014
• 年：2014
• 卷：287
• 期：1
• 页码：68-95
• 全文大小：1,377 KB
• 参考文献：1. J. Amendinger, D. Becherer, and M. Schweizer, 鈥淎 monetary value for initial information in portfolio optimization,鈥?Finance Stoch. 7(1), 29鈥?6 (2003). dx.doi.org/10.1007/s007800200075" target="_blank" title="It opens in new window">CrossRef
  2. S. Biagini, M. Frittelli, and M. Grasselli, 鈥淚ndifference price with general semimartingales,鈥?Math. Finance 21(3), 423鈥?46 (2011). dx.doi.org/10.1111/j.1467-9965.2010.00443.x" target="_blank" title="It opens in new window">CrossRef
  3. G. Callegaro, M. Jeanblanc, and B. Zargari, 鈥淐arthaginian enlargement of filtrations,鈥?ESAIM, Probab. Stat. 17, 550鈥?66 (2013). dx.doi.org/10.1051/ps/2011162" target="_blank" title="It opens in new window">CrossRef
  4. S. Cawston and L. Vostrikova, 鈥淟茅vy preservation and associated properties for / f-divergence minimal equivalent martingale measures,鈥?in / Prokhorov and Contemporary Probability Theory, Ed. by A. N. Shiryaev et al. (Springer, Berlin, 2013), pp. 163鈥?96. dx.doi.org/10.1007/978-3-642-33549-5_9" target="_blank" title="It opens in new window">CrossRef
  5. S. Cawston and L. Vostrikova, 鈥淎n / f-divergence approach for optimal portfolios in exponential L茅vy models,鈥?in / Inspired by Finance: The Musiela Festschrift, Ed. by Yu. Kabanov et al. (Springer, Cham, 2014), pp. 83鈥?01. dx.doi.org/10.1007/978-3-319-02069-3_5" target="_blank" title="It opens in new window">CrossRef
  6. T. Choulli and C. Stricker, 鈥淢inimal entropy-Hellinger martingale measure in incomplete markets,鈥?Math. Finance 15(3), 465鈥?90 (2005). dx.doi.org/10.1111/j.1467-9965.2005.00229.x" target="_blank" title="It opens in new window">CrossRef
  7. T. Choulli, C. Stricker, and J. Li, 鈥淢inimal Hellinger martingale measures of order / q,鈥?Finance Stoch. 11(3), 399鈥?27 (2007). dx.doi.org/10.1007/s00780-007-0039-3" target="_blank" title="It opens in new window">CrossRef
  8. F. Delbaen, P. Grandits, T. Rheinl盲nder, D. Samperi, M. Schweizer, and C. Stricker, 鈥淓xponential hedging and entropic penalties,鈥?Math. Finance 12(2), 99鈥?23 (2002). dx.doi.org/10.1111/1467-9965.02001" target="_blank" title="It opens in new window">CrossRef
  9. F. Esche and M. Schweizer, 鈥淢inimal entropy preserves the L茅vy property: How and why,鈥?Stoch. Processes Appl. 115(2), 299鈥?27 (2005). dx.doi.org/10.1016/j.spa.2004.05.009" target="_blank" title="It opens in new window">CrossRef
  10. D. Gasbarra, E. Valkeila, and L. Vostrikova, 鈥淓nlargement of filtration and additional information in pricing models: Bayesian approach,鈥?in / From Stochastic Calculus to Mathematical Finance, Ed. by Yu. Kabanov et al. (Springer, Berlin, 2006), pp. 257鈥?85. dx.doi.org/10.1007/978-3-540-30788-4_13" target="_blank" title="It opens in new window">CrossRef
  11. T. Goll and L. R眉schendorf, 鈥淢inimax and minimal distance martingale measures and their relationship to portfolio optimization,鈥?Finance Stoch. 5(4), 557鈥?81 (2001). dx.doi.org/10.1007/s007800100052" target="_blank" title="It opens in new window">CrossRef
  12. F. Hubalek and C. Sgarra, 鈥淓sscher transforms and the minimal entropy martingale measure for exponential L茅vy models,鈥?Quant. Finance 6(2), 125鈥?45 (2006). dx.doi.org/10.1080/14697680600573099" target="_blank" title="It opens in new window">CrossRef
  13. / Indifference Pricing: Theory and Applications, Ed. by R. Carmona (Princeton Univ. Press, Princeton, NJ, 2009).
  14. J. Jacod, / Calcul stochastique et probl猫mes de martingales (Springer, Berlin, 1979), Lect. Notes Math. 714.
  15. J. Jacod, 鈥淕rossissement initial, hypoth猫se (H / -) et th茅or猫me de Girsanov,鈥?in / Grossissements de filtrations: exemples et applications: S茅minaire de calcul stochastique 1982/83, Univ. Paris VI, Ed. by T. Jeulin and M. Yor (Springer, Berlin, 1985), Lect. Notes Math. 1118, pp. 15鈥?5. dx.doi.org/10.1007/BFb0075768" target="_blank" title="It opens in new window">CrossRef
  16. J. Jacod and A. N. Shiryaev, / Limit Theorems for Stochastic Processes (Springer, Berlin, 2003). dx.doi.org/10.1007/978-3-662-05265-5" target="_blank" title="It opens in new window">CrossRef
  17. E. I. Kolomietz, 鈥淩elations between triplets of local characteristics of semimartingales,鈥?Usp. Mat. Nauk 39(4), 163鈥?64 (1984) [Russ. Math. Surv. 39 (4), 123鈥?24 (1984)].
  18. D. Kramkov and M. S卯rbu, 鈥淎symptotic analysis of utility-based hedging strategies for small number of contingent claims,鈥?Stoch. Processes Appl. 117(11), 1606鈥?620 (2007). dx.doi.org/10.1016/j.spa.2007.04.014" target="_blank" title="It opens in new window">CrossRef
  19. M. Mania and M. Schweizer, 鈥淒ynamic exponential utility indifference valuation,鈥?Ann. Appl. Probab. 15(3), 2113鈥?143 (2005). dx.doi.org/10.1214/105051605000000395" target="_blank" title="It opens in new window">CrossRef
  20. Y. Miyahara, 鈥淢inimal entropy martingale measures of jump type price processes in incomplete assets markets,鈥?Asia-Pac. Financ. Mark. 6(2), 97鈥?13 (1999). dx.doi.org/10.1023/A:1010062625672" target="_blank" title="It opens in new window">CrossRef
  21. M. Musiela and T. Zariphopoulou, 鈥淚ndifference prices and related measures,鈥?Tech. Rep. (Univ. Texas, Austin, 2001).
  22. M. Musiela and T. Zariphopoulou, 鈥淎n example of indifference prices under exponential preferences,鈥?Finance Stoch. 8(2), 229鈥?39 (2004). dx.doi.org/10.1007/s00780-003-0112-5" target="_blank" title="It opens in new window">CrossRef
  23. K. Ritter, 鈥淒uality for nonlinear programming in a Banach space,鈥?SIAM J. Appl. Math. 15(2), 294鈥?02 (1967). dx.doi.org/10.1137/0115029" target="_blank" title="It opens in new window">CrossRef
  24. R. Rouge and N. El Karoui, 鈥淧ricing via utility maximization and entropy,鈥?Math. Finance 10(2), 259鈥?76 (2000). dx.doi.org/10.1111/1467-9965.00093" target="_blank" title="It opens in new window">CrossRef
  25. A. N. Shiryaev, 鈥淎bsolute continuity and singularity of probability measures in functional spaces,鈥?in / Proc. Int. Congr. Math., Helsinki, 1978 (Finn. Acad. Sci., Helsinki, 1980), Vol. 1, pp. 209鈥?25.
  26. C. Stricker and M. Yor, 鈥淐alcul stochastique d茅pendant d鈥檜n param猫tre,鈥?Z. Wahrscheinlichkeitstheor. Verw. Geb. 45, 109鈥?33 (1978). dx.doi.org/10.1007/BF00715187" target="_blank" title="It opens in new window">CrossRef
  27. E. Valkeila and L. Vostrikova, 鈥淎n integral representation for the Hellinger distance,鈥?Math. Scand. 58, 239鈥?54 (1986).
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Russian Library of Science
  
• 出版者：MAIK Nauka/Interperiodica distributed exclusively by Springer Science+Business Media LLC.
• ISSN：1531-8605

文摘

We consider the utility maximisation problem for semi-martingale models and HARA (hyperbolic absolute risk aversion) utilities. Using specific properties of HARA utilities, we reduce the initial maximisation problem to the conditional one, which we solve by applying a dual approach. Then we express the solution of the conditional maximisation problem via conditional information quantities related to HARA utilities, like the Kullback-Leibler information and Hellinger-type integrals. In turn, we express the information quantities in terms of information processes, which is helpful in indifference price calculus. Finally, we give equations for indifference prices. We show that the indifference price for a seller and the minus indifference price for a buyer are risk measures. We apply the results to Black-Scholes models with correlated Brownian motions. Using the identity-in-law technique, we give an explicit expression for information quantities. Then the previous formulas for the indifference price can be applied.