Abstract
Stability of the utility maximization problem with random endowment and indifference prices is studied for a sequence of financial markets in an incomplete Brownian setting. Our novelty lies in the nonequivalence of markets, in which the volatility of asset prices (as well as the drift) varies. Degeneracies arise from the presence of nonequivalence. In the positive real line utility framework, a counterexample is presented showing that the expected utility maximization problem can be unstable. A positive stability result is proved for utility functions on the entire real line.
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Notes
Many thanks to an anonymous reviewer for making this keen observation.
References
Ansel, J.-P., Stricker, C.: Lois de martingale, densités et décomposition de Föllmer–Schweizer. Ann. Inst. Henri Poincaré 28, 375–392 (1992)
Ansel, J.-P., Stricker, C.: Unicité et existence de la loi minimale. In: Azéma, J., et al. (eds.) Séminaire de Probabilités XXVII. Lecture Notes in Mathematics, vol. 1557, pp. 22–29. Springer, Berlin (1993)
Avellaneda, M., Levy, A., Parás, A.: Pricing and hedging derivative securities in markets with uncertain volatilities. Appl. Math. Finance 2, 73–88 (1995)
Bayraktar, E., Kravitz, R.: Stability of exponential utility maximization with respect to market perturbations. In: Stochastic Processes and Their Applications, vol. 123, pp. 1671–1690 (2013)
Biagini, S., Frittelli, M.: A unified framework for utility maximization problems: an Orlicz space approach. Ann. Appl. Probab. 18, 929–966 (2008)
Cherny, A., Shiryaev, A.: Vector stochastic integrals and the fundamental theorems of asset pricing. Proc. Steklov Math. Inst. 237, 12–56 (2002)
Christensen, P.O., Larsen, K.: Incomplete continuous-time securities markets with stochastic income volatility. Rev. Asset Pricing Stud. 4, 247–285 (2014)
Cvitanić, J., Schachermayer, W., Wang, H.: Utility maximization in incomplete markets with random endowment. Finance Stoch. 5, 259–272 (2001)
Davis, M.H.A.: Optimal hedging with basis risk. In: Kabanov, Y., et al. (eds.) From Stochastic Calculus to Mathematical Finance. The Shiryaev Festschrift, pp. 169–187. Springer, Berlin (2006)
Delbaen, F., Grandits, P., Rheinländer, T., Samperi, D., Schweizer, M., Stricker, C.: Exponential hedging and entropic penalties. Math. Finance 12, 99–123 (2002)
Denis, L., Kervarec, M.: Optimal investment under model uncertainty in non dominated models. SIAM J. Control Optim. 51, 1803–1822 (2013)
Föllmer, H., Schweizer, M.: Minimal martingale measure. In: Cont, R. (ed.) Encyclopedia of Quantitative Finance, pp. 1200–1204. Wiley, New York (2010)
Frei, C.: Convergence results of the indifference value based on the stability of BSDEs. Stochastics 85, 464–488 (2013)
Frei, C., Schweizer, M.: Exponential utility indifference valuation in two Brownian settings with stochastic correlation. Adv. Appl. Probab. 40, 401–423 (2008)
Grandits, P., Rheinländer, T.: On the minimal entropy martingale measure. Ann. Probab. 30, 1003–1038 (2002)
Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms I: Part 1: Fundamentals. Springer, Berlin (1996)
Hugonnier, J., Kramkov, D.: Optimal investment with random endowments in incomplete markets. Ann. Appl. Probab. 14, 845–864 (2004)
Kabanov, Y.M., Stricker, C.: On the optimal portfolio for the exponential utility maximization: remarks to the six-author paper. Math. Finance 12, 125–134 (2002)
Karatzas, I., Shreve, S.: Brownian Motion and Stochastic Calculus, 1st edn. Springer, New York (1988)
Kardaras, C., Žitković, G.: Stability of the utility maximization problem with random endowment in incomplete markets. Math. Finance 21, 313–333 (2011)
El Karoui, N., Quenez, M.-C.: Dynamic programming and pricing of contingent claims in an incomplete market. SIAM J. Control Optim. 33, 29–66 (1995)
Kazamaki, N.: Continuous Exponential Martingales and BMO. Lecture Notes in Mathematics, vol. 1579. Springer, Berlin (1994)
Kraft, H.: Optimal portfolios and Heston’s stochastic volatility model: an explicit solution for power utility. Quant. Finance 5, 303–313 (2005)
Kramkov, D., Schachermayer, W.: The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Probab. 9, 904–950 (1999)
Larsen, K., Soner, H.M., Žitković, G.: Facelifting in utility maximization. Finance Stoch. 20, 99–121 (2016)
Larsen, K., Žitković, G.: Stability of utility-maximization in incomplete markets. In: Stochastic Processes and Their Applications, vol. 117, pp. 1642–1662 (2007)
Liptser, R.S., Shiryaev, A.N.: Statistics of Random Processes: I. General Theory, 2nd edn. Springer, Berlin (2001)
Lyons, T.J.: Uncertain volatility and the risk-free synthesis of derivatives. Appl. Math. Finance 2, 117–133 (1995)
Matoussi, A., Possamaï, D., Zhou, C.: Robust utility maximization in nondominated models with 2BSDE: the uncertain volatility model. Math. Finance 25, 258–287 (2015)
Monoyios, M.: Performance of utility-based strategies for hedging basis risk. Quant. Finance 4, 245–255 (2004)
Owen, M., Žitković, G.: Optimal investment with an unbounded random endowment and utility-based pricing. Math. Finance 19, 129–159 (2009)
Schachermayer, W.: Optimal investment in incomplete markets when wealth may become negative. Ann. Appl. Probab. 11, 694–734 (2001)
Tevzadze, R., Toronjadze, T., Uzunashvili, T.: Robust utility maximization for a diffusion market model with misspecified coefficients. Finance Stoch. 17, 535–563 (2013)
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The author would like to thank Dmitry Kramkov, Steve Shreve, Kasper Larsen and the anonymous reviewers for their constructive comments and suggestions.
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Weston, K. Stability of utility maximization in nonequivalent markets. Finance Stoch 20, 511–541 (2016). https://doi.org/10.1007/s00780-016-0289-z
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DOI: https://doi.org/10.1007/s00780-016-0289-z