Abstract
This paper considers utility indifference valuation of derivatives under model uncertainty and trading constraints, where the utility is formulated as an additive stochastic differential utility of both intertemporal consumption and terminal wealth, and the uncertain prospects are ranked according to a multiple-priors model of Chen and Epstein (2002). The price is determined by two optimal stochastic control problems (mixed with optimal stopping time in the case of American option) of forward-backward stochastic differential equations. By means of backward stochastic differential equation and partial differential equation methods, we show that both bid and ask prices are closely related to the Black-Scholes risk-neutral price with modified dividend rates. The two prices will actually coincide with each other if there is no trading constraint or the model uncertainty disappears. Finally, two applications to European option and American option are discussed.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Anderson E, Hansen L, Sargent T. Robustness, detection and the price of risk. Technical Report. Chicago: University of Chicago, 1999
Ankirchner S, Imkeller P, dos Reis G. Pricing and hedging of derivatives based on non-tradable underlyings. Math Finance, 2010, 20: 289–312
Becherer D. Rational hedging and valuation of integrated risks under constant absolute risk aversion. Insurance Math Econ, 2003, 33: 1–28
Chen Z, Epstein L. Ambiguity, risk and asset return in continuous time. Econometrica, 2002, 70: 1403–1443
Chen Z, Tian W, Zhao G. Optimal stopping rule under ambiguity in continuous time. Submitted
Cheridito P, Hu Y. Optimal consumption and investment in incomplete markets with general constraints. Stoch Dyn, 2011, 283: 283–299
Cont R. Model uncertainty and its impact on the pricing of derivative instruments. Math Finance, 2006, 16: 519–547
Davis M. Optimal Hedging with Basis Risk. Berlin: Springer-Verlag, 2006
Duffie D, Epstein L. Stochastic differential utility. Econometrica, 1992, 60: 353–394
El Karoui N, Kapoudjian C, Pardoux E, et al. Reflected solutions of backward SDEs, and related obstacle problems for PDEs. Ann Probab, 1997, 25: 702–737
El Karoui N, Peng S, Quenez M. Backward stochastic differential equations in finance. Math Finance, 1997, 7: 1–71
Friedman A. Variational Principles and Free-Boundary Problems. New York: John Wiley & Sons, 1982
Guo D, Song B, Wang S. Backward stochastic differential equations and nonlinear pricing Parisian (Parasian) options (in Chinese). Sci Sin Math, 2013, 43: 91–103
Henderson V. Valuation of claims on nontraded assets using utility maximization. Math Finance, 2002, 12: 351–373
Henderson V, Liang G. Pseudo linear pricing rule for utility indifference valuation. Finance Stoch, in press
Hodges S, Neuberger A. Optimal replication of contingent claims under transaction costs. Rev Future Markets, 1989, 8: 222–239
Hu Y, Imkeller P, Müller M. Utility maximization in incomplete markets. Ann Appl Probab, 2005, 15: 1691–1712
Jaimungal S, Sigloch G. Incorporating risk and ambiguity aversion into a hybrid model of default Math. Finance, 2012, 22: 57–81
Karatzas I, Shreve S. Methods of Mathematical Finance. New York: Springer, 1998
Ladyzenskaja O A, Solonnikov V A, Ural’ceva N M. Linear and Quasi-linear Equations of Parabolic Type. Providence, RI: Amer Math Soc, 1968
Lieberman G M. Second Order Parabolic Differential Equations. Singapore: World Scientific, 1996
Mania M, Schweizer M. Dynamic exponential utility indifference valuation. Ann Appl Probab, 2005, 15: 2113–2143
Musiela M, Zariphopoulou T. An example of indifference prices under exponential preferences. Finance Stoch, 2004, 8: 229–239
Sircar R, Zariphopoulou T. Bounds and asymptotic approximations for utility prices when volatility is random. SIAM J Control Optim, 2005, 43: 1328–1353
Tao K. On an Aleksandrov-Bakel’Man type maximum principle for second-order parabolic equations. Comm Partial Differential Equations, 1985, 10: 543–553
Yi F, Yang Z. A variational inequality arising from European option pricing with transaction costs. Sci China Ser A, 2008, 51: 935–954
Yi F, Yang Z, Wang X. A variational inequality arising from European installment call options pricing. SIAM J Math Anal, 2008, 40: 306–326
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Yan, H., Liang, G. & Yang, Z. Indifference pricing and hedging in a multiple-priors model with trading constraints. Sci. China Math. 58, 689–714 (2015). https://doi.org/10.1007/s11425-014-4885-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-014-4885-0
Keywords
- indifference pricing
- stochastic differential utility
- trading constraints
- ambiguity
- variational inequality
- American option