The indifference valuation problem in incomplete binomial models is analyzed. The model is more general than the ones studied so far, because the stochastic factor, which generates the market incompleteness, may affect the transition propabilities and/or the values of the traded asset as well as the claim’s payoff. Two pricing algorithms are constructed which use, respectively, the minimal martingale and the minimal entropy measures. We study in detail the interplay among the different kinds of market incompleteness, the pricing measures and the price functionals. The dependence of the prices on the choice of the trading horizon is discussed. The family of “almost complete” (reduced) binomial models is also studied. It is shown that the two measures and the associated price functionals coincide, and that the effects of the horizon choice dissipate.

@article{MSIA_2010__3_2_1_0, author = {M. Musiela and E. Sokolova and T. Zariphopoulou}, title = {Indifference valuation in incomplete binomial models}, journal = {MathematicS In Action}, publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles}, volume = {3}, number = {2}, year = {2010}, pages = {1-36}, doi = {10.5802/msia.4}, mrnumber = {2725844}, zbl = {1204.91137}, language = {en}, url = {msia.centre-mersenne.org/item/MSIA_2010__3_2_1_0/} }

M. Musiela; E. Sokolova; T. Zariphopoulou. Indifference valuation in incomplete binomial models. MathematicS In Action, Volume 3 (2010) no. 2, pp. 1-36. doi : 10.5802/msia.4. https://msia.centre-mersenne.org/item/MSIA_2010__3_2_1_0/

[1] Becherer, D.: Rational hedging and valuation of integrated risks under constant absolute risk aversion, Insurance: Mathematics and Economics, 33, 1-28 (2003). | Article | MR 2011941 | Zbl 1072.91025

[2] Carmona, R. Indifference Pricing: Theory and Applications, Princeton University Press (2009). | Article | Zbl 1155.91008

[3] Delbaen, F. and W. Schachermayer: The variance-optimal martingale measure for continuous processes, Bernoulli, 2(1), 81-105 (1996). | Article | MR 1394053 | Zbl 0849.60042

[4] Delbaen, F., Grandits, P., Rheinlander, T., Samperi, D., Schweizer, M. and Stricker, C.: Exponential hedging and entropic penalties, Mathematical Finance, 12, 99-123 (2002). | Article | MR 1891730 | Zbl 1072.91019

[5] Elliott, R. and J. van der Hoek: Indifference paradigm for discrete models: Duality Methods, Indifference pricing, R. Carmona ed., Princeton University Press, 321-381 (2009) (abbreviated title in volume: Duality methods). | Article

[6] Foellmer, H. and Schweizer, M.: Hedging of contingent claims under incomplete information, In Applied Stochastic Analysis, Stochastic Monographs, Vol. 5, M.H.A. Davis and R. J. Elliott eds., Gordon and Breach, 389-414 (1991). | Zbl 0738.90007

[7] Frittelli, M.: The minimal entropy martingale measure and the valuation problem in incomplete markets, Mathematical Finance, 10, 39-52 (2000). | Article | MR 1743972 | Zbl 1013.60026

[8] Grandits, P. and Rheinlander, T.: On the minimal entropy measure, Annals of Probability, 30 (3), 1003-1038 (2002). | Article | MR 1920099 | Zbl 1049.60035

[9] Henderson, V.: Analytical comparison of option prices in stochastic volatility models, Mathematical Finance, 15(1), 49-59 (2005). | Article | MR 2116796 | Zbl 1109.91025

[10] Henderson, V.: Utility indifference pricing - An overview, Indifference Pricing: Theory and Applications, R. Carmona (ed.), Princeton University Press, 44-72 (2009). | Article | Zbl 1158.91379

[11] Henderson, V., Hobson, D., Howison, S. and T. Kluge: A comparison of option prices under different pricing measures in a stochastic volatility model with correlation, Review of Derivatives Research, 8, 5-25 (2005). | Article | Zbl 1134.91423

[12] Hobson, D.: Stochastic volatility models, correlation and the q-optimal measure, Mathematical Finance, 14, 537-556 (2004). | Article | MR 2092922 | Zbl 1169.60317

[13] Hu Y., Imkeller, P. and M. Müller, utility maximization in incomplete markets, Annals of Applied Probability, 15, 1691-1712 (2005). | Article | MR 2152241 | Zbl 1083.60048

[14] Kramkov, D. and W. Schachermayer: Necessary and sufficient conditions in the problem of optimal investment in incomplete markets, Annals of Applied Probability, 13(4), 1504-1516 (2003). | Article | MR 2023886 | Zbl 1091.91036

[15] Li, P. and J. Xia: Minimal martingale measures for discrete-time incomplete markets, Acta Mathematicae Applicatae Sinica, English Series, 18(2), 349-352 (2002). | Article | MR 2011450 | Zbl 1013.60024

[16] Malamud, S., Trubowitz, E. and Wuethrich, M.: Indifference pricing for power utilities, working paper (2008). | Article | MR 3067590

[17] Musiela, M. and Zariphopoulou, T.: An example of indifference prices under exponential preferences, Finance and Stochastics, 8, 229-239 (2004). | Article | MR 2048829 | Zbl 1062.93048

[18] Musiela, M. and Zariphopoulou, T.: A valuation algorithm for indifference pricing in incomplete markets, Finance and Stochastics, 8, 339-414 (2004). | Article | MR 2213258 | Zbl 1097.91046

[19] Musiela, M. and Zariphopoulou, T.: Derivatives management, investment pricing and the term structure of exponential utilities: The case of binomial model, Indifference pricing: Theory and Applications, R. Carmona ed., Princeton University Press, 5-44 (2009) (abbreviated title in volume: The single period model). | Article

[20] Musiela, M. and Zariphopoulou, T.: The backward and forward dynamic utilities and the associated pricing systems: The case study of the binomial model, preprint (2003). | Article

[21] Musiela, M., Sokolova, E. and Zariphopoulou, T.: Indifference pricing under forward investment performance criteria: The case study of the binomial model, submitted for publication (2010). | Article | MR 2725844 | Zbl 1204.91137

[22] Musiela, M. and Zariphopoulou, T.: Portfolio choice under dynamic investment performance criteria, Quantitative Finance, 9, 161-170 (2009). | Article | MR 2512986 | Zbl 1158.91387

[23] Rouge, R. and El Karoui, N.: Pricing via utility maximization and entropy, Mathematical Finance, 10, 259-276 (2000). | Article | MR 1802922 | Zbl 1052.91512

[24] Schachermayer, W.: A supermartingale property of the optimal portfolio process, Finance and Stochastics, 7(4), 433-456 (2003). | Article | MR 2014244 | Zbl 1039.91030

[25] Schweizer, M.: On the minimal martingale measure and the Foellmer-Schweizer decomposition, Stochastic Processes and Applications, 13, 573-599 (1995). | Article | Zbl 0837.60042

[26] Schweizer, M.: Variance-optimal hedging in discrete time, Mathematics of Operations Research, 20, 1-32(1995). | Article | MR 1320445 | Zbl 0835.90008

[27] Schweizer, M.: Approximation pricing and the variance-optimal measure, Annals of Probability, 24(1), 206-236 (1996). | Article | MR 1387633 | Zbl 0854.60045

[28] Schweizer, M.: A minimality property of the minimal martingale measure, Statistics and Probability Letters, 42, 27-31 (1999). | Article | MR 1671761 | Zbl 0984.60049

[29] Smith, J.E. and Nau, R.F.: Valuing risky projects: option pricing and analysis, Management Science, 41(5), 795-816 (1995). | Article | Zbl 0843.90015

[30] Smith, J. and McCardle, K.F.: Valuing oil properties: integrating option pricing and decision analysis approaches, Operations Research, 46(2), 198-217 (1998). | Article | Zbl 1032.91631

[31] Stoikov, S. and Zariphopoulou, T.: Optimal investment in the presence of unhedgeable risks and under CARA preferences, preprint (2005)

[32] Zariphopoulou, T.: Optimal asset allocation in a stochastic factor model - an overview and open problems, Advanced Financial Modelling, Radon Series of Computational and Applied Mathematics, 8, 427-453 (2009). | Article | Zbl 1181.91306