# American Institute of Mathematical Sciences

January  2017, 2: 13 doi: 10.1186/s41546-017-0024-5

## Good deal hedging and valuation under combined uncertainty about drift and volatility

 1 Institut für Mathematik, Humboldt Universität, Unter den Linden 6, 10099 Berlin, Germany; 2 Institut für Mathematik, Goethe-Universität, D-60054 Frankfurt am Main, Germany

Received  April 08, 2017 Revised  November 09, 2017 Published  June 2017

We study robust notions of good-deal hedging and valuation under combined uncertainty about the drifts and volatilities of asset prices. Good-deal bounds are determined by a subset of risk-neutral pricing measures such that not only opportunities for arbitrage are excluded but also deals that are too good, by restricting instantaneous Sharpe ratios. A non-dominated multiple priors approach to model uncertainty (ambiguity) leads to worst-case good-deal bounds. Corresponding hedging strategies arise as minimizers of a suitable coherent risk measure. Good-deal bounds and hedges for measurable claims are characterized by solutions to secondorder backward stochastic differential equations whose generators are non-convex in the volatility. These hedging strategies are robust with respect to uncertainty in the sense that their tracking errors satisfy a supermartingale property under all a-priori valuation measures, uniformly over all priors.
Citation: Dirk Becherer, Klebert Kentia. Good deal hedging and valuation under combined uncertainty about drift and volatility. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 13-. doi: 10.1186/s41546-017-0024-5
##### References:
 [1] Artzner, P, Delbaen, F, Eber, JM, Heath, D:Coherent measures of risk. Math. Finance. 9(3), 203-228(1999) [2] Avellaneda, M, Levy, A, Paras, A:Pricing and hedging derivative securities in markets with uncertain volatilities. Appl. Math. Finance. 2(2), 73-88 (1995) [3] Barrieu, P, El Karoui, N:Pricing, hedging and optimally designing derivatives via minimization of risk measures. In:Carmona, R (ed.) Indifference Pricing:Theory and Applications, pp. 77-146. Princeton University Press, Princeton (2009) [4] Becherer, D:From bounds on optimal growth towards a theory of good-deal hedging. In:Albrecher, H, Runggaldier, W, Schachermayer, W (eds.) Advanced Financial Modelling, Radon Series on Computational and Applied Mathematics, vol 8, pp. 27-52. De Gruyter, Berlin (2009) [5] Becherer, D, Kentia, K:Hedging under generalized good-deal bounds and model uncertainty. Math. Meth.Oper. Res. 86(1), 171-214 (2017) [6] Bertsekas, DP, Shreve, SE:Stochastic Optimal Control:The Discrete Time Case. Academic Press, New York (1978) [7] Biagini, S, Pınar, MÇ:The robust Merton problem of an ambiguity averse investor. Math. Financ. Econ. 11(1), 1-24 (2017) [8] Biagini, S, Bouchard, B, Kardaras, C, Nutz, M:Robust fundamental theorem for continuous processes.Math. Finance. 27(4), 963-987 (2017) [9] Bielecki, T, Cialenco, I, Pitera, M:A survey of time consistency of dynamic risk measures and dynamic performance measures in discrete time:LM-measure perspective. Probab. Uncertain. Quant. Risk. 2:52, paper no.3 (2017). doi:10.1186/s41546-017-0012-9 [10] Bielecki, TR, Cialenco, I, Zhang, Z:Dynamic coherent acceptability indices and their applications to finance. Math. Finance. 24(3), 411-441 (2014) [11] Björk, T, Slinko, I:Towards a general theory of good-deal bounds. Rev. Finance. 10(2), 221-260 (2006) [12] Cerný, A, Hodges, SD:The theory of good-deal pricing in financial markets. In:Geman, H, DP M, Plinska, S, Vorst, T (eds.) Mathematical Finance-Bachelier Congress 2000, pp. 175-202. Springer, Berlin(2002) [13] Chen, Z, Epstein, LG:Ambiguity, risk and asset returns in continuous time. Econometrica. 70(4), 1403-1443 (2002) [14] Cochrane, J, Saá-Requejo, J:Beyond arbitrage:good deal asset price bounds in incomplete markets. J. Polit. Econ. 108(1), 79-119 (2000) [15] Delbaen, F:The structure of m-stable sets and in particular of the set of risk neutral measures. Séminaire de Probabilités XXXIX, Lecture Notes in Math. 1874, pp. 215-258. Springer, Berlin (2006) [16] Delbaen, F, Schachermayer, W:A general version of the fundamental theorem of asset pricing. Math.Ann. 300(1), 463-520 (1994) [17] Denis, L, Martini, C:A theoretical framework for the pricing of contingent claims in the presence of model uncertainty. Ann. Appl. Probab. 16(2), 827-852 (2006) [18] Denis, L, Hu, M, Peng, S:Function spaces and capacity related to a sublinear expectation:application to G-Brownian motion paths. Potential Anal. 34(2), 139-161 (2011) [19] Ekeland, I, Temam, R:Convex Analysis and Variational Problems. SIAM, Philadelphia (1999) [20] El Karoui, N, Jeanblanc-Picqué, M, Shreve, SE:Robustness of the Black and Scholes formula. Math. Finance. 8(2), 93-126 (1998) [21] Epstein, LG, Ji, S:Ambiguous volatility and asset pricing in continuous time. Rev. Financ. Stud. 26(7), 1740-1786 (2013) [22] Epstein, LG, Ji, S:Ambiguous volatility, possibility and utility in continuous time. J. Math. Econom. 50, 269-282 (2014) [23] Garlappi, L, Uppal, R, Wang, T:Portfolio selection with parameter and model uncertainty:A multi-prior approach. Rev. Financ. Stud. 20(1), 41-81 (2007) [24] Gilboa, I, Schmeidler, D:Maxmin expected utility with non-unique prior. J. Math. Econom. 18(2), 141-153 (1989) [25] Hu, M, Ji, S, Peng, S, Song, Y:Backward stochastic differential equations driven by G-Brownian motion.Stoch. Process. Appl. 124(1), 759-784 (2014a) [26] Hu, M, Ji, S, Yang, S:A stochastic recursive optimal control problem under the G-expectation framework.Appl. Math. Optim. 70(2), 253-278 (2014b) [27] Karandikar, RL:On path-wise stochastic integration. Stoch. Process. Appl. 57(1), 11-18 (1995) [28] Klöppel, S, Schweizer, M:Dynamic utility-based good-deal bounds. Stat. Dec. 25(4), 285-309 (2007) [29] Kramkov, D:Optional decomposition of supermartingales and hedging in incomplete security markets.Probab. Theory Relat. Fields. 105(4), 459-479 (1996) [30] Lyons, TJ:Uncertain volatility and the risk-free synthesis of derivatives. Appl. Math. Finance. 2(2), 117-133 (1995) [31] Madan, D, Cherny, A:Markets as a counterparty:an introduction to conic finance. Int. J. Theor. Appl.Finance. 13(8), 1149-1177 (2010) [32] Matoussi, A, Possamaï, D, Zhou, C:Robust utility maximization in nondominated models with 2BSDE:the uncertain volatility model. Math. Finance. 25(2), 258-287 (2015) [33] Neufeld, A, Nutz, M:Superreplication under volatility uncertainty for measurable claims. Electron. J.Probab. 18(48), 1-14 (2013) [34] Neufeld, A, Nutz, M:Robust utility maximization with Lévy processes. Forthcom. Math. Finance (2016). doi:10.1111/mafi.12139 [35] Nutz, M:Path-wise construction of stochastic integrals. Electron. Commun. Probab. 17(24), 1-7 (2012a) [36] Nutz, M:A quasi-sure approach to the control of non-Markovian stochastic differential equations.Electron. J. Probab. 17(23), 1-23 (2012b) [37] Nutz, M, van Handel, R:Constructing sublinear expectations on path space. Stoch. Process. Appl. 123(8), 3100-3121 (2013) [38] Nutz, M, Soner, M:Superhedging and dynamic risk measures under volatility uncertainty. SIAM J.Control. Optim. 50(4), 2065-2089 (2012) [39] Øksendal, B, Sulem, A:Forward-backward stochastic differential games and stochastic control under model uncertainty. J. Optim. Theory Appl. 161(1), 22-55 (2014) [40] Peng, S:G-expectation, G-Brownian motion and related stochastic calculus of Itô type. Stochastic anal-ysis and applications. The Abel symposium 2005, Abel Symposia book series, vol 2, pp. 541-567.Springer, Berlin (2007) [41] Possamaï, D, Tan, X, Zhou, C:Stochastic control for a class of nonlinear kernels and applications. ArXiv e-print arXiv:1510.08439v1. To appear in Ann Prob (to be published in 2018).arxiv.org/pdf/1510.08439v1 [42] Quenez, MC:Optimal portfolio in a multiple-priors model. In:Dalang, R, Dozzi, M, Russo, F (eds.)Seminar on Stochastic Analysis, Random Fields and Applications IV, Progress in Probability, vol 58, pp. 291-321. Birkhäuser, Basel (2004) [43] Rockafellar, RT:Integral functionals, normal integrands and measurable selections. In:Waelbroeck, L (ed.) Nonlinear Operators and Calculus of Variations, Lecture Notes in Mathematics 543, pp. 157-207. Springer, Berlin (1976) [44] Rosazza Gianin, E, Sgarra, C:Acceptability indexes via g-expectations:an application to liquidity risk.Math. Financ. Econ. 7(4), 457-475 (2013) [45] Schied, A:Optimal investments for risk-and ambiguity-averse preferences:a duality approach. Finance.Stoch. 11(1), 107-129 (2007) [46] Schweizer, M:A guided tour through quadratic hedging approaches. In:Jouini, E, Cvitanić, J, Musiela, M (eds.) Option Pricing, Interest Rates and Risk Management, pp. 538-574. Cambridge University Press, Cambridge (2001) [47] Soner, HM, Touzi, N, Zhang, J:Quasi-sure stochastic analysis through aggregation. Electron. J. Probab. 16(67), 1844-1879 (2011) [48] Soner, HM, Touzi, N, Zhang, J:Wellposedness of second order backward SDEs. Probab. Theory Relat.Fields. 153(1-2), 149-190 (2012) [49] Soner, HM, Touzi, N, Zhang, J:Dual formulation of second order target problems. Ann. Appl. Probab. 23(1), 308-347 (2013) [50] Tevzadze, R, Toronjadze, T, Uzunashvili, T:Robust utility maximization for a diffusion market model with misspecified coefficients. Financ. Stoch. 17(3), 535-563 (2013) [51] Vorbrink, J:Financial markets under volatility uncertainty. J Math. Econom. 53, 64-78 (2014). special Section:Economic Theory of Bubbles (I)

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##### References:
 [1] Artzner, P, Delbaen, F, Eber, JM, Heath, D:Coherent measures of risk. Math. Finance. 9(3), 203-228(1999) [2] Avellaneda, M, Levy, A, Paras, A:Pricing and hedging derivative securities in markets with uncertain volatilities. Appl. Math. Finance. 2(2), 73-88 (1995) [3] Barrieu, P, El Karoui, N:Pricing, hedging and optimally designing derivatives via minimization of risk measures. In:Carmona, R (ed.) Indifference Pricing:Theory and Applications, pp. 77-146. Princeton University Press, Princeton (2009) [4] Becherer, D:From bounds on optimal growth towards a theory of good-deal hedging. In:Albrecher, H, Runggaldier, W, Schachermayer, W (eds.) Advanced Financial Modelling, Radon Series on Computational and Applied Mathematics, vol 8, pp. 27-52. De Gruyter, Berlin (2009) [5] Becherer, D, Kentia, K:Hedging under generalized good-deal bounds and model uncertainty. Math. Meth.Oper. Res. 86(1), 171-214 (2017) [6] Bertsekas, DP, Shreve, SE:Stochastic Optimal Control:The Discrete Time Case. Academic Press, New York (1978) [7] Biagini, S, Pınar, MÇ:The robust Merton problem of an ambiguity averse investor. Math. Financ. Econ. 11(1), 1-24 (2017) [8] Biagini, S, Bouchard, B, Kardaras, C, Nutz, M:Robust fundamental theorem for continuous processes.Math. Finance. 27(4), 963-987 (2017) [9] Bielecki, T, Cialenco, I, Pitera, M:A survey of time consistency of dynamic risk measures and dynamic performance measures in discrete time:LM-measure perspective. Probab. Uncertain. Quant. Risk. 2:52, paper no.3 (2017). doi:10.1186/s41546-017-0012-9 [10] Bielecki, TR, Cialenco, I, Zhang, Z:Dynamic coherent acceptability indices and their applications to finance. Math. Finance. 24(3), 411-441 (2014) [11] Björk, T, Slinko, I:Towards a general theory of good-deal bounds. Rev. Finance. 10(2), 221-260 (2006) [12] Cerný, A, Hodges, SD:The theory of good-deal pricing in financial markets. In:Geman, H, DP M, Plinska, S, Vorst, T (eds.) Mathematical Finance-Bachelier Congress 2000, pp. 175-202. Springer, Berlin(2002) [13] Chen, Z, Epstein, LG:Ambiguity, risk and asset returns in continuous time. Econometrica. 70(4), 1403-1443 (2002) [14] Cochrane, J, Saá-Requejo, J:Beyond arbitrage:good deal asset price bounds in incomplete markets. J. Polit. Econ. 108(1), 79-119 (2000) [15] Delbaen, F:The structure of m-stable sets and in particular of the set of risk neutral measures. Séminaire de Probabilités XXXIX, Lecture Notes in Math. 1874, pp. 215-258. Springer, Berlin (2006) [16] Delbaen, F, Schachermayer, W:A general version of the fundamental theorem of asset pricing. Math.Ann. 300(1), 463-520 (1994) [17] Denis, L, Martini, C:A theoretical framework for the pricing of contingent claims in the presence of model uncertainty. Ann. Appl. Probab. 16(2), 827-852 (2006) [18] Denis, L, Hu, M, Peng, S:Function spaces and capacity related to a sublinear expectation:application to G-Brownian motion paths. Potential Anal. 34(2), 139-161 (2011) [19] Ekeland, I, Temam, R:Convex Analysis and Variational Problems. SIAM, Philadelphia (1999) [20] El Karoui, N, Jeanblanc-Picqué, M, Shreve, SE:Robustness of the Black and Scholes formula. Math. Finance. 8(2), 93-126 (1998) [21] Epstein, LG, Ji, S:Ambiguous volatility and asset pricing in continuous time. Rev. Financ. Stud. 26(7), 1740-1786 (2013) [22] Epstein, LG, Ji, S:Ambiguous volatility, possibility and utility in continuous time. J. Math. Econom. 50, 269-282 (2014) [23] Garlappi, L, Uppal, R, Wang, T:Portfolio selection with parameter and model uncertainty:A multi-prior approach. Rev. Financ. Stud. 20(1), 41-81 (2007) [24] Gilboa, I, Schmeidler, D:Maxmin expected utility with non-unique prior. J. Math. Econom. 18(2), 141-153 (1989) [25] Hu, M, Ji, S, Peng, S, Song, Y:Backward stochastic differential equations driven by G-Brownian motion.Stoch. Process. Appl. 124(1), 759-784 (2014a) [26] Hu, M, Ji, S, Yang, S:A stochastic recursive optimal control problem under the G-expectation framework.Appl. Math. Optim. 70(2), 253-278 (2014b) [27] Karandikar, RL:On path-wise stochastic integration. Stoch. Process. Appl. 57(1), 11-18 (1995) [28] Klöppel, S, Schweizer, M:Dynamic utility-based good-deal bounds. Stat. Dec. 25(4), 285-309 (2007) [29] Kramkov, D:Optional decomposition of supermartingales and hedging in incomplete security markets.Probab. Theory Relat. Fields. 105(4), 459-479 (1996) [30] Lyons, TJ:Uncertain volatility and the risk-free synthesis of derivatives. Appl. Math. Finance. 2(2), 117-133 (1995) [31] Madan, D, Cherny, A:Markets as a counterparty:an introduction to conic finance. Int. J. Theor. Appl.Finance. 13(8), 1149-1177 (2010) [32] Matoussi, A, Possamaï, D, Zhou, C:Robust utility maximization in nondominated models with 2BSDE:the uncertain volatility model. Math. Finance. 25(2), 258-287 (2015) [33] Neufeld, A, Nutz, M:Superreplication under volatility uncertainty for measurable claims. Electron. J.Probab. 18(48), 1-14 (2013) [34] Neufeld, A, Nutz, M:Robust utility maximization with Lévy processes. Forthcom. Math. Finance (2016). doi:10.1111/mafi.12139 [35] Nutz, M:Path-wise construction of stochastic integrals. Electron. Commun. Probab. 17(24), 1-7 (2012a) [36] Nutz, M:A quasi-sure approach to the control of non-Markovian stochastic differential equations.Electron. J. Probab. 17(23), 1-23 (2012b) [37] Nutz, M, van Handel, R:Constructing sublinear expectations on path space. Stoch. Process. Appl. 123(8), 3100-3121 (2013) [38] Nutz, M, Soner, M:Superhedging and dynamic risk measures under volatility uncertainty. SIAM J.Control. Optim. 50(4), 2065-2089 (2012) [39] Øksendal, B, Sulem, A:Forward-backward stochastic differential games and stochastic control under model uncertainty. J. Optim. Theory Appl. 161(1), 22-55 (2014) [40] Peng, S:G-expectation, G-Brownian motion and related stochastic calculus of Itô type. Stochastic anal-ysis and applications. The Abel symposium 2005, Abel Symposia book series, vol 2, pp. 541-567.Springer, Berlin (2007) [41] Possamaï, D, Tan, X, Zhou, C:Stochastic control for a class of nonlinear kernels and applications. ArXiv e-print arXiv:1510.08439v1. To appear in Ann Prob (to be published in 2018).arxiv.org/pdf/1510.08439v1 [42] Quenez, MC:Optimal portfolio in a multiple-priors model. In:Dalang, R, Dozzi, M, Russo, F (eds.)Seminar on Stochastic Analysis, Random Fields and Applications IV, Progress in Probability, vol 58, pp. 291-321. Birkhäuser, Basel (2004) [43] Rockafellar, RT:Integral functionals, normal integrands and measurable selections. In:Waelbroeck, L (ed.) Nonlinear Operators and Calculus of Variations, Lecture Notes in Mathematics 543, pp. 157-207. Springer, Berlin (1976) [44] Rosazza Gianin, E, Sgarra, C:Acceptability indexes via g-expectations:an application to liquidity risk.Math. Financ. Econ. 7(4), 457-475 (2013) [45] Schied, A:Optimal investments for risk-and ambiguity-averse preferences:a duality approach. Finance.Stoch. 11(1), 107-129 (2007) [46] Schweizer, M:A guided tour through quadratic hedging approaches. In:Jouini, E, Cvitanić, J, Musiela, M (eds.) Option Pricing, Interest Rates and Risk Management, pp. 538-574. Cambridge University Press, Cambridge (2001) [47] Soner, HM, Touzi, N, Zhang, J:Quasi-sure stochastic analysis through aggregation. Electron. J. Probab. 16(67), 1844-1879 (2011) [48] Soner, HM, Touzi, N, Zhang, J:Wellposedness of second order backward SDEs. Probab. Theory Relat.Fields. 153(1-2), 149-190 (2012) [49] Soner, HM, Touzi, N, Zhang, J:Dual formulation of second order target problems. Ann. Appl. Probab. 23(1), 308-347 (2013) [50] Tevzadze, R, Toronjadze, T, Uzunashvili, T:Robust utility maximization for a diffusion market model with misspecified coefficients. Financ. Stoch. 17(3), 535-563 (2013) [51] Vorbrink, J:Financial markets under volatility uncertainty. J Math. Econom. 53, 64-78 (2014). special Section:Economic Theory of Bubbles (I)
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