Efficient hedging under ambiguity in continuous time

Ludovic Tangpi

doi:10.1186/s41546-020-00048-9

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January 2020, 5: 6 doi: 10.1186/s41546-020-00048-9

Efficient hedging under ambiguity in continuous time

Ludovic Tangpi ^,

Princeton University, Department of Operations Research and Financial Engineering, Princeton 08540, NJ, USA

Received March 13, 2019 Published August 2020

Abstract
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It is well known that the minimal superhedging price of a contingent claim is too high for practical use. In a continuous-time model uncertainty framework, we consider a relaxed hedging criterion based on acceptable shortfall risks. Combining existing aggregation and convex dual representation theorems, we derive duality results for the minimal price on the set of upper semicontinuous discounted claims.

Keywords: Superhedging, model ambiguity, Acceptance set, Risk measure, Optimized certainty equivalent, Volatility uncertainty.

Mathematics Subject Classification: 91B30;91G80;60H30;60G48.

Citation: Ludovic Tangpi. Efficient hedging under ambiguity in continuous time. Probability, Uncertainty and Quantitative Risk, 2020, 5 (0) : 6-. doi: 10.1186/s41546-020-00048-9

References:

[1]	Acciaio, B., Beiglböck, M., Penkner, F., Schachermayer, W.: A model-free version of the fundamental theorem of asset pricing and the super-replication theorem. Math. Finance. 26(2), 233–251 (2016) Google Scholar
[2]	Aliprantis, C.D., Border, K.C.:: Infinite Dimensional Analysis: a Hitchhiker’s Guide, 3rd ed. Springer(2006) Google Scholar
[3]	Arai, T.: Convex risk measures on orlicz spaces: inf-convolution and shortfall. Math. Finan. Econ. 3, 73– 88 (2010) Google Scholar
[4]	Bartl, D., Drapeau, S., Tangpi, L.: Computational aspects of robust optimized certainty equivalents and option pricing. Math. Finance. 30(1), 287–09 (2020) Google Scholar
[5]	Bartl, D., Kupper, M., Prömel, D.J., Tangpi, L.: Duality for pathwise superhedging in continuous time. Finance Stoch. 23(3), 697–728 (2019) Google Scholar
[6]	Bartl, D., Kupper, M., Neufeld, A.: Pathwise superhedging on prediction sets. Finance Stoch. 24, 215–48(2020) Google Scholar
[7]	Becherer, D., Kentia, K.: Good deal hedging and valuation under combined uncertainty about drift and volatility. Probab. Uncertain. Quant. Risk. 2(13) (2017) Google Scholar
[8]	Beiglböck, M., Henry-Labordère, P., Penkner, F.: Model-independent bounds for option prices – a mass transport approach. Finance Stoch. 17(3), 477–501 (2013) Google Scholar
[9]	Ben-Tal, A., Taboulle, M.: An old-new concept of convex risk measures: The optimized certainty equivalent. Math. Finance. 17, 449–476 (2007) Google Scholar
[10]	Bion-Nadal, J., Di Nunno, G.: Dynamic no-good-deal pricing measures and extension theorems for linear operators on L∞. Finance Stoch. 17(3), 587–613 (2013) Google Scholar
[11]	Bion-Nadal, J., Kervarec, M.: Risk Measuring under Model Uncertainty. Ann. Appl. Probab. 22(1), 213– 238 (2012) Google Scholar
[12]	Burzoni, M.: Arbitrage and hedging in model independent markets with frictions. SIAM J. Financial Math. 7(1), 812–844 (2016) Google Scholar
[13]	Burzoni, M., Frittelli, M., Maggis, M.: Model-free superhedging duality. Ann. Appl. Probab. 27(3), 1452– 1477 (2017) Google Scholar
[14]	Cheridito, P., Kupper, M., Tangpi, L.: Representation of increasing convex functionals with countably additive measures. Preprint (2015) Google Scholar
[15]	Cheridito, P., Kupper, M., Tangpi, L.: Duality formulas for robust pricing and hedging in discrete time.SIAM J. Financial Math. 8(1), 738–765 (2017) Google Scholar
[16]	Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of asset pricing. Math.Ann. 300(3), 463–520 (1994) Google Scholar
[17]	Dellacherie, C., Meyer, P.-A.:: Probabilities and Potential. B. North-Holland Mathematics Studies, vol. 72, p. 463. North-Holland Publishing Co., Amsterdam (1982). Theory of martingales, Translated from the French by J. P. Wilson Denis, L., Martini, C.: A theoretical framework for the pricing of contingent claims in the presence of model uncertainty. Ann. Appl. Probab. 16, 827–852 (2006) Google Scholar
[18]	Dolinsky, Y., Soner, H.M.: Martingale optimal transport and robust hedging in continuous time. Probab.Theory Relat. Fields. 160(1-2), 391–427 (2014) Google Scholar
[19]	Fan, K.: Minimax theorems. Proc. Nat. Acad. Sci. U.S.A. 39, 42–47 (1953) Google Scholar
[20]	Föllmer, H., Leukert, P.: Quantile hedging. Finance Stoch. 3(3), 251–273 (1999) Google Scholar
[21]	Föllmer, H., Leukert, P.: Efficient hedging: Cost versus shortfall risk. Finance Stoch. 4, 117–146 (2000) Google Scholar
[22]	Hou, Z., Obłój, J.: On robust pricing-hedging duality in continuous time. Finance Stoch. 22(3), 511–567(2018) Google Scholar
[23]	Jouini, E., Schachermayer, W., Touzi, N.: Law invariant risk measures have the fatou property. Adv. Math.Econ. 9, 49–71 (2006) Google Scholar
[24]	Kaina, M., Rüschendorf, L.: On convex risk measures on l^p-spaces. Math. Meth. Oper. Res. 69(3), 475– 495 (2009) Google Scholar
[25]	Karandikar, R.L.: On quadratic variation process of a continuous martingale. Illinois J. Math. 27, 178–181(1983) Google Scholar
[26]	Karatzas, I., Shreve, S.E.:: Brownian Motion and Stochastic Calculus (Graduate Texts in Mathematics).Springer, New York (2004) Google Scholar
[27]	Kramkov, D., Schachermayer, W.: The Asymptotic Elasticity of Utility Functions and Optimal Investment in Incomplete Market. Ann. Appl. Probab. 9(3), 904–950 (1999) Google Scholar
[28]	Kupper, M., Schachermayer, W.: Representation results for law invariant time consistent functions. Math.Financ. Econ. 2(3), 189–210 (2009) Google Scholar
[29]	Neufeld, A., Nutz, M.: Superreplication under Volatility Uncertainty for Measurable Claims. Electron. J.Probab. 18(48), 1–14 (2013) Google Scholar
[30]	Peng, S.: Nonlinear expectations and stochastic calculus under uncertainty. arXiv Preprint 1002.4546(2010) Google Scholar
[31]	Rudloff, B.: Convex hedging in incomplete markets. Appl. Math. Finance. 14(5), 437–452 (2007) Google Scholar
[32]	Soner, H.M., Touzi, N., Zhang, J.: Martingale representation theorem for the G-expectation. Stoch. Proc.Appl. 121(2), 265–287 (2011a) Google Scholar
[33]	Soner, M.H., Touzi, N., Zhang, J.: Quasi-sure stochastic analysis through aggregation. Electron. J. Probab. 16(67) (2011b) Google Scholar
[34]	Soner, H.M., Touzi, N., Zhang, J.: Dual formulation of second order target problems. Ann. Appl. Probab. 23(1), 308–347 (2013) Google Scholar
[35]	Tangpi, L.: Dual Representation of Convex Increasing Functionals with Applications to Finance. PhD thesis, University of Konstanz (2015) Google Scholar
[36]	Wisniewski, A.: The structure of measurable mappings on metric spaces. Proc. A.M.S. 122(1), 147–150(1994) Google Scholar

show all references

References:

[1]	Acciaio, B., Beiglböck, M., Penkner, F., Schachermayer, W.: A model-free version of the fundamental theorem of asset pricing and the super-replication theorem. Math. Finance. 26(2), 233–251 (2016) Google Scholar
[2]	Aliprantis, C.D., Border, K.C.:: Infinite Dimensional Analysis: a Hitchhiker’s Guide, 3rd ed. Springer(2006) Google Scholar
[3]	Arai, T.: Convex risk measures on orlicz spaces: inf-convolution and shortfall. Math. Finan. Econ. 3, 73– 88 (2010) Google Scholar
[4]	Bartl, D., Drapeau, S., Tangpi, L.: Computational aspects of robust optimized certainty equivalents and option pricing. Math. Finance. 30(1), 287–09 (2020) Google Scholar
[5]	Bartl, D., Kupper, M., Prömel, D.J., Tangpi, L.: Duality for pathwise superhedging in continuous time. Finance Stoch. 23(3), 697–728 (2019) Google Scholar
[6]	Bartl, D., Kupper, M., Neufeld, A.: Pathwise superhedging on prediction sets. Finance Stoch. 24, 215–48(2020) Google Scholar
[7]	Becherer, D., Kentia, K.: Good deal hedging and valuation under combined uncertainty about drift and volatility. Probab. Uncertain. Quant. Risk. 2(13) (2017) Google Scholar
[8]	Beiglböck, M., Henry-Labordère, P., Penkner, F.: Model-independent bounds for option prices – a mass transport approach. Finance Stoch. 17(3), 477–501 (2013) Google Scholar
[9]	Ben-Tal, A., Taboulle, M.: An old-new concept of convex risk measures: The optimized certainty equivalent. Math. Finance. 17, 449–476 (2007) Google Scholar
[10]	Bion-Nadal, J., Di Nunno, G.: Dynamic no-good-deal pricing measures and extension theorems for linear operators on L∞. Finance Stoch. 17(3), 587–613 (2013) Google Scholar
[11]	Bion-Nadal, J., Kervarec, M.: Risk Measuring under Model Uncertainty. Ann. Appl. Probab. 22(1), 213– 238 (2012) Google Scholar
[12]	Burzoni, M.: Arbitrage and hedging in model independent markets with frictions. SIAM J. Financial Math. 7(1), 812–844 (2016) Google Scholar
[13]	Burzoni, M., Frittelli, M., Maggis, M.: Model-free superhedging duality. Ann. Appl. Probab. 27(3), 1452– 1477 (2017) Google Scholar
[14]	Cheridito, P., Kupper, M., Tangpi, L.: Representation of increasing convex functionals with countably additive measures. Preprint (2015) Google Scholar
[15]	Cheridito, P., Kupper, M., Tangpi, L.: Duality formulas for robust pricing and hedging in discrete time.SIAM J. Financial Math. 8(1), 738–765 (2017) Google Scholar
[16]	Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of asset pricing. Math.Ann. 300(3), 463–520 (1994) Google Scholar
[17]	Dellacherie, C., Meyer, P.-A.:: Probabilities and Potential. B. North-Holland Mathematics Studies, vol. 72, p. 463. North-Holland Publishing Co., Amsterdam (1982). Theory of martingales, Translated from the French by J. P. Wilson Denis, L., Martini, C.: A theoretical framework for the pricing of contingent claims in the presence of model uncertainty. Ann. Appl. Probab. 16, 827–852 (2006) Google Scholar
[18]	Dolinsky, Y., Soner, H.M.: Martingale optimal transport and robust hedging in continuous time. Probab.Theory Relat. Fields. 160(1-2), 391–427 (2014) Google Scholar
[19]	Fan, K.: Minimax theorems. Proc. Nat. Acad. Sci. U.S.A. 39, 42–47 (1953) Google Scholar
[20]	Föllmer, H., Leukert, P.: Quantile hedging. Finance Stoch. 3(3), 251–273 (1999) Google Scholar
[21]	Föllmer, H., Leukert, P.: Efficient hedging: Cost versus shortfall risk. Finance Stoch. 4, 117–146 (2000) Google Scholar
[22]	Hou, Z., Obłój, J.: On robust pricing-hedging duality in continuous time. Finance Stoch. 22(3), 511–567(2018) Google Scholar
[23]	Jouini, E., Schachermayer, W., Touzi, N.: Law invariant risk measures have the fatou property. Adv. Math.Econ. 9, 49–71 (2006) Google Scholar
[24]	Kaina, M., Rüschendorf, L.: On convex risk measures on l^p-spaces. Math. Meth. Oper. Res. 69(3), 475– 495 (2009) Google Scholar
[25]	Karandikar, R.L.: On quadratic variation process of a continuous martingale. Illinois J. Math. 27, 178–181(1983) Google Scholar
[26]	Karatzas, I., Shreve, S.E.:: Brownian Motion and Stochastic Calculus (Graduate Texts in Mathematics).Springer, New York (2004) Google Scholar
[27]	Kramkov, D., Schachermayer, W.: The Asymptotic Elasticity of Utility Functions and Optimal Investment in Incomplete Market. Ann. Appl. Probab. 9(3), 904–950 (1999) Google Scholar
[28]	Kupper, M., Schachermayer, W.: Representation results for law invariant time consistent functions. Math.Financ. Econ. 2(3), 189–210 (2009) Google Scholar
[29]	Neufeld, A., Nutz, M.: Superreplication under Volatility Uncertainty for Measurable Claims. Electron. J.Probab. 18(48), 1–14 (2013) Google Scholar
[30]	Peng, S.: Nonlinear expectations and stochastic calculus under uncertainty. arXiv Preprint 1002.4546(2010) Google Scholar
[31]	Rudloff, B.: Convex hedging in incomplete markets. Appl. Math. Finance. 14(5), 437–452 (2007) Google Scholar
[32]	Soner, H.M., Touzi, N., Zhang, J.: Martingale representation theorem for the G-expectation. Stoch. Proc.Appl. 121(2), 265–287 (2011a) Google Scholar
[33]	Soner, M.H., Touzi, N., Zhang, J.: Quasi-sure stochastic analysis through aggregation. Electron. J. Probab. 16(67) (2011b) Google Scholar
[34]	Soner, H.M., Touzi, N., Zhang, J.: Dual formulation of second order target problems. Ann. Appl. Probab. 23(1), 308–347 (2013) Google Scholar
[35]	Tangpi, L.: Dual Representation of Convex Increasing Functionals with Applications to Finance. PhD thesis, University of Konstanz (2015) Google Scholar
[36]	Wisniewski, A.: The structure of measurable mappings on metric spaces. Proc. A.M.S. 122(1), 147–150(1994) Google Scholar

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