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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

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)

[2]

Aliprantis, C.D., Border, K.C.:: Infinite Dimensional Analysis: a Hitchhiker’s Guide, 3rd ed. Springer(2006)

[3]

Arai, T.: Convex risk measures on orlicz spaces: inf-convolution and shortfall. Math. Finan. Econ. 3, 73– 88 (2010)

[4]

Bartl, D., Drapeau, S., Tangpi, L.: Computational aspects of robust optimized certainty equivalents and option pricing. Math. Finance. 30(1), 287–09 (2020)

[5]

Bartl, D., Kupper, M., Prömel, D.J., Tangpi, L.: Duality for pathwise superhedging in continuous time. Finance Stoch. 23(3), 697–728 (2019)

[6]

Bartl, D., Kupper, M., Neufeld, A.: Pathwise superhedging on prediction sets. Finance Stoch. 24, 215–48(2020)

[7]

Becherer, D., Kentia, K.: Good deal hedging and valuation under combined uncertainty about drift and volatility. Probab. Uncertain. Quant. Risk. 2(13) (2017)

[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)

[9]

Ben-Tal, A., Taboulle, M.: An old-new concept of convex risk measures: The optimized certainty equivalent. Math. Finance. 17, 449–476 (2007)

[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)

[11]

Bion-Nadal, J., Kervarec, M.: Risk Measuring under Model Uncertainty. Ann. Appl. Probab. 22(1), 213– 238 (2012)

[12]

Burzoni, M.: Arbitrage and hedging in model independent markets with frictions. SIAM J. Financial Math. 7(1), 812–844 (2016)

[13]

Burzoni, M., Frittelli, M., Maggis, M.: Model-free superhedging duality. Ann. Appl. Probab. 27(3), 1452– 1477 (2017)

[14]

Cheridito, P., Kupper, M., Tangpi, L.: Representation of increasing convex functionals with countably additive measures. Preprint (2015)

[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)

[16]

Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of asset pricing. Math.Ann. 300(3), 463–520 (1994)

[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)

[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)

[19]

Fan, K.: Minimax theorems. Proc. Nat. Acad. Sci. U.S.A. 39, 42–47 (1953)

[20]

Föllmer, H., Leukert, P.: Quantile hedging. Finance Stoch. 3(3), 251–273 (1999)

[21]

Föllmer, H., Leukert, P.: Efficient hedging: Cost versus shortfall risk. Finance Stoch. 4, 117–146 (2000)

[22]

Hou, Z., Obłój, J.: On robust pricing-hedging duality in continuous time. Finance Stoch. 22(3), 511–567(2018)

[23]

Jouini, E., Schachermayer, W., Touzi, N.: Law invariant risk measures have the fatou property. Adv. Math.Econ. 9, 49–71 (2006)

[24]

Kaina, M., Rüschendorf, L.: On convex risk measures on lp-spaces. Math. Meth. Oper. Res. 69(3), 475– 495 (2009)

[25]

Karandikar, R.L.: On quadratic variation process of a continuous martingale. Illinois J. Math. 27, 178–181(1983)

[26]

Karatzas, I., Shreve, S.E.:: Brownian Motion and Stochastic Calculus (Graduate Texts in Mathematics).Springer, New York (2004)

[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)

[28]

Kupper, M., Schachermayer, W.: Representation results for law invariant time consistent functions. Math.Financ. Econ. 2(3), 189–210 (2009)

[29]

Neufeld, A., Nutz, M.: Superreplication under Volatility Uncertainty for Measurable Claims. Electron. J.Probab. 18(48), 1–14 (2013)

[30]

Peng, S.: Nonlinear expectations and stochastic calculus under uncertainty. arXiv Preprint 1002.4546(2010)

[31]

Rudloff, B.: Convex hedging in incomplete markets. Appl. Math. Finance. 14(5), 437–452 (2007)

[32]

Soner, H.M., Touzi, N., Zhang, J.: Martingale representation theorem for the G-expectation. Stoch. Proc.Appl. 121(2), 265–287 (2011a)

[33]

Soner, M.H., Touzi, N., Zhang, J.: Quasi-sure stochastic analysis through aggregation. Electron. J. Probab. 16(67) (2011b)

[34]

Soner, H.M., Touzi, N., Zhang, J.: Dual formulation of second order target problems. Ann. Appl. Probab. 23(1), 308–347 (2013)

[35]

Tangpi, L.: Dual Representation of Convex Increasing Functionals with Applications to Finance. PhD thesis, University of Konstanz (2015)

[36]

Wisniewski, A.: The structure of measurable mappings on metric spaces. Proc. A.M.S. 122(1), 147–150(1994)

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)

[2]

Aliprantis, C.D., Border, K.C.:: Infinite Dimensional Analysis: a Hitchhiker’s Guide, 3rd ed. Springer(2006)

[3]

Arai, T.: Convex risk measures on orlicz spaces: inf-convolution and shortfall. Math. Finan. Econ. 3, 73– 88 (2010)

[4]

Bartl, D., Drapeau, S., Tangpi, L.: Computational aspects of robust optimized certainty equivalents and option pricing. Math. Finance. 30(1), 287–09 (2020)

[5]

Bartl, D., Kupper, M., Prömel, D.J., Tangpi, L.: Duality for pathwise superhedging in continuous time. Finance Stoch. 23(3), 697–728 (2019)

[6]

Bartl, D., Kupper, M., Neufeld, A.: Pathwise superhedging on prediction sets. Finance Stoch. 24, 215–48(2020)

[7]

Becherer, D., Kentia, K.: Good deal hedging and valuation under combined uncertainty about drift and volatility. Probab. Uncertain. Quant. Risk. 2(13) (2017)

[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)

[9]

Ben-Tal, A., Taboulle, M.: An old-new concept of convex risk measures: The optimized certainty equivalent. Math. Finance. 17, 449–476 (2007)

[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)

[11]

Bion-Nadal, J., Kervarec, M.: Risk Measuring under Model Uncertainty. Ann. Appl. Probab. 22(1), 213– 238 (2012)

[12]

Burzoni, M.: Arbitrage and hedging in model independent markets with frictions. SIAM J. Financial Math. 7(1), 812–844 (2016)

[13]

Burzoni, M., Frittelli, M., Maggis, M.: Model-free superhedging duality. Ann. Appl. Probab. 27(3), 1452– 1477 (2017)

[14]

Cheridito, P., Kupper, M., Tangpi, L.: Representation of increasing convex functionals with countably additive measures. Preprint (2015)

[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)

[16]

Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of asset pricing. Math.Ann. 300(3), 463–520 (1994)

[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)

[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)

[19]

Fan, K.: Minimax theorems. Proc. Nat. Acad. Sci. U.S.A. 39, 42–47 (1953)

[20]

Föllmer, H., Leukert, P.: Quantile hedging. Finance Stoch. 3(3), 251–273 (1999)

[21]

Föllmer, H., Leukert, P.: Efficient hedging: Cost versus shortfall risk. Finance Stoch. 4, 117–146 (2000)

[22]

Hou, Z., Obłój, J.: On robust pricing-hedging duality in continuous time. Finance Stoch. 22(3), 511–567(2018)

[23]

Jouini, E., Schachermayer, W., Touzi, N.: Law invariant risk measures have the fatou property. Adv. Math.Econ. 9, 49–71 (2006)

[24]

Kaina, M., Rüschendorf, L.: On convex risk measures on lp-spaces. Math. Meth. Oper. Res. 69(3), 475– 495 (2009)

[25]

Karandikar, R.L.: On quadratic variation process of a continuous martingale. Illinois J. Math. 27, 178–181(1983)

[26]

Karatzas, I., Shreve, S.E.:: Brownian Motion and Stochastic Calculus (Graduate Texts in Mathematics).Springer, New York (2004)

[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)

[28]

Kupper, M., Schachermayer, W.: Representation results for law invariant time consistent functions. Math.Financ. Econ. 2(3), 189–210 (2009)

[29]

Neufeld, A., Nutz, M.: Superreplication under Volatility Uncertainty for Measurable Claims. Electron. J.Probab. 18(48), 1–14 (2013)

[30]

Peng, S.: Nonlinear expectations and stochastic calculus under uncertainty. arXiv Preprint 1002.4546(2010)

[31]

Rudloff, B.: Convex hedging in incomplete markets. Appl. Math. Finance. 14(5), 437–452 (2007)

[32]

Soner, H.M., Touzi, N., Zhang, J.: Martingale representation theorem for the G-expectation. Stoch. Proc.Appl. 121(2), 265–287 (2011a)

[33]

Soner, M.H., Touzi, N., Zhang, J.: Quasi-sure stochastic analysis through aggregation. Electron. J. Probab. 16(67) (2011b)

[34]

Soner, H.M., Touzi, N., Zhang, J.: Dual formulation of second order target problems. Ann. Appl. Probab. 23(1), 308–347 (2013)

[35]

Tangpi, L.: Dual Representation of Convex Increasing Functionals with Applications to Finance. PhD thesis, University of Konstanz (2015)

[36]

Wisniewski, A.: The structure of measurable mappings on metric spaces. Proc. A.M.S. 122(1), 147–150(1994)

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