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January  2016, 1: 3 doi: 10.1186/s41546-016-0003-2

Pathwise no-arbitrage in a class of Delta hedging strategies

Alexander Schied , and  Iryna Voloshchenko

Department of Mathematics, University of Mannheim, 68131 Mannheim, Germany

Received  January 07, 2016 Revised  June 10, 2016 Published  August 2016

We consider a strictly pathwise setting for Delta hedging exotic options, based on Föllmer's pathwise Itô calculus. Price trajectories are d-dimensional continuous functions whose pathwise quadratic variations and covariations are determined by a given local volatility matrix. The existence of Delta hedging strategies in this pathwise setting is established via existence results for recursive schemes of parabolic Cauchy problems and via the existence of functional Cauchy problems on path space. Our main results establish the nonexistence of pathwise arbitrage opportunities in classes of strategies containing these Delta hedging strategies and under relatively mild conditions on the local volatility matrix.
Keywords: Pathwise hedging, Exotic options, Pathwise arbitrage, Pathwise Itô calculus, Föllmer integral, Local volatility, Functional Itô formula, Functional Cauchy problem on path space.
Citation: Alexander Schied, Iryna Voloshchenko. Pathwise no-arbitrage in a class of Delta hedging strategies. Probability, Uncertainty and Quantitative Risk, 2016, 1 (0) : 3-. doi: 10.1186/s41546-016-0003-2
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). doi:10.1111/mafi.12060 Google Scholar

[2]

Alvarez, A, Ferrando, S, Olivares, P:Arbitrage and hedging in a non probabilistic framework. Math.Financ. Econ 7(1), 1-28 (2013). doi:10.1007/s11579-012-0074-5 Google Scholar

[3]

Bender, C, Sottinen, T, Valkeila, E:Pricing by hedging and no-arbitrage beyond semimartingales. Finance Stoch 12(4), 441-468 (2008). doi:10.1007/s00780-008-0074-8 Google Scholar

[4]

Biagini, S, Bouchard, B, Kardaras, C, Nutz, M:Robust fundamental theorem for continuous processes.Mathematical Finance (2015). doi:10.1111/mafi.12110 Google Scholar

[5]

Bick, A, Willinger, W:Dynamic spanning without probabilities. Stochastic Process. Appl 50(2), 349-374(1994). doi:10.1016/0304-4149(94)90128-7 Google Scholar

[6]

Bouchard, B, Nutz, M:Arbitrage and duality in nondominated discrete-time models. Ann. Appl. Probab 25(2), 823-859 (2015). doi:10.1214/14-AAP1011 Google Scholar

[7]

Cont, R, Fournié, D-A:Change of variable formulas for non-anticipative functionals on path space. J. Funct. Anal 259(4), 1043-1072 (2010). doi:10.1016/j.jfa.2010.04.017 Google Scholar

[8]

Davis, M, Obłój, J, Raval, V:Arbitrage bounds for prices of weighted variance swaps. Math. Finance 24(4), 821-854 (2014). doi:10.1111/mafi.12021 Google Scholar

[9]

Dudley, RM:Wiener functionals as Itô integrals. Ann. Probability 5(1), 140-141 (1977) Google Scholar

[10]

Dupire, B:Pricing and hedging with smiles. Mathematics of Derivative Securities (Cambridge, 1995), Publ. Newton Inst, vol. 15, pp. 103-111. Cambridge Univ. Press, Cambridge (1997) Google Scholar

[11]

Dupire, B:Functional Itô calculus. Bloomberg Portfolio Research Paper (2009) Google Scholar

[12]

El Karoui, N, Jeanblanc-Picqué, M, Shreve, SE:Robustness of the Black and Scholes formula. Math. Finance 8(2), 93-126 (1998). doi:10.1111/1467-9965.00047 Google Scholar

[13]

Föllmer, H:Calcul d'Itô sans probabilités. Seminar on Probability, XV (Univ. Strasbourg, Strasbourg, 1979/1980), Lecture Notes in Math, vol. 850, pp. 143-150. Springer, Berlin (1981) Google Scholar

[14]

Föllmer, H:Probabilistic aspects of financial risk. European Congress of Mathematics, Vol. I (Barcelona, 2000), Progr. Math, vol. 201, pp. 21-36. Birkhäuser, Basel (2001) Google Scholar

[15]

Föllmer, H, Schied, A Stochastic Finance. An Introduction in Discrete Time, 3rd edn., p. 544. Google Scholar

[16]

Freedman, D Brownian Motion and Diffusion, 2nd edn., p. 231. Springer, New York (1983) Google Scholar

[17]

Hobson, DG:Robust hedging of the lookback option. Finance Stoch 2(4), 329-347 (1998) Google Scholar

[18]

Janson, S, Tysk, J:Preservation of convexity of solutions to parabolic equations. J. Differential Equations 206(1), 182-226 (2004). doi:10.1016/j.jde.2004.07.016 Google Scholar

[19]

Ji, S, Yang, S:Classical solutions of path-dependent PDEs and functional forward-backward stochastic systems. Math. Probl. Eng, 423101-11 (2013). downloads.hindawi.com/journals/mpe/2013/423101.pdf Google Scholar

[20]

Lyons, TJ:Uncertain volatility and the risk-free synthesis of derivatives. Appl. Math. Finance 2(2), 117-133 (1995) Google Scholar

[21]

Peng, S, Wang, F:BSDE, path-dependent PDE and nonlinear Feynman-Kac formula. F. Sci. China Math 59, 19 (2016). doi:10.1007/s11425-015-5086-1 Google Scholar

[22]

Riedel, F:Financial economics without probabilistic prior assumptions. Decisions Econ. Finan 38, 75-91(2015). doi:10.1007/s10203-014-0159-0 Google Scholar

[23]

Schied, A:Model-free CPPI. J. Econom. Dynam. Control 40, 84-94 (2014). doi:10.1016/j.jedc.2013.12.010 Google Scholar

[24]

Schied, A:On a class of generalized Takagi functions with linear pathwise quadratic variation. J. Math.Anal. Appl 433, 974-990 (2016) Google Scholar

[25]

Schied, A, Stadje, M:Robustness of delta hedging for path-dependent options in local volatility models.J. Appl. Probab 44(4), 865-879 (2007). doi:10.1239/jap/1197908810 Google Scholar

[26]

Schied, A, Speiser, L, Voloshchenko, I:Model-free portfolio theory and its functional master formula(2016). arXiv preprint 1606.03325 Google Scholar

[27]

Sondermann, D:Introduction to Stochastic Calculus for Finance. Lecture Notes in Economics and Mathematical Systems, vol. 579, p. 136. Springer, Berlin (2006) Google Scholar

[28]

Stroock, DW, Varadhan, SRS:Diffusion processes with continuous coefficients. Comm. Pure Appl. Math 22, 345-400479530 (1969) Google Scholar

[29]

Stroock, DW, Varadhan, SRS:On the support of diffusion processes with applications to the strong maximum principle. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. III:Probability theory, pp. 333-359. Univ. California Press, Berkeley, Calif (1972) Google Scholar

[30]

Vovk, V:Rough paths in idealized financial markets. Lith. Math. J 51(2), 274-285 (2011).doi:10.1007/s10986-011-9125-5 Google Scholar

[31]

Vovk, V:Continuous-time trading and the emergence of probability. Finance Stoch 16(4), 561-609 (2012). doi:10.1007/s00780-012-0180-5 Google Scholar

[32]

Vovk, V:Itô calculus without probability in idealized financial markets. Lith. Math. J 55(2), 270-290 (2015). doi:10.1007/s10986-015-9280-1 Google Scholar

[33]

Widder, DV:The Heat Equation. Pure and Applied Mathematics, vol. 67, pp. 267, New York and London(1975) 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). doi:10.1111/mafi.12060 Google Scholar

[2]

Alvarez, A, Ferrando, S, Olivares, P:Arbitrage and hedging in a non probabilistic framework. Math.Financ. Econ 7(1), 1-28 (2013). doi:10.1007/s11579-012-0074-5 Google Scholar

[3]

Bender, C, Sottinen, T, Valkeila, E:Pricing by hedging and no-arbitrage beyond semimartingales. Finance Stoch 12(4), 441-468 (2008). doi:10.1007/s00780-008-0074-8 Google Scholar

[4]

Biagini, S, Bouchard, B, Kardaras, C, Nutz, M:Robust fundamental theorem for continuous processes.Mathematical Finance (2015). doi:10.1111/mafi.12110 Google Scholar

[5]

Bick, A, Willinger, W:Dynamic spanning without probabilities. Stochastic Process. Appl 50(2), 349-374(1994). doi:10.1016/0304-4149(94)90128-7 Google Scholar

[6]

Bouchard, B, Nutz, M:Arbitrage and duality in nondominated discrete-time models. Ann. Appl. Probab 25(2), 823-859 (2015). doi:10.1214/14-AAP1011 Google Scholar

[7]

Cont, R, Fournié, D-A:Change of variable formulas for non-anticipative functionals on path space. J. Funct. Anal 259(4), 1043-1072 (2010). doi:10.1016/j.jfa.2010.04.017 Google Scholar

[8]

Davis, M, Obłój, J, Raval, V:Arbitrage bounds for prices of weighted variance swaps. Math. Finance 24(4), 821-854 (2014). doi:10.1111/mafi.12021 Google Scholar

[9]

Dudley, RM:Wiener functionals as Itô integrals. Ann. Probability 5(1), 140-141 (1977) Google Scholar

[10]

Dupire, B:Pricing and hedging with smiles. Mathematics of Derivative Securities (Cambridge, 1995), Publ. Newton Inst, vol. 15, pp. 103-111. Cambridge Univ. Press, Cambridge (1997) Google Scholar

[11]

Dupire, B:Functional Itô calculus. Bloomberg Portfolio Research Paper (2009) Google Scholar

[12]

El Karoui, N, Jeanblanc-Picqué, M, Shreve, SE:Robustness of the Black and Scholes formula. Math. Finance 8(2), 93-126 (1998). doi:10.1111/1467-9965.00047 Google Scholar

[13]

Föllmer, H:Calcul d'Itô sans probabilités. Seminar on Probability, XV (Univ. Strasbourg, Strasbourg, 1979/1980), Lecture Notes in Math, vol. 850, pp. 143-150. Springer, Berlin (1981) Google Scholar

[14]

Föllmer, H:Probabilistic aspects of financial risk. European Congress of Mathematics, Vol. I (Barcelona, 2000), Progr. Math, vol. 201, pp. 21-36. Birkhäuser, Basel (2001) Google Scholar

[15]

Föllmer, H, Schied, A Stochastic Finance. An Introduction in Discrete Time, 3rd edn., p. 544. Google Scholar

[16]

Freedman, D Brownian Motion and Diffusion, 2nd edn., p. 231. Springer, New York (1983) Google Scholar

[17]

Hobson, DG:Robust hedging of the lookback option. Finance Stoch 2(4), 329-347 (1998) Google Scholar

[18]

Janson, S, Tysk, J:Preservation of convexity of solutions to parabolic equations. J. Differential Equations 206(1), 182-226 (2004). doi:10.1016/j.jde.2004.07.016 Google Scholar

[19]

Ji, S, Yang, S:Classical solutions of path-dependent PDEs and functional forward-backward stochastic systems. Math. Probl. Eng, 423101-11 (2013). downloads.hindawi.com/journals/mpe/2013/423101.pdf Google Scholar

[20]

Lyons, TJ:Uncertain volatility and the risk-free synthesis of derivatives. Appl. Math. Finance 2(2), 117-133 (1995) Google Scholar

[21]

Peng, S, Wang, F:BSDE, path-dependent PDE and nonlinear Feynman-Kac formula. F. Sci. China Math 59, 19 (2016). doi:10.1007/s11425-015-5086-1 Google Scholar

[22]

Riedel, F:Financial economics without probabilistic prior assumptions. Decisions Econ. Finan 38, 75-91(2015). doi:10.1007/s10203-014-0159-0 Google Scholar

[23]

Schied, A:Model-free CPPI. J. Econom. Dynam. Control 40, 84-94 (2014). doi:10.1016/j.jedc.2013.12.010 Google Scholar

[24]

Schied, A:On a class of generalized Takagi functions with linear pathwise quadratic variation. J. Math.Anal. Appl 433, 974-990 (2016) Google Scholar

[25]

Schied, A, Stadje, M:Robustness of delta hedging for path-dependent options in local volatility models.J. Appl. Probab 44(4), 865-879 (2007). doi:10.1239/jap/1197908810 Google Scholar

[26]

Schied, A, Speiser, L, Voloshchenko, I:Model-free portfolio theory and its functional master formula(2016). arXiv preprint 1606.03325 Google Scholar

[27]

Sondermann, D:Introduction to Stochastic Calculus for Finance. Lecture Notes in Economics and Mathematical Systems, vol. 579, p. 136. Springer, Berlin (2006) Google Scholar

[28]

Stroock, DW, Varadhan, SRS:Diffusion processes with continuous coefficients. Comm. Pure Appl. Math 22, 345-400479530 (1969) Google Scholar

[29]

Stroock, DW, Varadhan, SRS:On the support of diffusion processes with applications to the strong maximum principle. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. III:Probability theory, pp. 333-359. Univ. California Press, Berkeley, Calif (1972) Google Scholar

[30]

Vovk, V:Rough paths in idealized financial markets. Lith. Math. J 51(2), 274-285 (2011).doi:10.1007/s10986-011-9125-5 Google Scholar

[31]

Vovk, V:Continuous-time trading and the emergence of probability. Finance Stoch 16(4), 561-609 (2012). doi:10.1007/s00780-012-0180-5 Google Scholar

[32]

Vovk, V:Itô calculus without probability in idealized financial markets. Lith. Math. J 55(2), 270-290 (2015). doi:10.1007/s10986-015-9280-1 Google Scholar

[33]

Widder, DV:The Heat Equation. Pure and Applied Mathematics, vol. 67, pp. 267, New York and London(1975) Google Scholar

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