# American Institute of Mathematical Sciences

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

## Pathwise no-arbitrage in a class of Delta hedging strategies

 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.
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 [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 [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 [4] Biagini, S, Bouchard, B, Kardaras, C, Nutz, M:Robust fundamental theorem for continuous processes.Mathematical Finance (2015). doi:10.1111/mafi.12110 [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 [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 [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 [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 [9] Dudley, RM:Wiener functionals as Itô integrals. Ann. Probability 5(1), 140-141 (1977) [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) [11] Dupire, B:Functional Itô calculus. Bloomberg Portfolio Research Paper (2009) [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 [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) [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) [15] Föllmer, H, Schied, A Stochastic Finance. An Introduction in Discrete Time, 3rd edn., p. 544. [16] Freedman, D Brownian Motion and Diffusion, 2nd edn., p. 231. Springer, New York (1983) [17] Hobson, DG:Robust hedging of the lookback option. Finance Stoch 2(4), 329-347 (1998) [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 [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 [20] Lyons, TJ:Uncertain volatility and the risk-free synthesis of derivatives. Appl. Math. Finance 2(2), 117-133 (1995) [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 [22] Riedel, F:Financial economics without probabilistic prior assumptions. Decisions Econ. Finan 38, 75-91(2015). doi:10.1007/s10203-014-0159-0 [23] Schied, A:Model-free CPPI. J. Econom. Dynam. Control 40, 84-94 (2014). doi:10.1016/j.jedc.2013.12.010 [24] Schied, A:On a class of generalized Takagi functions with linear pathwise quadratic variation. J. Math.Anal. Appl 433, 974-990 (2016) [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 [26] Schied, A, Speiser, L, Voloshchenko, I:Model-free portfolio theory and its functional master formula(2016). arXiv preprint 1606.03325 [27] Sondermann, D:Introduction to Stochastic Calculus for Finance. Lecture Notes in Economics and Mathematical Systems, vol. 579, p. 136. Springer, Berlin (2006) [28] Stroock, DW, Varadhan, SRS:Diffusion processes with continuous coefficients. Comm. Pure Appl. Math 22, 345-400479530 (1969) [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) [30] Vovk, V:Rough paths in idealized financial markets. Lith. Math. J 51(2), 274-285 (2011).doi:10.1007/s10986-011-9125-5 [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 [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 [33] Widder, DV:The Heat Equation. Pure and Applied Mathematics, vol. 67, pp. 267, New York and London(1975)

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##### 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 [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 [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 [4] Biagini, S, Bouchard, B, Kardaras, C, Nutz, M:Robust fundamental theorem for continuous processes.Mathematical Finance (2015). doi:10.1111/mafi.12110 [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 [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 [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 [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 [9] Dudley, RM:Wiener functionals as Itô integrals. Ann. Probability 5(1), 140-141 (1977) [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) [11] Dupire, B:Functional Itô calculus. Bloomberg Portfolio Research Paper (2009) [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 [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) [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) [15] Föllmer, H, Schied, A Stochastic Finance. An Introduction in Discrete Time, 3rd edn., p. 544. [16] Freedman, D Brownian Motion and Diffusion, 2nd edn., p. 231. Springer, New York (1983) [17] Hobson, DG:Robust hedging of the lookback option. Finance Stoch 2(4), 329-347 (1998) [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 [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 [20] Lyons, TJ:Uncertain volatility and the risk-free synthesis of derivatives. Appl. Math. Finance 2(2), 117-133 (1995) [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 [22] Riedel, F:Financial economics without probabilistic prior assumptions. Decisions Econ. Finan 38, 75-91(2015). doi:10.1007/s10203-014-0159-0 [23] Schied, A:Model-free CPPI. J. Econom. Dynam. Control 40, 84-94 (2014). doi:10.1016/j.jedc.2013.12.010 [24] Schied, A:On a class of generalized Takagi functions with linear pathwise quadratic variation. J. Math.Anal. Appl 433, 974-990 (2016) [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 [26] Schied, A, Speiser, L, Voloshchenko, I:Model-free portfolio theory and its functional master formula(2016). arXiv preprint 1606.03325 [27] Sondermann, D:Introduction to Stochastic Calculus for Finance. Lecture Notes in Economics and Mathematical Systems, vol. 579, p. 136. Springer, Berlin (2006) [28] Stroock, DW, Varadhan, SRS:Diffusion processes with continuous coefficients. Comm. Pure Appl. Math 22, 345-400479530 (1969) [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) [30] Vovk, V:Rough paths in idealized financial markets. Lith. Math. J 51(2), 274-285 (2011).doi:10.1007/s10986-011-9125-5 [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 [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 [33] Widder, DV:The Heat Equation. Pure and Applied Mathematics, vol. 67, pp. 267, New York and London(1975)
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