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)

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

[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)

[1]

Rainer Buckdahn, Ingo Bulla, Jin Ma. Pathwise Taylor expansions for Itô random fields. Mathematical Control and Related Fields, 2011, 1 (4) : 437-468. doi: 10.3934/mcrf.2011.1.437

[2]

Giuseppe Da Prato, Franco Flandoli. Some results for pathwise uniqueness in Hilbert spaces. Communications on Pure and Applied Analysis, 2014, 13 (5) : 1789-1797. doi: 10.3934/cpaa.2014.13.1789

[3]

Xingchun Wang, Yongjin Wang. Variance-optimal hedging for target volatility options. Journal of Industrial and Management Optimization, 2014, 10 (1) : 207-218. doi: 10.3934/jimo.2014.10.207

[4]

María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters $H\in (1/3,1/2]$. Discrete and Continuous Dynamical Systems - B, 2015, 20 (8) : 2553-2581. doi: 10.3934/dcdsb.2015.20.2553

[5]

Nathan Glatt-Holtz, Roger Temam, Chuntian Wang. Martingale and pathwise solutions to the stochastic Zakharov-Kuznetsov equation with multiplicative noise. Discrete and Continuous Dynamical Systems - B, 2014, 19 (4) : 1047-1085. doi: 10.3934/dcdsb.2014.19.1047

[6]

Caibin Zeng, Xiaofang Lin, Jianhua Huang, Qigui Yang. Pathwise solution to rough stochastic lattice dynamical system driven by fractional noise. Communications on Pure and Applied Analysis, 2020, 19 (2) : 811-834. doi: 10.3934/cpaa.2020038

[7]

Bahareh Akhtari, Esmail Babolian, Andreas Neuenkirch. An Euler scheme for stochastic delay differential equations on unbounded domains: Pathwise convergence. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 23-38. doi: 10.3934/dcdsb.2015.20.23

[8]

Hakima Bessaih, María J. Garrido–Atienza, Björn Schmalfuss. Pathwise solutions and attractors for retarded SPDEs with time smooth diffusion coefficients. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 3945-3968. doi: 10.3934/dcds.2014.34.3945

[9]

Yiju Chen, Xiaohu Wang, Kenan Wu. Wong-Zakai approximations and pathwise dynamics of stochastic fractional lattice systems. Communications on Pure and Applied Analysis, 2022, 21 (8) : 2529-2560. doi: 10.3934/cpaa.2022059

[10]

Nicolas Lerner, Yoshinori Morimoto, Karel Pravda-Starov, Chao-Jiang Xu. Phase space analysis and functional calculus for the linearized Landau and Boltzmann operators. Kinetic and Related Models, 2013, 6 (3) : 625-648. doi: 10.3934/krm.2013.6.625

[11]

Fabrizio Colombo, Graziano Gentili, Irene Sabadini and Daniele C. Struppa. A functional calculus in a noncommutative setting. Electronic Research Announcements, 2007, 14: 60-68. doi: 10.3934/era.2007.14.60

[12]

Ian Knowles, Ajay Mahato. The inverse volatility problem for American options. Discrete and Continuous Dynamical Systems - S, 2020, 13 (12) : 3473-3489. doi: 10.3934/dcdss.2020235

[13]

Vladimir V. Kisil. Mobius transformations and monogenic functional calculus. Electronic Research Announcements, 1996, 2: 26-33.

[14]

Gregory Berkolaiko, Cónall Kelly, Alexandra Rodkina. Sharp pathwise asymptotic stability criteria for planar systems of linear stochastic difference equations. Conference Publications, 2011, 2011 (Special) : 163-173. doi: 10.3934/proc.2011.2011.163

[15]

Daoyi Xu, Weisong Zhou. Existence-uniqueness and exponential estimate of pathwise solutions of retarded stochastic evolution systems with time smooth diffusion coefficients. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 2161-2180. doi: 10.3934/dcds.2017093

[16]

Bixiang Wang. Stochastic bifurcation of pathwise random almost periodic and almost automorphic solutions for random dynamical systems. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3745-3769. doi: 10.3934/dcds.2015.35.3745

[17]

John A. D. Appleby, Alexandra Rodkina, Henri Schurz. Pathwise non-exponential decay rates of solutions of scalar nonlinear stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 667-696. doi: 10.3934/dcdsb.2006.6.667

[18]

Lorenzo Brasco, Filippo Santambrogio. An equivalent path functional formulation of branched transportation problems. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 845-871. doi: 10.3934/dcds.2011.29.845

[19]

Hassan Emamirad, Arnaud Rougirel. A functional calculus approach for the rational approximation with nonuniform partitions. Discrete and Continuous Dynamical Systems, 2008, 22 (4) : 955-972. doi: 10.3934/dcds.2008.22.955

[20]

Yong Chen, Hongjun Gao, María J. Garrido–Atienza, Björn Schmalfuss. Pathwise solutions of SPDEs driven by Hölder-continuous integrators with exponent larger than $1/2$ and random dynamical systems. Discrete and Continuous Dynamical Systems, 2014, 34 (1) : 79-98. doi: 10.3934/dcds.2014.34.79

 Impact Factor: 

Metrics

  • PDF downloads (28)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]