Static hedging of barrier options with a smile : an inverse problem
ESAIM: Control, Optimisation and Calculus of Variations, Volume 8  (2002), p. 127-142

Let L be a parabolic second order differential operator on the domain Π ¯=0,T×. Given a function u ^:R and x ^>0 such that the support of u ^ is contained in (-,-x ^], we let y ^:Π ¯ be the solution to the equation: Ly ^=0,y ^| {0}× =u ^. Given positive bounds 0<x 0 <x 1 , we seek a function u with support in x 0 ,x 1 such that the corresponding solution y satisfies: y(t,0)=y ^(t,0)t0,T. We prove in this article that, under some regularity conditions on the coefficients of L, continuous solutions are unique and dense in the sense that y ^| [0,T]×{0} can be C 0 -approximated, but an exact solution does not exist in general. This result solves the problem of almost replicating a barrier option in the generalised Black-Scholes framework with a combination of European options, as stated by Carr et al. in [6].

DOI : https://doi.org/10.1051/cocv:2002040
Classification:  93C20,  65M32,  62P05,  91B28
Keywords: inverse problems, Carleman estimates, barrier option hedging, replication
@article{COCV_2002__8__127_0,
     author = {Bardos, Claude and Douady, Rapha\"el and Fursikov, Andrei},
     title = {Static hedging of barrier options with a smile : an inverse problem},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {8},
     year = {2002},
     pages = {127-142},
     doi = {10.1051/cocv:2002040},
     zbl = {1063.91028},
     mrnumber = {1932947},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2002__8__127_0}
}
Bardos, Claude; Douady, Raphaël; Fursikov, Andrei. Static hedging of barrier options with a smile : an inverse problem. ESAIM: Control, Optimisation and Calculus of Variations, Volume 8 (2002) , pp. 127-142. doi : 10.1051/cocv:2002040. http://www.numdam.org/item/COCV_2002__8__127_0/

[1] L. Andersen, J. Andreasen and D. Eliezer, Static Replication of Barrier Options: Some General Results. Preprint Gen. Re Fin. Prod. (2000).

[2] M. Avellaneda and A. Paras, Managing the Volatility Risk of Portfolio of Derivative Securities: The Lagrangian Uncertain Volatility Model. Appl. Math. Finance 3 (1996) 21-52. | Zbl 1097.91514

[3] C. Bardos, R. Douady and A. Fursikov, Static Hedging of Barrier Options with a Smile: An Inverse Problem, Preprint CMLA No. 9810. École Normale Supérieure de Cachan (1998).

[4] F. Black and M. Scholes, The Pricing of Options and Corporate Liabilities. J. Polit. Econ. 81 (1973) 637-654. | Zbl 1092.91524

[5] P. Carr and A. Chou, Breaking Barriers. RISK (1997) 139-145.

[6] P. Carr, K. Ellis and V. Gupta, Static Hedging of Exotic Options. J. Finance (1998) 1165-1190.

[7] M.H. Davis, V.G. Panas and T. Zariphopoulou, European Option Pricing with Transaction Costs. SIAM J. Control Optim. 3 (1993) 470-493. | MR 1205985 | Zbl 0779.90011

[8] E. Derman and I. Kani, Riding on a Smile. Risk Mag. (1994) 32-39.

[9] E. Derman and I. Kani, Stochastic Implied Trees: Arbitrage Pricing with Stochastic Term and Strike Structure of Volatility. Int. J. Theor. Appl. Finance 1 (1998) 61-110. | Zbl 0908.90009

[10] R. Douady, Closed Form Formulas for Exotic Options and their Lifetime Distribution. Int. J. Theor. Appl. Finance 2 (1998) 17-42. | MR 1713819 | Zbl pre05320080

[11] N. Dubourg, Couverture dynamique en présence d'imperfections, Ph.D. Thesis. Univ. Paris I (1997).

[12] B. Dupire, Pricing and Hedging with Smiles in Mathematics of Derivative Securities, edited by M.A.H. Dempster and S.R. Pliska. Cambridge Univ. Press, Cambridge (1997) 103-111. | MR 1491370 | Zbl 0913.90012

[13] B. Dupire, A Unified Theory of Volatility, Preprint. Paribas Capital Markets (1995).

[14] A. Friedman, Partial differential equations of parabolic type. Prentice-hall, Inc. Englewood Cliffs, N.Y. (1964). | MR 181836 | Zbl 0144.34903

[15] A.V. Fursikov, Lagrange principle for problems of optimal control of ill-posed or singular distributed systems. J. Math. Pures Appl. 71 (1992) 139-194. | MR 1170249 | Zbl 0829.49001

[16] A.V. Fursikov and O.Yu. Imanuvilov, On approximate controllability of the Stokes system. Ann. Fac. Sci. Toulouse 11 (1993) 205-232. | Numdam | Zbl 0925.93416

[17] A.V. Fursikov and O.Yu. Imanuvilov, Local exact controllability of two dimensional Navier-Stokes system with control on the part of the boundary. Math. Sbornik. 187 (1996).

[18] L. Hörmander, Linear partial differential operators. Springer-Verlag, Berlin (1963). | MR 404822 | Zbl 0108.09301

[19] N. El Karoui, Évaluation et couverture des options exotiques, Working paper. Univ. Paris VI (1997).

[20] R. Lattès and J.-L. Lions, Méthode de quasi-réversibilité et applications. Dunod, Paris (1967). | MR 232549 | Zbl 0159.20803

[21] R.C. Merton, Theory of Rational Option Pricing. Bell J. Econ. Manag. Sci. 4 (1973) 141-183. | MR 496534

[22] J.-L. Lions, Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles. Dunod Gauthier-Villars, Paris (1968). | MR 244606 | Zbl 0179.41801

[23] M. Rubinstein, Exotic Options, Finance Working Paper No. 220. U.C. Berkeley (1991).

[24] P.O. Shorygin, On the Controllability Problem Arising in Financial Mathematics. J. Dynam. Control. Syst. 6 (2000) 353-363. | MR 1763674 | Zbl 1063.93009

[25] N. Taleb, Dynamic Hedging: Managing Vanilla and Exotic Options. J. Wiley & Sons, New York (1997).

[26] D. Tataru, Carleman estimates and unique continuation for solutions to boundary value problems. J. Math. Pures Appl. 75 (1996) 367-408. | MR 1411157 | Zbl 0896.35023

[27] M.E. Taylor, Partial Differential Equations II. Springer-Verlag, Berlin (1991). Related papers not cited in the article | Zbl 0869.35004

[28] P. Acworth, Pricing and Hedging Barrier and Forward Start Options Using Static Replication, Working paper. ING Barings (1997).

[29] S. Allen and O. Padovani, Risk Management Using Static Hedging, Working paper. Courant Institute, N.Y.U. (2001).

[30] L. Andersen and J. Andreasen, Static Barriers. RISK (2000) 120-122.

[31] S. Aparicio and L. Clewlow, A Comparison of Alternative Methods for Hedging Exotic Options, Working paper. FORC (1997).

[32] A. Bhandari, Static Hedging: A Genetic Algorithms Approach. Working paper (1999).

[33] J. Bowie and P. Carr, Static Simplicity. RISK (1994) 44-50.

[34] H. Brown, D. Hobson and C. Rogers, Robust Hedging of Barrier Options. Math. Finance 11 (2000) 285-314. | MR 1839367 | Zbl 1047.91024

[35] P. Carr and J. Picron, Static Hedging of Timing Risk. J. Derivatives (1999) 57-66.

[36] P. Carr and A. Chou, Static Hedging of Complex Barrier Options, Working paper. Courant Institute, N.Y.U. (1998).

[37] A. Chou and G. Grigoriev, A Uniform Approach to Static Replication. J. Risk Fall (1998) 73-86.

[38] M. Davis, W. Schachermayer and R. Tompkins, Pricing, No-arbitrage Bounds and Robust Hedging of Installment Options, Working paper. Tech. Univ. Vienna, Austria (2000). | MR 1870017

[39] E. Derman, D. Ergener and I. Kani, Forever Hedged RISK (1995) 139-145.

[40] E. Derman, D. Ergener and I. Kani, Static Option Replication. J. Derivatives 2 (1995) 78-85.

[41] N. El Karoui and M. Jeanblanc-Piqué Exotic Options Without Mathematics, Working paper. Univ. Paris VII (1997).

[42] E. Haug, First...Then...Knock-out Options. Wilmott Mag. (2001).

[43] E. Haug, Barrier Put-Call Transformations, Working paper. Paloma Partners (1999).

[44] E. Herzfeld and H. Konishi, Static Replication of Interest Rate Contingent Claims, Master Thesis. M.I.T. (1997).

[45] D. Hobson, Robust Hedging of the Lookback Option. Finance and Stochastics 2 (1998) 329-347. | Zbl 0907.90023

[46] P. Jaeckel and R. Rebonato, An Efficien and General Method to Value American-style Equity and FX Options in the Presence of User-defined Smiles and Time-dependent Volatility, Working paper. NatWest (1999).

[47] G. Koutmos, Financial Risk Management: Dynamic vs. Static Hedging. Global Bus. Econ. Rev. I (1999) 60-75.

[48] G. Peccati, A Time-space Hedging Theory, Working paper. Univ. Paris VI (2001).

[49] A. Sbuelz, A General Treatment of Barrier Options and Semi-static Hedges of Double Barrier Options, Working paper. Tilburg Univ. (2000).

[50] A. Sbuelz, Semi-static Hedging of Double Barrier Options, Working paper. Tilburg Univ. (2000).

[51] B. Thomas, Exotic Options II in Handbbok of Risk Management, Chap. 4, edited by C. Alexander (1998).

[52] H. Thomsen, Barrier Options: Evaluation and Hedging, Dissertation. Aarhus Univ. (1998).

[53] K. Toft and C. Xuan, How Well Can Barrier Options be Hedged by a Static Portfolio of Standard Options? J. Fin. Engrg. 7 (1998) 147-175.

[54] R. Tompkins, Static vs. Dynamic Hedging of Exotic Options: An Evaluation of Hedge Performance via Simulation. Net Exposure 2 (1997) 1-36.