Option hedging for small investors under liquidity costs

Finance and Stochastics

Volumn 14, Issue 3, 2010, Pages 317-341

  Çetin, Umut ^a   Soner, H Mete ^b   Touzi, Nizar ^c  

^a LONDON SCHOOL OF ECONOMICS   (United Kingdom)

^b ETH ZURICH   (Switzerland)

^c CENTRE DE MATHÉMATIQUES APPLIQUÉES   (France)

Author keywords

Gamma process; Liquidity cost; Parabolic majorant; PDE valuation; Super replication

Indexed keywords

EID: 77954535783     PISSN: 09492984     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00780-009-0116-x     Document Type: Article

References (32)

1. Bank, P., Baum, D.: Hedging and portfolio optimization in financial markets with a large trader. Math. Finance 14, 1-18 (2004).
2. Barles, G.: Soner, H. M, Option pricing with transaction costs and a nonlinear Black-Scholes equation. Finance Stoch. 2, 369-397 (1998).
3. Broadie, M., Cvitanić, J., Soner, H. M.: Optimal replication of contingent claims under portfolio constraints. Rev. Financ. Stud. 11, 59-79 (1998).
4. Çetin, U., Jarrow, R., Protter, P.: Liquidity risk and arbitrage pricing theory. Finance Stoch. 8, 311-341 (2004).
5. Çetin, U., Jarrow, R., Protter, P., Warachka, M.: Pricing options in an extended Black-Scholes economy with illiquidity: Theory and empirical evidence. Rev. Financ. Stud. 19, 493-529 (2006).
6. Çetin, U., Rogers, L. C. G.: Modelling liquidity effects in discrete time. Math. Finance 17, 15-29 (2006).
7. Cheridito, P., Soner, H. M., Touzi, N.: The multi-dimensional super-replication problem under gamma constraints. Ann. Henri Poincaré (C) Non Linear Anal. 22, 633-666 (2005).
8. Cheridito, P., Soner, H. M., Touzi, N.: Small time path behavior of double stochastic integrals and applications to stochastic control. Ann. Appl. Probab. 15, 2472-2495 (2005).
9. Cheridito, P., Soner, H. M., Touzi, N., Victoir, N.: Second order backward stochastic differential equations and fully non-linear parabolic PDEs. Commun. Pure Appl. Math. 60, 1081-1110 (2007).
10. Crandall, M. G., Ishii, H., Lions, P. L.: User's guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27, 1-67 (1992).
11. Cvitanić, J., Karatzas, I.: Hedging and portfolio optimization under transaction costs: A martingale approach. Math. Finance 6, 133-165 (1996).
12. Cvitanić, J., Ma, J.: Hedging options for a large investor and forward-backward SDEs. Ann. Appl. Prob. 6, 370-398 (1996).
13. Cvitanić, J., Pham, H., Touzi, N.: Super-replication in stochastic volatility models under portfolio constraints. J. Appl. Probab. 36, 523-545 (1999).
14. Davis, M., Panas, V. G., Zariphopoulou, T.: European option pricing with transaction fees. SIAM J. Control. Opt. 31, 470-493 (1993).
15. Evans, L. C.: Classical solutions of fully nonlinear, convex, second-order elliptic equations. Commun. Pure Appl. Math. 35, 333-363 (1982).
16. Fleming, W. H., Soner, H. M.: Controlled Markov Processes and Viscosity Solutions. Applications of Mathematics, vol. 25. Springer, New York (1993).
17. Frey, R.: Perfect option hedging for a large trader. Finance Stoch. 2, 115-141 (1998).
18. Frey, R.: Market illiquidity as a source of model risk in dynamic hedging. In: Gibson, R. (ed.) Model Risk, pp. 125-136. RISK Publications, London (2000).
19. Gökay, S., Soner, H. M.: Çetin-Jarrow-Protter model of liquidity in a binomial market and its limit. Preprint.
20. Jarrow, R.: Derivative security markets, market manipulation, and option pricing theory. J. Financ. Quant. Anal. 29, 241-261 (1994).
21. Longstaff, F. A.: Optimal portfolio choice and the valuation of illiquid securities. Rev. Financ. Stud. 14, 407-431 (2001).
22. Krylov, N. V., Safanov, M. V.: A certain property of solutions of parabolic equations with measurable coefficients. Math. USSR Izv. 16, 151-164 (1981).
23. Krylov, N. V.: Nonlinear Elliptic and Parabolic Equations of the Second Order. Reidel, Dordrecht (1981).
24. Ladyzhenskaya, O. A., Solonnikov, V. A., Uralceva, N. N.: Linear and Quasilinear Parabolic Equations. Am. Math. Soc., Providence (1968).
25. Platen, E., Schweizer, M.: On feedback effects from hedging derivatives. Math. Finance 8, 67-84 (1998).
26. Rogers, L. C. G., Singh, S.: Option pricing in an illiquid market. Technical Report, University of Cambridge, Cambridge (2005).
27. Schönbucher, P. J., Wilmott, P.: The feedback effects of hedging in illiquid markets. SIAM J. Appl. Math. 61, 232-272 (2000).
28. Sircar, K. R., Papanicolaou, G.: Generalized Black-Scholes models accounting for increased market volatility from hedging strategies, Appl. Math. Finance 5, 45-82 (1998).
29. Soner, H. M., Touzi, N.: Super-replication under gamma constraints. SIAM J. Control Optim. 39, 73-96 (2000).
30. Soner, H. M., Touzi, N.: Stochastic target problems, dynamic programming and viscosity solutions. SIAM J. Control Optim. 41, 404-424 (2002).
31. Soner, H. M., Touzi, N.: Dynamic programming for stochastic target problems and geometric flows. J. Eur. Math. Soc. 4, 201-236 (2002).
32. Soner, H. M., Touzi, N.: The dynamic programming equation for second order stochastic target problems. SIAM J. Control Optim. 48, 2344-2365 (2009).