Hedging with a correlated asset: Solution of a nonlinear pricing PDE

Journal of Computational and Applied Mathematics

Volumn 200, Issue 1, 2007, Pages 86-115

  Windcliff, H ^a   Wang, J ^b   Forsyth, P A ^b   Vetzal, K R ^b  

^a Morgan Stanley   (United States)

^b UNIVERSITY OF WATERLOO   (Canada)

Author keywords

Basis risk; Contingent claim; Hedging; Monotone discretization; Viscosity solution

Indexed keywords

ALGORITHMS; NONLINEAR EQUATIONS; PARAMETER ESTIMATION; PARTIAL DIFFERENTIAL EQUATIONS; PROBLEM SOLVING; RISK ASSESSMENT; STATISTICAL METHODS;

CONTINGENT CLAIM; MONOTONE DISCRETIZATION; STANDARD DEVIATION; UNHEDGEABLE RESIDUAL RISK; VISCOSITY SOLUTION;

CORRELATION METHODS;

EID: 33751098024     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cam.2005.12.008     Document Type: Article

References (38)

1. C. Albanese, S. Tompaidis, Small transaction cost asymptotics and dynamic hedging, Working paper, Imperial College, Department of Mathematics, to appear in European Journal of Operational Research, 2004.
2. Amadori A.L. Nonlinear integro-differential evolution problems arising in option pricing: a viscosity solution approach. Differential Integral Equations 16 (2003) 787-811
3. Andersen L.B.G., Andreasen J., and Brotherton-Ratcliffe R. The passport option. J. Comput. Finance 1 3 (1998) 15-36
4. Avellaneda M., Levy A., and Parás A. Pricing and hedging derivative securities in markets with uncertain volatilities. Appl. Math. Finance 2 (1995) 73-88
5. Barles G. Convergence of numerical schemes for degenerate parabolic equations arising in finance. In: Rogers L.C.G., and Talay D. (Eds). Numerical Methods in Finance (1997), Cambridge University Press, Cambridge 1-21
6. Barles G., and Burdeau J. The Dirichlet problem for semilinear second-order degenerate elliptic equations and applications to stochastic exit time control problems. Comm. Partial Differential Equations 20 (1995) 129-178
7. Barles G., and Rouy E. A strong comparison result for the Bellman equation arising in stochastic exit time control problems and applications. Comm. Partial Differential Equations 23 (1998) 1995-2033
8. Barles G., and Souganidis P.E. Convergence of approximation schemes for fully nonlinear equations. Asymptotic Anal. 4 (1991) 271-283
9. Bertsimas D., Kogan L., and Lo A.W. Hedging derivative securities in incomplete markets: an ε{lunate} arbitrage approach. Oper. Res. 49 (2001) 372-397
10. Bouchaud J.P., Iori G., and Sornette D. Real world options: smile and residual risk. Risk 9 3 (1996) 61-65
11. S. Chaumont, A strong comparison result for viscosity solutions to Hamilton-Jacobi-Bellman equations with Dirichlet conditions on a non-smooth boundary, Working paper, Institute Elie Cartan, Université Nancy I, 2004.
12. T.F. Coleman, Y. Li, M.C. Patron, Hedging guarantees in variable annuities, Working paper, Cornell Theory Center, 2004.
13. Cont R., and Tankov P. Financial Modelling with Jump Processes (2004), Chapman & Hall, London
14. R. Cont, E. Voltchkova, A finite difference scheme for option pricing in jump diffusion and Lévy process models, SIAM Journal on Numerical Analysis 43 (2005) 1596-1626.
15. Crandall M.G., Ishii H., and Lions P.L. User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27 (1992) 1-67
16. M.A.H. Davis, Optimal hedging with basis risk, Working Paper, Vienna University of Technology, 2000.
17. d'Halluin Y., Forsyth P.A., and Labahn G. A penalty method for American options with jump diffusion processes. Numer. Math. 97 (2004) 321-352
18. d'Halluin Y., Forsyth P.A., and Labahn G. A semi-Lagrangian approach for American Asian options under jump diffusion. SIAM J. Sci. Comput. 27 (2005) 315-345
19. Embrechts P. Actuarial versus financial pricing of insurance. Risk Finance 1 4 (2000) 17-26
20. Föllmer H., and Schweizer M. Hedging by sequential regression: an introduction to the mathematics of option trading. ASTIN Bull. 18 (1988) 147-160
21. Forsyth P.A., and Vetzal K.R. Quadratic convergence of a penalty method for valuing American options. SIAM J. Sci. Comput. 23 (2002) 2096-2123
22. Heath D., Platen E., and Schweizer M. A comparison of two quadratic approaches to hedging in incomplete markets. Math. Finance 11 (2001) 385-413
23. Leland H.E. Option pricing and replication with transaction costs. J. Finance 40 (1985) 1283-1301
24. Møler T. Risk minimizing strategies for unit linked life insurance contracts. ASTIN Bull. 28 (1998) 17-47
25. Møller T. On transformations of actuarial valuation principles. Insurance Math. Econom. 28 (2001) 281-303
26. Møller T. On valuation and risk management at the interface of insurance and finance. British Actuarial J. 8 (2002) 787-827
27. Oberman A., and Zariphopoulou T. Pricing early exercise contracts in incomplete markets. Comput. Manage. Sci. 1 (2003) 75-107
28. M. Otaka, Y. Kawaguchi, Hedging and pricing of real estate securities under market incompleteness, Quantitative Methods in Finance Conference, Cairns, Australia, 2002.
29. D.M. Pooley, Numerical methods for nonlinear equations in option pricing, Ph.D. Thesis, School of Computer Science, University of Waterloo, 2003.
30. Pooley D.M., Forsyth P.A., and Vetzal K.R. Numerical convergence properties of option pricing PDEs with uncertain volatility. IMA J. Numer. Anal. 23 (2003) 241-267
31. Qi L., and Sun J. A nonsmooth version of Newton's method. Math. Program. 58 (1993) 353-367
32. Qi L., and Zhou G. A smoothing Newton method for minimizing a sum of Euclidean. SIAM J. Optim. 11 (2000) 389-410
33. Rannacher R. Finite element solution of diffusion problems with irregular data. Numer. Math. 43 (1984) 309-327
34. Schal M. On quadratic cost criteria for option hedging. Math. Oper. Res. 19 (1994) 121-131
35. Schweizer M. Variance optimal hedging in discrete time. Math. Oper. Res. 20 (1995) 1-31
36. Schweizer M. From actuarial to financial valuation principles. Insurance Math. Econom. 28 (2001) 31-47
37. Wilmott P. Derivatives: The Theory and Practice of Financial Engineering (1998), Wiley, West Sussex, England
38. Windcliff H., Forsyth P.A., and Vetzal K.R. Segregated funds: shout options with maturity extensions. Insurance Math. Econom. 29 (2001) 1-21