Yip, Wing and Stephens, David and Olhede, Sofia (2008): Hedging strategies and minimal variance portfolios for European and exotic options in a Levy market. Forthcoming in: Mathematical Finance

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Abstract
This paper presents hedging strategies for European and exotic options in a Levy market. By applying Taylor's Theorem, dynamic hedging portfolios are con structed under different market assumptions, such as the existence of power jump assets or moment swaps. In the case of European options or baskets of European options, static hedging is implemented. It is shown that perfect hedging can be achieved. Delta and gamma hedging strategies are extended to higher moment hedging by investing in other traded derivatives depending on the same underlying asset. This development is of practical importance as such other derivatives might be readily available. Moment swaps or power jump assets are not typically liquidly traded. It is shown how minimal variance portfolios can be used to hedge the higher order terms in a Taylor expansion of the pricing function, investing only in a riskfree bank account, the underlying asset and potentially variance swaps. The numerical algorithms and performance of the hedging strategies are presented, showing the practical utility of the derived results.
Item Type:  MPRA Paper 

Original Title:  Hedging strategies and minimal variance portfolios for European and exotic options in a Levy market 
Language:  English 
Keywords:  Hedging Strategies; Levy processes; Variance Gamma; Choatic Representation Property; Power Jump Processs; Variance Swaps; Moment Swaps 
Subjects:  C  Mathematical and Quantitative Methods > C6  Mathematical Methods; Programming Models; Mathematical and Simulation Modeling C  Mathematical and Quantitative Methods > C5  Econometric Modeling > C51  Model Construction and Estimation C  Mathematical and Quantitative Methods > C0  General 
Item ID:  11176 
Depositing User:  Wing Yan Yip 
Date Deposited:  19. Oct 2008 07:41 
Last Modified:  16. Feb 2013 08:09 
References:  Benth, F., Nunno, G. Di, Løkka, A., Øksendal, B., & Proske, F. 2003. Explicit representation of the minimal variance portfolio in markets driven by L´evy processes. Mathematical Finance, 13(1), 55–72. Carr, P., Geman, H., & Madan, D. 2001. Pricing and hedging in incomplete markets. Journal of Financial Economics, 62, 131–167. Cont, R., Tankov, P., & Voltchkova, E. 2005. Hedging with options in models with jumps. Abel Symposium 2005 on Stochastic Analysis and Applications. Corcuera, J. M., Nualart, D., & Schoutens, W. 2005. Completion of a L´evy market by powerjump assets. Finance and Stochastics, 9, 109–127. Corcuera, J. M., J. Guerra, D. Nualart, & Schoutens, W. 2006. Optimal investment in a L´evy market. Applied Mathematics and Optimization, 53(3), 279–309. Demeterfi, K., Derman, E., Kamal, M., & Zou, J. 1999. A guide to volatility and variance swaps. The Journal of Derivatives, 9–32. Derman, E., Ergener, D., & Kani, I. 1995. Static options replication. Journal of Derivatives, 2(4). Dritschel, M., & Protter, P. 1999. Complete markets with discontinuous security price. Finance and Stochastics, 3, 203–214. He, C., Kennedy, J., Coleman, T., Forsyth, P., Li, Y., & Vetzal, K. 2005. Calibration and hedging under jump diffusion. Working paper. Hull, J. 2003. Options, futures, and other derivatives. 5th edn. Prentice Hall Finance Series. Ikeda, N., & Watanabe, S. 1989. Stochastic differential equations and diffusion processes. Amsterdam: NorthHolland. Itˆo, K. 1956. Spectral type of the shift transformation of differential processes with stationary increments. Transactions of the American Mathematical Society, 81, 253–263. Khan, I., & Ohba, R. 2003. Taylor series based finite difference approximations of higherdegree derivatives. Journal of Computational and Applied Mathematics, 154, 115–124. Kijima, M. 2002. Stochastic processes with applications to finance. Chapman and Hall. Løkka, A. 2004. Martingale representation of functionals of L´evy processes. Stochastic Analysis and Applications, 22(4), 867–892. Monat, P., & Stricker, C. 1995. FollmerSchweizer decomposition and meanvariance hedging for general claims. The Annals of Probability, 23(2), 605–628. Nualart, D., & Schoutens, W. 2000. Chaotic and predictable representations for L´evy processes. Stochastic Processes and their Applications, 90, 109–122. Protter, P. 2004. Stochastic integration and differential equations. 2nd edn. Springer. Sato, K. 1999. L´evy processes and infinitely divisible distribution. Vol. 68. Cambridge University Studies in Advanced Mathematics, Cambridge University Press, Cambridge. Schoutens, W. 2000. Stochastic processes and orthogonal polynomials. Springer. Schoutens, W. 2003. L´evy processes in finance: pricing financial derivatives,. Chichester, New York, N.Y. : J. Wiley. Schoutens, W. 2005. Moment swaps. Quantitative Finance, 5(6), 525–530. Windcliff, H., Forsyth, P., & Vetzal, K. 2006. Pricing methods and hedging strategies for volatility derivatives. Journal of Banking and Finance, 30(2), 409–431. Yip, W., Stephens, D., & Olhede, S. 2007. The explicit chaotic representation of powers of increments of L´evy processes. http://arxiv.org/abs/0706.1698. 
URI:  http://mpra.ub.unimuenchen.de/id/eprint/11176 