The Pricing and Hedging of Basket-124

The Pricing and Hedging of Basket Derivatives Based on the Rank One Approximation Method Jing Lia aBonn Graduate School of Economics University of Bonn Adenauerallee 24-42, 53111 Bonn, Germany. Tel: 0049-228-739225. lijing@uni-bonn.de ABSTRACT I propose a new approximation method, called the rank one approximation method, to price basket deriva- tives. As a first step, I approximate the covariance structure of the uncertain parts in the single assets’ price processes with a rank one covariance matrix which delivers a vector of stochastic processes driven by the same standard normally distributed random variable. The second step is to introduce several ad- justment parameters into the price process of the synthetic underlying basket to correct the distributional distortion resulting from the first step. Both, the dynamic and the static hedging of the basket deriva- tives are discussed. I show that the rank one approximation method allows for building an effective delta hedging procedure. The safe crossing hedging strategy is also touched upon with regard to the famous “cross-gamma” effect between the underlying assets in the basket. The idea of static hedging is to obtain a portfolio of simple options with single underlying assets to hedge against the risk exposure. The efficiency of the rank one approximation pricing helps to assess the performance of the static hedging more quickly. A numerical study is carried out to observe the performance of the rank one approximation method for both pricing and hedging. Keywords: Basket derivatives, Rank one approximation, Hedging strategies. References [1] Ashraff, J., Tarczon, J. and Wu, W.Q., “Safe Crossing”, Risk , v.8, p.56-57, 1995. [2] Beißer, J., “Tropics in Finance–A Conditional Expectation Approach to Value Asian, Basket and Spread Options”, Phd Thesis , Johannes Gutenberg-Universita¨t Mainz, 2001. [3] Brace, A. and Musiela, M.,“A Multifactor Gauss Markov Implementation of Heath, Jarrow and Morton”, Mathematical Finance, v.4, p.259-283, 1994. [4] Brigo, D., Mercurio, F., Rapisarda, F. and Scotti, R., “Approximated moment-matching dynamics for basket-options simulation”, Working Paper , 2001. [5] Carr, P., Ellis, K. and Gupta, V., “Static Hedging of Exotic Options”, The Journal of Finance, v.53, p.1165-1190, 1998. [6] Curran, M., “Valuing Asian and Portfolio Options by Conditioning on the Geometric Mean Price”, Management Science v.40, p.1705-1711, 1994. [7] Deelstra, G., Liinev, J. and Vanmaele, M., “Pricing of Arithmetic Basket Options by Conditioning”, Insurance: Mathematics and Economics v.34, p.1-23, 2004. [8] Geske, R. and Johnson, H.E., “The American Put Option Valued Analytically”, The Journal of Finance, v.39, p.1511-1524, 1984. [9] Joy, C., Boyle, P.P. and Tang, K.S., “Quasi-Monte Carlo Methods in Numerical Finance”, Management Science, v.42, p.926-938, 1996. [10] Hobson, D., Laurence, P. and Wang, T., “Static Arbitrage Upper Bounds for Basket Options”, Quan- titative Finance v.5, p.329-342, 2005. [11] Hull, J.C., “Options, Futures and Other Derivatives”, 6th Edition, Prentice Hall International, London, 2006. [12] Laurence, P. and Wang, T., “Sharp Upper and Lower Bounds for Basket Options”, Applied Mathe- matical Finance v.12, 253-282, 2005. [13] Milevsky, M. A. and Posner, S. E., “A Closed Form Approximation for Valuing Basket Options”, The Journal of Derivatives v.5, p.54-61, 1998. [14] Musiela, M. and Rutkowski, M., “Martingale Methods in Financial Modelling”, 2nd Edition, Springer- verlag Berlin Heidelberg, 2005. [15] Nielsen, J.A. and Sandmann, K., “Pricing of Asian Exchange Rate Options Under Stochastic Interest Rates as a Sum of Options”, Finance and Stochastic, v.3, p.355-770, 2002. [16] Nielsen, J.A. and Sandmann, K., “Pricing Bounds on Asian Options”, Journal of Financial and Quantitative Analyse, v.38, p.449-473, 2003. [17] Su,X., “Hedging Basket Options by Using a Subset of Underlying Assets”, Bonn Econ Discussion Paper 14/2006, University of Bonn, 2006.