JMF  Vol.2 No.4 , November 2012
The Malliavin Derivative and Application to Pricing and Hedging a European Exchange Option
Author(s) Sure Mataramvura*
ABSTRACT
The exchange option was introduced by Margrabe in [1] and its price was explicitly computed therein, albeit with some small variations to the models considered here. After that important introduction of an option to exchange one commodity for another, a lot more work has been devoted to variations of exchange options with attention focusing mainly on pricing but not hedging. In this paper, we demonstrate the efficiency of the Malliavin derivative in computing both the price and hedging portfolio of an exchange option. For that to happen, we first give a preview of white noise analysis and theory of distributions.

Cite this paper
S. Mataramvura, "The Malliavin Derivative and Application to Pricing and Hedging a European Exchange Option," Journal of Mathematical Finance, Vol. 2 No. 4, 2012, pp. 280-290. doi: 10.4236/jmf.2012.24031.
References
[1]   W. Margrabe, “The Value of an Option to Exchange One Asset for Another,” Journal of Finance, Vol. 33, No. 1, 1978, pp. 177-186. doi:10.1111/j.1540-6261.1978.tb03397.x

[2]   T. Hida, H.-H. Kuo, J. Potthoff and L. Streit, “White Noise,” Kluwer, Dordrecht, 1993.

[3]   T. Hida and J. Potthof, “White Noise Analysis—An Overview, White Noise Analysis: Mathematics and Applications,” World Scientific, Singapore, 1989.

[4]   H. H. Kuo, “White Noise Distribution Theory,” CRC Press, Boca Raton, 1996.

[5]   N. Obata, “White Noise Calculus and Fock Space,” Springer-Verlag, Berlin, 1994.

[6]   K. Aase, B. Oksendal, N. Privault and J. Uboe, “White Noise Generalizations of the Clark-Haussmann-Ocone Theorem, With Application to Mathematical Finance,” Finance and Stochastics, Vol. 4, No. 4, 2000, pp. 465-496. doi:10.1007/PL00013528

[7]   B. Oksendal, “An introduction to Malliavin Calculus with Applications to Economics,” Working paper 3/96, Institute of Finance and Management Science, Norwegian School of Economics and Business Administration, Bergen, 1996.

[8]   I. Karatzas and D. Ocone, “A Generalized Clark Representation Formula, with Application to Optimal Portfolios,” Stochastics and Stochastic Reports, Vol. 34, 1991, pp. 187-220.

[9]   B. Oksendal, “Stochastic Differential Equations,” 5th Edition, Springer-Verlag, Berlin, 2000.

 
 
Top