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A Backward Stochastic Differential Equation (BSDE) is a stochastic differential equation for which a terminal condition has been specified. In Ruijter and Oosterlee (A Fourier-cosine method for an efficient computation of solutions to BSDEs, 2013) a Fourier-cosine method to solve BSDEs is developed. This technique is known as BCOS method and consists of the approximation of the BSDE’s solution backwards in time by the use of the COS method developed in Fang and Oosterlee (SIAM J Sci Comput 31(2):826–848, 2008) to compute the conditional expectations that rise after the discretization by means of a θ-method for the time-integration.
In this work, the methodology is extended to the case in which there are more than one source of uncertainty or the terminal condition depends on more than one process, allowing the pricing of derivatives contracts such as rainbow options. The extension of the BCOS technique can be done taking into account some ideas developed in Ruijter and Oosterlee (SIAM J Sci Comput 34(5):B642–B671, 2012). We present some results concerning to derivatives on two processes without jumps. We also apply our extended method to solve the BSDEs that rise with the use of quadratic hedging techniques for pricing in incomplete markets without or with jumps (Lim, Math Oper Res 29(1):132–161, 2004; Lim, SIAM J Sci Comput 44(5):1893–1922, 2005). Problems in which the randomness of the terminal condition depends not only on the risky asset but also on the insurance risk or the counterparty default risk can be introduced in this framework (Delong, Backward Stochastic Differential Equations with Jumps and Their Actuarial and Financial Applications. Springer, London, 2013).
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Margrabe, W.: The value of an option to exchange one asset for another. J. Financ. 33(1),177–186 (1997) CrossRef
Pardoux, E., Peng, S.: Backward stochastic differential equations and quasilinear parabolic partial differential equations. In: Stochastic Partial Differential Equations and Their Applications. Lectures Notes in Control and Information Sciences, vol. 176, pp. 200–217. Springer, Berlin (1992)
Pellegrino, T., Sabino, P.: Pricing and hedging multi-asset spread options by a three-dimensional Fourier cosine series expansion method. Available at SSRN: http://ssrn.com/abstract=2410176or/link?doi=10.2139/ssrn.2410176 (2014)
Stulz, R.M.: Options on the minimum or the maximum of two risky assets: analysis and applications. J. Financ. Econ. 10(2), 161–185 (1982) CrossRef
- Extension of a Fourier-Cosine Method to Solve BSDEs with Higher Dimensions
M. R. Ruijter
C. W. Oosterlee
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