Abstract
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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Notes
- 1.
{{}{∑}′} denotes that the first term in the sum is multiplied by \(\frac{1} {2}\).
- 2.
Since the terminal conditions Z M and U m are not known, we set θ Y = θ Z 1 = … = θ Z d = θ U 1 = … = θ U c = 1 in the first time iteration.
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Acknowledgements
This work has been funded by the ITN Research Project STRIKE. The ITN Research Project STRIKE is supported by the European Union in the FP7-PEOPLE-2012-ITN Program under Grant Agreement Number 304617 (FP7 Marie Curie Action, Project Multi-ITN STRIKE—Novel Methods in Computational Finance).
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Pou, M., Ruijter, M.R., Oosterlee, C.W. (2016). Extension of a Fourier-Cosine Method to Solve BSDEs with Higher Dimensions. In: Russo, G., Capasso, V., Nicosia, G., Romano, V. (eds) Progress in Industrial Mathematics at ECMI 2014. ECMI 2014. Mathematics in Industry(), vol 22. Springer, Cham. https://doi.org/10.1007/978-3-319-23413-7_12
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