# Hedging and Pricing European-type, Early-Exercise and Discrete Barrier Options using Algorithm for the Convolution of Legendre Series

Tat Lung Chan and Nicholas Hale

Papers from arXiv.org

Abstract: This paper applies the $\mathcal{O}(N^2)$ algorithm for the convolution of compactly supported Legendre series (the CONLeg method), proposed by Hale and Townsend (2014a), to pricing/hedging European-type, early-exercise and discrete-monitored barrier options under the L\'evy process. The current paper takes advantage of Chebfun (cf. Trefethen et al. 2014) in computational finance and provides a non-quadrature approach by applying the Chebyshev series in financial modelling. The new approach is clearly different to the quadrature-based methodology (Clenshaw--Curtis quadrature) promoted in the previous literature (e.g. Ga$\ss$ et al. 2018, Pachon 2018). The main purpose of using the CONLeg method is to formulate option pricing and option Greek curves rather than individual prices/Greek values. Moreover, the CONLeg method can retain the global spectral convergence rate in option pricing and hedging when the risk-free smooth probability density function (PDF) is smooth. When the PDF is non-smooth, we also provide a solution to allow the method to gain the accurate algebraic rate. Finally, we show that our method requires a small number of terms to yield fast error convergence and is able to accurately price/hedge any options deep in/out of the money and with very long/short maturities. Compared with existing techniques, this new method performs either favourably or comparably in numerical experiments.

New Economics Papers: this item is included in nep-cmp
Date: 2018-11, Revised 2018-11
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