  This paper applies an algorithm for the convolution of compactly supported
Legendre series (the CONLeg method) (cf. Hale and Townsend 2014a), to
pricing/hedging European-type, early-exercise and discrete-monitored barrier
options under a Levy process. The paper employs Chebfun (cf. Trefethen et al.
2014) in computational finance and provides a quadrature-free approach by
applying the Chebyshev series in financial modelling. A significant advantage
of using the CONLeg method is to formulate option pricing and option Greek
curves rather than individual prices/values. Moreover, the CONLeg method can
yield high accuracy in option pricing and hedging when the risk-free smooth
probability density function (PDF) is smooth/non-smooth. Finally, we show that
our method can accurately price/hedge options deep in/out of the money and with
very long/short maturities. Compared with existing techniques, the CONLeg
method performs either favourably or comparably in numerical experiments.
