Pricing European-type, early-exercise and discrete barrier options using an algorithm for the convolution of Legendre series

Abstract

This paper applies an algorithm for the convolution of compactly supported Legendre series (the CONLeg method) (cf. Hale and Townsend, An algorithm for the convolution of Legendre series. SIAM J. Sci. Comput., 2014, 36, A1207–A1220), to pricing European-type, early-exercise and discrete-monitored barrier options under a Lévy process. The paper employs Chebfun (cf. Trefethen et al., Chebfun Guide, 2014 (Pafnuty Publications: Oxford), Available online at: http://www.chebfun.org/) 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 when the risk-free smooth probability density function (PDF) is smooth/non-smooth. Finally, we show that our method can accurately price 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.

Keywords:

• Convolution
• Legendre series
• European options
• Early-exercise options
• Discrete-monitored barrier options
• Lévy process

Jel Classification:

• C6

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Since convolution is a commutative operation, we consider that only f and g are in the same intervals.

2 The algorithm in $conv$ actually computes the convolution between two Chebyshev series by using fast Chebyshev-Legendre transform (Townsend et al. 2018) implemented in cheb2leg.

3 The expression of ${\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}U(e^y,K)g(y-x+\log K)\mathrm{d}y$ in (21(22) $\begin{array}{rl}V(x,K,t)& =e^{-r(T-t)}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}U(e^y,K)g\left(y-x+\log K\right)\mathrm{d}y\\ & =e^{-r(T-t)}K{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}f\left(y\right)g^R\left(\tilde{x}-y\right)\mathrm{d}y,\end{array}$(22) ) is indeed a cross-correlation integral; however, since we introduce the idea of the reflected function $g^R\left(\tilde{x}\right):=g(-\tilde{x})$, we can turn (21(22) $\begin{array}{rl}V(x,K,t)& =e^{-r(T-t)}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}U(e^y,K)g\left(y-x+\log K\right)\mathrm{d}y\\ & =e^{-r(T-t)}K{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}f\left(y\right)g^R\left(\tilde{x}-y\right)\mathrm{d}y,\end{array}$(22) ) into a convolution integral instead.

4 We refer a singularity as a point at which is not defined, or a point which fails to be well-behaved after differentiability

5 One should note that when $y\le 0,max(e^y-1,0)=0$.

6 To achieve this, we can set a flag–‘same’–in conv. See Appendix 4 for details.

7 A Toeplitz matrix or diagonal-constant matrix is an invertible matrix in which each descending diagonal from left to right is constant.