Barycentric spectral domain decomposition methods for valuing a class of infinite activity Lévy models

Edson Pindza; Francis Youbi; Eben Maré; Matt Davison

doi:10.3934/dcdss.2019040

Article Contents

2019, Volume 12, Issue 3: 625-643. Doi: 10.3934/dcdss.2019040

This issue Previous Article The discrete homotopy perturbation Sumudu transform method for solving partial difference equations Next Article Couette flows of a viscous fluid with slip effects and non-integer order derivative without singular kernel

Barycentric spectral domain decomposition methods for valuing a class of infinite activity Lévy models

1.
Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 002, Republic of South Africa
2.
Department of Statistical and Actuarial Sciences, University of Western Ontario, London, Ont., Canada

* Corresponding author: E. Pindza
* Corresponding author: E. Pindza

Received: July 2017

Revised: November 2017

Published: October 2018

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Abstract

A new barycentric spectral domain decomposition methods algorithm for solving partial integro-differential models is described. The method is applied to European and butterfly call option pricing problems under a class of infinite activity Lévy models. It is based on the barycentric spectral domain decomposition methods which allows the implementation of the boundary conditions in an efficient way. After the approximation of the spatial derivatives, we obtained the semi-discrete equations. The computation of these equations is performed by using the barycentric spectral domain decomposition method. This is achieved with the implementation of an exponential time integration scheme. Several numerical tests for the pricing of European and butterfly options are given to illustrate the efficiency and accuracy of this new algorithm. We also show that Greek options, such as Delta and Gamma sensitivity, are computed with no spurious oscillation.

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Mathematics Subject Classification: 76M22, 41A10, 41A20, 91G80.

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Figure 1. Spectral domain decomposition method matrix structures

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Figure 2. Numerical valuation of European call options for the CGMY, Meixner and GH model with their Greeks for the parameters in Table 2

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Figure 3. Convergence of the SDDM and FDM for European vanilla call options for the parameters in Table 2

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Figure 4. Numerical valuation of European butterfly call options for the CGMY, Meixner, and GH model with $N = 16, K_{1} = 40, K_{2} = 50, K_{3} = 60$ for the parameters in Table 2

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Figure 5. Convergence of the SDDM and FDM for European vanilla butterfly call options for the parameters in Table 2

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Table 1. Density functions for Lévy Processes

Model	Lévy density function
CGMY	$f(y)=\frac{C_{-}e^{-G\|y\|}}{\|y\|^{1+Y}}{\bf{1}}_{y<0}+ \frac{C_{+}e^{-M\|y\|}}{\|y\|^{1+Y}}{\bf{1}}_{y>0}$
Meixner	$f(y)=\frac{Ae^{-ay}}{y\sinh(by)}$
GH process	$f(y)=\frac{e^{\beta y}}{\|y\|}\left(\int_{0}^{\infty}\frac{e^{-\sqrt{2\zeta+\alpha^{2}}\|y\|}}{\pi^{2}\zeta \left(J^{2}_{\|\lambda\|}\left(\delta\sqrt{2\zeta}\right)+Y^{2}_{\|\lambda\|}\left(\delta\sqrt{2\zeta}\right)\right)}d\zeta+\max(0,\lambda)e^{-\alpha\|y\|}\right)$

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Table 2. The parameters for Lévy models used in both examples

Model	Parameters
GBM (Black-Scholes)	$K =50, \ r = 0.05, \sigma = 0.2, q=0$ and $T =0.5. $
CGMY	$C_{-} = 0.3, \ C_{+} = 0.1, \ \ G = 15,\ M = 25$ and $Y = 20. $
Meixner	$A=15, \ a=-1.5$ and $b=50$
GH process	$\alpha=4, \ \beta=-3.2, \ \delta=1.4775$ and $\lambda=-3$

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Table 3. The benchmark European call option values under Lévy processes with different values of S and $N = 150$ for the parameters in Table 2

Model	$S$
Model	40	50	60
CGMY	0.2210443864	3.3785900783	11.3681462140
Meixner	1.3420365535	5.4934725848	12.7780678851
GH processes	0.3237597911	3.8485639686	11.9164490861

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Table 4. Absolute errors (${\bf{AE}}$) of the benchmark and the European call option apply to the CGMY, Meixner and GH processes models with different values of $N$ and $S$ for the parameters in Table 2

		$SDDM $				$FDM $
	$S $	$40 $	50	60		$40 $	50	60
	$N$	${\bf{AE}}$	${\bf{AE}}$	${\bf{AE}}$	CPU	${\bf{AE}}$	${\bf{AE}}$	${\bf{AE}}$	CPU
CGMY	10	$1.15e^{-4}$	$1.28e^{-4}$	$1.35e^{-4}$	$0.30$	$1.14e^{-2}$	$5.72e^{-2}$	$9.55e^{-3}$	0.6
	15	$1.23e^{-5}$	$1.73e^{-5}$	$1.45e^{-5}$	$0.34$	$2.03e^{-3}$	$1.55e^{-2}$	$1.75e^{-3}$	0.72
	20	$2.75e^{-7}$	$2.13e^{-7}$	$2.34e^{-7}$	$0.40$	$6.25e^{-4}$	$4.09e^{-3}$	$6.06e^{-4}$	0.87
	25	$3.33e^{-10}$	$3.15e^{-10}$	$3.24e^{-10}$	$0.53$	$2.75e^{-4}$	$1.91e^{-3}$	$2.37e^{-4}$	1.32
Meixner	10	$2.12e^{-4}$	$2.45e^{-4}$	$2.35e^{-4}$	$0.32$	$1.46e^{-2}$	$4.35e^{-2}$	$8.51e^{-3}$	0.65
	15	$2.78e^{-5}$	$2.65e^{-5}$	$2.67e^{-5}$	$0.35$	$2.66e^{-3}$	$1.16e^{-2}$	$2.13e^{-3}$	0.78
	20	$3.40e^{-7}$	$3.23e^{-7}$	$3.14e^{-7}$	0.41	$6.45e^{-4}$	$3.95e^{-3}$	$5.45e^{-4}$	0.86
	25	$4.77e^{-10}$	$4.65e^{-10}$	$4.33e^{-10}$	$0.55$	$2.35e^{-4}$	$1.02e^{-3}$	$2.14e^{-4}$	1.41
GH processes	10	$3.33e^{-4}$	$3.29e^{-4}$	$3.17e^{-4}$	$0.65$	$1.45e^{-2}$	$5.33e^{-2}$	$7.13e^{-3}$	1.33
	15	$4.55e^{-5}$	$4.370e^{-5}$	$4.14e^{-5}$	$0.82$	$2.15e^{-3}$	$1.04e^{-2}$	$5.72e^{-2}$	1.61
	20	$5.14e^{-7}$	$5.21e^{-7}$	$5.14e^{-7}$	$1.41$	$5.61e^{-4}$	$4.02e^{-3}$	$6.12e^{-4}$	2.94
	25	$7.11e^{-10}$	$7.25e^{-10}$	$7.33e^{-10}$	$1.82$	$2.36e^{-4}$	$1.21e^{-3}$	$3.01e^{-4}$	3.51

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Table 5. The benchmark values of the European butterfly call option values under Lévy processes with different values of S and $N = 100$ for the parameters in Table 2

Model	$S$
Model	40	50	60
CGMY	2.2845953002	4.6814621409	2.1592689295
Meixner	2.2689295039	3.7101827676	2.3159268929
GH processes	2.3942558746	4.2898172323	1.7989556135

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Table 6. Absolute errors (${\bf{AE}}$) of the benchmark and the European call option apply to the CGMY, Meixner, and GH processes models with different values of $N$ and $S$ for the parameters in Table 2

		$SDDM $				$FDM $
	$S $	$40 $	50	60		$40 $	50	60
	$N$	${\bf{AE}}$	${\bf{AE}}$	${\bf{AE}}$	CPU	${\bf{AE}}$	${\bf{AE}}$	${\bf{AE}}$	CPU
CGMY	07	$1.88e^{-4}$	$1.76e^{-4}$	$1.76e^{-4}$	$0.20$	$1.24e^{-2}$	$5.81e^{-2}$	$8.98e^{-3}$	0.62
	10	$1.45e^{-5}$	$1.81e^{-5}$	$1.71e^{-5}$	$0.28$	$1.98e^{-3}$	$1.63e^{-2}$	$2.05e^{-3}$	0.73
	13	$2.91e^{-7}$	$2.33e^{-7}$	$2.25e^{-7}$	$0.33$	$5.99e^{-4}$	$4.29e^{-3}$	$5.66e^{-4}$	0.87
	16	$3.72e^{-10}$	$3.34e^{-10}$	$3.81e^{-10}$	$0.44$	$2.83e^{-4}$	$2.11e^{-3}$	$2.48e^{-4}$	1.41
Meixner	07	$2.45e^{-4}$	$3.32e^{-4}$	$3.22e^{-4}$	$0.24$	$1.56e^{-2}$	$4.28e^{-2}$	$8.88e^{-3}$	0.63
	10	$2.72e^{-5}$	$2.75e^{-5}$	$2.46e^{-5}$	$0.24$	$1.99e^{-3}$	$1.23e^{-2}$	$2.34e^{-3}$	0.75
	13	$3.54e^{-7}$	$3.28e^{-7}$	$3.69e^{-7}$	0.34	$6.25e^{-4}$	$4.05e^{-3}$	$5.32e^{-4}$	0.85
	16	$5.68e^{-10}$	$5.55e^{-10}$	$5.88e^{-10}$	$0.42$	$2.44e^{-4}$	$1.24e^{-3}$	$2.51e^{-4}$	1.42
GH processes	07	$3.12e^{-4}$	$3.02e^{-4}$	$3.45e^{-4}$	$0.51$	$1.25e^{-2}$	$5.71e^{-2}$	$6.97e^{-3}$	1.32
	10	$5.51e^{-5}$	$6.1e^{-5}$	$5.92e^{-5}$	$0.68$	$2.13e^{-3}$	$1.21e^{-2}$	$5.87e^{-2}$	1.63
	13	$7.11e^{-7}$	$7.25e^{-7}$	$7.33e^{-7}$	$0.82$	$5.11e^{-4}$	$4.14e^{-3}$	$6.22e^{-4}$	2.91
	16	$8.25e^{-10}$	$9.12e^{-10}$	$9.23e^{-10}$	$1.32$	$2.53e^{-4}$	$1.31e^{-3}$	$3.12e^{-4}$	3.48

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