Isogeometric analysis in option pricing

ABSTRACT

Isogeometric analysis is a recently developed computational approach that integrates finite element analysis directly into design described by non-uniform rational B-splines (NURBS). In this paper, we show that price surfaces that occur in option pricing can be easily described by NURBS surfaces. For a class of stochastic volatility models, we develop a methodology for solving corresponding pricing partial integro-differential equations numerically by isogeometric analysis tools and show that a very small number of space discretization steps can be used to obtain sufficiently accurate results. Presented solution by finite element method is especially useful for practitioners dealing with derivatives where closed-form solution is not available.

KEYWORDS:

• Isogeometric analysis
• option pricing
• NURBS
• finite element method
• stochastic volatility models

AMS CLASSIFICATIONS:

• 35R09
• 65M60
• 65D07
• 91G20
• 91G60

JEL CLASSIFICATIONS:

• C58
• C63
• G12

Acknowledgments

Computational resources were provided by the CESNET LM2015042 and the CERIT Scientific Cloud LM2015085, provided under the programme ‘Projects of Large Research, Development, and Innovations Infrastructure’.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Jan Pospíšil  http://orcid.org/0000-0002-4288-1614

Vladimír Švígler  http://orcid.org/0000-0003-0063-3564

Notes

1 The space $W_D^{1,2}(0,\overline{s})$ is the space of functions satisfying

• $f(\tau ,s)=h_D\left(\tau \right),s\in {\mathrm{\Gamma }}_D$ (Dirichlet boundary condition),

• $\parallel f\left(t\right){\parallel }_{L^2\left(\right[0,\overline{s}\left]\right)}^2+\parallel f_s\left(t\right){\parallel }_{L^2\left(\right[0,\overline{s}\left]\right)}^2<+\mathrm{\infty }$ where the derivative is considered in the weak sense.

2 Sobolev space of functions satisfying Dirichlet boundary condition in Equation (42(42) $\begin{array}{cc}{f}_{\tau }-& \\ -\frac{1}{2}v{s}^2{f}_{ss}-\rho q\left(v\right)\sqrt{v}s{f}_{sv}-\frac{1}{2}{q}^2\left(v\right)f_{vv}& \tau \in \left(0,T\right),\\ -(r-\lambda \beta )s{f}_s-p\left(v\right)f_v+rf-& s\in (0,\overline{s}),\\ -\lambda {\int }_0^{+\mathrm{\infty }}\left[f(\tau ,sy,v)-f(\tau ,s,v)\right]\varphi \left(y\right)\mathrm{d}y=0,& v\in (0,\overline{v}),\\ f(\tau ,s,v)=h_D(\tau ,s,v),& (s,v)\in {\mathrm{\Gamma }}_D,\\ \mathrm{\nabla }f(\tau ,s,v)\cdot \overrightarrow{n_P}=h_N(\tau ,s,v),& (s,v)\in {\mathrm{\Gamma }}_N,\\ f(0,s,v)=\phi (s,v).\end{array}$(42) ) in the sense of traces.

Additional information

Funding

This work was partially supported by the Grantová Agentura České Republiky (GACR) [grant number 14-11559S] Analysis of Fractional Stochastic Volatility Models and their Grid Implementation.