hp-FEM solutions for option price Bates’ model

Andrés Ávila Barrera (Universidad de La Frontera), Cecilia Rapimán

For valuating options, several stochastic models have been developed, where several assumptions on the market are imposed. For example, Black-Scholes’ model considers constant volatility and local small changes. To overcome these simplifications, Bates’ model includes stochastic volatility and jumps, which corresponds to the following system of stochastic differential equations \[\begin{cases} $$ dS_t = (\alpha-\frac{1}{2} Y_t)dt+ \sqrt{Y_t} dW_1+ dq$$,\\ $$ dY_t= \xi(\eta-Y_t) dt`+ \theta \sqrt{Y_t} dW_2$$ \end{cases}\] which can be reduced to a partial integro-differential equation on \(\Omega\times (0,T)=(0,S_0)\times (0,1)\times (0,T)\) \[\begin{aligned} \frac{\partial C}{\partial t}+ \left(r-q-\kappa \left( 1\right) \right)S \frac{\partial C }{\partial S}+\frac{1}{2}yS^{2}\frac{\partial ^{2}C}{\partial S^{2}}+\left[\xi \left( \eta -y\right)-\frac{1}{2}\left( \theta^{2}-\rho\theta y\right) \right]\frac{\partial C}{\partial y} \\ \frac{1}{2}\theta^{2}y\frac{\partial^{2}F}{\partial y^{2}}+\rho \theta y S \frac{\partial ^{2} C}{\partial y \partial S}+\lambda \int_{-\infty}^{\infty} C\left( S \exp \left( x\right),y,t\right)W\left(dx \right)=\left( r+\lambda \right)C.\end{aligned}\] with boundary conditions \(C(0,y,t)=0\), \(C(S_0,y,t)=S_0-K\) and final condition \(C(S,y,T)=(S-K)^+\). The conditions on \(y\) are undefined.

Based on Achdou & Tchou [1], Hilber et al. [5,6], and Reich et al. [9], we show the variational formulation and prove a Gårding type inequality. Also localization error is obtained. We base our numerical studies on Almendral & OOsterlee [2], Ballestra & Sgarra [3] and Miglio & Sgarra [8]. We propose that \(hp\)-FEM methods [7], as special method of singularly elliptic problems, can be used to improve unstabilities of the FEM methods detected in the simplification of the splitting. Some studies on the parameters on the effect of convective part over the diffusion part are also considered.

Keywords: Stochastic option pricing models, \(hp\)-finite element method, degenerate partial integro-differential equations

Mathematics Subject Classifications (2000):35K65, 65M15, 65M60, 65N30.

References

[1] Achdou, Y., Tchou, N. ESAIM: Mat. Mod. and Num. Analysis, 36/3, 373-395 (2002).

[2] Almendral, A., OOsterlee, C.W. Appl. Num. Math., 53, 1-18 (2005).

[3] Ballestra, L.V. & Sgarra, C. Comput. Math. Appl. 60 (2010), no. 6, 1571–1590.

[4] D. Bates Rev. Fin. Studies 9 (1996) 69–107.

[5] N. Hilber, Matache, A.-M., Scwab, C. J. Comp. Finance, 8/4, 1-42 (2005).

[6] N. Hilber, N. Reich, C. Schwab & C. Winter Finance Stoch (2009) 13: 471–500

[7] J. M. Melenk, Lecture Notes in Mathematics 1796, Springer, 2002.

[8] E. Miglio, E. & C. Sgarra A finite element discretization method for option pricing with the Bates model S\(\vec{{\rm e}}\)MA J. No. 55 (2011), 23–40.

[9] N. Reich, C. Schwab & C.Winter Finance Stoch (2010) 14: 527–567.