Finite difference scheme versus piecewise binomial lattice for interest rates under the skew CEV model

Abstract

Interest rates frequently exhibit regulated or controlled characteristics, for example, the prevailing zero interest rate policy, or the leading role of central banks in short rate markets. In order to capture the regulated dynamics of interest rates, we introduce the skew constant-elasticity-of-variance (skew CEV) model. We then propose two numerical approaches: an improved finite difference scheme and a piecewise binomial lattice to evaluate bonds and European/American bond options. Numerical simulations show that both of these two approaches are efficient and satisfactory, with the finite difference scheme being more superior.

Keywords:

• Skew CEV model
• Bond pricing
• Finite difference
• Binomial lattice
• Regulated market

Acknowledgments

We thank an anonymous referee for suggesting the comparison between the finite difference approach and the binomial tree lattice proposed in this article. All authors contributed equally to this work.

Disclosure statement

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

Notes

1 This is skew-extended version of the CKLS model (Chan et al. 1992).

2 Equation (20(20) ${\tilde{f}}_{2i,j}^{\mathrm{\Delta }t}=2{\tilde{f}}_{2i,j}^{\mathrm{\Delta }t}-{\tilde{f}}_{i,j}^{2\mathrm{\Delta }t},$(20) ) can be obtained as follows: By using Taylor's expansion, we have the following expression $f_{i+1,j}=f_{i,j}+\mathrm{\Delta }t{f}_t(i,j)+\frac{1}{2}\left(\mathrm{\Delta }t{)}^2{f}_{tt}\right(i,j)+\frac{1}{3!}(\mathrm{\Delta }t{)}^3\frac{{\mathrm{\partial }}^3}{\mathrm{\partial }{t}^3}f(i,j)+\cdots ,$ Then we have $f_t(i,j)=\frac{f_{i+1,j}-f_{i,j}}{\mathrm{\Delta }t}-\frac{1}{2}\mathrm{\Delta }t{f}_{tt}(i,j)-\frac{1}{3!}\left(\mathrm{\Delta }t{)}^2\frac{{\mathrm{\partial }}^3}{\mathrm{\partial }{t}^3}f\right(i,j)-\cdots .$ We write it in a simpler form: $f_t(i,j)=\frac{f_{i+1,j}-f_{i,j}}{\mathrm{\Delta }t}+a_1\mathrm{\Delta }t+a_2(\mathrm{\Delta }t{)}^2+a_3(\mathrm{\Delta }t{)}^3+\cdots ,$ Define $\varphi \left(\mathrm{\Delta }t\right)=\frac{f_{i+1,j}-f_{i,j}}{\mathrm{\Delta }t}$, then $\begin{array}{rl}& \varphi \left(\mathrm{\Delta }t\right)=f_t(i,j)-a_1\mathrm{\Delta }t-a_2(\mathrm{\Delta }t{)}^2-a_3(\mathrm{\Delta }t{)}^3-\cdots ,\\ & \varphi \left(2\mathrm{\Delta }t\right)=f_t(i,j)-a_1\cdot 2\mathrm{\Delta }t-a_2\cdot 4(\mathrm{\Delta }t{)}^2-a_3\cdot 8(\mathrm{\Delta }t{)}^3-\cdots .\end{array}$ Multiplying the first equation by 2 and subtracting it from the second equation yields, $f_t(i,j)=2\varphi \left(\mathrm{\Delta }t\right)-\varphi \left(2\mathrm{\Delta }t\right)-2a_2(\mathrm{\Delta }t{)}^2-6a_3(\mathrm{\Delta }t{)}^3-\cdots .$ Note that a second-order precision is achieved in time in this way.

3 If the down node $y_d=y_t-J\hat{\sigma }\left(y_t\right)\sqrt{\mathrm{\Delta }t}$ is less than $(1-p)\tilde{a}$, in other words, if $y_t$ jumps down and passes through the lower bound, the node $y_t$ would become the truncation line $y_{min}$ as in step 3.

4 Notably, we do not replace $(1-p)\tilde{a}$ (the lower bound of the domain of definition) with $(1-p)c$ for $\hat{\sigma }\left(y_t\right)$.

5 The skew CIR bond prices obtained from the spectral expansion method are computed using 35 roots.

6 The definition of notion of solutions can be found in Engelbert and Schmidt (1991).

Additional information

Funding

The project on which this publication is based has been carried out with funding provided by the National Natural Science Foundation of China [grant numbers 11701085,72001024]; the Alexander von Humboldt Foundation, under the programme financed by the German Federal Ministry of Education and Research entitled ‘German Research Chair’, [grant number 01DG15010]; Beijing Institute of Technology Research Fund Program for Young Scholars; and the Excellent Young Scholars Program in the University of International Business and Economics [grant number 19YQ14].