Convergence, non-negativity and stability of a new Lobatto IIIC-Milstein method for a pricing option approach based on stochastic volatility model

Abstract

Recently, stochastic differential equation (SDE) has been used for many applications in option pricing models which satisfy the non-negativity. So, constructing new numerical method preserves non-negativity for solving SDE is very important. This paper investigates the numerical analyses; convergence, non-negativity and stability of the multi-step Milstein method for SDE. We derive the new general s-stage Milstein method; the Lobatto IIIC-Milstein method for nonlinear SDE and show that the numerical solution preserves non-negativity. Moreover, we prove the strong convergence order 1.0 of the numerical method. The unconditional stability results are proven for SDE. In order to get insight into the numerical analysis of the proposed method; the Black–Scholes model is considered to explain that the exact mean square stability region is totally contained in the numerical region (i.e. the numerical method is stochastically A-stable). In addition, the accuracy and computational cost are discussed. Finally, the Lobatto IIIC-Milstein method was compared with existing Milstein type methods, Monte-Carlo and finite difference methods to examine the efficiency of the proposed method to value the price.

Background of the Lobatto IIIC methods

The fundamental analysis contain convergence and stability for numerical methods for differential equations are provided in [8, 12, 16, 17, 33, 34]. The families of Runge–Kutta (RK) methods based on Lobatto quadrature formulas are one of several classes of fully implicit RK methods possessing good stability properties for ODE. The number III is usually found in the literature associated to Lobatto methods. Ehle [7] introduced Lobatto IIIA, IIIB, IIIC classes, and the general definition of the Lobatto IIIC methods. For more information about the fundamental properties of Lobatto methods, we recommend [16, 17].

The classes of s-stage Lobatto methods are given in [17]

$$\begin{aligned} Y_{ni}= & y_n + h \sum _{j=1}^s a_{ij} f(t_n+c_jh,Y_{nj}), i=1,2,\ldots ,s, \end{aligned}$$

(87)

$$\begin{aligned} y_{n+1}= & y_n + h \sum _{j=1}^s b_j f(t_n+c_jh,Y_{nj}), \end{aligned}$$

(88)

where the stage value s satisfies $s\ge 2$ and the coefficients $a_{ij},b_j$ and $c_j$ characterized the Lobatto methods. The s intermediate values $Y_{nj}$ for $j=1,\ldots ,s$ are called the internal stages and can be considered as approximations to the solution at $t_n+c_jh$. The main numerical approximation at $t_{n+1} = t_n+h$ is given by $y_{n+1}$. Lobatto methods are characterized by $c_1=0$ and $c_s=1$. For a fixed value of s the various families of Lobatto methods share the same coefficients $b_j$ and $c_j$. In addition, the coefficients $a_{ij}$ are vary depending on the classes of Lobatto methods. For Lobatto IIIC class, the $a_{ij}$ defined as

$$\begin{aligned} a_{i1}=b_1, \quad i=1,\ldots ,s, \end{aligned}$$

(89)

and determined the remaining $a_{ij}$ by $C(s-1)$. The coefficients of the Lobatto IIIC methods can be displayed by the Butcher-tabelau in Table 7.

Table 7 The Lobatto IIIC methods of orders 2 (left) and order 4 (right)

The stability properties of numerical methods for deterministic ODE are reported in [17]. In the following, we present the well-known results for Lobatto methods in a way that helps to motivate the SDE analysis.

Proposition 1

(see [17]) The s-stage Lobatto IIIC methods (87–88) applied to the scalar test equation

$$\begin{aligned} \mathrm {d}X(t)=\lambda X(t) \mathrm {d}t, \quad t>0, \quad X(0)=X_0\ne 0, \end{aligned}$$

(90)

where $\lambda \in \mathbb {C}$ is a constant, yields

$$\begin{aligned} y_{n+1}=R(\lambda , h) y_n, \end{aligned}$$

(91)

with

$$\begin{aligned} R(Z)=1+Zb^T(I-ZA)^{-1} \mathbf {1}, \end{aligned}$$

(92)

where $b^T=(b_1,\ldots ,b_s)$, $A=(a_{ij})^s_{i,j=1}$, $\mathbf {1}=(1,\ldots ,1)^T$, and I is the identity matrix. R(Z) is called the stability function of the numerical method, which can be written for implicit methods as rational function with numerator and demonstrator of degree $\le s$ as follows

$$\begin{aligned} R(Z)=\frac{P(Z)}{Q(Z)}, \quad deg P=k, \quad deg Q =j. \end{aligned}$$

(93)

Let $S_L$ is the stability domain for the Lobatto IIIC methods (87–88), Then the method with stability function (93) is A-stable if and only if $|R(iy)|\le 1$ for all real y and R(Z) is analytic for $Re Z<0$. In addition, using the definition of the method coefficients (89), and (Proposition 3.8 in [17]), we find that the method also is L-stable. Furthermore, the Lobatto IIIC methods are characterized by non-stiff order $(2s-2)$, not symmetric, algebraically stable, B-stable, and the stability function R(z) is given by $(s-2,s)$ Padé approximation to $e^z$. So, the the Lobatto IIIC methods (87–88) are described as excellent methods for stiff problems.