Convergence, non-negativity and stability of a new Lobatto IIIC-Milstein method for a pricing option approach based on s
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Convergence, non‑negativity and stability of a new Lobatto IIIC‑Milstein method for a pricing option approach based on stochastic volatility model Mahmoud A. Eissa1,2 · Qiang Ye1 Received: 4 February 2020 / Revised: 24 July 2020 / Accepted: 4 September 2020 © The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature 2020
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. Keywords Stochastic differential equations · Black–Scholes model · Milstein methods · Convergence · Non-negativity · Stability Mathematics Subject Classification 60H10 · 60H30 · 65C30
* Mahmoud A. Eissa [email protected]; [email protected] 1
School of Economics and Management, Harbin Institute of Technology, Harbin 150001, China
2
Department of Mathematics and Computer Science, Faculty of Science, Menoufia University, Shebin El‑Kom 32511, Egypt
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M. A. Eissa, Q. Ye
1 Introduction Stochastic differential equation (SDE) has widely used to simulate many real-world phenomena in many fields of science; such as biology, chemistry, physics and financial mathematics [10, 11, 43, 47–49]. Recently, there has been an increasing interest in considering SDE as a standard model for financing, especially in the derivative of interest rates and asset prices. This work is concerned with Itô SDE in order to simulate the financial diffusion process as follows
dX(t) = f (X(t))dt + g(X(t))dW(t), t ∈ [t0 , T],
(1)
where X(t0 ) = X0 , f(X(t)) is the drift coefficient and g(X(t)) is the diffusion coefficient and the Wiener process W(t) is defined on a given probability space (Ω, F, P) with a filtration {Ft }t≥0 which satisfies the usual conditions. In practice, numerical solutions of SDE have attracted a lot of attention because the analytic solutions are usually not available. Recently, there is increasing interest on deriving numerical methods pr
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