Discrete Dynamics in Nature and Society

Volume 2018, Article ID 3093708, 9 pages

https://doi.org/10.1155/2018/3093708

## Reconstruction of the Time-Dependent Volatility Function Using the Black–Scholes Model

^{1}Department of Mathematics, Jilin Institute of Chemical Technology, Jilin 132022, China^{2}Department of Mathematics, Korea University, Seoul 02841, Republic of Korea^{3}Department of Financial Engineering, Korea University, Seoul 02841, Republic of Korea^{4}Major in Mathematics Education, Hankuk University of Foreign Studies, Seoul 02450, Republic of Korea^{5}Department of Mathematics, Kangwon National University, Gangwon-do 24341, Republic of Korea

Correspondence should be addressed to Darae Jeong; rk.ca.aerok@oyoyanit

Received 19 December 2017; Accepted 1 April 2018; Published 7 May 2018

Academic Editor: Francisco R. Villatoro

Copyright © 2018 Yuzi Jin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We propose a simple and robust numerical algorithm to estimate a time-dependent volatility function from a set of market observations, using the Black–Scholes (BS) model. We employ a fully implicit finite difference method to solve the BS equation numerically. To define the time-dependent volatility function, we define a cost function that is the sum of the squared errors between the market values and the theoretical values obtained by the BS model using the time-dependent volatility function. To minimize the cost function, we employ the steepest descent method. However, in general, volatility functions for minimizing the cost function are nonunique. To resolve this problem, we propose a predictor-corrector technique. As the first step, we construct the volatility function as a constant. Then, in the next step, our algorithm follows the prediction step and correction step at half-backward time level. The constructed volatility function is continuous and piecewise linear with respect to the time variable. We demonstrate the ability of the proposed algorithm to reconstruct time-dependent volatility functions using manufactured volatility functions. We also present some numerical results for real market data using the proposed volatility function reconstruction algorithm.

#### 1. Introduction

The accurate calibration of models using market option data is one of the most important problems in finance [1]. The reason is for accurate pricing and accordingly hedging strategy [2]. According to [3], the authors proposed the accurate computations for Greeks using the numerical solutions of the Black–Scholes partial differential equation. The well-known standard Black–Scholes (BS) model is not adequate for calibrating the market option data because it uses the constant volatility [4–6]. As an alternative to the BS equation with constant volatility [7], local volatility models were introduced to explain the volatility smiles or skews observed in the market. There has been much research carried out regarding the reconstruction of local volatility functions from market data [8, 9]. For example, in [10], radial basis functions were used to construct local volatility surfaces. In [1], the authors proposed a regularized optimization formulation, using spline kernels to ensure both accuracy and stability in the local volatility function calibration. In [9], the authors mentioned the calibration of a local volatility surface for European options using a nonparametric approach by employing a second-order Tikhonov regularization.

The main goal of this study is to propose a new simple and robust numerical method for the construction of a time-dependent volatility function, using the BS partial differential equation with nonconstant volatility [11, 12]:for , where is the option value of the underlying price and time , is the volatility function of time , and is the riskless interest rate. The final condition is the payoff function at expiry . Typically, the local volatility function is given as a function of an underlying asset and time, that is, [13, 14]. However, for simplicity and robustness of the solution algorithm, we assume that the local volatility function only depends on the time and is a piecewise linear function with respect to the time variable.

The outline of this paper is as follows. In Section 2, we describe our numerical algorithm for constructing the time-dependent volatility function. In Section 3, numerical experiments are presented. Finally, conclusions are drawn in Section 4.

#### 2. Numerical Algorithm

In this section, we present a new numerical algorithm for constructing the time-varying volatility function.

##### 2.1. Numerical Solution

Let be the value of the underlying asset price and be the time to expiry, then (1) can be given as the following initial-value problem:for with an initial condition , where the infinite domain is truncated to a finite computational domain [15]. Now, to solve (2) numerically, we apply a finite difference method (FDM). Let us denote the numerical approximation of the solution of (2) by , for and . Here, and are uniform spatial and temporal step sizes, respectively. is the number of grid points and is the number of time steps. Figure 1 illustrates the uniform grid with a spatial step size . Furthermore, the variable volatility is defined similarly as .