Efficient Option Pricing in Crisis Based on Dynamic Elasticity of Variance Model

Discrete Dynamics in Nature and Society, Mar 2016

Market crashes often appear in daily trading activities and such instantaneous occurring events would affect the stock prices greatly. In an unstable market, the volatility of financial assets changes sharply, which leads to the fact that classical option pricing models with constant volatility coefficient, even stochastic volatility term, are not accurate. To overcome this problem, in this paper we put forward a dynamic elasticity of variance (DEV) model by extending the classical constant elasticity of variance (CEV) model. Further, the partial differential equation (PDE) for the prices of European call option is derived by using risk neutral pricing principle and the numerical solution of the PDE is calculated by the Crank-Nicolson scheme. In addition, Kalman filtering method is employed to estimate the volatility term of our model. Our main finding is that the prices of European call option under our model are more accurate than those calculated by Black-Scholes model and CEV model in financial crashes.

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Efficient Option Pricing in Crisis Based on Dynamic Elasticity of Variance Model

Efficient Option Pricing in Crisis Based on Dynamic Elasticity of Variance Model Congyin Fan, Kaili Xiang, and Peimin Chen School of Economics and Mathematics, Southwestern University of Finance and Economics, Chengdu, Sichuan 611130, China Received 11 December 2015; Accepted 8 March 2016 Academic Editor: Leonid Shaikhet Copyright © 2016 Congyin Fan 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 Market crashes often appear in daily trading activities and such instantaneous occurring events would affect the stock prices greatly. In an unstable market, the volatility of financial assets changes sharply, which leads to the fact that classical option pricing models with constant volatility coefficient, even stochastic volatility term, are not accurate. To overcome this problem, in this paper we put forward a dynamic elasticity of variance (DEV) model by extending the classical constant elasticity of variance (CEV) model. Further, the partial differential equation (PDE) for the prices of European call option is derived by using risk neutral pricing principle and the numerical solution of the PDE is calculated by the Crank-Nicolson scheme. In addition, Kalman filtering method is employed to estimate the volatility term of our model. Our main finding is that the prices of European call option under our model are more accurate than those calculated by Black-Scholes model and CEV model in financial crashes. 1. Introduction Nowadays, more and more researchers focus on stock option pricing problems. In different environments, such as bull markets or bear markets, the returns of stock prices have different properties and distributions to follow, based on which many different models (see [1–7]) are proposed and some analytic formulae or approximations are provided. For some classical models without analytic solutions, lots of efficient numerical methods (see [8–14]) are presented in detail. In addition, by employing some existing model and numerical methods, some researches (see [15, 16]) focus on the empirical tests on actual data, and some great findings are observed in the real derivative markets. In recent two decades, the events of financial crisis, such as the stock market crash in 1987 and the subprime crisis in 2008, show that extreme events have great effects on financial markets and cannot be negligible in option pricing problems. So, more and more researches focus on the dynamic markets that experience sharp crashes, and lots of contributions on option pricing models are presented. For instance, from an empirical view the dynamic financial markets following the occurrence of a financial crash have been studied in [17, 18]. In 2003, Sornette in [19] finds that the postcrash stock prices follow a converging oscillatory motion through a nonparameter method. In addition, Lillo and Mantenga in [18] show that ex-postfinancial markets of crisis have characteristics of a power-law relaxation decay. By using results of [19], El-Khatib et al. in [20] study European option pricing model with postcrash relaxation times in 2007. In [7], Zhu and Galbraith present an evidence that stock returns can be fitted by Student’s -distribution during postcrash relaxation times, which is consistent with the results in [17, 18]. In 2011, Markose and Alentorn in [6] employ the generalized extreme value distribution to model the implied risk neutral density function and provide a flexible framework that captures the negative skewness and excess kurtosis of returns in turbulent financial markets. If markets are in heavy crisis, the volatility of financial assets changes sharply and is much bigger than that in bull markets. To grasp unstable volatility, many kinds of stochastic volatility models, such as Heston model, Hull-White model, GARCH model, and CEV model, are proposed and option pricing formula under risk neutral measure is given or approximated. In these models, the volatility terms are constructed by parametric processes, such as Cox-Ingersoll-Ross (CIR) process or Ornstein-Uhlenbeck (OU) process, and so forth, of which the parameters are assumed to be constants. Further, Yoon et al. in [21, 22] study problems of option pricing under stochastic elasticity of variance model and their works enhance the existing option price structures in view of flexibility and applicability through the market prices of elasticity risk. For these processes, there exist some certain distributions they followed. But, in the real stock markets, crashes have different shapes and properties without any fixed rules to obey. Moreover, lots of extreme values of returns accumulate on tails of return distributions. These facts lead to the point that it is impossible to use a certain distribution or process to describe the information hidden in volatility term. In (...truncated)


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Congyin Fan, Kaili Xiang, Peimin Chen. Efficient Option Pricing in Crisis Based on Dynamic Elasticity of Variance Model, Discrete Dynamics in Nature and Society, 2016, 2016, DOI: 10.1155/2016/7496539