Good jump, bad jump, and option valuation

• Date: 01 September 2018
• Author: Xinglin Yang
• Published date: 01 September 2018
• DOI: http://doi.org/10.1002/fut.21929

Received: 23 September 2017

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Revised: 9 April 2018

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Accepted: 17 April 2018

DOI: 10.1002/fut.21929

RESEARCH ARTICLE

Good jump, bad jump, and option valuation

Xinglin Yang

Institute of Chinese Financial Studies,

Southwestern University of Finance and

Economics, Chengdu, China

Correspondence

Xinglin Yang, Institute of Chinese

Financial Studies, Southwestern

University of Finance and Economics,

555 Liutai Avenue, 611130 Chengdu,

Sichuan, China.

Email: xinglinyang@126.com

Funding information

National Natural Science Foundation of

China, Grant/Award Number: 71473200

I develop a new class of closed‐form option pricing models that incorporate

variance risk premium and symmetric or asymmetric double exponential jump

diffusion. These models decompose the jump component into upward and

downward jumps using two independent exponential distributions and thus

capture the impact of good and bad news on asset returns and option prices.

The empirical results show that the model with an asymmetric double

exponential jump diffusion improves the fit on Shanghai Stock Exchange 50ETF

returns and options and provides relatively better in‐and out‐of‐sample pricing

performance.

KEYWORDS

downward jump, GARCH, option valuation, upward jump, variance‐dependent pricing kernel

JEL CLASSIFICATION

G13

1

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INTRODUCTION

To adequately function in the market for options, it is important that option valuation is built on a reliable pricing

formula. Studies of option valuations have expanded considerably since the classic Black and Scholes (1973) model was

proposed. Considering that the return process consists of a linear drift, a Brownian motion, and a compound Poisson

process, Merton (1976) models the underlying asset price of options with a jump‐diffusion process in which jump size

follows the normal distribution. Bates (2000) and Pan (2002) develop option pricing models under the theoretical

framework of stochastic volatility and time‐varying jump intensity. Christoffersen, Feunou, and Jeon (2015) build a

class of discrete‐time option valuation models that are driven by generalized autoregressive conditional heteroskedastic

(GARCH) type dynamic volatility and jump intensity. All of these empirical results suggest that jump and dynamic

jump intensity can offer substantial benefits for asset returns fitting and option prices valuation.

In the real market, the upward (downward) jumps in prices are usually caused by good (bad) news. However,

once jump size is simply assumed to follow the normal distribution, the jump component has a potential limitation

because the type of jumps cannot be distinguished. Motivated by overcoming this limitation, Kou (2002) and Kou

and Wang (2004) use two independent exponential distributions generating the up and down jump magnitudes to

price options. Their empirical findings indicate that asymmetric exponential jump‐diffusion models can

significantly improve evaluations of European and American options. Also, the empirical results of Ramezani

and Zeng (2007) suggest that the asymmetric double exponential jump‐diffusion model performs better than the

normal jump‐diffusion model for fitting both indexes and individual stocks. Here, the upward and downward

jumps follow the binomial distribution with parameters specified by probability.

Overwhelming empirical evidenceconfirmsthatmodelingtime‐varying volatility and clustering i s critically

important in modeling returns and pricing options. The jump‐GARCH models were recently proposed to capture

J Futures Markets. 2018;38:1097–1125. wileyonlinelibrary.com/journal/fut © 2018 Wiley Periodicals, Inc.

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the time‐varying volatility and dynamic jump intensity in asset returns, following the path‐breaking work of Engle

(1982) and Bollerslev (1986). In addition, the discrete‐time GARCH‐type models are relatively straightforward to

implement and less computationally demanding. This paper combines GARCH dynamic variance and jump

intensity to fit asset returns and to evaluate option prices based on the particle filter technique.

The empirical findings of Babaoglu, Christoffersen, Heston, and Jacobs (2016) and Byun, Jeon, Min, and Yoon (2015)

address the importance of a negative variance risk premium for explaining option prices considering the stylized fact that

option‐implied risk‐neutralvarianceexceedsphysicalvarianceonaverage.Accordingly,themodelsinthispaper

incorporate variance risk premium to capture the wedge between physical and risk‐neutral variance. However, the jump‐

GARCH models in Byun et al. (2015) are not affine, making the inference challenging. Ornthanalai (2014) presents a

closed‐form option pricing formula, but the dynamics of variance and jump intensity do not allow for jumps. It has been

shown in the index option literature that jumps in volatility are useful for explaining option volatility smiles and smirks

(e.g.,seeBroadie,Chernov,&Johannes,2007; Christoffersen, Jacobs, & Ornthanalai, 2012; Eraker, 2004; Eraker, Johannes,

& Polson, 2003). To incorporate jump variance and intensity, Christoffersen, Jacobs, and Li (2016) assume that the time‐

varying process for variance and intensity are driven by thesamedynamicswhentheyevaluatetheoptionsoncrudeoil

futures. In contrast, I use a more general specification in which variance and intensity are governed by separate processes,

and in which the jump type includes a class of asymmetric double exponential jump diffusion.

I estimate six models using daily 50ETF returns and options. Under the physical measure, the estimation results on

50ETF returns show that time‐varying volatility and jump intensity are significant and that asymmetrical double

exponential jump diffusion effectively captures the properties of asset returns. Calibrating models directly to option

prices, I identify the market risk price of normal and jump components, which are difficult to estimate precisely using

the information of returns (Byun et al., 2015; Christoffersen et al., 2012). Furthermore, a negative and significant

variance risk premium is identified across all of the models based on the option’s information. To further analyze the

performance of option valuation, I test the pricing performance of these models based on in‐sample mispricing and out‐

of‐sample forecastability. All of the findings suggest that the asymmetrical double exponential jump‐diffusion model

should be the default model of choice in asset pricing models.

The remainder of the paper proceeds as follows: Section 2 introduces the asymmetric double exponential

jump‐diffusion models and benchmark models; Section 3 presents the empirical results from estimating daily returns;

Section 4 illustrates option valuation theory; Section 5 estimates models on options and analyzes pricing performance;

and Section 6 provides the summary and conclusion.

2

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THE KOU MODEL

2.1

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The return process

The most general asymmetrical double exponential jump diffusion model is henceforth referred to as the Kou model.

The asset dynamics under the physical measure are specified by

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YANG