Monte Carlo analysis of methods for extracting risk‐neutral densities with affine jump diffusions

• Author: Shan Lu
• Date: 01 December 2019
• DOI: http://doi.org/10.1002/fut.22049
• Published date: 01 December 2019

J Futures Markets. 2019;39:1587–1612. wileyonlinelibrary.com/journal/fut © 2019 Wiley Periodicals, Inc.

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1587

Received: 1 February 2019

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Accepted: 31 July 2019

DOI: 10.1002/fut.22049

RESEARCH ARTICLE

Monte Carlo analysis of methods for extracting

risk‐neutral densities with affine jump diffusions

Shan Lu

School of Management, University of

Bradford, Bradford, UK

Correspondence

Shan Lu, School of Management,

University of Bradford, Emm Lane,

Bradford BD9 4JL, UK.

Email: s.lu4@bradford.ac.uk

Abstract

This article compares several widely used and recently developed methods to

extract risk‐neutral densities (RNDs) from option prices in terms of estimation

accuracy. It shows that the positive convolution approximation method

consistently yields the most accurate RND estimates, and is insensitive to the

discreteness of option prices. RND methods are less likely to produce accurate

RND estimates when the underlying process incorporates jumps and when

estimations are performed on sparse data, especially for short time‐to‐

maturities, though sensitivity to the discreteness of the data differs across

different methods.

KEYWORDS

affine jump diffusions, Monte Carlo simulation, risk‐neutral density

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INTRODUCTION

This article compares the performance of various estimation methods to extract risk‐neutral densities (RNDs) from

option prices. As the true RND is latent, a pseudoprice‐based methodology is used to evaluate the performance of the

RND methods in terms of estimation accuracy and consistency. The pseudo‐based method begins with an assumed

affine jump‐diffusion model and a set of estimated model parameters, the hypothetical affine jump‐diffusion model is

then used to generate artificial option data by using the model parameters; the “true”RND is obtained analytically

from the model according to the result of Breeden and Litzenberger (1978), whereas the RND estimate is extracted

by RND methods from the artificial option data. The performance of the RND methods is then tested by comparing the

“true”RND and the RND estimate through a goodness‐of‐fit test. The procedure is repeated 1,000 times to provide

the distributional information about the test statistic, which answers the question of how likely these RND

methods produce statistically accurate estimates in practice. The procedure is repeated for artificial option data with

different maturities.

The study differs from prior research on the comparison of RND methods in several aspects of methodology. First,

the assumed hypothetical underlying process for price and volatility takes both price and volatility jumps into

consideration, making the comparison more realistic than prior comparison studies, though no‐jump and single‐jump

models are also employed for comparison purposes. Most of the prior studies on the comparison of RND methods

assume either a single parametric form of the RND or a stochastic process of the underlying asset price and volatility.

For example, Söderlind (2000) starts with an assumed parametric form of the RND, and employs a Monte Carlo

simulation to fit option prices based on the assumed error distribution; Bu and Hadri (2007) and Santos and Guerra

(2015) start with the Heston model. Furthermore, in the case of assuming stochastic processes for the underlying asset

price and volatility, prior studies assume stochastic volatility processes that account for either no jumps or only price

jumps (Lai, 2014; Santos & Guerra, 2015), and no prior studies have employed the double‐jump model for the

comparison of RND methods. It should be noted that the inclusion of both price and volatility jumps in the hypothetical

underlying asset process is consistent with the findings in equity price dynamics and option pricing literature. For

example, Broadie, Chernov, and Johannes (2007) find strong evidence of the presence of both price and volatility jumps

in a time series of equity prices, and both jumps are important components for option pricing. Second, we adopt a noise

perturbation procedure proposed by Bondarenko (2003a) which is consistent with the options exchange’s regulation on

maximum bid–ask differentials to simulate market frictions, such as the bid–ask spread; whereas prior studies do not

consider the regulation on maximum bid–ask differentials when adding noises to the simulated option prices: For

example, Lai (2014) sets the price noise to a proportional white noise to the simulated option prices; Bliss and

Panigirtzoglou (2002) and Santos and Guerra (2015) add a uniformly distributed random noise, whose size is between

minus half and half of the option price tick size.

The eight RND methods compared in this study consist of both parametric and nonparametric, widely used and

recently developed methods, spanning a wide range of categories. These methods include: a mixture of two lognormals

(LN2; Ritchey, 1990), the Hermite polynomial with Gram–Charlier expansion (HPGC; Jondeau & Rockinger, 2001), the

generalized beta distribution of the second kind (GB2; Bookstaber & MacDonald, 1987), the generalized extreme value

distribution (GEV; Markose & Alentorn, 2011), curve‐fitting with quadratic polynomial (QP; Shimko, 1993), curve‐

fitting with cubic smoothing spline (CS; Bliss & Panigirtzoglou, 2004), positive convolution approximation (PCA;

Bondarenko, 2000, 2003a), and the spectral recovery method (SRM; Monnier, 2013).

The evaluation of estimation methods for RNDs is important for both market participants and policymakers, as

RNDs embedded in option prices have important financial and economic applications. Risk preferences of market

agents embedded in RNDs play important roles in the modeling and determination of insurance policies, pension plans,

and tax regulations. Higher moments of RNDs contain predictive content about stock returns and returns of option

portfolios (Bali & Murray, 2013). Policymakers use RND estimates to access the credibility of monetary policy (Bahra,

2007; Olijslagers, Petersen, de Vette, & van Wijnbergen, 2018), gauge market sentiment and access market beliefs about

economic and political events (e.g., Birru & Figlewski, 2012). Option traders over‐the‐counter rely on RND estimates to

price exotic options. In addition, RNDs are also used to test market rationality (Bondarenko, 2003b), access bankruptcy

probabilities of financial institutions (Taylor, Tzeng, & Widdicks, 2014), measure risk premiums (Ivanova & Gutiérrez,

2014), and study volatility pricing kernels (Völkert, 2015).

We find that first, PCA consistently yields the most accurate RND estimates and is insensitive to the discreteness of

option prices, whereas the LN2 method performs the worst. HPGC is the best performer among parametric methods;

HPGC, GB2, and GEV in the parametric domain outperforms the QP in the nonparametric domain. Second, RND

methods are less likely to produce accurate RND estimates when the underlying process incorporates jumps, especially

for short time‐to‐maturities. Third, the discreteness of option prices negatively affects most of the RND methods,

especially for the short time‐to‐maturities and when the generating process incorporates double jumps, though

sensitivity to the discreteness of the data differs across different methods. Finally, double‐jump models outperform

models with no‐jump or single‐jump models, reinforcing the finding in the equity price dynamics and option pricing

literature that both price and volatility jumps are important components for option pricing.

The rest of the article is organized as follows: Section 2 reviews the RND methods in the literature and briefly

introduces and discusses the methods compared in the article. Section 3 presents the affine jump‐diffusion models and

the model‐derived RND functions. Section 4 presents the data. The methodologies for the analysis of RND methods are

presented in Section 5. Section 6 presents the results. Section 7 concludes the article.

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LITERATURE REVIEW

2.1

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RND methods

Numerous methods to extract RNDs from option prices have been developed (see, Figlewski, 2018; Jackwerth, 1999,

2004, for a review). There are mainly two strands of methods: parametric and nonparametric methods. On the one

hand, parametric methods fit the RND to a parametric form of selected densities; they often estimate the parameters by

minimizing the sum of pricing errors. Nonparametric methods, on the other hand, directly estimate the RND from

linear/nonlinear segments or by pointwise fitting. Compared to parametric methods which assume a functional form of

distributions, nonparametric methods are data‐driven and more flexible.

There are three major categories within parametric methods: the mixture methods, the expansion methods, and

the generalized distribution methods. Mixture methods use the weighted averages of several simple distributions

to add flexibility to the estimated probability distribution. A typical example is the mixture of LN2 distributions

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