Estimating Option Pricing Models Using a Characteristic Function-Based Linear State Space Representation H. Peter Boswijk

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Estimating Option Pricing Models Using a Characteristic

Function-Based Linear State Space Representation∗

H. Peter Boswijk

Amsterdam School of Economics

University of Amsterdam

and Tinbergen Institute

Roger J. A. Laeven

Amsterdam School of Economics

University of Amsterdam, EURANDOM

and CentER

Evgenii Vladimirov

Amsterdam School of Economics

University of Amsterdam

and Tinbergen Institute

October 13, 2022

Abstract

We develop a novel ﬁltering and estimation procedure for parametric option pricing

models driven by general aﬃne jump-diﬀusions. Our procedure is based on the comparison

between an option-implied, model-free representation of the conditional log-characteristic

function and the model-implied conditional log-characteristic function, which is functionally

aﬃne in the model’s state vector. We formally derive an associated linear state space

representation and establish the asymptotic properties of the corresponding measurement

errors. The state space representation allows us to use a suitably modiﬁed Kalman ﬁltering

technique to learn about the latent state vector and a quasi-maximum likelihood estimator

of the model parameters, which brings important computational advantages. We analyze

the ﬁnite-sample behavior of our procedure in Monte Carlo simulations. The applicability of

our procedure is illustrated in two case studies that analyze S&P 500 option prices and the

impact of exogenous state variables capturing Covid-19 reproduction and economic policy

uncertainty.

Keywords: Options; Characteristic Function; Aﬃne Jump-Diﬀusion; State Space Representation.

JEL Classiﬁcation: Primary: C13; C58; G13; Secondary: C32; G01.

∗We are very grateful to Torben Andersen, Kris Jacobs, Frank Kleibergen, Siem Jan Koopman, Olivier Scaillet, George

Tauchen, Viktor Todorov, Fabio Trojani, and conference and seminar participants at the 2021 SoFiE Financial Econo-

metrics Summer School at Northwestern University, the 2022 Quantitative Finance and Financial Econometrics (QFFE)

Conference at Aix-Marseille University, the 2022 Annual SoFiE Conference at the University of Cambridge, the 2022 Dyn-

stoch meeting at the Institut Henri Poincar´e in Paris, the 74th European Meeting of the Econometric Society (ESEM) at

Bocconi University in Milan, the University of Amsterdam, Kellogg School of Management at Northwestern University, the

Center for Econometrics and Business Analytics at St. Petersburg State University, and the Tinbergen Institute for helpful

comments and suggestions. Julia code to implement the estimation procedure developed in this paper is available from

https://github.com/evladimirov/OptionModels-cKF-ccf. This research was funded in part by the Netherlands Organi-

zation for Scientiﬁc Research (NWO) under grant NWO-Vici 2019/2020 (Laeven). Email addresses: H.P.Boswijk@uva.nl,

R.J.A.Laeven@uva.nl, and E.Vladimirov@uva.nl.

arXiv:2210.06217v1 [econ.EM] 12 Oct 2022

1 Introduction

Over the past decades, explosive growth in the trading of option contracts has attracted the

attention of academics and practitioners to the development and estimation of increasingly

sophisticated option pricing models. The building blocks of many continuous-time option pricing

models are semimartingale stochastic processes that govern the dynamics of the underlying asset.

These processes are often latent with stochastic diﬀusive volatility as the prototypical example,

as in the classical Heston (1993) model. The literature also suggests the need to allow for a

discontinuous jump component, both in the asset price dynamics and in its volatility process,

potentially with a time-varying stochastic jump intensity.

An important econometric challenge lies in estimating the parameters of these continuous-

time models and in ﬁltering their unobserved and time-varying components, since option prices

are highly nonlinear functions of the state vector. This stands in contrast to, for instance,

term structure models, where bond yields can be represented as a linear function of the states,

at least within the aﬃne framework (see, e.g., Piazzesi, 2010, for a review of the aﬃne term

structure literature). To evaluate option prices as a function of the state vector, one typically

needs to apply either Fourier-based methods or simulation-based approaches, in both cases at a

substantial computational cost. This is one of the reasons why in much of the empirical research

on option pricing, only a subset of the available option price data is used, such as at-the-money

contracts or weekly (typically Wednesday) options data.

In this paper, we develop a new latent state ﬁltering and parameter estimation procedure

for option pricing models governed by general aﬃne jump-diﬀusion processes. Our procedure

leverages the linear relationship between the logarithm of an option-implied, model-free span-

ning formula for the conditional characteristic function of the underlying asset return on the

one hand, and the state vector induced by parametric model speciﬁcation on the other hand.

From this relationship, we formally derive a linear state space representation, and establish the

asymptotic properties of the corresponding measurement errors. Linearity of the measurement

and state updating equations that make up the state space representation, with coeﬃcient and

variance matrices that are (semi-)closed-form functions of the parameters, allows us to exploit

Kalman ﬁltering techniques. The proposed estimation procedure is fast and easy to implement,

circumventing the typical computational burden in conducting inference on option pricing mod-

els.

Exploiting the option-spanning formula of Carr and Madan (2001) for European-style payoﬀ

functions, we replicate the risk-neutral conditional characteristic function (CCF) of the under-

lying log-asset price at the expiration date in a completely model-independent way. In other

words, we imply information about the CCF from the option prices without imposing any

parametric assumptions on the underlying asset price dynamics. A similar option-spanning ap-

proach for the CCF is used by Todorov (2019) to develop an option-based nonparametric spot

volatility estimator. On the other hand, a large stream of literature is devoted to parametric

option pricing models belonging to the general aﬃne jump-diﬀusion (AJD) family; canonical

examples are Heston (1993), Duﬃe, Pan, and Singleton (2000), Pan (2002), and Bates (2006).1

The deﬁning property of the AJD class is the exponential-aﬃne joint CCF, which is available in

semi-closed form. By comparing the two option pricing representations—model-free and model-

implied—we can obtain a linear relation between the logarithm of the option-implied CCF and

the model-dependent CCF within the aﬃne framework.

The state vector in AJD option pricing models typically contains both observable processes

and latent factors. We address the ﬁltering of such latent factors by developing a linear state

space representation for this model class. The development includes an asymptotic analysis

of the measurement error components, consisting of observation, truncation and discretization

errors, under a double asymptotic scheme in the moneyness dimension. The state space repre-

sentation allows us to employ suitably modiﬁed Kalman ﬁltering techniques to learn about the

unobserved intrinsic components of the model and estimate the model parameters using quasi-

maximum likelihood (QML). QML approaches based on Kalman ﬁltering are often used in the

aﬃne term structure literature, where the yields themselves are linear functions of the state

vector (see, e.g., Duﬀee, 1999, de Jong, 2000, Driessen, 2005). Besides the possibility to exploit

Kalman ﬁltering and QML estimation techniques, another advantage of our approach is that,

once the model-free CCF has been obtained from the data, no further numerical option pricing

methods, such as the FFT approach of Carr and Madan (1999) or simulation-based methods,

are needed. Therefore, our method reduces computational costs considerably relative to many

existing approaches in the option pricing literature. We note that, whereas the parametric CCF

is used to price options in Fourier-based methods, here we use the CCF to directly learn about

the latent factors and model parameters.

We analyze the developed estimation procedure in Monte Carlo simulations based on several

AJD speciﬁcations. We consider a one-factor AJD option pricing model, with the stochastic

volatility and jump intensity both being aﬃne functions of a single latent process, and a two-

factor AJD model speciﬁcation with an observable exogenous factor. We ﬁnd good ﬁnite-sample

performance in both cases, notwithstanding the challenging nature of the econometric problem.

Finally, we illustrate our new ﬁltering and estimation approach in an empirical application

to S&P 500 index options. In particular, we ﬁlter and estimate the latent volatility and jump

intensity from a stochastic volatility model with co-jumps in returns and volatility. We also

investigate the impact of the Covid-19 propagation rate on the stock market within this model,

by embedding the associated reproduction number as an exogenous factor into the volatility

and jump intensity dynamics. Our results show that while the reproduction number has only

a mild eﬀect on total diﬀusive volatility, it contributes substantially to the likelihood of jumps.

By contrast, when we consider an Economic Policy Uncertainty index as exogenous factor, the

jump intensity process is not aﬀected, but the exogenous factor contributes signiﬁcantly to

diﬀusive volatility.

Various estimation and ﬁltering strategies for option pricing models have been developed in

the literature. These include the (penalized) nonlinear least squares methods in, for instance,

1See also, e.g., Broadie, Chernov, and Johannes (2007), A¨ıt-Sahalia, Cacho-Diaz, and Laeven (2015), Andersen,

Fusari, and Todorov (2017) and the references therein.

Bakshi, Cao, and Chen (1997), Broadie et al. (2007), Andersen, Fusari, and Todorov (2015a); the

eﬃcient method of moments of Gallant and Tauchen (1996) as applied in Chernov and Ghysels

(2000) and Andersen, Benzoni, and Lund (2002); the implied-state methods initiated by Pan

(2002) and further analyzed by Santa-Clara and Yan (2010); the Markov Chain Monte Carlo

method in Eraker (2004) and Eraker, Johannes, and Polson (2003); and the particle ﬁltering

method, see Johannes, Polson, and Stroud (2009) and Bardgett, Gourier, and Leippold (2019).

Most of these estimation methods use as inputs option prices or a monotonic transformation

thereof, such as implied volatilities. By contrast, we propose an estimation procedure based on

the prices of spanning option portfolios that by the bijection between CCFs and conditional

distributions, in principle, contain all probabilistic information about the stochastic process

governing the dynamics of the underlying asset.

In general, estimation strategies based on the transform space of conditional characteristic

functions are, of course, not new to the literature. For instance, Carrasco and Florens (2000)

develop a generalized method of moments (GMM) estimator with a continuum of moment

conditions based on the CCF; see also Singleton (2001), Carrasco, Chernov, Florens, and Ghysels

(2007). In applications to option prices, Boswijk, Laeven, and Lalu (2015) and Boswijk, Laeven,

Lalu, and Vladimirov (2021) propose to imply the latent state vector from a panel of options

and then estimate the model via GMM with a continuum of moments. Bates (2006) develops

maximum likelihood estimation and ﬁltering using CCFs. In particular, he proposes a recursive

likelihood evaluation by updating the CCF of a latent variable conditional upon observed data.

However, unlike our approach, these methods require numerical integration over the dimension

of the state vector, thus suﬀering from a ‘curse of dimensionality’.

Our work is also related to Feunou and Okou (2018), who exploit the linear relation between

the ﬁrst four risk-neutral cumulants of the log-asset price and latent factors. They obtain these

cumulants via a portfolio of options and employ the Kalman ﬁlter to estimate the latent factors.

The main diﬀerence with our approach is that we exploit the CCF, and the corresponding

state space representation we develop, instead of the ﬁrst four moments. The CCF contains

much richer information, leading to more eﬃcient inference. Another diﬀerence is in dimension

reduction: Feunou and Okou (2018) use a two-step principal components analysis (PCA) to

reduce the dimension of the risk-neutral cumulants observed at diﬀerent maturities. Instead,

we use a modiﬁed version of a so-called collapsed Kalman ﬁltering approach, originally developed

by Jungbacker and Koopman (2015), which does not suﬀer from information losses relative to

the full-dimensional setting.

The paper is organized as follows. Section 2 provides the theoretical framework for aligning

the option-implied and model-implied CCFs. In Section 3, we develop the state space represen-

tation, and establish the main result about the orders of measurement errors, under a double

asymptotic scheme. This allows us to next develop the ﬁltering approach and corresponding

estimation procedure. Section 4 presents the Monte Carlo simulation results. We describe the

data in Section 5 and the empirical applications in Section 6. Conclusions are in Section 7. In

supplementary material, four appendices provide details on (i) the proof of Proposition 1, (ii)

the computation of conditional moments, (iii) the inter- and extrapolation scheme for option

prices and the measurement errors in the CCF replication, and (iv) additional simulation and

empirical results.

2 Theoretical Framework

In this section, we provide the theoretical framework for our approach. We start with extracting

information about the CCF from option prices allowing for general underlying dynamics. Next,

we consider the CCF within the AJD class, which is exponentially aﬃne in the model’s state

variables. Finally, we discuss how to align the two CCFs—option-implied model-free and AJD

model-implied—in order to conduct inference about the model parameters and the latent state

variables.

2.1 Option-implied CCF

Throughout the paper, we ﬁx a ﬁltered probability space (Ω,F,{Ft}t≥0,P). On this proba-

bility space, we consider the dynamics of an arbitrage-free ﬁnancial market. The no-arbitrage

assumption guarantees the existence of a risk-neutral probability measure Q, locally equivalent

to P. Since we are interested in exploiting information from options, we formulate the model

dynamics under Q.

Let us denote by Ftthe futures price at time tfor a stock or an index futures contract

with some ﬁxed maturity. The absence of arbitrage implies that the futures price process is a

semimartingale. In this subsection, we assume the following general dynamics for Ftunder Q:

dFt

=vtdWt+ZR

x˜µ(dt, dx), F0>0,(1)

where vtis an adapted, locally bounded, but otherwise unspeciﬁed stochastic volatility process;

Wtis a standard Brownian motion; µis a counting random measure with compensator νt(dx)dt

such that ˜µ(dt, dx) := µ(dt, dx)−νt(dx)dtis the associated martingale measure and R(x2∧

1)νt(dx)<∞.

We further denote out-of-the-money (OTM) European-style option prices at time twith

time-to-maturity τ > 0 and strike price K > 0 by Ot(τ, K). Under the no-arbitrage assumption,

the option prices equal the risk-neutral conditional expectations of the corresponding discounted

payoﬀ functions:

Ot(τ, K) = 





EQ[e−rτ (Ft+τ−K)+|Ft],if K > Ft,

EQ[e−rτ (K−Ft+τ)+|Ft],if K≤Ft.

The OTM price Ot(τ, K) is a call option price if K > Ftand a put option price if K≤Ft. For

simplicity, we assume a constant interest rate r.

Following Carr and Madan (2001), any twice continuously diﬀerentiable European-style

payoﬀ function g(Ft+τ), with ﬁrst and second derivatives gFand gF F , can be spanned via

a position in risk-free bonds, futures (or stocks) and options with a continuum of strikes, as

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EstimatingOptionPricingModelsUsingaCharacteristicFunction-BasedLinearStateSpaceRepresentation*H.PeterBoswijkAmsterdamSchoolofEconomicsUniversityofAmsterdamandTinbergenInstituteRogerJ.A.LaevenAmsterdamSchoolofEconomicsUniversityofAmsterdam,EURANDOMandCentEREvgeniiVladimirovAmsterdamSchoolofEconomicsU...

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