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Journal ArticleDOI

American options with stochastic dividends and volatility: A nonparametric investigation

01 Jan 2000-Journal of Econometrics (Elsevier BV)-Vol. 94, Iss: 1, pp 53-92

Abstract: In this paper, we consider American option contracts when the underlying asset has stochastic dividends and stochastic volatility. We provide a full discussion of the theoretical foundations of American option valuation and exercise boundaries. We show how they depend on the various sources of uncertainty which drive dividend rates and volatility, and derive equilibrium asset prices, derivative prices and optimal exercise boundaries in a general equilibrium model. The theoretical models identify the relevant factors underlying option prices but yield fairly complex expressions which are difficult to estimate. We therefore adopt a nonparametric approach in order to investigate the reduced forms suggested by the theory. Indeed, we use nonparametric methods to estimate call prices and exercise boundaries conditional on dividends and volatility. Since the latter is a latent process, we propose several approaches, notably using EGARCH filtered estimates, implied and historical volatilities. The nonparametric approach allows us to test whether call prices and exercise decisions are primarily driven by dividends, as has been advocated by Harvey and Whaley (1992a. Journal of Financial Economics 30, 33–73; 1992b. Journal of Futures Markets 12, 123–137) and Fleming and Whaley (1994. Journal of Finance 49, 215–236) for the OEX contract, or whether stochastic volatility complements dividend uncertainty. We find that dividends alone do not account for all aspects of option pricing and exercise decisions, suggesting a need to include stochastic volatility.
Topics: Implied volatility (68%), Volatility smile (68%), Stochastic volatility (66%), Volatility swap (63%), Valuation of options (58%)

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Série Scientifique
Scientific Series
Montréal
Octobre 1996
96s-26
American Options with Stochastic
Dividends and Volatility: A
Nonparametric Investigation
Mark Broadie, Jérôme Detemple,
Eric Ghysels, Olivier Torrès

Ce document est publié dans lintention de rendre accessibles les résultats préliminaires de la
recherche effectuée au CIRANO, afin de susciter des échanges et des suggestions. Les idées et les
opinions émises sont sous lunique responsabilité des auteurs, et ne représentent pas nécessairement
les positions du CIRANO ou de ses partenaires.
This paper presents preliminary research carried out at CIRANO and aims to encourage
discussion and comment. The observations and viewpoints expressed are the sole responsibility
of the authors. They do not necessarily represent positions of CIRANO or its partners.
CIRANO
L e CIRA NO e st une corporation privée à but non lucratif constituée en vertu de la Loi des
compagnies du Québec. Le financement de son infrastructure et de ses activités de recherche
pro vient des cotisations de ses organisations-membres, dune subvention dinfrastructure du
ministère de lIndustrie, du Commerce, de la Science et de la Technologie, de même que des
sub venti o ns et mandats obtenus par ses équipes de recherche. La
Série Scientifique
est la
réal isati on dune des missions que sest données le CIRANO, soit de développer lanalyse
scientifique des organisations et des comportements stratégiques.
CIRANO is a private non-profit organization incorporated under the Québec Companies Act.
Its infrastructure and research activities are funded through fees paid by member
organizations, an infrastructure grant from the Ministère de lIndustrie, du Commerce, de la
Science e t de l a Technologie, and grants and research mandates obtained by its research
teams. The
Scientific Series
fulfils one of the missions of CIRANO: to develop the scientific
analysis of organizations and strategic behaviour.
Les organisations-partenaires / The Partner Organizations
École des Hautes Études Commerciales.
École Polytechnique.
McGill University.
Université de Montréal.
Université du Québec à Montréal.
Université Laval.
MEQ.
MICST.
Avenor.
Banque Nationale du Canada.
Bell Québec.
Fédération des caisses populaires de Montréal et de lOuest-du-Québec.
Hydro-Québec.
La Caisse de dépôt et de placement du Québec.
Raymond, Chabot, Martin, Paré.
Société délectrolyse et de chimie Alcan Ltée.
Téléglobe Canada.
Ville de Montréal.
ISSN 1198-8177

Correspondence Address: Eric Ghysels, CIRANO, 2020 University Street, 25th floor, Montréal, Qc,
*
Canada H3A 2A5 Tel: (514) 985-4025 Fax: (514) 985-4039 e-mail: ghyselse@cirano.umontreal.ca
We would like to thank Tim Hankes from the OCC and Tom Finucane from Syracuse University for
supplying us OEX exercise data. We are equally grateful to Cam Harvey for providing us dividend series
and to Jeff Flemming for the implied volatility data. Early versions of this paper were presented at
CIRANO, the Fields Institute, Ohio State University, UC San Diego, the LIFE/METEOR Third Workshop
on Financial Modelling and Econometric Analysis in Maastricht, the Cornell-Queens Derivative Securities
Conference, the Journées de lOptimisation in Montréal, the University of Aarhus, McGill University, the
FFA meetings in Geneva, the CIFO conference in Montréal, the Workshop on Neural Networks in Montréal
and ESEM96 in Istanbul. We thank the participants for helpful comments and suggestions. Part of this
work was funded by the Social Sciences and Humanities Research Council of Canada under Strategic Grant
804-96-0027 and the TMR Work Programme of the European Commission under grant
Nr ERB4001GT950641.
Columbia University
McGill University and CIRANO
Pennsylvania State University and CIRANO
§
C.O.R.E., Université Catholique de Louvain
American Options with Stochastic
Dividends and Volatility: A
Nonparametric Investigation
*
Mark Broadie , Jérôme Detemple , Eric Ghysels ,
§
Olivier Torrès
Résumé / Abstract
Cet article examine les contrats optionnels de type américain lorsque
lactif sous-jacent paie des dividendes et a une volatilité stochastiques. Nous
présentons une discussion complète des fondations théoriques de lévaluation des
options américaines et de leurs frontières dexercice. Nous démontrons leur
dépendance par rapport aux diverses sources dincertitude qui déterminent le taux
de dividendes et la volatilité, et dérivons les prix déquilibre des actifs, titres dérivés
ainsi que les politiques optimales dexercice dans un modèle déquilibre général.
Les modèles théoriques conduisent à des expressions complexes qui sont difficiles
à estimer. Cest pourquoi nous adoptons une approche non-paramétrique qui permet
dexaminer des formes réduites. Nous utilisons des méthodes non-paramétriques
pour estimer les prix doptions à lachat et les frontières dexercice conditionnelles
aux dividendes et à la volatilité. Puisque cette dernière est un processus latent nous
proposons plusieurs approches, fondées en particulier sur des estimateurs-filtres
EGARCH, des volatilités implicites et historiques. Lapproche non-paramétrique
nous permet de tester si les prix doptions et les décisions dexercice sont
principalement déterminés par les dividendes, comme suggéré par Harvey et

Whaley (1992a, b) et Fleming et Whaley (1994) pour le contrat OEX, ou si la
volatilité stochastique complémente lincertitude sur les dividendes. Nous
établissons que les dividendes seuls ne rendent pas compte de tous les aspects de
lévaluation de ces options et des décisions dexercice, ce qui suggère la nécessité
dinclure la volatilité stochastique.
In this paper, we consider American option contracts when the
underlying asset has stochastic dividends and stochastic volatility. We provide
a full discussion of the theoretical foundations of American option valuation and
exercise boundaries. We show how they depend on the various sources of
uncertainty which drive dividend rates and volatility, and derive equilibrium
asset prices, derivative prices and optimal exercise boundaries in a general
equilibrium model. The theoretical models yield fairly complex expressions
which are difficult to estimate. We therefore adopt a nonparametric approach
which enables us to investigate reduced forms. Indeed, we use nonparametric
methods to estimate call prices and exercise boundaries conditional on dividends
and volatility. Since the latter is a latent process, we propose several
approaches, notably using EGARCH filtered estimates, implied and historical
volatilities. The nonparametric approach allows us to test whether call prices
and exercise decisions are primarily driven by dividends, as has been advocated
by Harvey and Whaley (1992a,b) and Fleming and Whaley (1994) for the OEX
contract, or whether stochastic volatility complements dividend uncertainty. We
find that dividends alone do not account for all aspects of call option pricing and
exercise decisions, suggesting a need to include stochastic volatility.
Mots Clés
:
Prix doptions, titres dérivés, contrat OEX, estimation par méthode
de noyau
Keywords :
Option Pricing, Derivative Securities, OEX Contract, Kernel
Estimation
JEL : C14, C51, D52, G13

1 Intro duction
The early exercise feature of American option contracts considerably
complicates their valuation. Even the relatively simple case of an un-
derlying asset with a Geometric Brownian Motion (GBM) price pro cess
and constant dividend rate requires numerical algorithms to value the
option and determine the optimal exercise policy.
1
Two critical assumptions, namely (1) a constant dividend rate and
(2) constantvolatility, are often cited as restrictive and counter-factual.
For the OEX contract, the most widely traded American-type option
written on the S&P100 Sto ck Index, Harvey and Whaley (1992a, b) and
Fleming and Whaley (1994) underline the importance of the amount
and the timing of dividends. To account for discrete dividend payments
on the S&P100 index portfolio they use a mo dication of the Cox, Ross
and Rubinstein (1979) binomial metho d which reduces the index level by
the discounted ow of dividends during the lifetime of the option. Using
this approach they show that ignoring dividends has a signicant impact
on pricing errors. The computations are already so demanding in this
simple case with constant volatility that the stochastic volatility case
appears to be beyond reach. It is interesting to note that for Europ ean-
type options, like the SPX contract on the S&P500 Sto ck Index, there
has been far more interest in studying the sto chastic volatility case.
2
One may therefore wonder whether it is either sto chastic volatility, or
stochastic dividends,
or both
, which determine American as well as Eu-
ropean options. The purpose of our pap er is to address this question.
We study the case of American options as it is considerably more di-
cult than the European-type contract. The approachwe take, however,
1
A whole range of numerical pro cedures have b een prop osed, including nite dif-
ferences, binomial, multinomial, quasi-analytical, quadratic metho ds as well as the
method of lines and Richardson extrap olations. A partial list of contributions in-
cludes Brennan and Schwartz (1977), Cox, Ross and Rubinstein (1979), Geske (1979),
Whaley (1981), Geske and Johnson (1984), Barone-Adesi and Whaley (1987), Boyle
(1988), Breen (1991), Yu (1993), Broadie and Detemple (1996) and Carr and Faguet
(1994), among others. For a review of these procedures, see Broadie and Detemple
(1996).
2
See for instance Hull and White (1987), Johnson and Shanno (1987), Scott (1987),
Wiggins (1987), Chesney and Scott (1989), Stein and Stein (1991), Heston (1993),
among others.
1

Citations
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09 May 2001
Abstract: This article presents a simple yet powerful new approach for approximating the value of American options by simulation. The key to this approach is the use of least squares to estimate the conditional expected payoff to the optionholder from continuation. This makes this approach readily applicable in path-dependent and multifactor situations where traditional finite difference techniqes cannot be used. We illustrate this technique with several realistic examples including valuing an option when the underlying asset follows a jump-diffusion process and valuing an American swaption in a 20-factor string model of the term structure.

2,602 citations


Journal ArticleDOI
Abstract: This article presents a simple yet powerful new approach for approximating the value of American options by simulation. The key to this approach is the use of least squares to estimate the conditional expected payoff to the optionholder from continuation. This makes this approach readily applicable in path-dependent and multifactor situations where traditional finite difference techniques cannot be used. We illustrate this technique with several realistic examples including valuing an option when the underlying asset follows a jump-diffusion process and valuing an American swaption in a 20-factor string model of the term structure.

2,409 citations


Journal ArticleDOI
Mikhail Chernov1, Eric Ghysels2, Eric Ghysels3Institutions (3)
Abstract: The purpose of this paper is to bridge two strands of the literature, one pertaining to the objective or physical measure used to model an underlying asset and the other pertaining to the risk-neutral measure used to price derivatives. We propose a generic procedure using simultaneously the fundamental price, St, and a set of option contracts [(σitI)i=1,m] where m⩾1 and σitI is the Black–Scholes implied volatility. We use Heston's (1993. Review of Financial Studies 6, 327–343) model as an example, and appraise univariate and multivariate estimation of the model in terms of pricing and hedging performance. Our results, based on the S&P 500 index contract, show dominance of univariate approach, which relies solely on options data. A by-product of this finding is that we uncover a remarkably simple volatility extraction filter based on a polynomial lag structure of implied volatilities. The bivariate approach, involving both the fundamental security and an option contract, appears useful when the information from the cash market reflected in the conditional kurtosis provides support to price long term.

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Abstract: Traditional models of portfolio choice assume that investors can continuously trade unlimited amounts of securities. In reality, investors face liquidity constraints. I analyze a model where investors are restricted to trading strategies that are of bounded variation. An investor facing this type of illiquidity behaves very differently from an unconstrained investor. A liquidity-constrained investor endogenously acts as if facing borrowing and short-selling constraints, and one may take riskier positions than in liquid markets. I solve for the shadow cost of illiquidity and show that large price discounts can be sustained in a rational model. Article published by Oxford University Press on behalf of the Society for Financial Studies in its journal, The Review of Financial Studies.

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Abstract: Do long-term and short-term options contain differential information? If so, can long-term options better differentiate among alternative models? To answer these questions, we first demonstrate analytically that differences among alternative models usually may not surface when applied to short-term options, but do so when applied to long-term contracts. Using S&P 500 options and LEAPS, we find that short- and long-term contracts indeed contain different information. While the data suggest little gains from modeling stochastic interest rates or random jumps (beyond stochastic volatility) for pricing LEAPS, incorporating stochastic interest rates can nonetheless enhance hedging performance in certain cases involving long-term contracts.

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8,616 citations


Journal ArticleDOI
Steven L. Heston1Institutions (1)
Abstract: I use a new technique to derive a closed-form solution for the price of a European call option on an asset with stochastic volatility. The model allows arbitrary correlation between volatility and spotasset returns. I introduce stochastic interest rates and show how to apply the model to bond options and foreign currency options. Simulations show that correlation between volatility and the spot asset’s price is important for explaining return skewness and strike-price biases in the BlackScholes (1973) model. The solution technique is based on characteristi c functions and can be applied to other problems.

7,295 citations


"American options with stochastic di..." refers background in this paper

  • ...2See for instance Hull and White (1987), Johnson and Shanno (1987), Scott (1987), Wiggins (1987), Chesney and Scott (1989), Stein and Stein (1991), Heston (1993), among others....

    [...]

  • ...We noted in the Introduction that models often encountered in the literature on European options feature stochastic volatility, see Hull and White (1987), Johnson and Shanno (1987), Scott (1987), Wiggins (1987), Chesney and Scott (1989), Stein and Stein (1991), Heston (1993), among others....

    [...]


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