Results 1  10
of
105
The Dynamics of Stochastic Volatility: Evidence from Underlying and Option Markets
, 2000
"... This paper proposes and estimates a more general parametric stochastic variance model of equity index returns than has been previously considered using data from both underlying and options markets. The parameters of the model under both the objective and riskneutral measures are estimated simultane ..."
Abstract

Cited by 81 (2 self)
 Add to MetaCart
This paper proposes and estimates a more general parametric stochastic variance model of equity index returns than has been previously considered using data from both underlying and options markets. The parameters of the model under both the objective and riskneutral measures are estimated simultaneously. I conclude that the square root stochastic variance model of Heston (1993) and others is incapable of generating realistic returns behavior and find that the data are more accurately represented by a stochastic variance model in the CEV class or a model that allows the price and variance processes to have a timevarying correlation. Specifically, I find that as the level of market variance increases, the volatility of market variance increases rapidly and the correlation between the price and variance processes becomes substantially more negative. The heightened heteroskedasticity in market variance that results generates realistic crash probabilities and dynamics and causes returns to display values of skewness and kurtosis much more consistent with their sample values. While the model dramatically improves the fit of options prices relative to the square root process, it falls short of explaining the implied volatility smile for shortdated options.
Maximum likelihood estimation for stochastic volatility models
 JOURNAL OF FINANCIAL ECONOMICS
, 2007
"... We develop and implement a method for maximum likelihood estimation in closedform of stochastic volatility models. Using Monte Carlo simulations, we compare a full likelihood procedure, where an option price is inverted into the unobservable volatility state, to an approximate likelihood procedure ..."
Abstract

Cited by 51 (3 self)
 Add to MetaCart
We develop and implement a method for maximum likelihood estimation in closedform of stochastic volatility models. Using Monte Carlo simulations, we compare a full likelihood procedure, where an option price is inverted into the unobservable volatility state, to an approximate likelihood procedure where the volatility state is replaced by proxies based on the implied volatility of a shortdated atthemoney option. The approximation results in a small loss of accuracy relative to the standard errors due to sampling noise. We apply this method to market prices of index options for several stochastic volatility models, and compare the characteristics of the estimated models. The evidence for a general CEV model, which nests both the affine Heston model and a GARCH model, suggests that the elasticity of variance of volatility lies between that assumed by the two nested models.
Option Pricing by Transform Methods: Extensions, Unification, and Error Control
 Journal of Computational Finance
"... We extend and unify Fourieranalytic methods for pricing a wide class of options on any underlying state variable whose characteristic function is known. In this general setting, we bound the numerical pricing error of discretized transform computations, such as DFT/FFT. These bounds enable algorith ..."
Abstract

Cited by 36 (3 self)
 Add to MetaCart
We extend and unify Fourieranalytic methods for pricing a wide class of options on any underlying state variable whose characteristic function is known. In this general setting, we bound the numerical pricing error of discretized transform computations, such as DFT/FFT. These bounds enable algorithms to select efficient quadrature parameters and to price with guaranteed numerical accuracy.
Multiscale stochastic volatility asymptotics
 SIAM J. MULTISCALE MODELING AND SIMULATION
, 2003
"... In this paper we propose to use a combination of regular and singular perturbations to analyze parabolic PDEs that arise in the context of pricing options when the volatility is a stochastic process that varies on several characteristic time scales. The classical BlackScholes formula gives the pri ..."
Abstract

Cited by 31 (13 self)
 Add to MetaCart
In this paper we propose to use a combination of regular and singular perturbations to analyze parabolic PDEs that arise in the context of pricing options when the volatility is a stochastic process that varies on several characteristic time scales. The classical BlackScholes formula gives the price of call options when the underlying is a geometric Brownian motion with a constant volatility. The underlying might be the price of a stock or an index say and a constant volatility corresponds to a fixed standard deviation for the random fluctuations in the returns of the underlying. Modern market phenomena makes it important to analyze the situation when this volatility is not fixed but rather is heterogeneous and varies with time. In previous work, see for instance [5], we considered the situation when the volatility is fast mean reverting. Using a singular perturbation expansion we derived an approximation for option prices. We also provided a calibration method using observed option prices as represented by the socalled term structure of implied volatility. Our analysis of market data, however, shows the need for introducing also a slowly varying factor in the model for the stochastic volatility. The combination of regular and singular perturbations approach that we set forth in this paper deals with this case. The resulting approximation is still independent of the particular details of the volatility model and gives more flexibility in the parametrization of the
The Econometrics of Option Pricing
"... The growth of the option pricing literature parallels the spectacular developments of derivative securities and the rapid expansion of markets for derivatives in the last three decades. Writing a survey of option pricing models appears therefore like a formidable task. To delimit our focus we will ..."
Abstract

Cited by 13 (1 self)
 Add to MetaCart
The growth of the option pricing literature parallels the spectacular developments of derivative securities and the rapid expansion of markets for derivatives in the last three decades. Writing a survey of option pricing models appears therefore like a formidable task. To delimit our focus we will put emphasis on the more recent contributions since there are
Pricing options on scalar diffusions: an eigenfunction expansion approach
 Management Science
"... This paper develops an eigenfunction expansion approach to pricing options on scalar diffusion processes. All derivative securities are unbundled into portfolios of primitive securities termed eigensecurities. Eigensecurities are eigenvectors of the pricing operator (present value operator). Pricing ..."
Abstract

Cited by 11 (5 self)
 Add to MetaCart
This paper develops an eigenfunction expansion approach to pricing options on scalar diffusion processes. All derivative securities are unbundled into portfolios of primitive securities termed eigensecurities. Eigensecurities are eigenvectors of the pricing operator (present value operator). Pricing is then immediate by the linearity property of the pricing operator and the eigenvector property of eigensecurities. To illustrate the computational power of the method, we develop two applications: pricing vanilla, single and doublebarrier options under the constant elasticity of variance (CEV) process and interest rate knockout options in the CoxIngersollRoss (CIR) termstructure model.
Equivalent and absolutely continuous measure changes for jumpdiffusion processes” to appear in the Annals of Applied Probability
"... We provide explicit sufficient conditions for absolute continuity and equivalence between the distributions of two jumpdiffusion processes that can explode and be killed by a potential. ..."
Abstract

Cited by 11 (2 self)
 Add to MetaCart
We provide explicit sufficient conditions for absolute continuity and equivalence between the distributions of two jumpdiffusion processes that can explode and be killed by a potential.
IMPLIED AND LOCAL VOLATILITIES UNDER STOCHASTIC VOLATILITY
, 2001
"... For asset prices that follow stochasticvolatility diffusions, we use asymptotic methods to investigate the behavior of the local volatilities and Black–Scholes volatilities implied by option prices, and to relate this behavior to the parameters of the stochastic volatility process. We also give app ..."
Abstract

Cited by 11 (3 self)
 Add to MetaCart
For asset prices that follow stochasticvolatility diffusions, we use asymptotic methods to investigate the behavior of the local volatilities and Black–Scholes volatilities implied by option prices, and to relate this behavior to the parameters of the stochastic volatility process. We also give applications, including riskpremiumbased explanations of the biases in some naïve pricing and hedging schemes. We begin by reviewing option pricing under stochastic volatility and representing option prices and local volatilities in terms of expectations. In the case that fluctuations in price and volatility have zero correlation, the expectations formula shows that local volatility (like implied volatility) as a function of logmoneyness has the shape of a symmetric smile. In the case of nonzero correlation, we extend Sircar and Papanicolaou’s asymptotic expansion of implied volatilities under slowlyvarying stochastic volatility. An asymptotic expansion of local volatilities then verifies the rule of thumb that local volatility has the shape of a skew with roughly twice the slope of the implied volatility skew. Also we compare the slowvariation asymptotics against what we call smallvariation asymptotics, and against Fouque, Papanicolaou, and Sircar’s rapidvariation
LinearQuadratic JumpDiffusion Modeling with Application to Stochastic Volatility
, 2004
"... We aim at accommodating the existing affine jumpdiffusion and quadratic models under the same roof, namely the linearquadratic jumpdiffusion (LQJD) class. We give a complete characterization of the dynamics of this class of models by stating explicitly a list of structural constraints, and comput ..."
Abstract

Cited by 10 (0 self)
 Add to MetaCart
We aim at accommodating the existing affine jumpdiffusion and quadratic models under the same roof, namely the linearquadratic jumpdiffusion (LQJD) class. We give a complete characterization of the dynamics of this class of models by stating explicitly a list of structural constraints, and compute standard and extended transforms relevant to asset pricing. We show that the LQJD class can be embedded into the affine class through use of an augmented state vector, and further establish that a onetoone equivalence relationship holds between both classes in terms of transform analysis. An option pricing application to multifactor stochastic volatility models reveals that adding nonlinearity into the model would reduce pricing errors and yield parameter estimates that are more in line with sensible economic interpretation.
BUBBLES, CONVEXITY AND THE BLACK–SCHOLES EQUATION
, 908
"... A bubble is characterized by the presence of an underlying asset whose discounted price process is a strict local martingale under the pricing measure. In such markets, many standard results from option pricing theory do not hold, and in this paper we address some of these issues. In particular, we ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
A bubble is characterized by the presence of an underlying asset whose discounted price process is a strict local martingale under the pricing measure. In such markets, many standard results from option pricing theory do not hold, and in this paper we address some of these issues. In particular, we derive existence and uniqueness results for the Black–Scholes equation, and we provide convexity theory for option pricing and derive related ordering results with respect to volatility. We show that American options are convexity preserving, whereas European options preserve concavity for general payoffs and convexity only for bounded contracts. 1. Introduction. Recently