Results 1  10
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36
TimeChanged Lévy Processes and Option Pricing
, 2002
"... As is well known, the classic BlackScholes option pricing model assumes that returns follow Brownian motion. It is widely recognized that return processes differ from this benchmark in at least three important ways. First, asset prices jump, leading to nonnormal return innovations. Second, return ..."
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Cited by 89 (12 self)
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As is well known, the classic BlackScholes option pricing model assumes that returns follow Brownian motion. It is widely recognized that return processes differ from this benchmark in at least three important ways. First, asset prices jump, leading to nonnormal return innovations. Second, return volatilities vary stochastically over time. Third, returns and their volatilities are correlated, often negatively for equities. We propose that timechanged Lévy processes be used to simultaneously address these three facets of the underlying asset return process. We show that our framework encompasses almost all of the models proposed in the option pricing literature. Despite the generality of our approach, we show that it is straightforward to select and test a particular option pricing model through the use of characteristic function technology.
A Study towards a Unified Approach to the Joint Estimation of Objective and Risk Neutral Measures for the Purpose of Options Valuation
, 1999
"... The purpose of this paper is to bridge two strands of the literature, one pertaining to the objectiveorphysical measure used to model the underlying asset and the other pertaining to the riskneutral measure used to price derivatives. We propose a generic procedure using simultaneously the fundame ..."
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Cited by 74 (4 self)
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The purpose of this paper is to bridge two strands of the literature, one pertaining to the objectiveorphysical measure used to model the underlying asset and the other pertaining to the riskneutral measure used to price derivatives. We propose a generic procedure using simultaneously the fundamental price S t and a set of option contracts ### I it # i=1;m # where m # 1 and # I it is the BlackScholes implied volatility.We use Heston's #1993# 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 that the univariate approach only involving options by and large dominates. Abyproduct of this #nding is that we uncover a remarkably simple volatility extraction #lter based on a polynomial lag structure of implied volatilities. The bivariate approachinvolving both the fundamental and an option appears useful when the information from the cash market ...
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 ..."
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Cited by 72 (1 self)
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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 ..."
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Cited by 48 (3 self)
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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.
Stochastic Volatility, Smile & Asymptotics
, 1998
"... We consider the pricing and hedging problem for options on stocks whose volatility is a random process. Traditional approaches, such as that of Hull & White, have been successful in accounting for the much observed smile curve, and the success of a large class of such models in this respect is guara ..."
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Cited by 13 (9 self)
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We consider the pricing and hedging problem for options on stocks whose volatility is a random process. Traditional approaches, such as that of Hull & White, have been successful in accounting for the much observed smile curve, and the success of a large class of such models in this respect is guaranteed by a theorem of Renault & Touzi, for which we present a simplified proof. We also present new asymptotic formulas that describe the geometry of smile curves and can be used for interpolation of implied volatility data. Motivated by the robustness of the smile effect to specific modelling of the unobserved volatility process, we present a new approach to stochastic volatility modelling starting with the BlackScholes pricing PDE with a random volatility coefficient. We identify and exploit distinct time scales of fluctuation for the stock price and volatility processes yielding an asymptotic approximation that is a BlackScholes type price or hedging ratio plus a Gaussian random variable quantifying the risk from the uncertainty in the volatility. These lead us to translate volatility risk into pricing and hedging bands for the derivative securities, without needing to estimate the market's value of risk. For some special cases, we can give explicit formulas. We outline
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 ..."
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Cited by 12 (1 self)
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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
Risk neutral compatibility with option prices
, 2006
"... Abstract A common problem is to choose a “risk neutral ” measure in an incomplete market in asset pricing models. We show in this paper that in some circumstances it is possible to choose a unique “equivalent local martingale measure ” by completing the market with option prices. We do this by model ..."
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Cited by 9 (1 self)
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Abstract A common problem is to choose a “risk neutral ” measure in an incomplete market in asset pricing models. We show in this paper that in some circumstances it is possible to choose a unique “equivalent local martingale measure ” by completing the market with option prices. We do this by modeling the behavior of the stock price X, together with the behavior of the option prices for a relevant family of options which are (or can theoretically be) effectively traded. In doing so, we need to ensure a kind of ’‘compatibility ” between X and the prices of our options, and this poses some significant mathematical difficulties.
A comparison of qoptimal option prices in a stochastic volatility model with correlation. Oxford Financial Research Centre Preprint 2003MF02
, 2003
"... This paper investigates option prices in an incomplete stochastic volatility model with correlation. In a general setting, we prove an ordering result which says that prices for European options with convex payoffs are decreasing in the market price of volatility risk. As an example, and as our main ..."
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Cited by 5 (3 self)
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This paper investigates option prices in an incomplete stochastic volatility model with correlation. In a general setting, we prove an ordering result which says that prices for European options with convex payoffs are decreasing in the market price of volatility risk. As an example, and as our main motivation, we investigate option pricing under the class of qoptimal pricing measures. Using the ordering result, we prove comparison theorems between option prices under the minimal martingale, minimal entropy and varianceoptimal pricing measures. If the Sharpe ratio is deterministic, the comparison collapses to the well known result that option prices computed under these three pricing measures are the same. As a concrete example, we specialise to a variant of the Heston model for which the Sharpe ratio is increasing in volatility. For this example we are able to deduce option prices are decreasing in the parameter q. Numerical solution of the pricing pde corroborates the theory and shows the magnitude of the differences in option price due to varying q. Choice of q is shown to influence the level of the implied volatility smile for options of varying maturity. We would like to thank participants at the OxfordPrinceton Mathematical Finance Workshop,
Using Implied Volatility to Measure Uncertainty About Interest Rates.” Federal Reserve
 Bank of St. Louis Review, May/June
"... Option prices can be used to infer the level of uncertainty about future asset prices. The first two parts of this article explain such measures (implied volatility) and how they can differ from the market’s true expectation of uncertainty. The third then estimates the implied volatility of threemon ..."
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Cited by 4 (2 self)
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Option prices can be used to infer the level of uncertainty about future asset prices. The first two parts of this article explain such measures (implied volatility) and how they can differ from the market’s true expectation of uncertainty. The third then estimates the implied volatility of threemonth eurodollar interest rates from 1985 to 2001 and evaluates its ability to predict realized volatility. Implied volatility shows that uncertainty about shortterm interest rates has been falling for almost 20 years, as the levels of interest rates and inflation have fallen. And changes in implied volatility are usually coincident with major news about the stock market, the real economy, and monetary policy. Federal Reserve Bank of St. Louis Review, May/June 2005, 87(3), pp. 40725. Economists often use asset prices along with models of their determination to derive financial markets ’ expectations of events. For example, monetary economists use federal funds futures prices to measure expectations of interest rates (Krueger and Kuttner, 1995; Pakko and Wheelock, 1996). Similarly, a large literature on fixed and target zone exchange rates has used forward exchange rates to measure the credibility of exchange rate regimes or to predict their collapse (Svensson,
A new approach for option pricing under stochastic volatility
 Review of Derivatives Research
, 2007
"... Abstract We develop a new approach for pricing Europeanstyle contingent claims written on the time T spot price of an underlying asset whose volatility is stochastic. Like most of the stochastic volatility literature, we assume continuous dynamics for the price of the underlying asset. In contrast ..."
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Cited by 4 (1 self)
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Abstract We develop a new approach for pricing Europeanstyle contingent claims written on the time T spot price of an underlying asset whose volatility is stochastic. Like most of the stochastic volatility literature, we assume continuous dynamics for the price of the underlying asset. In contrast to most of the stochastic volatility literature, we do not directly model the dynamics of the instantaneous volatility. Instead, taking advantage of the recent rise of the variance swap market, we directly assume continuous dynamics for the time T variance swap rate. The initial value of this variance swap rate can either be directly observed, or inferred from option prices. We make no assumption concerning the real world drift of this process. We assume that the ratio of the volatility of the variance swap rate to the instantaneous volatility of the underlying asset just depends on the variance swap rate and on the variance swap maturity. Since this ratio is assumed to be independent of calendar time, we term this key assumption the stationary volatility ratio hypothesis (SVRH). The instantaneous volatility of the futures follows an unspecified stochastic process, so both the underlying futures price and the variance swap rate have unspecified stochastic volatility. Despite this, we show that the payoff to a pathindependent contingent claim can be perfectly replicated by dynamic trading in futures contracts and variance swaps of the same maturity. As a result, the contingent claim is uniquely valued relative to its underlying’s futures price and the assumed observable variance swap rate. In contrast to standard models of stochastic volatility, our approach does not require specifying the market price of volatility risk or observing the initial level of instantaneous volatility.