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281
Power and Bipower Variation with Stochastic Volatility and Jumps
 Journal of Financial Econometrics
, 2004
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A JumpDiffusion Model for Option Pricing
 Management Science
, 2002
"... Brownian motion and normal distribution have been widely used in the Black–Scholes optionpricing framework to model the return of assets. However, two puzzles emerge from many empirical investigations: the leptokurtic feature that the return distribution of assets may have a higher peak and two (as ..."
Abstract

Cited by 149 (5 self)
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Brownian motion and normal distribution have been widely used in the Black–Scholes optionpricing framework to model the return of assets. However, two puzzles emerge from many empirical investigations: the leptokurtic feature that the return distribution of assets may have a higher peak and two (asymmetric) heavier tails than those of the normal distribution, and an empirical phenomenon called “volatility smile ” in option markets. To incorporate both of them and to strike a balance between reality and tractability, this paper proposes, for the purpose of option pricing, a double exponential jumpdiffusion model. In particular, the model is simple enough to produce analytical solutions for a variety of optionpricing problems, including call and put options, interest rate derivatives, and pathdependent options. Equilibrium analysis and a psychological interpretation of the model are also presented.
Stochastic Volatility for Lévy Processes
, 2001
"... Three processes re°ecting persistence of volatility are initially formulated by evaluating three L¶evy processes at a time change given by the integral of a mean reverting square root process. The model for the mean reverting time change is then generalized to include NonGaussian models that are so ..."
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Cited by 141 (10 self)
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Three processes re°ecting persistence of volatility are initially formulated by evaluating three L¶evy processes at a time change given by the integral of a mean reverting square root process. The model for the mean reverting time change is then generalized to include NonGaussian models that are solutions to OU (OrnsteinUhlenbeck) equations driven by one sided discontinuous L¶evy processes permitting correlation with the stock. Positive stock price processes are obtained by exponentiating and mean correcting these processes, or alternatively by stochastically exponentiating these processes. The characteristic functions for the log price can be used to yield option prices via the fast Fourier transform. In general, mean corrected exponentiation performs better than employing the stochastic exponential. It is observed that the mean corrected exponential model is not a martingale in the ¯ltration in which it is originally de¯ned. This leads us to formulate and investigate the important property of martingale marginals where we seek martingales in altered ¯ltrations consistent with the one dimensional marginal distributions of the level of the process at each future date. 1
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 124 (20 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.
The Finite Moment Log Stable Process and Option Pricing
, 2002
"... We document a surprising pattern in market prices of S&P 500 index options. When implied volatilities are graphed against a standard measure of moneyness, the implied volatility smirk does not flatten out as maturity increases up to the observable horizon of two years. This behavior contrasts sh ..."
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Cited by 69 (12 self)
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We document a surprising pattern in market prices of S&P 500 index options. When implied volatilities are graphed against a standard measure of moneyness, the implied volatility smirk does not flatten out as maturity increases up to the observable horizon of two years. This behavior contrasts sharply with the implications of many pricing models and with the asymptotic behavior implied by the central limit theorem (CLT). We develop a parsimonious model which deliberately violates the CLT assumptions and thus captures the observed behavior of the volatility smirk over the maturity horizon. Calibration exercises demonstrate its superior performance against several widely used alternatives.
Pricing and Hedging in Incomplete Markets
 Journal of Financial Economics
, 2001
"... We present a new approach for positioning, pricing, and hedging in incomplete markets that bridges standard arbitrage pricing and expected utility maximization. Our approach for determining whether an investor should undertake a particular position involves specifying a set of probability measures a ..."
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Cited by 66 (8 self)
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We present a new approach for positioning, pricing, and hedging in incomplete markets that bridges standard arbitrage pricing and expected utility maximization. Our approach for determining whether an investor should undertake a particular position involves specifying a set of probability measures and associated °oors which expected payo®s must exceed in order for the investor to consider the hedged and ¯nanced investment to be acceptable. By assuming that the liquid assets are priced so that each portfolio of assets has negative expected return under at least one measure, we derive a counterpart to the ¯rst fundamental theorem of asset pricing. We also derive a counterPricing and Hedging in Incomplete Markets 2 part to the second fundamental theorem, which leads to unique derivative security pricing and hedging even though markets are incomplete. For products that are not spanned by the liquid assets of the economy, we show how our methodology provides more realistic bidask spreads.
Stock Return Characteristics, Skew Laws, and the Differential Pricing of Individual Equity Options
, 2001
"... This article provides several new insights into the economic sources of skewness. First, we document the differential pricing of individual equity options versus the market index, and relate it to variations in return skewness. Second, we show how risk aversion introduces skewness in the riskneutra ..."
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Cited by 66 (9 self)
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This article provides several new insights into the economic sources of skewness. First, we document the differential pricing of individual equity options versus the market index, and relate it to variations in return skewness. Second, we show how risk aversion introduces skewness in the riskneutral density. Third, we derive laws that decompose individual return skewness into a systematic component and an idiosyncratic component. Empirical analysis of OEX options and 30 stocks demonstrates that individual riskneutral distributions differ from that of the market index by being far less negatively skewed. This paper explains the presence and evolution of riskneutral skewness over time and in the crosssection of individual stocks.