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Derivative asset analysis in models with leveldependent and stochastic volatility
 CWI QUARTERLY
, 1996
"... In this survey we discuss models with leveldependent and stochastic volatility from the viewpoint of derivative asset analysis. Both classes of models are generalisations of the classical BlackScholes model; they have been developed in an effort to build models that are flexible enough to cope wit ..."
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Cited by 56 (1 self)
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In this survey we discuss models with leveldependent and stochastic volatility from the viewpoint of derivative asset analysis. Both classes of models are generalisations of the classical BlackScholes model; they have been developed in an effort to build models that are flexible enough to cope with the known deficits of the classical BlackScholes model. We start by briefly recalling the standard theory for pricing and hedging derivatives in complete frictionless markets and the classical BlackScholes model. After a review of the known empirical contradictions to the classical BlackScholes model we consider models with leveldependent volatility. Most of this survey is devoted to derivative asset analysis in stochastic volatility models. We discuss several recent developments in the theory of derivative pricing under incompleteness in the context of stochastic volatility models and review analytical and numerical approaches to the actual computation of option values.
MeanReverting Stochastic Volatility
, 2000
"... We present derivative pricing and estimation tools for a class of stochastic volatility models that exploit the observed "bursty" or persistent nature of stock price volatility. An empirical analysis of highfrequency S&P 500 index data confirms that volatility reverts slowly to its me ..."
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Cited by 40 (10 self)
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We present derivative pricing and estimation tools for a class of stochastic volatility models that exploit the observed "bursty" or persistent nature of stock price volatility. An empirical analysis of highfrequency S&P 500 index data confirms that volatility reverts slowly to its mean in comparison to the tickbytick fluctuations of the index value, but it is fast meanreverting when looked at over the time scale of a derivative contract (many months). This motivates an asymptotic analysis of the partial differential equation satisfied by derivative prices, utilizing the distinction between these time scales. The analysis yields pricing and implied volatility formulas, and the latter is used to "fit the smile" from European index option prices. The theory identifies the important group parameters that are needed for the derivative pricing and hedging problem for Europeanstyle securities, namely the average volatility and the slope and intercept of the implied volatility line, plotted as a function of the logmoneynesstomaturityratio. The results considerably simplify the estimation procedure, and the data produces estimates
Financial Modelling and Option Theory with the Truncated Levy Process
, 1997
"... In recent studies the truncated Levy process (TLP) has been shown to be very promising for the modeling of financial dynamics. In contrast to the Levy process, the TLP has finite moments and can account for both the previously observed excess kurtosis at short timescales, along with the slow converg ..."
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Cited by 39 (0 self)
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In recent studies the truncated Levy process (TLP) has been shown to be very promising for the modeling of financial dynamics. In contrast to the Levy process, the TLP has finite moments and can account for both the previously observed excess kurtosis at short timescales, along with the slow convergence to Gaussian at longer timescales. I further test the truncated Levy paradigm using high frequency data from the Australian All Ordinaries share market index. I then consider, for the early Levy dominated regime, the issue of option hedging for two different hedging strategies that are in some sense optimal. These are compared with the usual delta hedging approach and found to differ significantly. I also derive the natural generalization of the BlackScholes option pricing formula when the underlying security is modeled by a geometric TLP. This generalization would not be possible without the truncation.
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 ..."
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Cited by 26 (3 self)
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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
Some Results on Quadratic Hedging with Insider Trading
"... We consider the hedging problem in an arbitragefree financial market, where there are two kinds of investors with di#erent levels of information about the future price evolution, described by two filtrations F and G = F #(G) where G is a given r.v. ..."
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Cited by 5 (0 self)
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We consider the hedging problem in an arbitragefree financial market, where there are two kinds of investors with di#erent levels of information about the future price evolution, described by two filtrations F and G = F #(G) where G is a given r.v.
Derivative Asset Analysis in Models with LevelDependent and Stochastic Volatility
, 1997
"... email: frey @ math.ethz.ch ..."