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81
Pricing And Hedging Derivative Securities In Markets With Uncertain Volatilities
 Applied Mathematical Finance
, 1995
"... We present a model for pricing and hedging derivative securities and option portfolios in an environment where the volatility is not known precisely, but is assumed instead to lie between two extreme values oe min and oe max . These bounds could be inferred from extreme values of the implied volatil ..."
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Cited by 109 (3 self)
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We present a model for pricing and hedging derivative securities and option portfolios in an environment where the volatility is not known precisely, but is assumed instead to lie between two extreme values oe min and oe max . These bounds could be inferred from extreme values of the implied volatilities of liquid options, or from highlow peaks in historical stock or optionimplied volatilities. They can be viewed as defining a confidence interval for future volatility values. We show that the extremal nonarbitrageable prices for the derivative asset which arise as the volatility paths vary in such a band can be described by a nonlinear PDE, which we call the BlackScholesBarenblatt equation. In this equation, the "pricing" volatility is selected dynamically from the two extreme values oe min ,oe max , according to the convexity of the valuefunction. A simple algorithm for solving the equation by finitedifferencing or a trinomial tree is presented. We show that this model capture...
All in the Family: Nesting Symmetric and Asymmetric GARCH Models
 Journal of Financial Economics
, 1995
"... This paper develops a parametric family of models of generalized autoregressive heteroskedasticity (GARCH). The family nests the most popular symmetric and asymmetric GARCH models, thereby highlighting the relation between the models and their treatment of asymmetry. Furthermore, the structure perm ..."
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Cited by 90 (0 self)
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This paper develops a parametric family of models of generalized autoregressive heteroskedasticity (GARCH). The family nests the most popular symmetric and asymmetric GARCH models, thereby highlighting the relation between the models and their treatment of asymmetry. Furthermore, the structure permits nested tests of different ypes of asymmetry and functional forms. Daily U.S. stock return data reject all standard GARCH models in favor of a model in which, roughly speaking, the conditional standard deviation depends on the shifted absolute value of the shocks raised to the power three halves and past standard deviations.
Estimating security price derivatives using simulation
 Management Science
, 1996
"... Simulation has proved to be a valuable tool for estimating security prices for which simple closed form solutions do not exist. In this paper we present two direct methods, a pathwise method and a likelihood ratio method, for estimating derivatives of security prices using simulation. With the direc ..."
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Cited by 75 (3 self)
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Simulation has proved to be a valuable tool for estimating security prices for which simple closed form solutions do not exist. In this paper we present two direct methods, a pathwise method and a likelihood ratio method, for estimating derivatives of security prices using simulation. With the direct methods, the information from a single simulation can be used to estimate multiple derivatives along with a security’s price. The main advantage of the direct methods over resimulation is increased computational speed. Another advantage is that the direct methods give unbiased estimates of derivatives, whereas the estimates obtained by resimulation are biased. Computational results are given for both direct methods and comparisons are made to the standard method of resimulation to estimate derivatives. The methods are illustrated for a path independent model (European options), a path dependent model (Asian options), and a model with multiple state variables (options with stochastic volatility).
Complete Models with Stochastic Volatility
, 1996
"... The paper proposes an original class of models for the continuous time price process of a financial security with nonconstant volatility. The idea is to define instantaneous volatility in terms of exponentiallyweighted moments of historic logprice. The instantaneous volatility is therefore driven ..."
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Cited by 64 (4 self)
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The paper proposes an original class of models for the continuous time price process of a financial security with nonconstant volatility. The idea is to define instantaneous volatility in terms of exponentiallyweighted moments of historic logprice. The instantaneous volatility is therefore driven by the same stochastic factors as the price process, so that unlike many other models of nonconstant volatility, it is not necessary to introduce additional sources of randomness. Thus the market is complete and there are unique, preferenceindependent options prices. We find a partial differential equation for the price of a European Call Option. Smiles and skews are found in the resulting plots of implied volatility. Keywords: Option pricing, stochastic volatility, complete markets, smiles. Acknowledgement. It is a pleasure to thank the referees of an earlier draft of this paper whose perceptive comments have resulted in many improvements. 1 Research supported in part by Record Treasu...
Complications with Stochastic Volatility Models
 University of Cambridge
, 1995
"... We show a class of stochastic volatility price models for which the most natural candidates for martingale measures are only strictly local martingale measures, contrary to what is usually assumed in the finance literature. We also show the existence of martingale measures, however, and give explici ..."
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Cited by 25 (0 self)
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We show a class of stochastic volatility price models for which the most natural candidates for martingale measures are only strictly local martingale measures, contrary to what is usually assumed in the finance literature. We also show the existence of martingale measures, however, and give explicit examples. 1 Introduction In the mathematical finance literature there is a growing interest in the study of incomplete markets, and among the efforts in understanding their properties the stochastic volatility models have become increasingly popular: see for example Hull and White (87), Scott (87), Wiggins (87), Johnson and Shanno (87), Stein and Stein (91), Heston (93), Dupire (92), Hofmann, Platen and Schweizer (92) among others. They all start by imposing certain dynamics for a "volatility process" and construct asset prices as stochasticexponentials of integrals of this volatility with respect to a Brownian motion. If we want to apply results from arbitragepricing theory, which char...
Which GARCH Model for Option Valuation
 Management Science
, 2004
"... Characterizing asset return dynamics using volatility models is an important part of empirical finance. The existing literature on GARCH models favors some rather complex volatility specifications whose relative performance is usually assessed through their likelihood based on a timeseries of asset ..."
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Cited by 24 (7 self)
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Characterizing asset return dynamics using volatility models is an important part of empirical finance. The existing literature on GARCH models favors some rather complex volatility specifications whose relative performance is usually assessed through their likelihood based on a timeseries of asset returns. This paper compares a range of GARCH models along a different dimension, using option prices and returns under the riskneutral as well as the physical probability measure. We judge the relative performance of various models by evaluating an objective function based on option prices. In contrast with returnsbased inference, we find that our optionbased objective function favors a relatively parsimonious model. Specifically, when evaluated outofsample, our analysis favors a model that besides volatility clustering only allows for a standard leverage effect. JEL Classification: G12
Stochastic Volatility
, 2005
"... Stochastic volatility (SV) is the main concept used in the fields of financial economics and mathematical finance to deal with the endemic timevarying volatility and codependence found in financial markets. Such dependence has been known for a long time, early comments include Mandelbrot (1963) and ..."
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Cited by 17 (0 self)
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Stochastic volatility (SV) is the main concept used in the fields of financial economics and mathematical finance to deal with the endemic timevarying volatility and codependence found in financial markets. Such dependence has been known for a long time, early comments include Mandelbrot (1963) and Officer (1973). It was also clear to the founding fathers of modern continuous time finance that homogeneity was an unrealistic if convenient simplification, e.g. Black and Scholes (1972, p. 416) wrote “... there is evidence of nonstationarity in the variance. More work must be done to predict variances using the information available. ” Heterogeneity has deep implications for the theory and practice of financial economics and econometrics. In particular, asset pricing theory is dominated by the idea that higher rewards may be expected when we face higher risks, but these risks change through time in complicated ways. Some of the changes in the level of risk can be modelled stochastically, where the level of volatility and degree of codependence between assets is allowed to change over time. Such models allow us to explain, for example, empirically observed departures from BlackScholesMerton prices for options and understand why we should expect to see occasional dramatic moves in financial markets. The outline of this article is as follows. In section 2 I will trace the origins of SV and provide links with the basic models used today in the literature. In section 3 I will briefly discuss some of the innovations in the second generation of SV models. In section 4 I will briefly discuss the literature on conducting inference for SV models. In section 5 I will talk about the use of SV to price options. In section 6 I will consider the connection of SV with realised volatility. A extensive reviews of this literature is given in Shephard (2005). 2 The origin of SV models The origins of SV are messy, I will give five accounts, which attribute the subject to different sets of people.