# Option Pricing With Model-Guided Nonparametric Methods

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Abstract
Parametric option pricing models are largely used in Finance. These models capture several features of asset price dynamics. However, their pricing performance can be significantly enhanced when they are combined with nonparametric learning approaches that learn and correct empirically the pricing errors. In this paper, we propose a new nonparametric method for pricing derivatives assets. Our method relies on the state price distribution instead of the state price density because the former is easier to estimate nonparametrically than the latter. A parametric model is used as an initial estimate of the state price distribution. Then the pricing errors induced by the parametric model are fitted nonparametrically. This model-guided method estimates the state price distribution nonparametrically and is called Automatic Correction of Errors (ACE). The method is easy to implement and can be combined with any model-based pricing formula to correct the systematic biases of pricing errors. We also develop a nonparametric test based on the generalized likelihood ratio to document the efficacy of the ACE method. Empirical studies based on S&P 500 index options show that our method outperforms several competing pricing models in terms of predictive and hedging abilities.
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Under a generalized skew normal distribution we consider the problem of European option pricing. Existence of the martingale measure is proved. An explicit expression for a given European option price is presented in terms of the cumulative distribution function of the univariate skew normal and the bivariate standard normal distributions. Some special cases are investigated in a greater detail. To carry out the sensitivity of the option price to the skew parameters, numerical methods are applied. Some concluding remarks and further works are given. The results obtained are extensions of the results provided by .
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Using the wavelet-Galerkin method for solving partial integro-differential equations, we derive an implement computationally efficient formula for pricing European options on assets driven by multivariate jump-diffusions. This pricing formula is then used to solve the inverse problem of estimating the corresponding risk-neutral coefficient functions of the underlying jump-diffusions from observed option data. The ill-posedness of this estimation problem is proved, and a consistent estimation technique employing Tikhonov regularization is proposed. Using S&P 500 Index option data, it is shown that the coefficient functions in a stochastic volatility model with jumps are nonlinear, contrary to the affine specification widely used in the literature.
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We propose an efficient method for the construction of an arbitrage-free call option price function from observed call price quotes. The no-arbitrage theory of option pricing places various shape constraints on the option price function. For each available maturity on a given trading day, the proposed method estimates an option price function of strike price using a Bernstein polynomial basis. Using the properties of this basis, we transform the constrained functional regression problem to the least-squares problem of finite dimension and derive the sufficiency conditions of no-arbitrage pricing to a set of linear constraints. The resultant linearly constrained least square minimization problem can easily be solved using an efficient quadratic programming algorithm. The proposed method is easy to use and constructs a smooth call price function which is arbitrage-free in the entire domain of the strike price with any finite number of observed call price quotes. We empirically test the proposed method on S&P 500 option price data and compare the results with the cubic spline smoothing method to see the applicability.
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This paper presents a new warrants pricing method based on nonparametric estimation with respect to China warrants market and Hong Kong warrants market, and applies it to out-of-time prediction. The result shows that model-guided nonparametric correction method outperforms direct nonparametric method, semi-parametric model and parametric model. In addition, model-guided nonparametric correc- tion method has better performance than other models in terms of out-of-time prediction ability.
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A State Price Density (SPD) is the density function of a risk neutral equivalent martingale measure for option pricing, and is indispensable for exotic option pricing and portfolio risk management. Many approaches have been proposed in the last two decades to calibrate a SPD using financial options from the bond and equity markets. Among these, non and semiparametric methods were preferred because they can avoid model mis-specification of the underlying. However, these methods usually require a large data set to achieve desired convergence properties. One faces the problem in estimation by e.g., kernel techniques that there are not enough observations locally available. For this situation, we employ a Bayesian quadrature method because it allows us to incorporate prior assumptions on the model parameters and hence avoids problems with data sparsity. It is able to compute the SPD of both call and put options simultaneously, and is particularly robust when the market faces the data sparsity issue. As illustration, we calibrate the SPD for weather derivatives, a classical example of incomplete markets with financial contracts payoffs linked to non-tradable assets, namely, weather indices. Finally, we study related weather derivatives data and the dynamics of the implied SPDs.
• Article
We analyze the properties of the implied volatility, the commonly used volatility estimator by direct option price inversion. It is found that the implied volatility is subject to a systematic bias in the presence of pricing errors, which makes it inconsistent to the underlying volatility. We propose an estimator of the underlying volatility by first estimating nonparametrically the option price function, followed by inverting the nonparametrically estimated price. It is shown that the approach removes the adverse impacts of the pricing errors and produces a consistent volatility estimator for a wide range of option price models. We demonstrate the effectiveness of the proposed approach by numerical simulation and empirical analysis on S&P 500 option data.
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We suggest a semi-nonparametric estimator for the call-option price surface. The estimator is a bivariate tensor-product B-spline. To enforce no-arbitrage constraints across strikes and expiry dates, we establish sufficient no-arbitrage conditions on the control net of the B-spline surface. The conditions are linear and therefore allow for an implementation of the estimator by means of standard quadratic programming techniques. The consistency of the estimator is proved. By means of simulations, we explore the statistical efficiency benefits that are associated with estimating option price surfaces and state-price densities under the full set of no-arbitrage constraints. We estimate a call-option price surface, families of first-order strike derivatives, and state-price densities for S&P 500 option data.
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There is extensive empirical evidence that index option prices systematically differ from Black-Scholes prices. Out-of-the-money put prices (and in-the-money call prices) are relatively high compared to the Black-Scholes price. Motivated by these empirical facts, we develop a new discrete-time dynamic model of stock returns with Inverse Gaussian innovations. The model allows for conditional skewness as well as conditional heteroskedasticity and a leverage effect. We present an analytic option pricing formula consistent with this stock return dynamic. An extensive empirical test of the model using S&P500 index options shows that the new Inverse Gaussian GARCH model's performance is superior to a standard existing nested model for out-of-the money puts, thus demonstrating the importance of conditional skewness. The discrete-time Inverse Gaussian GARCH process has two interesting continuous-time limits. One limit is the standard stochastic volatility model of Heston (1993). The other is a pure jump process with stochastic intensity. Using these limit results, an equivalent motivation for our model is that it generalizes standard stochastic volatility models by allowing for "jumps" and other fat-tailed negative movements in stock returns. The empirical results therefore also demonstrate the importance of jumps for the pricing of out-of-the-money puts.
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Introduction.- Stationary Time Series.- Smoothing in Time Series.- ARMA Modeling and Forecasting.- Parametric Nonlinear Time Series Models.- Nonparametric Models.- Hypothesis Testing.- Continuous Time Models in Finance.- Nonlinear Prediction.
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In this article we study the method of nonparametric regression based on a weighted local linear regression. This method has advantages over other popular kernel methods. Moreover, such a regression procedure has the ability of design adaptation: It adapts to both random and fixed designs, to both highly clustered and nearly uniform designs, and even to both interior and boundary points. It is shown that the local linear regression smoothers have high asymptotic efficiency (i.e., can be 100% with a suitable choice of kernel and bandwidth) among all possible linear smoothers, including those produced by kernel, orthogonal series, and spline methods. The finite sample property of the local linear regression smoother is illustrated via simulation studies. Nonparametric regression is frequently used to explore the association between covariates and responses. There are many versions of kernel regression smoothers. Some estimators are not good for random designs, such as in observational studies, and others are not good for nonequispaced designs. Furthermore, most nonparametric regression smoothers have “boundary effects” and require modifications at boundary points. However, the local linear regression smoothers do not share these disadvantages. They adapt to almost all regression settings and do not require any modifications even at boundary. Besides, this method has higher efficiency than other traditional nonparametric regression methods.
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Local least squares kernel regression provides an appealing solution to the nonparametric regression, or "scatterplot smoothing," problem, as demonstrated by Fan, for example. The practical implementation of any scatterplot smoother is greatly enhanced by the availability of a reliable rule for automatic selection of the smoothing parameter. In this article we apply the ideas of plug-in bandwidth selection to develop strategies for choosing the smoothing parameter of local linear squares kernel estimators. Our results are applicable to odd-degree local polynomial fits and can be extended to other settings, such as derivative estimation and multiple nonparametric regression. An implementation in the important case of local linear fits with univariate predictors is shown to perform well in practice. A by-product of our work is the development of a class of nonparametric variance estimators, based on local least squares ideas, and plug-in rules for their implementation.
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Frequently, economic theory places shape restrictions on functional relationships between eco-nomic variables. This paper develops a method to constrain the values of the ÿrst and second derivatives of nonparametric locally polynomial estimators. We apply this technique to estimate the state price density (SPD), or risk-neutral density, implicit in the market prices of options. The option pricing function must be monotonic and convex. Simulations demonstrate that non-parametric estimates can be quite feasible in the small samples relevant for day-to-day option pricing, once appropriate theory-motivated shape restrictions are imposed. Using S&P 500 option prices, we show that unconstrained nonparametric estimators violate the constraints during more than half the trading days in 1999, unlike the constrained estimator we propose.
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In frictionless and rational markets, perfect substitutes must have the same price. In markets with trading costs, however, price differences may be as large as the costs of executing the arbitrage between markets. Moreover, if trading costs differ, trading activity will tend to be concentrated in the lowest-cost market. This study tests the differential trading cost hypothesis by examining the rate at which new information is incorporated in stock, index futures, and index option prices. The lead/lag return relations among markets are consistent with their relative trading costs. Prices in the index derivative markets appear to lead prices in the stock market. At the same time, index futures prices tend to lead index option prices, and the prices of index calls and index puts move together. The trading cost hypothesis reconciles the disparity found between the temporal relation in the stock index/index derivative markets versus the stock/stock option markets.
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Three processes reflecting persistence of volatility are initially formulated by evaluating three Lévy 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 non-Gaussian models that are solutions to Ornstein-Uhlenbeck equations driven by one-sided discontinuous Lévy 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 filtration in which it is originally defined. This leads us to formulate and investigate the important property of martingale marginals where we seek martingales in altered filtrations consistent with the one-dimensional marginal distributions of the level of the process at each future date.
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This article develops an option pricing model and its corresponding delta formula in the context of the generalized autoregressive conditional heteroskedastic (GARCH) asset return process. the development utilizes the locally risk-neutral valuation relationship (LRNVR). the LRNVR is shown to hold under certain combinations of preference and distribution assumptions. the GARCH option pricing model is capable of reflecting the changes in the conditional volatility of the underlying asset in a parsimonious manner. Numerical analyses suggest that the GARCH model may be able to explain some well-documented systematic biases associated with the Black-Scholes model. Copyright 1995 Blackwell Publishers.
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The purpose of this paper is to bridge two strands of the literature, one pertaining to the objective or physical measure used to model an underlying asset and the other pertaining to the risk-neutral measure used to price derivatives. We propose a generic procedure using simultaneously the fundamental price, St, and a set of option contracts [(σitI)i=1,m] where m⩾1 and σitI is the Black–Scholes implied volatility. We use Heston's (1993. Review of Financial Studies 6, 327–343) 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 dominance of univariate approach, which relies solely on options data. A by-product of this finding is that we uncover a remarkably simple volatility extraction filter based on a polynomial lag structure of implied volatilities. The bivariate approach, involving both the fundamental security and an option contract, appears useful when the information from the cash market reflected in the conditional kurtosis provides support to price long term.
• Article
Index option prices differ systematically from Black–Scholes prices. Out-of-the-money put prices (and in-the-money call prices) are relatively high compared to the Black–Scholes price. Motivated by these empirical facts, we develop a new discrete-time dynamic model of stock returns with inverse Gaussian innovations. The model allows for conditional skewness as well as conditional heteroskedasticity and a leverage effect. We present an analytic option pricing formula consistent with this stock return dynamic. An extensive empirical test of the model using S&P500 index options shows that the new inverse Gaussian GARCH model's performance is superior to a standard existing nested model for out-of-the money puts.
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We explore convenient analytic properties of distributions constructed as mixtures of scaled and shifted t-distributions. Particularly desirable for econometric applications are closed-form expressions for antiderivatives (e.g., the cumulative density function). We illustrate the usefulness of these distributions in two applications. In the first application, we produce density forecasts of U.S. inflation and show that these forecasts are more accurate, out-of-sample, than density forecasts obtained using normal or standard t-distributions. In the second application, we replicate the option-pricing exercise of Abadir and Rockinger [Density functionals, with an option-pricing application. Econometric Theory 19, 778–811.] and obtain comparably good results, while gaining analytical tractability.
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In this paper we introduce the Extended Method of Moments (XMM) estimator. This estimator accommodates a more general set of moment restrictions than the standard Generalized Method of Moments (GMM) estimator. More specifically, the XMM differs from the GMM in that it can handle not only uniform conditional moment restrictions (i.e. valid for any value of the conditioning variable), but also local conditional moment restrictions valid for a given fixed value of the conditioning variable. The local conditional moment restrictions are of special relevance in derivative pricing for reconstructing the pricing operator at a given day, by using the information in a few cross-sections of observed traded derivative prices and a time series of underlying asset returns. The estimated derivative prices are consistent for large time series dimension, but fixed number of cross-sectionally observed derivative prices. The asymptotic properties of the XMM estimator are nonstandard, since the combination of uniform and local conditional moment restrictions induces different rates of convergence (parametric and nonparametric) for the parameters.
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In this paper, we propose a combined regression estimator by using a parametric estimator and a nonparametric estimator of the regression function. The asymptotic distribution of this estimator is obtained for cases where the parametric regression model is correct, incorrect, and approximately correct. These distributional results imply that the combined estimator is superior to the kernel estimator in the sense that it can never do worse than the kernel estimator in terms of convergence rate and it has the same convergence rate as the parametric estimator in the case where the parametric model is correct. Unlike the parametric estimator, the combined estimator is robust to model misspecification. In addition, we also establish the asymptotic distribution of the estimator of the weight given to the parametric estimator in constructing the combined estimator. This can be used to construct consistent tests for the parametric regression model used to form the combined estimator.
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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. I conclude that the square root stochastic variance model of Heston (Rev. Financial Stud. 6 (1993) 327) is incapable of generating realistic returns behavior, and that the data are better represented by a stochastic variance model in the CEV class or a model with a time-varying leverage effect. As the level of market variance increases, the volatility of market variance increases rapidly and the leverage effect becomes substantially stronger. The heightened heteroskedasticity in market variance that results causes returns to display unconditional skewness and kurtosis much closer to their sample values, while the model falls short of explaining the implied volatility smile for short-dated options and conditional higher moments in returns.
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A novel methodology for the analysis of derivatives pricing in incomplete markets is tested empirically. The methodology generates hedge ratios and derivatives prices. They are estimated from the correlation structure between the local co-movements of securities prices. First, the hedge ratios from a parsimonious complete-market model are estimated by fitting locally the changes in the derivatives and the underlying securities prices. Second, derivatives prices are obtained from the locally estimated hedge ratios. The methodology, referred to as local parametric estimation, is tested on a dataset of DAX index options and futures transactions from the computerized German Futures Exchange.
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Fast closed form solutions for prices on European stock options are developed in a jump&dash;diffusion model with stochastic volatility and stochastic interest rates. The probability functions in the solutions are computed by using the Fourier inversion formula for distribution functions. The model is calibrated for the S and P 500 and is used to analyze several effects on option prices, including interest rate variability, the negative correlation between stock returns and volatility, and the negative correlation between stock returns and interest rates. Copyright Blackwell Publishers Inc 1997.
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The traditional kernel density estimator of an unknown density is by construction completely nonparametric in the sense that it has no preferences and will work reasonably well for all shapes. The present paper develops a class of semiparametric methods that are designed to work better than the kernel estimator in a broad nonparametric neighbourhood of a given parametric class of densities, for example, the normal, while not losing much in precision when the true density is far from the parametric class. The idea is to multiply an initial parametric density estimate with a kernel-type estimate of the necessary correction factor. This works well in cases where the correction factor function is less rough than the original density itself. Extensive comparisons with the kernel estimator are carried out, including exact analysis for the class of all normal mixtures. The new method, with a normal start, wins quite often, even in many cases where the true density is far from normal. Procedures for choosing the smoothing parameter of the estimator are also discussed. The new estimator should be particularly useful in higher dimensions, where the usual nonparametric methods have problems. The idea is also spelled out for nonparametric regression.
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If options are correctly priced in the market, it should not be possible to make sure profits by creating portfolios of long and short positions in options and their underlying stocks. Using this principle, a theoretical valuation formula for options is derived. Since almost all corporate liabilities can be viewed as combinations of options, the formula and the analysis that led to it are also applicable to corporate liabilities such as common stock, corporate bonds, and warrants. In particular, the formula can be used to derive the discount that should be applied to a corporate bond because of the possibility of default.
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This paper implements the time-state preference model in a multi-period economy, deriving the prices of primitive securities from the prices of call options on aggregate consumption. These prices permit an equilibrium valuation of assets with uncertain payoffs at many future dates. Furthermore, for any given portfolio, the price of a \$1.00 claim received at a future date, if the portfolio's value is between two given levels at that time, is derived explicitly from a second partial derivative of its call-option pricing function. An intertemporal capital asset pricing model is derived for payoffs that are jointly lognormally distributed with aggregate consumption. It is shown that using the Black-Scholes equation for options on aggregate consumption implies that individuals' preferences aggregate to isoelastic utility.
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Full-text available
The advance of technology facilitates the collection of statistical data. Flexible and refined statistical models are widely sought in a large array of statistical problems. The question arises frequently whether or not a family of parametric or nonparametric models fit adequately the given data. In this paper we give a selective overview on nonparametric inferences using generalized likelihood ratio (GLR) statistics. We introduce generalized likelihood ratio statistics to test various null hypotheses against nonparametric alternatives. The trade-off between the flexibility of alternative models and the power of the statistical tests is emphasized. Well-established Wilks’ phenomena are discussed for a variety of semi- and non-parametric models, which sheds light on other research using GLR tests. A number of open topics worthy of further study are given in a discussion section.
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The fundamental valuation equation of Cox, Ingersoll and Ross was expressed in terms of the indirect utility of wealth function. As closed-form solution for the indirect utility is generally unobtainable when investment opportunities are stochastic, existing contingent claims models involving general stochastic processes were almost all derived under the restrictive log utility assumption. An alternative valuation equation is proposed here that depends only on the direct utility function. This alternative valuation approach is applied to derive closed-form solutions for bonds, bond options, individual stocks, and stock options under both power utility and exponential utility functions. Allowable processes for aggregate output, firms' dividends, and state variables are quite general and empirically plausible. The resulting interest rate and stock price dynamics have many empirically plausible properties. Our bond and stock option pricing models with stochastic volatility and stochastic interest rates have most existing models nested. The stock option pricing model is also shown to have the ability to reconcile certain puzzling empirical regularities such as the volatility smile.
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An efficient method is developed for pricing American options on stochastic volatility/jump-diffusion processes under systematic jump and volatility risk. The parameters implicit in deutsche mark (DM) options of the model and various submodels are estimated over the period 1984 to 1991 via nonlinear generalized least squares, and are tested for consistency with \$/DM futures prices and the implicit volatility sample path. The stochastic volatility submodel cannot explain the "volatility smile" evidence of implicit excess kurtosis, except under parameters implausible given the time series properties of implicit volatilities. Article published by Oxford University Press on behalf of the Society for Financial Studies in its journal, The Review of Financial Studies.
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I use a new technique to derive a closed-form solution for the price of a European call option on an asset with stochastic volatility. The model allows arbitrary correlation between volatility and spot-asset returns. I introduce stochastic interest rates and show how to apply the model to bond options and foreign currency options. Simulations show that correlation between volatility and the spot asset's price is important for explaining return skewness and strike-price biases in the Black-Scholes (1973) model. The solution technique is based on characteristic functions and can be applied to other problems
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We propose a new method for pricing options based on GARCH models with filtered historical innovations. In an incomplete market framework, we allow for different distributions of historical and pricing return dynamics, which enhances the model's flexibility to fit market option prices. An extensive empirical analysis based on S&P 500 index options shows that our model outperforms other competing GARCH pricing models and ad hoc Black-Scholes models. We show that the flexible change of measure, the asymmetric GARCH volatility, and the nonparametric innovation distribution induce the accurate pricing performance of our model. Using a nonparametric approach, we obtain decreasing state-price densities per unit probability as suggested by economic theory and corroborating our GARCH pricing model. Implied volatility smiles appear to be explained by asymmetric volatility and negative skewness of filtered historical innovations.
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This paper develops a closed-form option valuation formula for a spot asset whose variance follows a GARCH(p, q) process that can be correlated with the returns of the spot asset. It provides the first readily computed option formula for a random volatility model that can be estimated and implemented solely on the basis of observables. The single lag version of this model contains Heston's (1993) stochastic volatility model as a continuous-time limit. Empirical analysis on S&P500 index options shows that the out-of-sample valuation errors from the single lag version of the GARCH model are substantially lower than the ad hoc Black-Scholes model of Dumas, Fleming and Whaley (1998) that uses a separate implied volatility for each option to fit to the smirk/smile in implied volatilities. The GARCH model remains superior even though the parameters of the GARCH model are held constant and volatility is filtered from the history of asset prices while the ad hoc Black-Scholes model is updated every period. The improvement is largely due to the ability of the GARCH model to simultaneously capture the correlation of volatility, with spot returns and the path dependence in volatility. Article published by Oxford University Press on behalf of the Society for Financial Studies in its journal, The Review of Financial Studies.
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This paper analyzes the relation of stock volatility with real and nominal macroeconomic volatility, economic activity, financial leverage, and stock trading activity using monthly data from 1857 to 1987. An important fact, previously noted by Robert R. Officer (1973), is that stock return variability was unusually high during the 1929-39 Great Depression. While aggregate leverage is significantly correlated with volatility, it explains a relatively small part of the movements in stock volatility. The amplitude of the fluctuations in aggregate stock volatility is difficult to explain using simple models of stock valuation, especially during the Great Depression. Copyright 1989 by American Finance Association.
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One option-pricing problem which has hitherto been unsolved is the pricing of European call on an asset which has a stochastic volatility. This paper examines this problem. The option price is determined in series form for the case in which the stochastic volatility is independent of the stock price. Numerical solutions are also produced for the case in which the volatility is correlated with the stock price. It is found that the Black-Scholes price frequently overprices options and that the degree of overpricing increases with the time to maturity. Copyright 1987 by American Finance Association.
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This paper, prepared for the Handbook of Statistics , vol.14, Statistical Methods in Finance, surveys the subject of Stochastic Volatility. The following subjects are covered : volatility in financial markets (instantaneous volatility of asset returns, implied volatilities in option prices and related stylized facts), statistical modelling in discrete and continuous time and finally statistical inference ( methods of moments, Quasi-Maximum-Likelihood, Likelihood based and Bayesian Methods and Indirect Inference). Cet article, préparé pour le Handbook of Statistics , vol. 14, Statistical Methods in Finance, passe en revue les modèles de volatilité stochastique. On traite les sujets suivants : volatilité des actifs financiers (volatilité instantanée des rendements d'actifs, volatilités implicites dans les prix d'options et régularités empiriques), modélisation statistique en temps discret et continu et enfin inférence statistique (méthodes de moments, pseudo-maximum de vraisemblance, méthodes bayesiennes et autres fondées sur la vraisemblance, inférence indirecte).
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Substantial progress has been made in developing more realistic option pricing models. Empirically, however, it is not known whether and by how much each generalization improves option pricing and hedging. The authors fill this gap by first deriving an option model that allows volatility, interest rates, and jumps to be stochastic. Using S&P 500 options, they examine several alternative models from three perspectives: (1) internal consistency of implied parameters/volatility with relevant time-series data, (2) out-of-sample pricing, and (3) hedging. Overall, incorporating stochastic volatility and jumps is important for pricing and internal consistency. But for hedging, modeling stochastic volatility alone yields the best performance. Copyright 1997 by American Finance Association.
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This article develops a new method for inferring risk-neutral probabilities (or state-contingent prices) from the simultaneously observed prices of European options. These probabilities are then used to infer a unique fully specified recombining binomial tree that is consistent with these probabilities (and, hence, consistent with all the observed option prices). A simple backwards recursive procedure solves for the entire tree. From the standpoint of the standard binomial option pricing model, which implies a limiting risk-neutral lognormal distribution for the underlying asset, the approach here provides the natural (and probably the simplest) way to generalize to arbitrary ending risk-neutral probability distributions. Copyright 1994 by American Finance Association.
• Article
The authors use an extension of the equilibrium framework of M. Rubinstein (1976) and M. Brennan (1979) to derive an option valuation formula when the stock return volatility is both stochastic and systematic. Their formula incorporates a stochastic volatility process as well as a stochastic interest rate process in the valuation of options. If the 'mean,'volatility, and 'covariance' processes for the stock return and the consumption growth are predictable, the authors' option valuation formula can be written in 'preference-free'form. Further, many popular option valuation formulae in the literature can be written as special cases of their general formula. Copyright 1993 by American Finance Association.