In the setting of ‘‘affine’ ’ jump-diffusion state processes, this paper provides an analytical treatment of a class of transforms, including various Laplace and Fourier transforms as special cases, that allow an analytical treatment of a range of valuation and econometric problems. Example applications include fixed-income pricing models, with a role for intensity-based models of default, as well as a wide range of option-pricing applications. An illustrative example examines the implications of stochastic volatility and jumps for option valuation. This example highlights the impact on option ‘smirks ’ of the joint distribution of jumps in volatility and jumps in the underlying asset price, through both jump amplitude as well as jump timing.

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Implicit in the prices of traded financial assets are Arrow-Debreu prices or, with continuous states, the state-price density (SPD). We construct a nonparametric estimator for the SPD implicit in option prices and derive its asymptotic sampling theory. This estimator provides an arbitrage-free method of pricing new, complex, or illiquid securities while capturing those features of the data that are most relevant from an asset-pricing perspective, e.g., negative skewness and excess kurtosis for asset returns, volatility "smiles" for option prices. We perform Monte Carlo experiments and extract the SPD from actual S&P 500 option prices.

Abstract: This paper examines the joint time series of the S&P 500 index and near-the-money short-dated option prices with an arbitrage-free model, capturing both stochastic volatility and jumps. Jump-risk premia uncovered from the joint data respond quickly to market volatility, becoming more prominent during volatile markets. This form of jump-risk premia is important not only in reconciling the dynamics implied by the joint data, but also in explaining the volatility “smirks” of cross-sectional options data.

Black and Scholes (1973) implied volatilities tend to be systematically related to the option's exercise price and time to expiration. Derman and Kani (1994), Dupire (1994), and Rubinstein (1994) attribute this behavior to the fact that the Black/Scholes constant volatility assumption is violated in practice. These authors hypothesize that the volatility of the underlying asset's return is a deterministic function of the asset price and time. Since the volatility function in their model has an arbitrary specification, the deterministic volatility (DV) option valuation model has the potential of fitting the observed cross-section of option prices exactly. Using a sample of S&P 500 index options during the period June 1988 and December 1993, we attempt to evaluate the economic significance of the implied volatility function by examining the predictive and hedging performance of the DV option valuation model. Discussion draft: September 8, 1995 ____________________________________________...

: A three parameter stochastic process, termed the variance gamma process, that generalizes Brownian motion is developed as a model for the dynamics of log stock prices. The process is obtained by evaluating Brownian motion with drift at a random time given by a gamma process. The two additional parameters are the drift of the Brownian motion and the volatility of the time change. These additional parameters provide control over the skewness and kurtosis of the return distribution. Closed forms are obtained for the return density and the prices of European options. The statistical and risk neutral densities are estimated for data on the S&P500 Index and the prices of options on this Index. It is observed that the statistical density is symmetric with some kurtosis, while the risk neutral density is negatively skewed with a larger kurtosis. The additional parameters also correct for pricing biases of the Black Scholes model that is a parametric special case of the option pricing model d...

Post-crash distributions inferred from S

This paper shows how the fast Fourier Transform may be used to value options when the characteristic function of the return is known analytically.

We propose using the price range in the estimation of stochastic volatility models. We show theoretically, numerically, and empirically that range-based volatility proxies are not only highly efficient, but also approximately Gaussian and robust to microstructure noise. Hence range-based Gaussian quasi-maximum likelihood estimation produces highly efficient estimates of stochastic volatility models and extractions of latent volatility. We use our method to examine the dynamics of daily exchange rate volatility and find the evidence points strongly toward two-factor models with one highly persistent factor and one quickly mean-reverting factor. VOLATILITY IS A CENTRAL CONCEPT in finance, whether in asset pricing, portfolio choice, or risk management. Not long ago, theoretical models routinely assumed constant volatility ~e.g., Merton ~1969!, Black and Scholes ~1973!!. Today, however, we widely acknowledge that volatility is both time varying and predictable ~e.g., Andersen and Bollerslev ~1997!!, andstochastic volatility models are commonplace. Discrete- and continuous-time stochastic volatility models are extensively used in theoretical finance, empirical finance, and financial econometrics, both in academe and industry ~e.g., Hull and

This paper examines a class of continuous-time models with stochastic volatility that incorporate jumps in returns and volatility. We develop a likelihood-based es- timation strategy and provide estimates of model parameters, spot volatility, jump times and jump sizes using S&P 500 and Nasdaq 100 index returns. Estimates of jump times, jump sizes and volatility are particularly useful for identifying the effects of these factors during periods of market stress, such as those in 1987, 1997 and 1998.