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.

This paper studies the empirical performance of jump-diffusion models that allow for stochastic volatility and correlated jumps affecting both prices and volatility. The results show that the models in question provide reasonable fit to both option prices and returns data in the in-sample estimation period. This contrasts previous findings where stochastic volatility paths are found to be too smooth relative to the option implied dynamics. While the models perform well during the high volatility estimation period, they tend to overprice long dated contracts out-of-sample. This evidence points towards a too simplistic specification of the mean dynamics of volatility.

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. The parameters of the model under both the objective and riskneutral measures are estimated simultaneously. I conclude that the square root stochastic variance model of Heston (1993) and others is incapable of generating realistic returns behavior and find that the data are more accurately represented by a stochastic variance model in the CEV class or a model that allows the price and variance processes to have a time-varying correlation. Specifically, I find that as the level of market variance increases, the volatility of market variance increases rapidly and the correlation between the price and variance processes becomes substantially more negative. The heightened heteroskedasticity in market variance that results generates realistic crash probabilities and dynamics and causes returns to display values of skewness and kurtosis much more consistent with their sample values. While the model dramatically improves the fit of options prices relative to the square root process, it falls short of explaining the implied volatility smile for short-dated options.

This paper develops an arbitrage-free time-series model of yields in continuous time that incorporates central bank policy. Policy-related events, such as FOMC meetings and releases of macroeconomic news the Fed cares about, are modeled as jumps. The model introduces a class of linear-quadratic jump-diffusions as state variables, which allows for a wide variety of jump types but still leads to tractable solutions for bond prices. I estimate a version of this model with U.S. interest rates, the Federal Reserve’s target rate, and key macroeconomic aggregates. The estimated model improves bond pricing, especially at short maturities. The “snake-shape ” of the volatility curve is linked to monetary policy inertia. A new monetary policy shock series is obtained by assuming that the Fed reacts to information available right before the FOMC meeting. According to the estimated policy rule, the Fed is mainly reacting to information contained in the yield-curve. Surprises in analyst forecasts turn out to be merely temporary components of macro variables, so that the “hump-shaped” yield response to these surprises is not consistent with a Taylor-type policy rule.

This paper characterizes the arbitrage-free dynamics of interest rates, in the presence of both jumps and diffusion, when the term structure is modeled through simple forward rates (i.e., through discretely compounded forward rates evolving continuously in time) or forward swap rates. Whereas instantaneous continuously compounded rates form the basis of most traditional interest rate models, simply compounded rates and their parameters are more directly observable in practice and are the basis of recent research on "market models." We consider very general types of jump processes, modeled through marked point processes, allowing randomness in jump sizes and dependence between jump sizes, jump times, and interest rates. We make explicit how jump and diffusion risk premia enter into the dynamics of simple forward rates.

We study the numerical approximation of viscosity solutions for integro-differential, possibly degenerate, parabolic problems. Similar models arise in option pricing, to generalize the celebrated Black-Scholes equation, when the processes which generate the underlying stock returns may contain both a continuous part and jumps. Convergence is proven for monotone schemes and numerical tests are presented and discussed.

This paper examines a particular class of continuous-time stochastic processes commonly known as af�ne diffusions (AD) and af�ne jump-diffusions (AJD) in which the drift, the diffusion and the jump coef�cients are all af�ne functions of the state variables. By deriv-ing the joint characteristic function associated with a vector of observed state variables for such models, we are able to examine the statistical properties of these diffusions and jump-diffusions as well as develop an ef�cient estimation technique based on empirical character-istic functions (ECF) and a GMM estimation procedure based on exact moment conditions. The estimators developed in this paper are in stark contrast to those available in the literature in the sense that our methods require neither discretization nor simulation. We demonstrate that our methods are in particular useful for the AD and AJD models with latent variables, i.e. the case where some of the state variables are unobserved. We illustrate our approach with a detailed examination of the continuous-time square-root stochastic volatility (SV) model, along with an empirical application using S&P 500 index returns.

Preliminary, comments are welcome. This paper is based on the essay 3 of my doctoral dissertation entitled “Essays in Financial

The unbiasedness tests of implied volatility as a forecast of future realized volatility have found implied volatility to be a biased predictor. We explain this puzzle by recognizing that option prices contain a market risk premium not only on the asset itself, but also on its volatility. We show using a stochastic volatility model, that a call option price can be represented as an expected value of the Black-Scholes formula evaluated at the average integrated volatility. If we allow volatility risk to be priced, this expectation should be taken under the risk-neutral probability measure, and can be decomposed into the expectation with respect to the physical measure and the risk-premium term. This term is just a linear function of the unobservable spot volatility. The decomposition explains the bias documented in the empirical literature and shows that the realized and historical volatility, which are used in the tests, are in fact the estimates of the unobserved quadratic variation and spot volatility of the stock-return generating process. Therefore, the use of these estimates generates the error-in-the-variables problem. We provide an empirical example based on the S&P 100 returns and the VIX index. We find, that when we take into an account the risk-premium and use efficient methods to estimate volatility, the unbiasedness hypothesis can not be rejected, and the point estimate of the slope in the traditional regression is exactly equal to 1.

We are grateful to Stephen Lynagh for excellent research assistance and to the Centre for Hedge