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This paper investigates a nonparametric modular neural network (MNN) model to price the S&P-500 European call options. The modules are based on time to maturity and moneyness of the options. The option price function of interest is homogeneous of degree one with respect to the underlying index price and the strike price. When compared to an array of parametric and nonparametric models, the MNN method consistently exerts superior out-of-sample pricing performance. We conclude that modularity improves the generalization properties of standard feedforward neural network option pricing models (with and without the homogeneity hint).

This article examines the implications of the existence of private information in the spot foreign exchange market. Our framework is a high-frequency version of a structural microstructure trade model that measures the market maker's beliefs directly. We find that the underpinnings for the time-varying pattern of the probability of informed trading are rooted in the strategic arrival of informed traders on a particular hour-of-day, day-of-week, and geographic location (market). Specifically, we document that informed traders not only pick the low activity hours, but also attach the largest market weight to a particular market. The distributions of the estimated arrival rates confirm the commitment of the informed traders to strategic trading activities. In our framework, we acknowledge that an expected loss of informed trading to the market maker is a function of both the probability of informed trading and its likely impact on the price. The impact of the uninformed traders arrival on the daily foreign exchange price volatility is about twice the magnitude of the one for informed traders. These effects are in stark contrast to the findings from the hourly data that indicate dominance of informed traders. Finally, the results relate the informational content of trading to the trade size and suggest that the probability of the informed large trading is significantly higher than the probability of uninformed large trading.

We develop a dynamic framework to identify aggregate market fears ahead of a major market crash through the skewness premium of European options. Our methodology is based on measuring the distribution of a skewness premium through a q-Gaussian density and a maximum entropy principle. Our findings indicate that the October 19th, 1987 crash was predictable from the study of the skewness premium of deepest out-of-the-money options about two months prior to the crash.

Conventional time series analysis, focusing exclusively on a time series at a given scale, lacks the ability to explain the nature of the data generating process. A process equation that successfully explains daily price changes, for example, is unable to characterize the nature of hourly price changes. On the other hand, statistical properties of monthly price changes are often not fully covered by a model based on daily price changes. In this paper, we simultaneously model regimes of volatilities at multiple time scales through wavelet-domain hidden Markov models. We establish an important stylized property of volatility across different time scales. We call this property asymmetric vertical dependence. It is asymmetric in the sense that a low volatility state (regime) at a long time horizon is most likely followed by low volatility states at shorter time horizons. On the other hand, a high volatility state at long time horizons does not necessarily imply a high volatility state at shorter time horizons. Our analysis provides evidence that volatility is a mixture of high and low volatility regimes, resulting in a distribution that is non-Gaussian. This result has important implications regarding the scaling behavior of volatility, and consequently, the calculation of risk at different time scales.