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One classic example of a binary classifier is one which employs the mean and standard deviation of the data set as a mechanism for classification. Indeed, principle component analysis has played a major role in this effort. In this paper, we propose that one should also include skew in order to make this method of classification a little more precise. One needs a simple probability distribution function which can be easily fit to a data set and use this pdf to create a classifier with improved error rates and comparable to other classifiers.

For asset prices that follow stochastic-volatility diffusions, we use asymptotic methods to investigate the behavior of the local volatilities and Black–Scholes volatilities implied by option prices, and to relate this behavior to the parameters of the stochastic volatility process. We also give applications, including risk-premium-based explanations of the biases in some naïve pricing and hedging schemes.

We begin by reviewing option pricing under stochastic volatility and representing option prices and local volatilities in terms of expectations. In the case that fluctuations in price and volatility have zero correlation, the expectations formula shows that local volatility (like implied volatility) as a function of log-moneyness has the shape of a symmetric smile. In the case of non-zero correlation, we extend Sircar and Papanicolaou's asymptotic expansion of implied volatilities under slowly-varying stochastic volatility. An asymptotic expansion of local volatilities then verifies the rule of thumb that local volatility has the shape of a skew with roughly twice the slope of the implied volatility skew. Also we compare the slow-variation asymptotics against what we call small-variation asymptotics, and against Fouque, Papanicolaou, and Sircar's rapid-variation asymptotics.

We apply the slow-variation asymptotics to approximate the biases of two naïve pricing strategies. These approximations shed some light on the signs and the relative magnitudes of the biases empirically observed in out-of-sample pricing tests of implied-volatility and local-volatility schemes. Similarly, we examine the biases of three different strategies for hedging under stochastic volatility, and we propose ways to implement these strategies without having to specify or estimate any particular stochastic volatility model. Our approximations suggest that a number of the empirical pricing and hedging biases may be explained by a positive premium for the portion of volatility risk that is uncorrelated with asset risk.

In this study, the seismic vulnerability of post-tensioned reinforced concrete box-girder highway bridges with moderate-to-large skew angles to various levels of ground motion intensity is investigated. The fragility curves are generated by performing incremental nonlinear dynamic analysis (IDA) on the bridges of skew angles of 0, 30, and 60°s. A total of 45 ground motion pairs are considered to develop the fragility curves. The damage states are presented and quantified based on the column rotational ductility and superstructure displacements at the abutments. Furthermore, the fragility curves constructed are compared against those recommended by HAZUS. It is demonstrated that as the skew angle increases, skew bridges become more vulnerable to seismically induced damages. It is also shown that the HAZUS fragility curves may not lead to a consistent prediction of the vulnerability of skewed bridges.

We present a framework for calibrating a pricing model to a prescribed set of option prices quoted in the market. Our algorithm yields an arbitrage-free diffusion process that minimizes the relative entropy distance to a prior diffusion. We solve a constrained (minimax) optimal control problem using a finite-difference scheme for a Bellman parabolic equation combined with a gradient-based optimization routine. The number of unknowns in the optimization step is equal to the number of option prices that need to be matched, and is independent of the mesh-size used for the scheme. This results in an efficient, non-parametric calibration method that can match an arbitrary number of option prices to any desired degree of accuracy. The algorithm can be used to interpolate, both in strike and expiration date, between implied volatilities of traded options and to price exotics. The stability and qualitative properties of the computed volatility surface are discussed, including the effect of the Bayesian prior on the shape of the surface and on the implied volatility smile/skew. The method is illustrated by calibrating to market prices of Dollar-Deutschemark over-the-counter options and computing interpolated implied-volatility curves.