We present a new approach to estimate the risk-neutral probability density function (pdf) of the future prices of an underlying asset from the prices of options written on the asset. The estimation is carried out in the space of cubic spline functions, yielding appropriate smoothness. The resulting optimization problem, used to invert the data and determine the corresponding density function, is a convex quadratic or semidefinite programming problem, depending on the formulation. Both of these problems can be efficiently solved by numerical optimization software.In the quadratic programming formulation the positivity of the risk-neutral pdf is heuristically handled by posing linear inequality constraints at the spline nodes. In the other approach, this property of the risk-neutral pdf is rigorously ensured by using a semidefinite programming characterization of nonnegativity for polynomial functions.We tested our approach using data simulated from Black-Scholes option prices and using market data for options on the S&P 500 Index. The numerical results we present show the effectiveness of our methodology for estimating the riskneutral probability density function.

In this study, features of the financial returns of the PSI20index, related to market efficiency, are captured using wavelet-and entropy-based techniques. This characterization includes the following points. First, the detection of long memory, associated with low frequencies, and a global measure of the time series: the Hurst exponent estimated by several methods, including wavelets. Second, the degree of roughness, or regularity variation, associated with the Hölder exponent, fractal dimension and estimation based on the multifractal spectrum. Finally, the degree of the unpredictability of the series, estimated by approximate entropy. These aspects may also be studied through the concepts of non-extensive entropy and distribution using, for instance, the Tsallis q-triplet. They allow one to study the existence of efficiency in the financial market. On the other hand, the study of local roughness is performed by considering wavelet leader-based entropy. In fact, the wavelet coefficients are computed from a multiresolution analysis, and the wavelet leaders are defined by the local suprema of these coefficients, near the point that we are considering. The resulting entropy is more accurate in that detection than the Hölder exponent. These procedures enhance the capacity to identify the occurrence of financial crashes.

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This study aims to describe the size distribution of Portuguese firms, as measured by annual sales and total assets, between 2006 and 2012, giving an economic interpretation for the evolution of the distribution along the time. Three distributions are fitted to data: the lognormal, the Pareto (and as a particular case Zipf) and the Simplified Canonical Law (SCL). We present the main arguments found in literature to justify the use of distributions, emphasizing the interpretation of SCL coefficients and its analogy with thermodynamics. Methods of estimation include Maximum Likelihood, modified Ordinary Least Squares in log-log scale and Nonlinear Least Squares considering the Levenberg-Marquardt algorithm. We apply these approaches to Portuguese firm data. In the sales case, the evolution of estimated parameters in both lognormal and SCL reflects the existence of a recession period more pronounced after 2008.

Option pricing theory determines the structure of call and put option pricing functions. In nonparametric risk‐neutral density estimation based on kernel functions, local constraints cannot induce a second derivative function that must integrate one. Convexity and monotonicity of pricing functions also cannot be enforced. A large‐scale (optimization) approach is proposed for the risk‐neutral density estimation, imposing an enlarged set of no‐arbitrage constraints. We considered simulations using Heston's model and hypergeometric functions. The method is applied to samples of intraday data from VIX and S&P500 indexes.