Trading on Deviations of Implied and Historical Densities

We propose a new method to estimate the empirical pricing kernel based on option data. We estimate the pricing kernel nonparametrically by using the ratio of the risk-neutral density estimator and the subjective density estimator. The risk-neutral density is approximated by a weighted kernel density estimator with varying unknown weights for different observations, and the subjective density is approximated by a kernel density estimator with equal weights. We represent the European call option price function by the second order integration of the risk-neutral density, so that the unknown weights are obtained through one-step penalized least squares estimation with the Kullback-Leibler divergence as the penalty function. Asymptotic results of the resulting estimators are established. The performance of the proposed method is illustrated empirically by simulation and real data application studies.

Implied volatility is an important element in risk management and option pricing. Black-Scholes model assumes a constant volatility, however, the evidence from financialmarket shows that the volatility is not constant but change with strike and time tomaturity. In this paper, the time to maturity is fixed and we will construct the implied volatility function of strike or moneyness. We can use regression method for estimation, but the data from financial market contains some noise and we need to apply smoothing techniques to estimate this implied volatility function. The standard non- and semiparametric regression methods don’t guarantee the resulting IV functions are arbitrage free, so we will insert our estimation result to Black and Scholes model and calculate the state price density (SPD). In a Black-Scholes model it is lognormal distribution with constant volatility parameter. In practice as volatility changes the distribution deviates from log-normality. We estimate volatilities and SPDs using EUREX option data on the DAX index by using different smoothing techniques. Our estimation will be carried out through the strike direction and moneyness direction. We will briefly introduce Local polynomials as one method. The most important smoothing techniques we will applied in this paper is B-splines, with the usage of roughness penalty, which allows a flexible choice on the degree of smoothness, and is promising for future research work on the arbitrage free constraint of implied volatility.

Pricing Kernel Modeling

Do option markets correctly price the probabilities of movement of the underlying asset?

A set of elementary axioms for stochastic finance is presented wherein a prominent role is played by the state–price density, which in turn determines the stochastic dynamics of the interest–rate term structure. The fact that the state–price density is a potential implies the existence of an asymptotic random variable X ∞ with the property that its conditional variance is the state–price density. The Wiener chaos expansion technique can then be applied to X ∞ , thus enabling us to ‘parametrize’ the dynamics of the discount–bond system in terms of the deterministic coefficients of the chaos expansion. Using this method, we find that there is a natural map from the space of all admissible term–structure trajectories to the symmetric Fock space F naturally associated with the space of square-integrable random variables on the underlying probability space. An element of F is either coherent or incoherent, and a stochastic bond–price system is necessarily represented by an incoherent element of F . Making use of the linearity of F we derive simple analytic formulae for the bond–price system, the volatility structure, the short rate, and the risk premium associated with an arbitrary admissible term–structure model. Extensions to foreign–exchange markets and general asset systems are also developed.

Option prices can be used to extract the implied risk-neutral density functions of the future underlying asset prices and returns. These market expectations provide valuable information that can be helpful to policy makers and investors. We tested the accuracy and stability of four nonstructural models in estimating the “true” risk-neutral density functions from option prices: the density functional based on the confluent hypergeometric function, the mixture of lognormal distributions, the smoothed implied volatility smile and Edgeworth expansions. The “true” risk-neutral density is unknown, so it was generated using the Carr–Geman–Madan–Yor (CGMY) Gamma-Ornstein–Uhlenbeck (Gamma-OU) model, a structural model able to generate flexible “true” risk-neutral densities. We observed that the density functional based on the confluent hypergeometric function and mixture of lognormal distributions outperformed the smoothed implied volatility smile and the Edgeworth expansion models in capturing the “true” risk-neutral density. The smoothed implied volatility smile had the best performance in terms of stability. This work aims to be more exhaustive than previous studies, by testing a wider variety of nonstructural models. In addition, we tested the effectiveness of these models in more demanding conditions that are closer to reality. In fact, the use of the CGMY Gamma-OU model to build the “true” risk-neutral density allows densities with a higher probability of extreme events and more leptokurtosis, and also generates sudden, discontinuous moves in prices, which has advantages over the diffusion models used in previous studies.

International Stochastic Discount Factors and Stochastic Correlation

Este artigo estima via componentes principais uma serie temporal para o Fator Estocastico de Desconto contendo informacoes sobre os retornos de fundos cambiais e de renda fixa operando no Brasil, visando utiliza-lo na modelagem do câmbio nominal mensal Real Brasileiro/Dolar Americano atraves de um Generalized Autoregressive Conditional Heteroskedasticity-in-Mean (GARCH-M) bivariado. Este exercicio empirico baseado na teoria de aprecamento de ativos segue metodologicamente Chong et alii (2002) e Da Costa et alii (2010), sendo possivel evidenciar um erro de previsao de 5,27% para depreciacao cambial e um acerto do sinal da variacao cambial em 57,5% dos meses, durante o periodo de janeiro de 2000 a dezembro de 2009. Estes resultados sugerem a literatura que nao se deve desprezar a variacao temporal dos momentos condicionais de segunda ordem.

The asset pricing behavior is studied in an overlapping generations model that is dynamically inefficient because of inefficient aggregate risk sharing. The inefficiency is linked with the stochastic distribution of income over agents within a period. The introduction of a clearing house may eliminate the dynamic inefficiency, but depending on how the clearing house operates, households may only have partial insurance against income risk. Hence, the clearing house affects the marginal rate of substitution (MRS) across generations within a period and also the intertemporal marginal rate of substitution faced by a household, which is the stochastic discount factor (SDF) applied to income streams over time. The opportunity to insure against income risk is related to different concepts of Pareto optimality. The implications for asset prices and the market price of risk are derived. In particular, the matrix of contingent claim prices supporting a Pareto optimal allocation must have a dominant root no greater than unity. This is equivalent to restricting the matrix of within-period MRS to have a dominant root no greater than unity. The sequence of powers of the pricing matrix determines the time series properties of the SDF, asset prices and asset returns.

The state price density is modeled as an exponential function of the underlying state variables, and the Esscher transform is used to specify the forward-risk-adjusted measure. With the aid of state price densities, Esscher transforms, and characteristic functions, this paper provides a consistent framework for pricing options on stocks, interest rates, and foreign exchange rates. The framework discussed is quite general and is related to many popular models.

Interest rate forecasts, state price densities and risk premium from Euribor options

Pricing kernels implicit in option prices play a key role in assessing the risk aversion over equity returns. We deal with nonparametric estimation of the pricing kernel (Empirical Pricing Kernel) given by the ratio of the risk-neutral density estimator and the subjective density estimator. The former density can be represented as the second derivative w.r.t. the European call option price function, which we estimate by nonparametric regression. The subjective density is estimated nonparametrically too. In this framework, we develop the asymptotic distribution theory of the EPK in the L1 sense. Particularly, to evaluate the overall variation of the pricing kernel, we develop a uniform confidence band of the EPK. Furthermore, as an alternative to the asymptotic approach, we propose a bootstrap confidence band. The developed theory is helpful for testing parametric specifications of pricing kernels and has a direct extension to estimating risk aversion patterns. The established results are assessed and compared in a Monte-Carlo study. As a real application, we test risk aversion over time induced by the EPK.

The establishment of the fractional Black–Scholes option pricing model is under a major condition with the normal distribution for the state price density (SPD) function. However, the fractional Brownian motion is deemed to not be martingale with a long memory effect of the underlying asset, so that the estimation of the state price density (SPD) function is far from simple. This paper proposes a convenient approach to get the fractional option pricing model by changing variables. Further, the option price is transformed as the integral function of the cumulative density function (CDF), so it is not necessary to estimate the distribution function individually by complex approaches. Finally, it encourages to estimate the fractional option pricing model by the way of nonparametric regression and makes empirical analysis with the traded 50 ETF option data in Shanghai Stock Exchange (SSE).

Purpose – The purpose of this paper is to understand that option pricing is the response of option implied volatility (IV) to macroeconomic announcements. Design/methodology/approach – The authors use high-frequency data on ASX SPI 200 index options to examine the response of option IV, as well as higher moments of the underlying return distribution, to macroeconomic announcements. Additionally, the authors identify the response of the moments as a function of moneyness of the options. Findings – The findings suggest that in-the-money and out-of-the money options have difference characteristics in their responses, leading to the conclusion that heterogeneity in investor beliefs and preferences affect option IV through the state price density (SPD) function. Originality/value – The research contributes to the literature that examines whether IV captures the beliefs of market participants about the likelihood of future states together with the preferences of market participants towards these states. In particular, the authors relate changes in option IV to changes in macroeconomic announcements, through the impact of these announcements on the moments of the SPD function.

The discovery of useful patterns embodied in a time series is of fundamental relevance in many real applications. Repetitive structures and common type of segments can also provide very useful information of patterns in financial time series. In this paper, we introduce a time series segmentation and characterization methodology combining a hybrid genetic algorithm and a clustering technique to automatically group common patterns from this kind of financial time series and address the problem of identifying stock market prices trends. This hybrid genetic algorithm includes a local search method aimed to improve the quality of the final solution. The local search algorithm is based on maximizing a likelihood ratio, assuming normality for the series and the subseries in which the original one is segmented. To do so, we select two stock market index time series: IBEX35 Spanish index (closing prices) and a weighted average time series of the IBEX35 (Spanish), BEL20 (Belgian), CAC40 (French) and DAX (German) indexes. These are processed to obtain segments that are mapped into a five dimensional space composed of five statistical measures, with the purpose of grouping them according to their statistical properties. Experimental results show that it is possible to discover homogeneous patterns in both time series.