Density functionals, with an option-pricing application

We present a method of estimating density-related functionals, without prior knowledge of the density’s functional form. The approach revolves around the specification of an explicit formula for a new class of distributions that encompasses many of the known cases in statistics, including the normal, gamma, inverse gamma, and mixtures thereof. The functionals are based on a couple of hypergeometric functions. Their parameters can be estimated, and the estimates then reveal both the functional form of the density and the parameters that determine centering, scaling, etc. The function to be estimated always leads to a valid density, by design, namely, one that is nonnegative everywhere and integrates to 1. Unlike fully nonparametric methods, our approach can be applied to small datasets. To illustrate our methodology, we apply it to finding risk-neutral densities associated with different types of financial options. We show how our approach fits the data uniformly very well. We also find that our estimated densities’ functional forms vary over the dataset, so that existing parametric methods will not do uniformly well

The Tail Behavior of Sotck Returns: Emerging Versus Mature Markets.

In the following paper the authors start with a review of theoretical elements of extreme value theory (evt). In the empirical section of this study they consider five mature markets, nine Asian, six Eastern European, and seven Latin American emerging markets. The tail-behavior of returns is found to be compatible with the existence of up to the third moment but not beyond. Using a subsample of countries they also demonstrate the limitations of evt. Finally they show that little can be learned from 19th century US data about presently emerging markets' tail behavior.Extreme value theory ; Generalized Pareto distribution ; Stock-market returns.

Conditional Dependency of Financial Series: An Application of Copulas.

We develop a new methodology that measures conditional dependency. We achieve this by using copula functions that link marginal distributions, here chosen to obey a GARCH-type model with time-varying skewness and kurtosis. We apply this model to daily returns of stock-market indices. We find strong evidence of persistence in dependency both for local currency and $ US denominated series. For European stock markets, we also find evidence that large simultaneous returns of either sign lead to higher subsequent dependency. We show that dependency changes through time, as well. For stock markets within Europe, dependency increased whereas it decreased since the mid 90s when involving the S&P 500 or the Nikkei. We also suggest extensions for conditional asset pricing models involving time variation of co-skewness and co-kurtosis.International correlation ; Market integration ; ARCH, Stock indices.

Conditional Volatility, Skewness, and Kurtosis: Existence and Persistence.

Recent portfolio choice asset pricing and option valuation models highlight the importance of skewness and kurtosis. Since skewness and kurtosis are related to extreme variations they are also important for Value-at-Risk measurements. Our framework builds on a GARCH model with a condi-tional generalized-t distribution for residuals. We compute the skewness and kurtosis for this model and compare the range of these moments with the maximal theoretical moments. Our model thus allows for time-varying conditional skewness and kurtosis.GARCH Stock indices Exchange rates Interest rates SNOPT VaR

Optimal Portfolio Allocation Under Higher Moments

We evaluate how departure from normality may affect the allocation of assets. A Taylor series expansion of the expected utility allows to focus on certain moments and to compute numerically the optimal portfolio allocation. A decisive advantage of this approach is that it remains operational even if a large number of assets is involved. We show that under moderate non-normality the mean-variance criterion provides a good approximation of the expected utility maximization. In contrast, under large departure from normality (as found in some stocks in mature markets or in some stock indices in emerging markets), the mean-variance criterion may fail to approximate the expected utility correctly. In such cases, the three-moment or four-moment optimization strategies may provide a good approximation of the expected utility.Asset allocation ; Stock returns ; Non-normality ; Utility function

Estimating Gram-Charlier Expansions with Positivity Constraints.

The Gram-Charlier expansion, where skewness and kurtosi directly appear as parameters, has become popular in Finance as a generalization of the normal density. We show how positivity constraints can be numerically implemented, thereby guaranteeing that the expansion defines a density. The constrained expansion can be referred to as a Gram-Charlier density. First, we apply our method to the estimation of risk neutral densities. Then, we assess the statistical properties of maximum-likelihood estimates of Gram-Charlier densities. Lastly, we apply the framework to the estimation of a GARCH model where the conditional density is a Gram-Charlier density.Hermite expansions ; Semi-nonparametric estimation ; Risk-neutral density ; GARCH model.

Asset Allocation in Transition Economies.

Designing an investment strategy in transition economies is a difficult task, because stock markets opened through time, time series are short, and there is little guidance how to obtain expected returns and covariance matrices necessary for mean-variance asset allocation. Moments of market returns can be expected to be time varying as structural changes occur in nascent market economies. We develop an ad-hoc optimal asset-allocation strategy with a flavor of Bayesian learning adapted to these various characteristics. Since an extreme event often heralds a new state of the economy, we re-initialize learning when unlikely returns materialize. By considering a Cornell benchmark, we show the usefulness of our strategy for certain types of re-initializations. Our model can also be used in situations when new industries emerge or when companies are subject to important restructuring.Emerging markets; mean-variance allocation; sequential Bayesian learning; structural breaks.

Reading the Smile: The Message Conveyed by Methods Which Infer Risk Neutral

In this study we compare the quality and information content of risk neutral densities obtained by various methods. We consider a non-parametric method based on a mixture of log-normal densities, the semi-parametric ones based on an Hermite approximation of Madan and Milne, or based on an Edgeworth expansion of Jarrow and Rudd, the parametric approach of Malz which assumes a jump-diffusion for the underlying process, and eventually Heston's approach assuming a stochastic volatility model. We apply those models on FRF/DEM exchange rate options for two dates, for various maturities. Models differ when important news hit the market (here the 1997 snap elections). The non-parametric model provides a good fit to options prices but is unable under critical circumstances to provide as much information about market participants expectations than Malz's jump-diffusion model.Risk neutral density ; Option pricing ; Exchange rate option.

The Bank Bias: Segmentation of French Fund Families

In this paper, we investigate the performance-growth relation of French mutual funds. Using panel techniques, we find that capital inflows to French past top performing funds are not as strong as expected. This result suggests that there exist barriers to investment, that may come from the fact that funds are mostly managed by banks and insurance companies and that there are high switching costs for an investor to transfer cash from one financial institution to another. We call this phenomenon ''bank bias'', because investors do not diversify enough across banks' funds. Furthermore, we provide a test of our conjecture and cannot reject it.Mutual funds ; Performance ; SICAV ; FCP