On the estimation of location parameters in the multivariate one sample and two sample problems

Summary In this paper we consider the problem of estimating the vectors of location parameters in the multivariate one sample and two sample problems. These estimators are obtained through the use of the multivariate rank order statistics such as theWilcoxon or the normal scores statistic considered by the authors inPuri, Sen [1966] andSen, Puri [1967] for the corresponding testing problems. The distribution of these estimators is shown to be symmetric with respect to the parameters being estimated. These estimators are translation invariant, robust and asymptotically normal. Their asymptotic relative efficiencies with respect to the estimators based on the vector of means and medians are discussed by applying the criterion ofWilks generalized variance [Anderson, p. 166]. In particular, it is shown that the estimators based on the multivariate normal scores statistics are asymptotically as efficient as the ones based on the method of least squares when the parent distributions are normal. Research sponsored by National Science Foundation Grant No. GP-12462, and by Research Grant, GM-12868 from the N.I.H., Public Health Service.     

Returns of claims on the upside and the viability of U-shaped pricing kernels

When the pricing kernel is U-shaped, then expected returns of claims with payout on the upside are negative for strikes beyond a threshold, determined by the slope of the U-shaped kernel in its increasing region, and have negative partial derivative with respect to strike in the increasing region of the kernel. Using returns of (i) S&P 500 index calls, (ii) calls on major international equity indexes, (iii) digital calls, (iv) upside variance contracts, and (v) a theoretical construct that we denote as kernel call, we find broad support for the implications of U-shaped pricing kernels. A possible theoretical reconciliation of our empirical findings is explored through a model that accommodates heterogeneity in beliefs about return outcomes and short-selling.     

Financial equilibrium with non-linear valuations

Classical Arrow Debreu equilibria employ budget feasibility to require individuals to ensure excess supplies to be nonnegative in value using the single equilibrium price system for valuation purposes. Yet by the selection of state contingent prices, they seek excess supplies that are nonnegative in each component, and not just the value. A financial equilibrium, on the other hand, defines acceptable economic risks as excess supplies that are nonnegative in value for a number of prespecified valuation price systems. The collection of prespecified valuation price systems may be referred to as features for which clearing is sought. The number of features will generally be less than the number of states. It is then shown that by also defining budget feasibility nonlinearly one may construct a financial equilibrium with fewer securities than there are features to be cleared.     

Instantaneous portfolio theory

Instantaneous risk is described by the arrival rate of jumps in log price relatives. As a consequence there is then no concept of a mean return compensating risk exposures, as zero is the only instantaneous risk-free return. From this perspective, all portfolios are subject to risk and there are only bad and better ways of holding risk. For the purpose of analysing portfolios, the univariate variance gamma model is extended to higher dimensions with an arrival rate function with full high-dimensional support and independent levels of marginal skewness and excess kurtosis. Investment objectives are given by concave lower price functionals formulated as measure distorted variations. Specific measure distortions are calibrated to data on S&P 500 index options and the time series of the index. The time series estimation is conducted by digital moment matching applied to uncentred data and it is shown that data centring is a noisy activity to be generally avoided. The evaluation of the instantaneous investment objective requires the computation of measure distorted integrals. This is done using Monte Carlo applied to gamma distributed ellipitical radii with a low shape parameter. The resulting risk reward frontiers are between finite variation as the reward and measure distorted variations as risk. In the absence of an instantaneous risk-free return, portfolios on the efficient frontier are characterized by differences in asset variations being given by differences in asset covariations with the risk charge differential of the efficient portfolio. Portfolio variations seen as the equivalent of excess returns, may optimally be negative. Lower price maximizing portfolios are presented in two, six and twenty five dimensions.     

Adapted hedging

Exponentials of squared returns in Gaussian densities, with their consequently thin tails, are replaced by the absolute return to form Laplacian and exponentially tilted Laplacian densities at unit time. Scaling provides densities at other maturities. Stochastic processes with these marginals are identified. In addition to a specific local volatility model the densities are consistent with the difference of compound exponential processes taken at log time and scaled by the square root of time. The underlying process has a single parameter, the constant variance rate of the process. Delta hedging using Laplacian and Asymmetric Laplacian implied volatilities are developed and compared with Black Merton Scholes implied volatility hedging.The hedging strategies are implemented for stylized businesses represented by dynamic volatility indexes. The Laplacian hedge is seen to be smoother for the skew trade. It also performs better through the financial crisis for the sale of strangles. The Laplacian and Gaussian models are then synthesized as special cases of a model allowing for other powers between unity and the square. Numerous hedging strategies may be run using different powers and biases in the probability of an up move. Adapted strategies that select the best performer on past quarterly data can dominate fixed strategies. Adapted hedging strategies can effectively reduce drawdowns in the marked to market value of businesses trading options.     

Asset pricing theory for two price economies

We show that nonlinearly discounted nonlinear martingales are related to no arbitrage in two price economies as linearly discounted martingales were related to no arbitrage in economies satisfying the law of one price. Furthermore, assuming risk acceptability requires a positive physical expectation, we demonstrate that expected rates of return on ask prices should be dominated by expected rates of return on bid prices. A preliminary investigation conducted here, supports this hypothesis. In general we observe that asset pricing theory in two price economies leads to asset pricing inequalities. A model incorporating both nonlinear discounting and nonlinear martingales is developed for the valuation of contingent claims in two price economies. Examples illustrate the interactions present between the severity of measure changes and their associated discount rates. As a consequence arbitrage free two price economies can involve unique discount curves and measure changes that are however specific to both the product being priced and the trade direction. Furthermore the developed valuation operators call into question the current practice of Debt Valuation Adjustments.     

Risk premia in option markets

Risk premia are related to price probability ratios or for continuous time pure jump processes the ratios of jump arrival rates under the pricing and physical measures. The variance gamma model is employed to synthesize densities with risk premia seen as the ratio of the three parameters. The premia are shown to be mean reverting, predictable, focused on crashes at shorter horizons and rallies at the longer horizon. Predicted premia may be used to adjust physical parameters to develop option prices based on time series data.     

CONTINGENT CLAIMS VALUED AND HEDGED BY PRICING AND INVESTING IN A BASIS

Contingent claims with payoffs depending on finitely many asset prices are modeled as elements of a separable Hilbert space. Under fairly general conditions, including market completeness, it is shown that one may change measure to a reference measure under which asset prices are Gaussian and for which the family of Hermite polynomials serves as an orthonormal basis. Basis pricing synthesizes claim valuation and basis investment provides static hedging opportunities. For claims written as functions of a single asset price we infer from observed option prices the implicit prices of basis elements and use these to construct the implied equivalent martingale measure density with respect to the reference measure, which in this case is the Black-Scholes geometric Brownian motion model. Data on S & P 500 options from the Wall Street Journal are used to illustrate the calculations involved. On this illustrative data set the equivalent martingale measure deviates from the Black-Scholes model by relatively discounting the larger price movements with a compensating premia placed on the smaller movements.