Dynamics of state price densities

State price densities (SPDs) are an important element in applied quantitative finance. In a Black–Scholes world they are lognormal distributions, but in practice volatility changes and the distribution deviates from log-normality. In order to study the degree of this deviation, we estimate SPDs using EUREX option data on the DAX index via a nonparametric estimator of the second derivative of the (European) call pricing function. The estimator is constrained so as to satisfy no-arbitrage constraints and corrects for the intraday covariance structure in option prices. In contrast to existing methods, we do not use any parametric or smoothness assumptions.     

Nonparametric state price density estimation using constrained least squares and the bootstrap

The economic theory of option pricing imposes constraints on the structure of call functions and state price densities. Except in a few polar cases, it does not prescribe functional forms. This paper proposes a nonparametric estimator of option pricing models which incorporates various restrictions (such as monotonicity and convexity) within a single least squares procedure. The bootstrap is used to produce confidence intervals for the call function and its first two derivatives and to calibrate a residual regression test of shape constraints. We apply the techniques to option pricing data on the DAX.     

Semi-parametric estimation of American option prices

We introduce a novel semi-parametric estimator of American option prices in discrete time. The specification is based on a parameterized stochastic discount factor and is nonparametric w.r.t. the historical dynamics of the Markovian state variables. The historical transition density estimator minimizes a distance built on the Kullback–Leibler divergence from a kernel transition density, subject to the no-arbitrage restrictions for a non-defaultable bond, the underlying asset and some American option prices. We use dynamic programming to make explicit the nonlinear restrictions on the Euclidean and functional parameters coming from option data. We study asymptotic and finite sample properties of the estimators.     

On implied volatility for options—Some reasons to smile and more to correct

We analyze the properties of the implied volatility, the commonly used volatility estimator by direct option price inversion. It is found that the implied volatility is subject to a systematic bias in the presence of pricing errors, which makes it inconsistent to the underlying volatility. We propose an estimator of the underlying volatility by first estimating nonparametrically the option price function, followed by inverting the nonparametrically estimated price. It is shown that the approach removes the adverse impacts of the pricing errors and produces a consistent volatility estimator for a wide range of option price models. We demonstrate the effectiveness of the proposed approach by numerical simulation and empirical analysis on S&P 500 option data.     

Link-save trading

Transaction costs involved while trading several assets may be described using bid-ask spread of the asset prices. We assume that the prices of several assets may be linked, so that transactions involving several assets have prices that are not necessarily equal to the sums of (bid or ask) prices of the individual assets. The family of possible price combinations forms a convex (random) set which changes in time and is called the set-valued price process. It is shown that the necessary and sufficient condition for no-arbitrage is the existence of a martingale selection, i.e. a martingale that takes values in the set-valued price process. Examples and applications to option pricing are discussed.     

A Simple Estimator of Two-Dimensional Copulas,with Applications1

Copulas are distributions with uniform marginals. Non-parametric copula estimates may violate the uniformity condition in finite samples. We look at whether it is possible to obtain valid piecewise linear copula densities by triangulation. The copula property imposes strict constraints on design points, making an equi-spaced grid a natural starting point. However, the mixed-integer nature of the problem makes a pure triangulation approach impractical on fine grids. As an alternative, we study the ways of approximating copula densities with triangular functions which guarantees that the estimator is a valid copula density. The family of resulting estimators can be viewed as a non-parametric MLE of B-spline coefficients on possibly non-equally spaced grids under simple linear constraints. As such, it can be easily solved using standard convex optimization tools and allows for a degree of localization. A simulation study shows an attractive performance of the estimator in small samples and compares it with some of the leading alternatives. We demonstrate empirical relevance of our approach using three applications. In the first application, we investigate how the body mass index of children depends on that of parents. In the second application, we construct a bivariate copula underlying the Gibson paradox from macroeconomics. In the third application, we show the benefit of using our approach in testing the null of independence against the alternative of an arbitrary dependence pattern.     

A Bayesian approach to bandwidth selection for multivariate kernel regression with an application to state-price density estimation

This paper presents a Bayesian approach to bandwidth selection for multivariate kernel regression. A Monte Carlo study shows that under the average squared error criterion, the Bayesian bandwidth selector is comparable to the cross-validation method and clearly outperforms the bootstrapping and rule-of-thumb bandwidth selectors. The Bayesian bandwidth selector is applied to a multivariate kernel regression model that is often used to estimate the state-price density of Arrow–Debreu securities with the S&P 500 index options data and the DAX index options data. The proposed Bayesian bandwidth selector represents a data-driven solution to the problem of choosing bandwidths for the multivariate kernel regression involved in the nonparametric estimation of the state-price density pioneered by Aït-Sahalia and Lo [Aït-Sahalia, Y., Lo, A.W., 1998. Nonparametric estimation of state-price densities implicit in financial asset prices. The Journal of Finance, 53, 499, 547.]     

Predictive density construction and accuracy testing with multiple possibly misspecified diffusion models

This paper develops tests for comparing the accuracy of predictive densities derived from (possibly misspecified) diffusion models. In particular, we first outline a simple simulation-based framework for constructing predictive densities for one-factor and stochastic volatility models. We then construct tests that are in the spirit of Diebold and Mariano (1995) and White (2000). In order to establish the asymptotic properties of our tests, we also develop a recursive variant of the nonparametric simulated maximum likelihood estimator of Fermanian and Salanié (2004). In an empirical illustration, the predictive densities from several models of the one-month federal funds rates are compared.     

Arbitrage, linear programming and martingales¶in securities markets with bid-ask spreads

In a general, finite-dimensional securities market model with bid-ask spreads, we characterize absence of arbitrage opportunities both by linear programming and in terms of martingales. We first show that absence of arbitrage is equivalent to the existence of solutions to the linear programming problems that compute the minimum costs of super-replicating the feasible future cashflows. Via duality, we show that absence of arbitrage is also equivalent to the existence of underlying frictionless (UF) state-prices. We then show how to transform the UF state-prices into state-price densities, and use them to characterize absence of arbitrage opportunities in terms of existence of a securities market with zero bid-ask spreads whose price process lies inside the bid-ask spread. Finally, we argue that our results extend those of Naik (1995) and Jouini and Kallal (1995) to the case of intermediate dividend payments and positive bid-ask spreads on all assets.     

The no-arbitrage condition and financial markets with heterogeneous information

This paper examines the role of the no-arbitrage condition in financial markets with heterogeneous expectations. We consider a single-period, state-contingent claims model, withM risky securities andS states. There exist two types of heterogeneously informed investors, where the information heterogeneity is defined with respect to either the security payoff matrix, the state probability vector, or state partitions. When the information heterogeneity is defined with respect to either the security payoff matrix or state partitions, the no-arbitrage condition imposes a constraint on the dispersion of information between informed and uninformed investors. Further, the no-arbitrage condition is useful in ascertaining the patterns of heterogeneity among investors that are consistent with equilibrium. However, when the information heterogeneity is defined with respect to state probabilities, the role of the no-arbitrage condition is severely restricted. Finally, the no-arbitrage condition may have important implications for the (necessary and sufficient) conditions for the existence of an equilibrium price vector in financial markets with heterogeneous expectations.     

Estimating quadratic variation when quoted prices change by a constant increment

For financial assets whose best quotes almost always change by jumping by the market’s price tick size (one cent, five cents, etc.), this paper proposes an estimator of Quadratic Variation which controls for microstructure effects. It measures the prevalence of alternations, where quotes jump back to their just-previous price. It defines a simple property called “uncorrelated alternation”, which under conditions implies that the estimator is consistent in an asymptotic limit theory, where jumps become very frequent and small. Feasible limit theory is developed, and in simulations works well.     

Density approximations for multivariate affine jump-diffusion processes

We introduce closed-form transition density expansions for multivariate affine jump-diffusion processes. The expansions rely on a general approximation theory which we develop in weighted Hilbert spaces for random variables which possess all polynomial moments. We establish parametric conditions which guarantee existence and differentiability of transition densities of affine models and show how they naturally fit into the approximation framework. Empirical applications in option pricing, credit risk, and likelihood inference highlight the usefulness of our expansions. The approximations are extremely fast to evaluate, and they perform very accurately and numerically stable.     

An efficient nonparametric estimator for models with nonlinear dependence

We provide a convenient econometric framework for the analysis of nonlinear dependence in financial applications. We introduce models with constrained nonparametric dependence, which specify the conditional distribution or the copula in terms of a one-dimensional functional parameter. Our approach is intermediate between standard parametric specifications (which are in general too restrictive) and the fully unrestricted approach (which suffers from the curse of dimensionality). We introduce a nonparametric estimator defined by minimizing a chi-square distance between the constrained densities in the family and an unconstrained kernel estimator of the density. We derive the nonparametric efficiency bound for linear forms and show that the minimum chi-square estimator is nonparametrically efficient for linear forms.     

Efficient estimation and inference in linear pseudo-panel data models

We consider pseudo-panel data models constructed from repeated cross sections in which the number of individuals per group is large relative to the number of groups and time periods. First, we show that, when time-invariant group fixed effects are neglected, the OLS estimator does not converge in probability to a constant but rather to a random variable. Second, we show that, while the fixed-effects (FE) estimator is consistent, the usual t statistic is not asymptotically normally distributed, and we propose a new robust t statistic whose asymptotic distribution is standard normal. Third, we propose efficient GMM estimators using the orthogonality conditions implied by grouping and we provide t tests that are valid even in the presence of time-invariant group effects. Our Monte Carlo results show that the proposed GMM estimator is more precise than the FE estimator and that our new t test has good size and is powerful.     

Exact computation of max weighted score estimators

We show that exact computation of a family of ‘max weighted score’ estimators, including Manski’s max score estimator, can be achieved efficiently by reformulating them as mixed integer programs (MIP) with disjunctive constraints. The advantage of our MIP formulation is that estimates are exact and can be computed using widely available solvers in reasonable time. In a classic work-trip mode choice application, our method delivers exact estimates that lead to a different economic interpretation of the data than previous heuristic estimates. In a small Monte Carlo study we find that our approach is computationally efficient for usual estimation problem sizes.     

On the use of semimartingales and stochastic integrals to model continuous trading

The continuous-time contingent claim valuation model is generalized to stopping times (random trading dates), giving substance to the question of what it means to have unlimited opportunities to trade while trading intercessions may occur only finitely, albeit arbitrarily, often. Robustness and stability of the contingent claim valuation formula are demonstrated, validating the use of continuous-time models for approximating discrete-time trading environments. The no-arbitrage value of a redundant contingent claim is shown to be a stochastic integral obtained as the limit of bounded arbitrage opportunities. As a corollary, we find that the hypothesis needed to value an American call option is significantly stronger than that sufficient to value a European call option.     

The geometry of arbitrage and the existence of competitive equilibrium

We present the basic geometry of arbitrage, and use this basic geometry to shed new light on the relationships between various no-arbitrage conditions found in the literature. For example, under very mild conditions, we show that the no-arbitrage conditions of Hart [Journal of Economic Theory 9 (1974) 293] and Werner [Econometrica 55 (1987) 1403] are equivalent and imply the compactness of the set of utility possibilities. Moreover, we show that if agents’ sets of useless net trades are linearly independent, then the Hart–Werner conditions are equivalent to the stronger condition of no-unbounded-arbitrage due to Page [Journal of Economic theory 41 (1987) 392]—and, in turn, all are equivalent to compactness of the set of rational allocations. We also consider the problem of existence of equilibrium. We show, for example, that under a uniformity condition on preferences weaker than Werner’s uniformity condition, the Hart–Werner no-arbitrage conditions are sufficient for existence. With an additional condition of weak no-half-lines—a condition weaker than Werner’s no-half-lines condition—we show that the Hart–Werner conditions are both necessary and sufficient for existence.     

Instrumental variable estimation of nonseparable models

There are many environments where knowledge of a structural relationship is required to answer questions of interest. Also, nonseparability of a structural disturbance is a key feature of many models. Here, we consider nonparametric identification and estimation of a model that is monotonic in a nonseparable scalar disturbance, which disturbance is independent of instruments. This model leads to conditional quantile restrictions. We give local identification conditions for the structural equations from those quantile restrictions. We find that a modified completeness condition is sufficient for local identification. We also consider estimation via a nonparametric minimum distance estimator. The estimator minimizes the sum of squares of predicted values from a nonparametric regression of the quantile residual on the instruments. We show consistency of this estimator.     

Local GMM estimation of time series models with conditional moment restrictions

This paper investigates statistical properties of the local generalized method of moments (LGMM) estimator for some time series models defined by conditional moment restrictions. First, we consider Markov processes with possible conditional heteroskedasticity of unknown forms and establish the consistency, asymptotic normality, and semi-parametric efficiency of the LGMM estimator. Second, we undertake a higher-order asymptotic expansion and demonstrate that the LGMM estimator possesses some appealing bias reduction properties for positively autocorrelated processes. Our analysis of the asymptotic expansion of the LGMM estimator reveals an interesting contrast with the OLS estimator that helps to shed light on the nature of the bias correction performed by the LGMM estimator. The practical importance of these findings is evaluated in terms of a bond and option pricing exercise based on a diffusion model for spot interest rate.     

How useful are no-arbitrage restrictions for forecasting the term structure of interest rates?

We develop a general framework for analyzing the usefulness of imposing parameter restrictions on a forecasting model. We propose a measure of the usefulness of the restrictions that depends on the forecaster’s loss function and that could be time varying. We show how to conduct inference about this measure. The application of our methodology to analyzing the usefulness of no-arbitrage restrictions for forecasting the term structure of interest rates reveals that: (1) the restrictions have become less useful over time; (2) when using a statistical measure of accuracy, the restrictions are a useful way to reduce parameter estimation uncertainty, but are dominated by restrictions that do the same without using any theory; (3) when using an economic measure of accuracy, the no-arbitrage restrictions are no longer dominated by atheoretical restrictions, but for this to be true it is important that the restrictions incorporate a time-varying risk premium.