Narrow Results

A new framework for asset price dynamics is introduced in which the concept of noisy information about future cash flows is used to derive the corresponding price processes. In this framework an asset is defined by its cash-flow structure. Each cash flow is modelled by a random variable that can be expressed as a function of a collection of independent random variables called market factors. With each such "X-factor" we associate a market information process, the values of which we assume are accessible to market participants. Each information process consists of a sum of two terms; one contains true information about the value of the associated market factor, and the other represents "noise". The noise term is modelled by an independent Brownian bridge that spans the interval from the present to the time at which the value of the factor is revealed. The market filtration is assumed to be that generated by the aggregate of the independent information processes. The price of an asset is given by the expectation of the discounted cash flows in the risk-neutral measure, conditional on the information provided by the market filtration. In the case where the cash flows are the dividend payments associated with equities, an explicit model is obtained for the share-price process. Dividend growth is taken into account by introducing appropriate structure on the market factors. The prices of options on dividend-paying assets are derived. Remarkably, the resulting formula for the price of a European-style call option is of the Black–Scholes–Merton type. We consider the case where the rate at which information is revealed to the market is constant, and the case where the information rate varies in time. Option pricing formulae are obtained for both cases. The information-based framework generates a natural explanation for the origin of stochastic volatility in financial markets, without the need for specifying on an ad hoc basis the dynamics of the volatility.

Dollar cost averaging (DCA) is a widely employed investment strategy in financial markets. At the same time it is also well documented that such gradual policy is sub-optimal from the point of view of risk averse decision makers with a fixed investment horizon T > 0. However, an explicit strategy that would be preferred by all risk averse decision makers did not yet appear in the literature. In this paper, we give a novel proof for the suboptimality of DCA when (log) returns are governed by Lévy processes and we construct a dominating strategy explicitly. The optimal strategy we propose is static and consists in purchasing a suitable portfolio of path-independent options. Next, we discuss a market governed by a Brownian motion in more detail. We show that the dominating strategy amounts to setting up a portfolio of power options. We provide evidence that the relative performance of DCA becomes worse in volatile markets, but also give some motivation to support its use. We also analyse DCA in presence of a minimal guarantee, explore the continuous setting and discuss the (non) uniqueness of the dominating strategy.

We propose a model where an algorithmic trader takes a view on the distribution of prices at a future date and then decides how to trade in the direction of their predictions using the optimal mix of market and limit orders. As time goes by, the trader learns from changes in prices and updates their predictions to tweak their strategy. Compared to a trader who cannot learn from market dynamics or from a view of the market, the algorithmic trader’s profits are higher and more certain. Even though the trader executes a strategy based on a directional view, the sources of profits are both from making the spread as well as capital appreciation of inventories. Higher volatility of prices considerably impairs the trader’s ability to learn from price innovations, but this adverse effect can be circumvented by learning from a collection of assets that comove. Finally, we provide a proof of convergence of the numerical scheme to the viscosity solution of the dynamic programming equations which uses new results for systems of PDEs.

We study spectral gaps w.r.t. marginals of pinned Wiener measures on spaces of discrete loops (or, more generally, pinned paths) on a compact Riemannian manifold M. The asymptotic behaviour of the spectral gap as the time parameter T of the underlying Brownian bridge goes to 0 is investigated. It turns out that depending on the choice of a Riemannian metric on the base manifold, very different asymptotic behaviours can occur. For example, on discrete loop spaces over sufficiently round ellipsoids the gap grows of order α/T as T ↓ 0. The strictly positive rate α stabilizes as the discretization approaches the continuum limit. On the other extreme, if there exists a closed geodesic γ : S¹ → M such that the sectional curvature on γ(S¹) is strictly negative, and the loop is pinned close to γ(S¹), then the gap decays of order exp(-β/T), and the decay rate β approaches +∞ as the discretization approaches the continuum limit.

We present lower estimates for the best constant appearing in the weak (1, 1) maximal inequality in the space (Rⁿ, ‖ · ‖_∞). We show that this constant grows to infinity faster than (log n)^1-o(1) when n tends to infinity. To this end, we follow and simplify the approach used by J. M. Aldaz. The new part of the argument relies on Donsker's theorem identifying the Brownian bridge as the limit object describing the statistical distribution of the coordinates of a point randomly chosen in the unit cube [0, 1]ⁿ (n large).

For each n, let $U_n$ be Haar distributed on the group of $n\times n$ unitary matrices. Let $x_{n,1},\dots ,x_{n,m}$ denote orthogonal nonrandom unit vectors in ${ℂ}^n$ and let $u_{n,k}={(u_k^1,\dots ,u_k^n)}^{\ast }=U_n^{\ast }{x}_{n,k}$, $k=1,\dots ,m$. Define the following functions on $[0,1]$: $X_n^{k,k}(t)=\sqrt{n}{\sum }_{i=1}^{[nt]}(|u_k^i{|}^2-\frac{1}{n})$, $X_n^{k,k^{\prime }}(t)=\sqrt{2n}{\sum }_{i=1}^{[nt]}{ū}_k^i{u}_{k^{\prime }}^i$, $k<k^{\prime }$. Then it is proven that $X_n^{k,k},\Re X_n^{k,k^{\prime }}$, $\Im X_n^{k,k^{\prime }}$, considered as random processes in $D[0,1]$, converge weakly, as $n\to \infty$, to $m^2$ independent copies of Brownian bridge. The same result holds for the $m(m+1)/2$ processes in the real case, where $O_n$ is real orthogonal Haar distributed and $x_{n,i}\in {ℝ}^n$, with $\sqrt{n}$ in $X_n^{k,k}$ and $\sqrt{2n}$ in $X_n^{k,k^{\prime }}$ replaced with $\sqrt{\frac{n}{2}}$ and $\sqrt{n}$, respectively. This latter result will be shown to hold for the matrix of eigenvectors of $M_n=(1/s)V_n{V}_n^T$ where $V_n$ is $n\times s$ consisting of the entries of $\{v_{ij}\},i,j=1,2,\dots$, i.i.d. standardized and symmetrically distributed, with each $x_{n,i}=\{\pm 1/\sqrt{n},\dots ,\pm 1/\sqrt{n}\}$ and $n/s\to y>0$ as $n\to \infty$. This result extends the result in [J. W. Silverstein, Ann. Probab. 18 (1990) 1174–1194]. These results are applied to the detection problem in sampling random vectors mostly made of noise and detecting whether the sample includes a nonrandom vector. The matrix $B_n=𝜃v_n{v}_n^{\ast }+S_n$ is studied where $S_n$ is Hermitian or symmetric and nonnegative definite with either its matrix of eigenvectors being Haar distributed, or $S_n=M_n$, $𝜃>0$ nonrandom and $v_n$ is a nonrandom unit vector. Results are derived on the distributional behavior of the inner product of vectors orthogonal to $v_n$ with the eigenvector associated with the largest eigenvalue of $B_n.$

A new framework for asset price dynamics is introduced in which the concept of noisy information about future cash flows is used to derive the corresponding price processes. In this framework an asset is defined by its cash-flow structure. Each cash flow is modelled by a random variable that can be expressed as a function of a collection of independent random variables called market factors. With each such “X-factor” we associate a market information process, the values of which we assume are accessible to market participants. Each information process consists of a sum of two terms; one contains true information about the value of the associated market factor, and the other represents “noise”. The noise term is modelled by an independent Brownian bridge that spans the interval from the present to the time at which the value of the factor is revealed. The market filtration is assumed to be that generated by the aggregate of the independent information processes. The price of an asset is given by the expectation of the discounted cash flows in the riskneutral measure, conditional on the information provided by the market filtration. In the case where the cash flows are the dividend payments associated with equities, an explicit model is obtained for the share-price process. Dividend growth is taken into account by introducing appropriate structure on the market factors. The prices of options on dividend-paying assets are derived. Remarkably, the resulting formula for the price of a European-style call option is of the Black-Scholes-Merton type. We consider the case where the rate at which information is revealed to the market is constant, and the case where the information rate varies in time. Option pricing formulae are obtained for both cases. The information-based framework generates a natural explanation for the origin of stochastic volatility in financial markets, without the need for specifying on an ad hoc basis the dynamics of the volatility.

A circularity statistic, based upon pairwise Mann-Whitney statistics, and measuring the non-transitivity effect A > B > C > A, was introduced in Brown & Hettmansperger (2002). In the present paper, its large sample null distribution is shown to be logistic. To test for non-transitivity, possibly indicating the presence of mixture terms, one of the components of the logistic limit variable is used as a regulator to prevent the circularity statistic being inflated by large transitive rather than non-transitive effects. An example is discussed.

This paper addresses the null distribution of the likelihood ratio statistic for threshold autoregression with normally distributed noise. The problem is non-standard because the threshold parameter is a nuisance parameter which is absent under the null hypothesis. We reduce the problem to the first-passage probability associated with a Gaussian process which, in some special cases, turns out to be a Brownian bridge. It is also shown that, in some specific cases, the asymptotic null distribution of the test statistic depends only on the ‘degrees of freedom’ and not on the exact null joint distribution of the time series.

A new framework for asset price dynamics is introduced in which the concept of noisy information about future cash flows is used to derive the corresponding price processes. In this framework an asset is defined by its cash-flow structure. Each cash flow is modelled by a random variable that can be expressed as a function of a collection of independent random variables called market factors. With each such "X-factor" we associate a market information process, the values of which we assume are accessible to market participants. Each information process consists of a sum of two terms; one contains true information about the value of the associated market factor, and the other represents "noise". The noise term is modelled by an independent Brownian bridge that spans the interval from the present to the time at which the value of the factor is revealed. The market filtration is assumed to be that generated by the aggregate of the independent information processes. The price of an asset is given by the expectation of the discounted cash flows in the riskneutral measure, conditional on the information provided by the market filtration. In the case where the cash flows are the dividend payments associated with equities, an explicit model is obtained for the share-price process. Dividend growth is taken into account by introducing appropriate structure on the market factors. The prices of options on dividend-paying assets are derived. Remarkably, the resulting formula for the price of a European-style call option is of the Black–Scholes–Merton type. We consider the case where the rate at which information is revealed to the market is constant, and the case where the information rate varies in time. Option pricing formulae are obtained for both cases. The information-based framework generates a natural explanation for the origin of stochastic volatility in financial markets, without the need for specifying on an ad hoc basis the dynamics of the volatility.