HEAT KERNEL INTEREST RATE MODELS WITH TIME-INHOMOGENEOUS MARKOV PROCESSES

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.

A heat kernel approach is proposed for the development of a novel method for asset pricing over a finite time horizon. We work in an incomplete market setting and assume the existence of a pricing kernel that determines the prices of financial instruments. The pricing kernel is modeled by a weighted heat kernel driven by a multivariate Markov process. The heat kernel is chosen so as to provide enough freedom to ensure that the resulting model can be calibrated to appropriate data, e.g. to the initial term structure of bond prices. A class of models is presented for which the prices of bonds, caplets, and swaptions can be computed in closed form. The dynamical equations for the price processes are derived, and explicit formulae are obtained for the short rate of interest, the risk premium, and for the stochastic volatility of prices. Several of the closed-form models presented are driven by combinations of Markovian jump processes with different probability laws. Such models provide a basis for consistent applications in various market sectors, including equity markets, fixed-income markets, commodity markets, and insurance. The flexible multidimensional and multivariate structure on which the resulting price models are based lends itself well to the modeling of dependence across asset classes. As an illustration, the impact of spiraling debt, a typical feature of a financial crisis, is modeled explicitly, and the contagion effects can be readily observed in the dynamics of the associated asset returns.

Let 𝒳 be a space of homogeneous type. Assume that L has a bounded holomorphic functional calculus on L 2 ( Ω ) and L generates a semigroup with suitable upper bounds on its heat kernels where Ω is a measurable subset of 𝒳 . For appropriate bounded holomorphic functions b , we can define the operators b ( L ) on L p ( Ω ) , 1 ≤ p ≤ ∞ . We establish conditions on positive weight functions u , v such that for each p , 1 < p < ∞ , there exists a constant c p such that ∫ Ω | b ( L ) f ( x ) | p u ( x ) d μ ( x ) ≤ c p | | b | | ∞ p ∫ Ω | f ( x ) | p v ( x ) d μ ( x ) for all f ∈ L p ( v d μ ) . Applications include two-weight L p inequalities for Schrodinger operators with non-negative potentials on R n and divergence form operators on irregular domains of R n .

We construct default-free interest rate models in the spirit of the well-known Markov functional models: our focus is analytic tractability of the models and generality of the approach. We work in the setting of state price densities and construct models by means of the so called propagation property. The propagation property can be found implicitly in all of the popular state price density approaches, in particular heat kernels share the propagation property (wherefrom we deduced the name of the approach). As a related matter, an interesting property of heat kernels is presented, too.

This paper builds on existing asset pricing models in an intertemporal capital asset pricing model framework to investigate the pricing of options on interest rate futures. It addresses the issues of selecting the preferred pricing kernel model by employing the second Hansen–Jagannathan distance criterion. This criterion restricts the set of admissible models to those with a positive stochastic discount factor that ensures the model is arbitrage-free. The results indicate that the three-term polynomial pricing kernel with three non-wealth-related state variables, namely the real interest rate, maximum Sharpe ratio, and implied volatility, clearly dominates the other candidates. This pricing kernel is always strictly positive and everywhere monotonically decreasing in market returns in conformity with economic theory.

A Pricing Kernel Approach to Valuing Options on Interest Rate Futures

On the Polynomials Orthogonal on Regular Polygons

Arbitrage, Good Deals and Stochastic Discount Factor Restrictions on Multi-Factor Models With Coskewness

Let G be a graph, and let w be a positive real-valued weight function on V(G). For every subset X of V(G), let $$w(X)=\sum _{v \in X} w(v)$$ . A non-empty subset $$S \subset V(G)$$ is a weighted safe set of (G, w) if, for every component C of the subgraph induced by S and every component D of $$G-S$$ , we have $$w(C) \ge w(D)$$ whenever there is an edge between C and D. If the subgraph of G induced by a weighted safe set S is connected, then the set S is called a connected weighted safe set of (G, w). The weighted safe number $$\mathrm{s}(G,w)$$ and connected weighted safe number $$\mathrm{cs}(G,w)$$ of (G, w) are the minimum weights w(S) among all weighted safe sets and all connected weighted safe sets of (G, w), respectively. It is easy to see that for any pair (G, w), $$\mathrm{s}(G,w) \le \mathrm{cs}(G,w)$$ by their definitions. In this paper, we discuss the possible equality when G is a path or a cycle. We also give an answer to a problem due to Tittmann et al. (Eur J Combin 32:954–974, 2011) concerning subgraph component polynomials for cycles and complete graphs.

The general procedure for constructing a set of orthonormal polynomials is given for an arbitrary positive definite weight function, w(x), in the interval [a, b]. The Lanczos method is used to generate the three‐term recursion relation, which is then used to produce the polynomial coefficients. A discrete variable representation (DVR) is constructed from Gaussian nodes and weights that result from the three‐term recursion relation. These are termed “designer polynomials” and the associated “designer DVRs.” It will be shown by construction that every such set of “synthetic polynomials” carries an associated DVR. The term “designer” derives from the fact that the interval [a, b] and the weight function w(x) are arbitrary (except that w(x) must be positive definite on [a, b] and must have continuous derivatives except at a finite number of isolated discontinuities) and may be adapted to the physical problem of interest. The difficulties of applying a DVR to a “bare” Coulomb problem will be illustrated on a “toy” model in one dimension (1‐D hydrogen atom). A solution for the 1‐D Coulomb problem will be given, thereby motivating the need for designer DVRs. In doing so, a new set of polynomials is defined with a weight function w(x) = |x|kexp(−α|x|), (such that k = −1, 0, +1, +2, …) between the symmetrical limits [−∞, +∞]. These are called “synthetic Cartesian exponential polynomials (SCEP).” These polynomials are then used in a spectral and pseudospectral (DVR) representation to solve the 1‐D hydrogen atom problem. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002

Coherent states are required to form a complete set of vectors in the Hilbert space by providing the resolution of identity. We study the completeness of coherent states for two different models in a noncommutative space associated with the generalised uncertainty relation by finding the resolution of unity with a positive definite weight function. The weight function, which is sometimes known as the Borel measure, is obtained through explicit analytic solutions of the Stieltjes and Hausdorff moment problem with the help of the standard techniques of inverse Mellin transform.

The choice of a portfolio optimization of shares has been for so long a subject of major interest in the field of investment and financial mathematics. This article aims to create a portfolio optimization which follows the objective of maximized return and minimized risk under given constraint among S&amp;P 500 and the ten stocks, utilizing data collected from Yahoo! Data. The article ought to elucidate the Markowitz Model and Single Index Model in terms of generating a better result via comparing their distinct mechanisms in calculating their return percentage, standard deviation and Sharpe ratio, alongside with the corresponding efficiency frontier to compare their sharp ratio to determine a better model for each scenario. The result illustrates a better performance of the Markowitz Model in the aspect of producing a better portfolio optimization contrasting the Index Model in all the constraint except constraint three. Eventually backtesting was utilized to verify the feasibility.

Let w(θ) be a positive weight function on the unit circle of the complex plane. For a sequence of points { αk }k = 1∞ included in a compact subset of the unit disk, we consider the orthogonal rational functions φn that are obtained by orthogonalization of the sequence { 1, z / π1, z2 / π2, ... } where \( {\pi }_k \left( {\rm Z} \right) = \prod\nolimits_{j^{ = 1} }^k {\left( {1 - \overline {\alpha } j{\rm Z}} \right)} \), with respect to the inner product \(\left\langle {f,g} \right\rangle = \frac{1}{{2{\pi }}}\int_{{ - \pi }}^{\pi } {f\left( {{e}^{i\theta } } \right)} \overline {g\left( {{e}^{i\theta } } \right)} w\left( {\theta } \right){d\theta }\) In this paper we discuss the behaviour of φn(t) for ∣ t ∣ = 1 and n → ∞ under certain conditions. The main condition on the weight is that it satisfies a Lipschitz–Dini condition and that it is bounded away from zero. This generalizes a theorem given by Szegő in the polynomial case, that is when all αk = 0.

We construct explicit families of coherent states whose parametrization permits an arbitrarily small deviation from conventional coherent states. They possess the resolution of unity with a positive weight function given as the solution of a Stieltjes power-moment problem, with moments analytically expressible through Tricomi's confluent hypergeometric function. These states are in general squeezed, can be either sub- or super-Poissonian in nature, and induce a deformation of the metric of the complex plane.

We consider the quantitative analysis problem for interprocedural control-flow graphs (ICFGs). The input consists of an ICFG, a positive weight function that assigns every transition a positive integer-valued number, and a labelling of the transitions (events) as good, bad, and neutral events. The weight function assigns to each transition a numerical value that represents a measure of how good or bad an event is. The quantitative analysis problem asks whether there is a run of the ICFG where the ratio of the sum of the numerical weights of good events versus the sum of weights of bad events in the long-run is at least a given threshold (or equivalently, to compute the maximal ratio among all valid paths in the ICFG). The quantitative analysis problem for ICFGs can be solved in polynomial time, and we present an efficient and practical algorithm for the problem. We show that several problems relevant for static program analysis, such as estimating the worst-case execution time of a program or the average energy consumption of a mobile application, can be modeled in our framework. We have implemented our algorithm as a tool in the Java Soot framework. We demonstrate the effectiveness of our approach with two case studies. First, we show that our framework provides a sound approach (no false positives) for the analysis of inefficiently-used containers. Second, we show that our approach can also be used for static profiling of programs which reasons about methods that are frequently invoked. Our experimental results show that our tool scales to relatively large benchmarks, and discovers relevant and useful information that can be used to optimize performance of the programs.