This paper studies the form of the instantaneous impact cost function in a financial market with transaction costs via an axiomatic approach. We show that several kinds of convexity of the cost function are equivalent to the corresponding properties of the price impact functions. The results clarify the implicit assumptions made when selecting a particular form of the cost function and can be used when choosing the correct portfolio optimization framework.
This paper considers super-replication in a guaranteed deterministic problem setting with discrete time. The aim of hedging a contingent claim is to ensure the coverage of possible payoffs under the option contract for all admissible scenarios. These scenarios are given by means of a priori given compacts that depend on the history of prices. The increments of the price at each moment in time must lie in the corresponding compacts. The absence of transaction costs is assumed. The game–theoretic interpretation of pricing American options implies that the corresponding Bellman–Isaacs equations hold for both pure and mixed strategies. In the present paper, we study some properties of the least favorable (for the “hedger”) mixed strategies of the “market” and of their supports in the special case of convex payoff functions.
We consider the issue of short term immunization of a bond-like obligation with respect to changes in interest rates using a portfolio of bonds. In the case that the zero-coupon yield curve belongs to a fixed low-dimensional manifold, the problem is widely known as parametric immunization. Parametric immunization aims to make the price sensitivity of the hedged portfolio for all parameters of the model zero. However, within a popular approach to estimating non-parametric (smoothing splines) structural terms, parametric hedging is not applied immediately. We present a non-parametric approach to hedging a bond-like obligation, allowing for a common form of assessment of the timing structure with possible smoothing. We show that our approach gives standard immunization on the basis of the maximum duration, when the degree of smoothing goes to infinity. We also reinstating the industry's best hedging approach, based on the length of the key rate, as another specific case. The hedging portfolio is easy to calculate using only the basic operations of linear algebra.
The article shows how the Bayesian approach to income adjustment can be implemented in a non-parametric structure with automatic smoothing obtained from data. It also briefly illustrates the benefits of this approach using real data.
The article uses an infinite-dimensional (functional space) approach to reverse problems. Numerical calculations are performed using the Markov-Monte Carlo chain algorithm with several settings to ensure good performance. The model clearly uses spreads between queries and sentences to account for observation errors and provides automatic smoothing based on them.
The non-parametric structure captures the complex forms of the zero-coupon curves of emerging markets. The Bayesian approach assesses the accuracy of estimates, which is crucial for some applications. Examples of valuation results are given for three different bond markets: liquid (German), medium liquid (Chinese) and illiquid (Russian).
The result shows that an infinite-dimensional Bayesian approach to evaluating the structure of the term is possible. Market practices can use this approach to better understand the timing of interest rates. For example, they could now supplement their non-parametric estimates of the timing structure with Bayesian confidence intervals to enable them to assess the statistical significance of their results.
The model does not require parameters to be set during the evaluation. It has its own parameters, but they must be selected during the model configuration.
In this short note we show that a weak version of Bernstein’s characterization of the normal distribution implies the local integrability of a measurable solution of the Cauchy functional equation; the linearity of a solution of the Cauchy functional equation is an easy consequence of its local integrability. In its turn, this weak version of Bernstein’s theorem can be derived from Cauchy’s theorem.
For the discrete-time superreplication problem, a guaranteed deterministic formulation is proposed: the problem is to ensure the cheapest coverage of the contingent claim on an American option under all admissible scenarios. These scenarios are set by a priori defined compacts depending on the price history; the price increment at each moment of time must lie in the corresponding compact. The market is considered without trading constraints and transaction costs. The problem statement is game-theoretic in nature and leads directly to the Bellman–Isaacs equations of a special form under the assumption of no trading constraints. In the present study, we estimate the modulus of continuity of uniformly continuous solutions, including the Lipschitz case.
We present a robust dynamic programming approach to the general portfolio selection problem in the presence of transaction costs and trading limits. We formulate the problem as a dynamic infinite game against nature and obtain the corresponding Bellman-Isaacs equation. Under~several additional assumptions, we get an alternative form of the equation, which is more feasible for a numerical solution. The framework covers a wide range of control problems, such as the estimation of the portfolio liquidation value, or portfolio selection in an adverse market. The~results can be used in the presence of model errors, non-linear transaction costs and a price impact.
The paper considers the parametric hedging of non-parallel shifts in the yield curve. In order to determine capital requirements and stress testing, Basel committee recommends taking into account the risk of non-parallel interest rate shifts. (Basel Committee on Banking Supervision, 2016). As of April 2017, only one Russian bank took this risk into account in calculating interest rate risk, and one was developing a methodology (Central bank of Russia, 2017). We use several term structure models for hedging non-parallel interest rate shifts. The study uses a 5-year span of Russian bond market data. We use VaR and MAE to assess the effectiveness of hedging approaches.
The novelty of the work lies in the application of different term structure models, most of which have not previously been used for parametric hedging. We also present an original methodology for assessing the effectiveness of hedging. For the first time a study is conducted on the Russian bond market.
Cross-validation shows that the Nelson-Siegel (and also its shortened version), Svensson and Cox-Ingersoll-Ross models within the parametric hedging problem give better results than the generally accepted Fisher-Weil duration model. The results of this work have practical significance for fixed income managers.
We use the linear programming approach to quantify quote inconsistencies in risk-free bond markets. We present an algorithm to identify whether an inconsistency is probably due to the insufficient framework flexibility, the insufficient data quality, or the non-homogeneity of the dataset. In the latter case we study the problem of filtering out some instruments so that the remaining dataset be homogeneous. We show that the traditional filtering approach performs unacceptably poor and propose new algorithms. We find that the bonds, which get mispriced the most by a fitting algorithm, surprisingly are not the bonds, which cause the inconsistencies.