Modelling Specific Interest Rate Risk with Estimation of Missing Data

For the treatment of specific interest rate risk, a risk model is suggested, quantifying and combining both market and credit risk components consistently. The market risk model is based on credit spreads derived from traded bond prices. Though traded bond prices reveal a maximum amount of issuer specific information, illiquidity problems do not allow for classical parameter estimation in this context. To overcome this difficulty an efficient multiple imputation method is proposed that also quantifies the amount of risk associated with missing data. The credit risk component is based on event risk caused by correlated rating migrations of individual bonds using a Copula function approach.     

An artificial boundary method for the Hull-White model of American interest rate derivatives

The Hull-White (HW) model is a widely used one-factor interest rate model because of its analytical tractability on liquidly traded derivatives, super-calibration ability to the initial term structure and elegant tree-building procedure. As an explicit finite difference scheme, lattice method is subject to some stability criteria, which may deteriorate the computational efficiency for early exercisable derivatives. This paper proposes an artificial boundary method based on the partial differential equations (PDEs) to price interest rate derivatives with early exercise (American) feature under the HW model. We construct conversion factors to extract the market information from the zero-coupon curve and then reduce the infinite computational domain into a finite one by using an artificial boundary on which an exact boundary condition is derived. We then develop an implicit θ-scheme with unconditional stability to solve the PDE in the reduced bounded domain. With a finite computational domain, the optimal exercise strategy can be determined efficiently. Our numerical examples show that the proposed scheme is accurate, robust to the truncation size, and more efficient than the popular lattice method for accurate derivative prices. In addition, the singularity-separating technique is incorporated into the artificial boundary method to enhance accuracy and flexibility of the numerical scheme.     

Bad news and Dow Jones make the Spanish stocks go round

This paper is a data-based attempt to analyse what kind of information basically affects close-to-open returns, open-to-close returns, volatility and volume in actively traded individual securities on the Spanish stock market. Specifically, we are interested in detecting how these variables react to specific pieces of news considered as exogenous information. However, as volume itself could be interpreted as a proxy of the information flow, we first apply the linear and nonlinear Granger causality tests from volume to return and to volatility. We do not find evidence supporting this latter hypothesis. Furthermore, we only find significant evidence of linear causality from volume to volatility. The other major finding is that both bad news and the Dow Jones play a significant informational role in explaining price changes and volatility. As a consequence of these findings, we also test the residual role of volume as a proxy for noise/liquidity trading after filtering for news, although we do not find evidence in favour of this argument.     

Hedging electricity swaptions using partial integro-differential equations

The basic contracts traded on energy exchanges are swaps involving the delivery of electricity for fixed-rate payments over a certain period of time. The main objective of this article is to solve the quadratic hedging problem for European options on these swaps, known as electricity swaptions. We consider a general class of Hilbert space valued exponential jump-diffusion models. Since the forward curve is an infinite-dimensional object, but only a finite set of traded contracts are available for hedging, the market is inherently incomplete. We derive the optimization problem for the quadratic hedging problem under the risk neutral measure and state a representation of its solution, which is the starting point for numerical algorithms.     

‘Model selection for generalized linear models with factor‐augmented predictors’

This paper considers generalized linear models in a data‐rich environment in which a large number of potentially useful explanatory variables are available. In particular, it deals with the case that the sample size and the number of explanatory variables are of similar sizes. We adopt the idea that the relevant information of explanatory variables concerning the dependent variable can be represented by a small number of common factors and investigate the issue of selecting the number of common factors while taking into account the effect of estimated regressors. We develop an information criterion under model mis‐specification for both the distributional and structural assumptions and show that the proposed criterion is a natural extension of the Akaike information criterion (AIC). Simulations and empirical data analysis demonstrate that the proposed new criterion outperforms the AIC and Bayesian information criterion. Copyright © 2009 John Wiley & Sons, Ltd.     

Optimal management and inflation protection for defined contribution pension plans

Due to the increasing risk of inflation and diminishing pension benefits, insurance companies have started selling inflation-linked products. Selling such products the insurance company takes over some or all of the inflation risk from their customers. On the other side financial derivatives which are linked to inflation such as inflation linked bonds are traded on financial markets and appear to be of increasing popularity. The insurance company can use these products to hedge its own inflation risk. In this article we study how to optimally manage a pension fund taking positions in a money market account, a stock and an inflation linked bond, while financing investments through a continuous stochastic income stream such as the plan member’s contributions. We use the martingale method in order to compute an analytic expression for the optimal strategy and express it in terms of observable market variables.     

An outer approximate subdifferential method for piecewise affine optimization

Piecewise affine functions arise from Lagrangian duals of integer programming problems, and optimizing them provides good bounds for use in a branch and bound method. Methods such as the subgradient method and bundle methods assume only one subgradient is available at each point, but in many situations there is more information available. We present a new method for optimizing such functions, which is related to steepest descent, but uses an outer approximation to the subdifferential to avoid some of the numerical problems with the steepest descent approach. We provide convergence results for a class of outer approximations, and then develop a practical algorithm using such an approximation for the compact dual to the linear programming relaxation of the uncapacitated facility location problem. We make a numerical comparison of our outer approximation method with the projection method of Conn and Cornuéjols, and the bundle method of Schramm and Zowe. Received September 10, 1998 / Revised version received August 1999?Published online December 15, 1999     

A multiobjective evolutionary approach for linearly constrained project selection under uncertainty

In the project selection problem a decision maker is required to allocate limited resources among an available set of competing projects. These projects could arise, although not exclusively, in an R&D, information technology or capital budgeting context. We propose an evolutionary method for project selection problems with partially funded projects, multiple (stochastic) objectives, project interdependencies (in the objectives), and a linear structure for resource constraints. The method is based on posterior articulation of preferences and is able to approximate the efficient frontier composed of stochastically nondominated solutions. We compared the method with the stochastic parameter space investigation method (PSI) and illustrate it by means of an R&D portfolio problem under uncertainty based on Monte Carlo simulation.     

Jump-diffusion international asset allocation

We examine international asset allocation with jump-diffusion assets in the presence of risky deviations of exchanges rates from purchasing power parity when investors consume both traded and nontraded goods. We show that adding new jump risks to existing diffusion assets does not alter investors’ original optimal portfolios of diffusion assets, as long as diffusion-risk premia remain unchanged. We also show that hedge portfolios against purchasing power parity deviations are integral parts of optimal portfolios for investors from different countries, and they can be constructed by using foreign and domestic inflation-indexed bonds. Moreover, country-specific demand for risky assets can arise from nontraded-good-specific inflation-rate-differential risks.     

Price oligopoly as a cooperative game

We consider an oligopolistic market as follows. In the market, one good is traded for money. Each oligopolist is a price setter and has the same linear cost function. Each buyer is a price taker and buys the good from oligopolists setting the lowest price. We formulate this market as a cooperative game, and consider two kinds of solution concepts, the core and a bargaining set of the game. First we show that in the monopolistic market, the core gives the monopoly price, but in the oligopolistic market, the core is empty. Second, we obtain the bargaining set of the oligopolistic market.     

A double-sided multiunit combinatorial auction for substitutes: Theory and algorithms

Combinatorial exchanges have existed for a long time in securities markets. In these auctions buyers and sellers can place orders on combinations, or bundles of different securities. These orders are conjunctive: they are matched only if the full bundle is available. On business-to-business (B2B) exchanges, buyers have the choice to receive the same product with different attributes; for instance the same product can be produced by different sellers. A buyer indicates his preference by submitting a disjunctive order, where he specifies the quantity he wants of each particular good and what limit price he is willing to pay for each good, thus providing a subjective valuation of each attribute. Only the goods with the best prices will be traded. This article considers a doubled-sided multiunit combinatorial auction for substitutes, that is, a uniform price auction where buyers and sellers place both types of orders, conjunctive (AND orders) and disjunctive (XOR orders). We show that linear competitive prices exist. We also propose an algorithm to clear the market, which is particularly efficient when the number of traders is large, and the goods are divisible.     

Incorporating order-flow into optimal execution

We provide an explicit closed-form strategy for an investor who executes a large order when market order-flow from all agents, including the investor’s own trades, has a permanent price impact. The strategy is found in closed-form when the permanent and temporary price impacts are linear in the market’s and investor’s rates of trading. We do this under very general assumptions about the stochastic process followed by the order-flow of the market. The optimal strategy consists of an Almgren–Chriss execution strategy adjusted by a weighted-average of the future expected net order-flow (given by the difference of the market’s rate of buy and sell market orders) over the execution trading horizon and proportional to the ratio of permanent to temporary linear impacts. We use historical data to calibrate the model to Nasdaq traded stocks and use simulations to show how the strategy performs.     

Pricing Fixed-Income Securities in an Information-Based Framework

Abstract

The purpose of this article is to introduce a class of information-based models for the pricing of fixed-income securities. We consider a set of continuous-time processes that describe the flow of information concerning market factors in a monetary economy. The nominal pricing kernel is assumed to be given at any specified time by a function of the values of information processes at that time. Using a change-of-measure technique, we derive explicit expressions for the prices of nominal discount bonds and deduce the associated dynamics of the short rate of interest and the market price of risk. The interest rate positivity condition is expressed as a differential inequality. An example that shows how the model can be calibrated to an arbitrary initial yield curve is presented. We proceed to model the price level, which is also taken at any specified time to be given by a function of the values of the information processes at that time. A simple model for a stochastic monetary economy is introduced in which the prices of the nominal discount bonds and inflation-linked notes can be expressed in terms of aggregate consumption and the liquidity benefit generated by the money supply.     

Term structure extrapolation and asymptotic forward rates

We investigate different inter- and extrapolation methods for term structures under different constraints in order to generate market-consistent estimates which describe the asymptotic behavior of forward rates. Our starting point is the method proposed by Smith and Wilson, which is used by the European insurance supervisor EIOPA. We use the characterization of the Smith–Wilson class of interpolating functions as the solution to a functional optimization problem to extend their approach in such a way that forward rates will converge to a value which is an outcome of the optimization process. Precise conditions are stated which guarantee that the optimization problems involved are well-posed on appropriately chosen function spaces. As a result, a well-defined optimal asymptotic forward rate can be derived directly from prices and cashflows of traded instruments. This allows practitioners to use raw market data to extract information about long term forward rates, as we will show in a study which analyzes historical EURIBOR swap data.     

Slack-based ranking method: an interpretation to the cross-efficiency method in DEA

The cross-efficiency method is generally utilized to rank decision-making units (DMUs) in data envelopment analysis (DEA) based on peer-evaluation logic. This brief note provides a method of using the available information from the linear program outputs to calculate the ranking of all DMUs with fewer computations and offers an alternative interpretation to the cross-efficiency method based on slack analysis in DEA.     

Nonlinear function approximation: Computing smooth solutions with an adaptive greedy algorithm

In contrast to linear schemes, nonlinear approximation techniques allow for dimension independent rates of convergence. Unfortunately, typical algorithms (such as, e.g., backpropagation) are not only computationally demanding, but also unstable in the presence of data noise. While we can show stability for a weak relaxed greedy algorithm, the resulting method has the drawback that it requires in practise unavailable smoothness information about the data.In this work we propose an adaptive greedy algorithm which does not need this information but rather recovers it iteratively from the available data. We show that the generated approximations are always at least as smooth as the original function and that the algorithm also remains stable, when it is applied to noisy data. Finally, the applicability of this algorithm is demonstrated by numerical experiments.     

Complexity of differential and integral equations

We are interested in the intrinsic difficulty (or complexity) of computing an approximate solution of the linear operator equation Lu = f. Practical examples of such problems include the cases where L is a known partial differential or integral operator. Problems of the form Lu = f are typically solved under the constraint that only partial information about f is available, such as the values of a finite number of inner products, or the values of f at a finite number of points. It is of interest to determine when algorithms which are in wide use are optimal algorithms, i.e., algorithms which produce an approximation with minimal cost. We are especially interested in determining conditions which are necessary and sufficient for the finite element method (FEM) to be optimal. For the cases of elliptic partial differential equations and of Fredholm integral equations of the second kind, we describe such a condition, in the form of an inequality involving the order of the problem and the degree of the finite element subspace. Suppose this inequality is violated; is the nonoptimality of the FEM inherent in the information used by the FEM, or is it because the FEM uses this information in a nonoptimal manner? The latter is the case; there always exists an algorithm using this information which is optimal. We also discuss the situation in which the information used by the finite element method (which consists of inner products) is not available. Suppose that the only admissible information about f consists of evaluations of f. In the case of the Fredholm problem of the second kind, this information is optimal; moreover, a finite element method in which the inner products are approximated by quadrature rules is an optimal algorithm. However there exist elliptic problems of positive order for which this new information is nonoptimal.     

On the efficient utilisation of duration

In this article we present a new approach to estimate the change of the present value of a given cashflow pattern caused by an interest rate shift. Our approximation is based on analysing the evolution of the present value function through a linear differential equation. The outcome is far more accurate than the standard approach achieved by a Taylor expansion. Furthermore, we derive an approximation formula of second order that produces nearly accurate results. In particular, we prove that our method is superior to any known alternative approximation formula based on duration. In order to demonstrate the power of this improved approximation we apply it to coupon bonds, level annuities, and level perpetuities. We finally generalise the approach to a non-flat term structure. As for applications in insurance, we estimate the change of the discounted value of future liabilities due to a proportional shift in the set of capital accumulation factors. These findings are of particular importance to capital adequacy calculations with respect to interest rate stress scenarios that are part of regulatory solvency requirements.     

Thin trading and estimation of systematic risk: An application of an error-correction model

Financial economics literature indicates that estimates for securities' systematic risk, i.e. the beta coefficients, are highly affected by infrequent trading. This is an especially serious problem in small security markets. In this study, the applicability of an error-correction model is investigated for modeling the risk behavior of thinly traded securities. The empirical results from a small stock market, i.e. the Helsinki Stock Exchange, indicate the estimated error-correction term to be highly dependent on the underlying trading frequency of the stock, while the direct effect is dependent merely on the market value of the firm. The model thus appears to produce useful information about the risk characteristics of thinly traded stocks.     

Convergence results for the indifference value based on the stability of BSDEs

We study the exponential utility indifference value h for a contingent claim H in an incomplete market driven by two Brownian motions. The claim H depends on a non-tradable asset variably correlated with the traded asset available for hedging. We provide an explicit sequence that converges to h, complementing the structural results for h known from the literature. Our study is based on a convergence result for quadratic backward stochastic differential equations. This convergence result, which we prove in a general continuous filtration under weak conditions, also yields that the indifference value in a setting with trading constraints enjoys a continuity property in the constraints.