Portfolio optimization under convex incentive schemes

We consider the terminal wealth utility maximization problem from the point of view of a portfolio manager who is paid by an incentive scheme, which is given as a convex function g of the terminal wealth. The manager’s own utility function U is assumed to be smooth and strictly concave; however, the resulting utility function U°g fails to be concave. As a consequence, the problem considered here does not fit into the classical portfolio optimization theory. Using duality theory, we prove wealth-independent existence and uniqueness of the optimal portfolio in general (incomplete) semimartingale markets as long as the unique optimizer of the dual problem has a continuous law. In many cases, this existence and uniqueness result is independent of the incentive scheme and depends only on the structure of the set of equivalent local martingale measures. As examples, we discuss (complete) one-dimensional models as well as (incomplete) lognormal mixture and popular stochastic volatility models. We also provide a detailed analysis of the case where the unique optimizer of the dual problem does not have a continuous law, leading to optimization problems whose solvability by duality methods depends on the initial wealth of the investor.     

Bounding the generalized convex call price

The present article introduces the concept of generalized calls (options whose value at expiry can be any function of the difference between the price of the underlying security and the striking price) and presents some of the properties of such options through the use of absence of stochastic dominance arguments. It deals with bounding relations of call premium applied to generalized options of the convex type, i.e. nonlinear convex options. These relations are obtained from the hypothesis of absence of second-order stochastic dominance between comparable strategies and without any hypothesis on the underlying security's distribution. The article presents economic justification of this method, some classical lemmas about stochastic dominance, and some bounds for convex calls.     

Satisfying convex risk limits by trading

A random variable, representing the final position of a trading strategy, is deemed acceptable if under each of a variety of probability measures its expectation dominates a floor associated with the measure. The set of random variables representing pre-final positions from which it is possible to trade to final acceptability is characterized. In particular, the set of initial capitals from which one can trade to final acceptability is shown to be a closed half-line . Methods for computing are provided, and the application of these ideas to derivative security pricing is developed.Received: May 2004, Mathematics Subject Classification (2000): 91B30, 60H30, 60G44JEL Classification: G10Steven E. Shreve: Work supported by the National Science Foundation under grants DMS-0103814 and DMS-0139911.Reha Tütüncü: Work supported by National Science Foundation under grants CCR-9875559 and DMS-0139911.     

Saddlepoint approximation of CreditRisk+

CreditRisk⁺ is an influential and widely implemented model of portfolio credit risk. As a close variant of models long used for insurance risk, it retains the analytical tractability for which the insurance models were designed. Value-at-risk (VaR) can be obtained via a recurrence-rule algorithm, so Monte Carlo simulation can be avoided. Little recognized, however, is that the algorithm is fragile. Under empirically realistic conditions, numerical error can accumulate in the execution of the recurrence rule and produce wildly inaccurate results for VaR.This paper provides new tools for users of CreditRisk⁺ based on the cumulant generating function (cgf) of the portfolio loss distribution. Direct solution for the moments of the loss distribution from the cgf is almost instantaneous and is computationally robust. Thus, the moments provide a convenient, quick and independent diagnostic on the implementation and execution of the standard solution algorithm. Better still, with the cgf in hand we have an alternative to the standard algorithm. I show how tail percentiles of the loss distribution can be calculated quickly and easily by saddlepoint approximation. On a large and varied sample of simulated test portfolios, I find a natural complementarity between the two algorithms: Saddlepoint approximation is accurate and robust in those situations for which the standard algorithm performs least well, and is less accurate in those situations for which the standard algorithm is fast and reliable.     

Update rules for convex risk measures

In the first part of the paper we investigate the properties that describe the intertemporal structure of dynamic convex risk measures. The usual backward approach to dynamic risk assessment leads to strong and weak versions of time consistency. As an alternative, we introduce a forward approach of consecutivity. In the second part we discuss the problem of how to update a convex risk measure when new information arrives. We analyse to what extent the above properties are appropriate update criteria.     

Nonlinear expectations of random sets

Finance and Stochastics - Sublinear functionals of random variables are known as sublinear expectations; they are convex homogeneous functionals on infinite-dimensional linear spaces. We extend...     

General convex order on risk aggregation

Using a general notion of convex order, we derive general lower bounds for risk measures of aggregated positions under dependence uncertainty, and this in arbitrary dimensions and for heterogeneous models. We also prove sharpness of the bounds obtained when each marginal distribution has a decreasing density. The main result answers a long-standing open question and yields an insight in optimal dependence structures. A numerical algorithm provides bounds for quantities of interest in risk management. Furthermore, our numerical results suggest that the bounds obtained in this paper are generally sharp for a broader class of models.     

Options on realized variance and convex orders

Realized variance option and options on quadratic variation normalized to unit expectation are analysed for the property of monotonicity in maturity for call options at a fixed strike. When this condition holds the risk-neutral densities are said to be increasing in the convex order. For Lévy processes, such prices decrease with maturity. A time series analysis of squared log returns on the S&P 500 index also reveals such a decrease. If options are priced to a slightly increasing level of acceptability, then the resulting risk-neutral densities can be increasing in the convex order. Calibrated stochastic volatility models yield possibilities in both directions. Finally, we consider modeling strategies guaranteeing an increase in convex order for the normalized quadratic variation. These strategies model instantaneous variance as a normalized exponential of a Lévy process. Simulation studies suggest that other transformations may also deliver an increase in the convex order.     

Optimal capital growth with convex shortfall penalties

The optimal capital growth strategy or Kelly strategy has many desirable properties such as maximizing the asymptotic long-run growth of capital. However, it has considerable short-run risk since the utility is logarithmic, with essentially zero Arrow–Pratt risk aversion. It is common to control risk with a Value-at-Risk (VaR) constraint defined on the end of horizon wealth. A more effective approach is to impose a VaR constraint at each time on the wealth path. In this paper, we provide a method to obtain the maximum growth while staying above an ex-ante discrete time wealth path with high probability, where shortfalls below the path are penalized with a convex function of the shortfall. The effect of the path VaR condition and shortfall penalties is a lower growth rate than the Kelly strategy, but the downside risk is under control. The asset price dynamics are defined by a model with Markov transitions between several market regimes and geometric Brownian motion for prices within a regime. The stochastic investment model is reformulated as a deterministic programme which allows the calculation of the optimal constrained growth wagers at discrete points in time.     

Multiresolution approximation for volatility processes

Abstract

We present an application of wavelet techniques to non-stationary time series with the aim of detecting the dependence structure which is typically found to characterize intraday stock index financial returns. It is particularly important to identify what components truly belong to the underlying volatility process, compared with those features appearing instead as a result of the presence of disturbance processes. The latter may yield misleading inference results when standard financial time series models are adopted. There is no universal agreement on whether long memory really affects financial series, or instead whether it could be that non-stationarity, once detected and accounted for, may allow for more power in detecting the dependence structure and thus suggest more reliable models. Wavelets are still a novel tool in the domain of applications in finance; thus, one goal is to try to show their potential use for signal decomposition and approximation of time-frequency signals. This might suggest a better interpretation of multi-scaling and aggregation effects in high-frequency returns. We show, by using special dictionaries of functions and ad hoc algorithms, that a pre-processing procedure for stock index returns leads to a more accurate identification of dependent and non-stationary features, whose detection results are improved compared with those obtained by other traditional Fourier-based methods. This allows generalized autoregressive conditional heteroscedastic models to be more effective for statistical estimation purposes.     

On a class of law invariant convex risk measures

We consider the class of law invariant convex risk measures with robust representation r_h,p(X)=sup_fò₀¹ [AV@R_s(X)f(s)-f^p(s)h(s)] ds\rho_{h,p}(X)=\sup_{f}\int_{0}^{1} [AV@R_{s}(X)f(s)-f^{p}(s)h(s)]\,ds, where 1≤p<∞ and h is a positive and strictly decreasing function. The supremum is taken over the set of all Radon–Nikodym derivatives corresponding to the set of all probability measures on (0,1] which are absolutely continuous with respect to Lebesgue measure. We provide necessary and sufficient conditions for the position X such that ρ _h,p (X) is real-valued and the supremum is attained. Using variational methods, an explicit formula for the maximizer is given. We exhibit two examples of such risk measures and compare them to the average value at risk.     

Nonlinearly weighted convex risk measure and its application

We propose a new class of risk measures which satisfy convexity and monotonicity, two well-accepted axioms a reasonable and realistic risk measure should satisfy. Through a nonlinear weight function, the new measure can flexibly reflect the investor’s degree of risk aversion, and can control the fat-tail phenomenon of the loss distribution. A realistic portfolio selection model with typical market frictions taken into account is established based on the new measure. Real data from the Chinese stock markets and American stock markets are used for empirical comparison of the new risk measure with the expected shortfall risk measure. The in-sample and out-of-sample empirical results show that the new risk measure and the corresponding portfolio selection model can not only reflect the investor’s risk-averse attitude and the impact of different trading constraints, but can find robust optimal portfolios, which are superior to the corresponding optimal portfolios obtained under the expected shortfall risk measure.     

An evaluation of computerized accounting practice sets

The purpose of this paper is to discuss computerized practice sets for accounting principles and a method for evaluating them. Criteria for selecting and evaluating computerized practice sets are identified. An evaluation of 10 actual computerized practice sets is presented.     

Consideration sets for financial services brands

This study examines the extent of consumer information search and consideration of financial services brands. It uses data from two surveys of purchasing behavior. This study finds a surprisingly low level of consumer consideration, either by personal enquiry or via the internet. The most common consideration set comprised only one brand, and this was the case for both high-value and low-value services. The managerial implication is that services marketers should make brand salience a top priority, with the competitiveness of their offer not being the primary driver of sales. If a financial services brand is salient to a consumer, there is a very high chance they will purchase that brand, without extensive comparison of the merits of alternatives.     

Statistical rehabilitation of improper correlation matrices

The simplest way to describe the dependence for a set of financial assets is their correlation matrix. This correlation matrix can be improper when it is specified element-wise. We describe a new method for obtaining a positive definite correlation matrix starting from an improper one. The expert's opinion and trust in each pairwise correlation is described by a beta distribution. Then, by combining these individual distributions, a joint distribution over the space of positive definite correlation matrices is obtained using Cholesky factorization, and its mode constitutes the new proper correlation matrix. The optimization is complemented by a visual representation of the entries that were most affected by the legalization procedure. We also sketch a Bayesian approach to the same problem.