Luis Seco

Portfolio optimization when asset returns have the Gaussian mixture distribution

Abstract In this paper we consider a portfolio optimization problem where the underlying asset returns are distributed as a mixture of two multivariate Gaussians; these two Gaussians may be associated with “distressed” and “tranquil” market regimes. In this context, the Sharpe ratio needs to be replaced by other non-linear objective functions which, in the case of many underlying assets, lead to optimization problems which cannot be easily solved with standard techniques. We obtain a geometric characterization of efficient portfolios, which reduces the complexity of the portfolio optimization problem.

Estimating the spectral measure of a multivariate stable distribution via spherical harmonic analysis

A new method is developed for estimating the spectral measure of a multivariate stable probability measure, by representing the measure as a sum of spherical harmonics.

Principal Component Value at Risk

Value at risk (VaR) is an industrial standard for monitoring financial risk in an investment portfolio. It measures potential losses within a given confidence interval. The implementation, calculation, and interpretation of VaR contains a wealth of mathematical issues that are not fully understood. In this paper we present a methodology for an approximation to value at risk that is based on the principal components of a sensitivity-adjusted covariance matrix. The result is an explicit expression in terms of portfolio deltas, gammas, and the variance/covariance matrix. It can be viewed as a nonlinear extension of the linear model given by the delta-normal VaR or Risk Metrics (J.P. Morgan, 1996).

Pricing of spread options on stochastically correlated underlyings

This paper proposes a method to price spread options on stochastically correlated underlying assets. Therefore it provides a more realistic approach towards correlation structure. We generalize a constant correlation tree model developed by Hull (2002) and extend it by the notion of stochastic correlation. The resulting tree model is recombining and easy to implement. Moreover, the numerical convergence of our model is very fast. Our sensitivity analysis with respect to the stochastic correlation parameters shows that the constant correlation model systematically overprices spread options on two stochastically correlated underlying assets. Furthermore, we use our model to derive hedging parameters for the correlation of a spread option and show that the constant correlation model also overprices the hedging parameters.

Interval Arithmetic in Quantum Mechanics

Quantum mechanics is an area which, over the last ten years or so, has sparked a respectable amount of rigorous computer assisted work (see, for example [9, 13, 20, 39, 38], and their applications in [10, 11, 12, 3, 4, 5,14,15,16,17,18,19, 20]). The purpose of this review is to select a piece of that body of work, and to give a more or less detailed account, both of the quantum mechanical problem surrounding the computer work and of the computer assisted proof itself. We hope this will be enlightening since much of the other computer assisted work in quantum mechanics shares many of the main features presented below.

Risk modeling in crude oil market: a comparison of Markov switching and GARCH models

– The purpose of this paper is to deal with the different phases of volatility behavior and the dependence of the variability of the time series on its own past, models allowing for heteroscedasticity like autoregressive conditional heteroscedasticity (ARCH), generalized autoregressive conditional heteroscedasticity (GARCH), or regime‐switching models have been suggested by reserachers. Both types of models are widely used in practice., – Both regime‐switching models and GARCH are used in this paper to model and explain the behavior of crude oil prices in order to forecast their volatility. In regime‐switching models, the oil return volatility has a dynamic process whose mean is subject to shifts, which is governed by a two‐state first‐order Markov process., – The GARCH models are found to be very useful in modeling a unique stochastic process with conditional variance; regime‐switching models have the advantage of dividing the observed stochastic behavior of a time series into several separate phases with different underlying stochastic processes., – The regime‐switching models show similar goodness‐of‐fit result to GARCH modeling, while has the advantage of capturing major events affecting the oil market. Daily data of crude oil prices are used from NYMEX Crude Oil market for the period 13 February 2006 up to 21 July 2009.

Introduction To The Special Issue On "Operational Research And Asia Risk Management"

Risks are traditionally defined as the combination of probability and severity, but are actually characterized by additional factors. We believe the characteristics of risks include uncertainties, dynamics, dependence, clusterings and complexities, which motivate the utilization of various operational research tools. The objective of this issue is to survey the practice of using operational research tools in risk management, especially Asian risk management.

A Fund of Hedge Funds under Regime Switching

This article investigates the use of a regime-switching model of returns for the asset allocation decision of a fund of hedge funds. In each time period, returns follow a multi-variate normal distribution from one of two possible regimes, corresponding to periods of “normal” and “distressed” markets. The prevailing regime in any given period is determined by the value of a two-state Markov chain. The case where serial correlation is absent and returns in different time periods are i.i.d. Gaussian mixture variables is also considered. The models are tested on empirical data and compared to a benchmark, assuming i.i.d. normally distributed returns. The results show that in a mean–variance framework, the use of regime switching can improve risk and performance measures. The importance of the sensitivity of optimal portfolio weights to the estimate of the probability of the distressed regime is discussed, and methods for calculating sensitivities are presented and illustrated on market data.

Algorithmic estimation of risk factors in financial markets with stochastic drift

Abstract We assume a financial market governed by a diffusion process reverting to a stochastic mean which is itself governed by an unobservable ergodic diffusion, similar to those observed in electricity and other energy markets. We develop a moment method algorithm for the estimation of the parameters of both the observable process and the unobservable stochastic mean. Our approach is contrasted with other methods for parameter estimation of partially observed diffusions, and applications to the modelling of interest rates and commodity prices are discussed.

The mathematics of risk transfer

In this paper we present a historical account of the evolution of mathematics and risk management over the last 20 years. In it, we will focus primarily on present credit market developments and we give an account of some new credit derivatives: collateralised fund obligations.