The Fox School of Business and Management

• Abstracts

Abstracts

Parsimonious Principle of GARCH Models:
A Monte Carlo Approach

Wu Jing
Division of Economics
Nanyang Technological University, Singapore

Abstract: This paper employs Monte Carlo simulation as well as real data to test the robustness of fitness of nested GARCH models. Likelihood family tests are used to test in sample fitness, while Mean Squared Prediction Error is employed for out sample prediction test. Parsimonious principle applies well in either criterion. However, there comes the conflict between the in sample and out sample fitness. In sample likelihood family tests pay more attention to conditional distribution, or they are more sensitive to fat tail effect; while the out sample criteria focus more on the accuracy of parameter estimation. Thus, a model which fits well for one group of criteria may fail to follow another.

Fuzzy Coefficient Volatility Models with Financial Applications

K. Thiagarajah
Department of Mathematics, Illinois State University, Normal, Illinois, USA

A. Thavaneswaran
Department of Statistics, University of Manitoba, Winnipeg, Manitoba, Canada

Abstract: Recently identification of random coefficient GARCH models has been studied by Thavaneswaran et. al (2005).In this presentation, following Buckley (2004), we introduce a new class of fuzzy coefficient volatility models and discuss the moment properties. Application to option pricing using fuzzy volatility based on observed data will also be discussed in some detail.

The Implied Liquidity Premium for Equities

E. Robert Fernholz
Chief Investment Officer, INTECH

Ioannis Karatzas
Mathematics Department, Columbia University

Abstract: Over the long term, the returns on smaller stocks are likely to be higher than the returns on larger stocks. This phenomenon has been called size effect, and a number of explanations have been proposed to account for it. In this talk we show that the difference in return between the larger and the smaller stocks liquidity premium for the smaller stocks, and we estimate the value of this premium using structural parameters for the capital distribution of the U.S. stock market during the 1990s.

A Benchmark Approach to Portfolio Optimization and Derivative Pricing

Eckhard Platen
University of Technology, Sydney, Australia

Abstract: The lecture derives in a general continuous market setting a benchmark approach to portfolio optimization, derivative pricing, financial modeling and risk management. It is based on the natural assumption that investors prefer more rather than less. Each investor is shown to hold an efficient portfolio in the sense of Markowitz with some wealth invested in the market portfolio and the remainder deposited in the savings account. The market portfolio is shown to equal the growth optimal portfolio. Without using Markovianity, expected utility or equilibrium assumptions the capital asset pricing model is recovered. The dynamics of the discounted market portfolio turns out to be that of a time transformed squared Bessel process. The corresponding time transformation is determined by the discounted underlying economic value of the market portfolio. An equivalent risk neutral martingale measure does not exist under the resulting parsimonious market model. Risk neutral derivative pricing is generalized by fair pricing, which uses the growth optimal portfolio as numeraire and the real world measure as pricing measure.

Momentum Strategies Using Risk-adjusted Stock Selection Criteria

Teo Jasic, Germany

Abstract: We study the extension of momentum strategies by applying reward-risk stock selection criteria to momentum portfolio construction using daily data. Our alternative criteria are in the form of reward-risk ratios with risk measure that belongs to the class of coherent risk measures and, in different form, capture the risk of the tail distribution. Alternative risk-adjusted stock selection criteria are applicable when stock returns are not normally distributed. Additionally, we utilize the ratio criterion as the objective function in the portfolio optimization problem and obtain optimal risky winner and loser portfolios. We find that risk-adjusted stock selection criteria are able to generate more profitable momentum strategies than those based on usual cumulative or total return criterion. Moreover, our results are robust to transaction costs for both equal-weighted and optimized-weighted strategies. In particular, our alternative ratios outperform the cumulative return and the Sharpe ratio across all strategies measured by total realized return and independent performance measures over the observed period.

Penalized Splines and Financial Data

David Ruppert
Cornell University

Abstract: The talk will give an introduction to semiparametric regression modeling using penalized splines. Penalized splines bypass the thorny problem of knot selection which is necessary if one fits a regression spline by ordinary least squares. Penalty splines use a reasonably fine grid of knots, typically between 5 and 20, placed evenly on the support of the predictor variable. A roughness penalty is used to prevent undersmoothing. Unlike with smoothing splines, the number of knots of a penalized spline can be controlled by the modeler and can be much less than the number observations. This makes computations more tractable, especially for complex models. Applications to financial data include modeling of term structure, modeling the drift and volatily functions in the dynamics of short-term interest rates, and modeling time-varying coefficients in factor models. There are several ways to select the penalty parameter, including cross-validation, REML, and Bayesian estimation.

OTC Interest Rate Derivatives: Valuation and Relative Values

Nandi, Saikat
Fannie Mae

Abstract: Over-the-counter (OTC) interest rate derivatives such as swaptions, caps, callable bonds, cancelable swaps are popular vehicles to manage the prepayment risk of a MBS (Mortgage Backed Security) or a CMO (Collateralized Mortgage Obligation) security. However, a particular interest rate model that can value an interest rate derivative such as a swaption or a cap may not be computationally tractable in the context of computing the value of a MBS/CMO security. Thus, not all available interest rate models are particularly suitable for a valuation framework that seeks to value both the MBS/CMO security and an interest rate derivative on a consistent basis. We will discuss the class of interest rate models that can be used to value both the interest rate derivative and the MBS/CMO security on a consistent basis, while maintaining computational tractability. Also, we will discuss how relative value comparisons are sometimes made in the somewhat opaque market for American/Bermudan swaptions.

Time-Varying Risk Exposure of Hedge Funds

Monica Billio, Mila Getmansky, Loriana Pelizzon

Abstract: This article aims to investigate risk exposure of hedge funds using switching regime beta models. This approach allows to analyze hedge fund tail event behavior and in particular the changes in hedge fund exposure conditional on different states of various risk factors. We and that in the normal state of the market, the exposure to risk factors could be very low but as soon as the market risk factor captured by S&P500 moves to a crisis state characterized by negative returns and high volatility, the exposure of hedge fund indexes to the S&P500 and other risk factors may change significantly. We further extend the regime switching model to allow for non-linearity in residuals and show that switching regime models are able to capture and forecast the evolution of the idiosyncratic risk factor in terms of changes from a low volatility regime to a distressed state that are not directly related to market risk factors.

Jumps in Rank and Expected Returns: Introducing Varying Cross-sectional Risk

Gloria González-Rivera
University of California, Riverside

Santosh Mishra
Oregon State University

Tae-Hwy Lee
University of California, Riverside

Abstract: We propose an extension of the meaning of volatility by introducing a measure, namely the Varying Cross-sectional Risk (VCR), that is based on a ranking of returns. VCR is defined as the conditional probability of a sharp jump in the position of an asset return within the cross-sectional return distribution of the assets that constitute the market, which is represented by the Standard and Poor’s 500 Index (SP500). We model the joint dynamics of the cross-sectional position and the asset return by analyzing (1) the marginal probability distribution of a sharp jump in the cross-sectional position within the context of a duration model, and (2) the probability distribution of the asset return conditional on a jump, for which we specify different dynamics in returns depending upon whether or not a jump has taken place. As a result, the marginal probability distribution of returns is a mixture of distributions. The performance of our model is assessed in an out-of-sample exercise. We design a set of trading rules that are evaluated according to their profitability and riskiness. A trading rule based on our VCR model is dominant providing superior mean trading returns and accurate Value-at-Risk forecasts.

Revision of the Continuous Limit of GARCH

Carol Alexander and Emese Lazar
University of Reading, United Kingdom

Abstract: In 1990 Dan Nelson published a paper that demonstrated that the continuous limit of GARCH(1,1) is a stochastic variance diffusion with zero price-volatility correlation. There has been some debate about this result, because it depends on a specific assumption about the limiting behaviour of the GARCH parameters as the time step decreases to zero. Under alternative assumptions the continuous limit would be a deterministic model for the variance dynamics, as demonstrated by Valentina Corradi in 2000. Which assumptions are correct? To find a definitive answer to this question we have been forced to use the definition of GARCH that satisfies the time-aggregation property. Following Drost and Nijman this is called the ‘weak’ GARCH model and we prove that its continuous limit is a stochastic variance diffusion with non-zero price-volatility correlation and where the diffusion and correlation parameters depend on the skewness and kurtosis of the physical returns density.

Deriving smooth zero yield curves and inflation expectations from bond market

Zvi Wiener
Finance Department
School of Business Administration
The Hebrew University of Jerusalem

Abstract: We develop and test a mathematical method of deriving zero yield curve from market prices of government bonds. The method is based on a forward curve approximated by a linear (or piecewise constant) spline and should be applicable even for markets with low liquidity. The best fitting curve is derived by minimizing the penalty function. The penalty is defined as a sum of squared price discrepancies (theoretical curve based price minus market closing price) weighted by trade volume and an additional penalty for non-smoothness of the yield curve. We apply this method to both nominal and CPI linked bonds traded in Israel (some segments of these markets have low liquidity). The resulting two yield curves are used for derivation of market expected inflation rate. The main problems are low liquidity of some bonds and imperfect linkage to inflation in the CPI linked market. Usage of forward curves as the state space for the minimization problem leads to a stable solution that fits the data very well and can be used for calculating forward rates.

Option pricing for some Stochastic Volatility Models

A. Thavaneswaran
Department of Statistics, University of Manitoba

Jagbir Singh
Department of Statistics, Temple University

Abstract: In financial modeling, the kurtosis of the distribution plays an important role and therefore this paper considers the derivation of the kurtosis for a number of popular volatility models, including the class of random coefficient GARCH models. Option pricing formula for various class of volatility models are derived by using the moment properties of truncated normal distribution. A simple proof of the option pricing formula for Black Scholes model is given as a special case. Option pricing formula for some fuzzy volatility models are also discussed.
Key words: Stochastic volatility, Random coefficients, Kurtosis, Sign-switching, Option pricing, Fuzzy volatility.

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