Donald R. Chambers

An Improved Approach to Computing Implied Volatility

A well-known problem in finance is the absence of a closed form solution for volatility in common option pricing models. Several approaches have been developed to provide closed form approximations to volatility. This paper examines Chances (1993, 1996) model, Corrado and Millers (1996) model and Bharadia, Christofides and Salkins (1996) model for approximating implied volatility. We develop a simplified extension of Chances model that has greater accuracy than previous models. Our tests indicate dramatically improved results.

A Tree Model for Pricing Convertible Bonds with Equity, Interest Rate, and Default Risk

Black and Scholes modeled stock options assuming the underlying stock was exposed to a single source of risk. Once it is recognized that both stocks and bonds are actually derivatives written on the underlying firm, many kinds of hybrid securities, like convertible bonds, are seen to have exposure to both stock market risk and interest rate risk. Further, as attention has turned to default as a critical risk, it is now apparent that properly valuing and risk-managing these instruments requires dealing formally with that third risk factor, as well. Developing tractable models for these cases seems like a daunting task, but Chambers and Lu present a lattice-based approach that does it. Their approach is easily implemented; it embeds important real world features of the problem, including correlation between stock price and interest rate changes; and it shares the advantage of other tree-based numerical methods, that increasing the number of time steps to improve accuracy does not cause execution time to explode in an unmanageable way. A real world example illustrates the use of the technique to price a convertible bond issued by Lucent.

The Limitations of Diversification Return

Diversification return is the amount by which the geometric mean return (i.e., average compounded return) of a portfolio exceeds the weighted average of the geometric means of the portfolio’s constituent assets. Diversification return has been touted as a source of added return, even if markets are informationally efficient. Portfolio rebalancing has been advocated as a valuable source of diversification return. The authors demonstrate that diversification return is not a source of increased expected value. However, portfolio rebalancing can be an effective mean-reverting strategy. Any enhanced expected value from rebalancing emanates from mean-reversion, rather than from diversification or variance reduction.

The Role of the Constant Recovery Assumption in the Subprime Bubble

Securitization of mortgages is believed to have contributed to the recent boom and bust in real estate. In particular, structured products with wide tranches of AAA-rated derivative securities are retrospectively vilified.A key issue has been to understand how such large tranches of securities could have been viewed as safe despite the apparent high risks of the underlying mortgages. In this article, the authors analyze the complex models and the models’parameters to estimate the risks of the structured securities.They find that the models themselves, not the inputted parameters, such as expected default rates and correlations, were responsible for inappropriate ratings. In particular, they find that the assumption of a constant recovery rate generated generous ratings for large tranches of securities even with reasonable parameter estimates.

CDO Squareds: The Case of Subprime Mortgages

Investment in residential subprime mortgages through structured products was at the heart of the financial crisis that began in 2007. Perhaps the most toxic such product was the CDO-squared. CDO-squareds are CDOs (collateralized debt obligations) with tranches of other CDOs serving as their collateral pool. CDO-squareds on mortgage-backed securities (ABS-CDOs) issued vast quantities of AAA rated securities that quickly fell to market values of a few cents on the dollar. This article examines ABS-CDOs with two primary purposes. First, it provides an intuitive explanation of ABS-CDOs, including their characteristics, risks, and models. Second, it performs a comprehensive sensitivity analysis of the AAA senior tranche width with respect to various parameters based on the Hull and White model of mortgage-backed securities. The authors find that ABSCDO senior tranche widths were unreasonable prior to the financial collapse that began in 2007, and they attribute the associated rating errors with the non-linearity, hyper-sensitivity, and complexity of the risks of the products with respect to their underlying parameters. Simply put, there is no reliable basis on which accurate measurements of risk can be made using existing data and theory.

Semivolatility of Returns as a Measure of Downside Risk

Within statistics, semistandard deviation is a well-known measure used to analyze the dispersion of probability distributions. In finance, semistandard deviation of returns is sometimes defined consistently with its statistical definition, but it is sometimes defined differently. The ambiguity emanates from whether the number of observations in its calculation is specified as T, the total number of observations in a sample, or T*, the number of negative deviations. The authors show that the use of T is consistent with the statistical definition but generates a measure that cannot be directly compared to standard deviation. Practitioners should be aware of the implications of using either T or T* both as a stand-alone risk measure and as the denominator of the Sortino ratio. The authors derive an alternative measure of downside risk based on T* that provides several advantages over semistandard deviation. They term that measure semivolatility and demonstrate its usefulness.

Understanding the Estimation Risks of Value at Risk

This article discusses estimation of value at risk (VaR) in the context of alternative investments. The diverse return distributions of alternative investments raise both opportunities and challenges for value at risk. The opportunities arise from the inadequacy of traditional risk measures such as volatility and beta to capture the risks of alternative investments. The challenges arise from the difficulty of estimating VaR when return distributions are not well understood and/or trading strategies are dynamic.

Capital Gains Taxation and Equity Returns: The Case of Mutual Savings Banks

IPOs of demutualized savings banks create tax sensitive shareholders with identical acquisition dates and tax bases. We investigate security volume and returns surrounding the one�?year anniversary of the IPOs when unrealized capital gains and losses for original shareholders become long�?term capital gains for taxation purposes. Trading volume levels confirmed our hypothesis that investors defer the recognition of capital gains, but we could not confirm that tax motivated trading affected security prices. The results have implications beyond taxation and US markets. For non�?institutional investors, presumed to exhibit non�?rationality, tax motivated trading exists and does not have a significant effect on prices.

Investigation of arbitrage opportunities in the Eurodollar futures market using neural networks

Financial analysis is based on two opposing views: market efficiency theory and technical or fundamental analysis. There are three forms of market efficiency: weak form, semi-strong form and strong form. The weak form of market efficiency precludes the trends and patterns that technical analysis attempts to exploit. This work investigates whether it is possible to detect or predict patterns underlying Eurodollar futures trading. Multilayer perceptrons and radial basis functions were chosen to predict three time series based on different financial models. The findings challenge the existing efficient market theory in the case of Eurodollar futures trading.