Barry Schachter

Unbiased estimation of the Black/Scholes formula

Abstract The Black/Scholes model gives the price of an option as a function of the true variance rate of the underlying stock and other parameters. Because the true variance rate is unobservable, an estimate of the variance rate is used in empirical tests. But, because the Black/Scholes formula is non-linear in the variance, option price estimates using an estimated variance are biased, even if the variance estimate itself is unbiased. This paper develops an unbiased estimator of the Black/Scholes formula from a Taylor series expansion of the formula and the properties of the pdf of the estimated variance.

Interday variations in volume, variance and participation of large speculators

Abstract We use data uniquely available from the Commodity Futures Trading Commission (CFTC) to document the intraweek trading patterns of large speculators in five futures markets. These markets include futures traded against the Standard and Poors 500 stock index, Treasury Bonds, gold, corn, and soybeans. We also examine the influence of large speculator trades on the patterns of volume and volatility for the contracts in our sample. Though we detect the familiar U-shaped and inverted U-shaped patterns across weekdays for volatility and aggregate volume, the association between volume and volatility becomes stronger when we separate large speculator volume from volume associated with other traders. The coefficient on large speculator volume is much larger than the coefficient on other volume in these regressions. Compared with total volume, large speculator volume is greater on Mondays than on the other days of the week in all five markets.

Derivatives Regulation and Financial Management: Lessons from Gibson Greetings

M On April 19, 1994, Gibson Greetings, Inc. (Gibson), a manufacturer of seasonal cards, wrapping paper, and related products with headquarters in Cincinnati, Ohio, filed an 8-K with the SEC, announcing that it had taken a

Open Interest In Stock-Options Around Quarterly Earnings Announcements

16.7 million charge against first quarter income resulting from losses on two swap transactions with BT Securities Corporation, a subsidiary of Bankers Trust New York Corporation (BT). The announcement stated that this loss was in addition to a

Risk Parity: Rewards, Risks, and Research Opportunities

In this paper I document the behavior of open interest in listed stock options around quarterly earnings announcements. Using pooled timeseries cross-section regressions, I find a significant decline in open interest prior to announcements. This behavior cannot be explained by the reactions of investors to good and bad news. I also find in several separate regressions that open interest behaves differently for options whose values are most sensitive to changes in stock volatility, suggesting the behavior of open interest is correlated with recently documented temporary increases in the volatility of stock returns around announcement dates. Many studies based on the volume and price behavior of equities have confirmed that investors respond to public announcements thought to contain information. Interpreting those responses is a more difficult task. For example, while Beaver [1968] has argued that consensus among investors is an important factor in investor responses, others have shown that it is not possible to make unambiguous inferences about consensus by examining price and volume alone (see Miller [1977], Verrecchia [1981], and Karpoff [1986]).

Robust Risk Estimation and Hedging: A Reverse Stress Testing Approach

Mean–variance optimization has recently come under great criticism based on the poor performance experienced by asset managers during the global financial crisis. In response, an alternative approach, called risk parity, which proceeds by equalizing risk contributions, has garnered much interest. The authors summarize the work of a group of leading researchers on risk parity chosen for this special issue. They survey more generally what is known about this approach. Although risk parity has intuitive appeal and has performed well over some historical time periods, it is premature to claim the superiority of risk parity over other asset allocation approaches. The authors raise several conceptual and practical questions about risk parity that they think are worthy of additional research.

LDI: Complexities, Challenges, and Opportunities

The stress test has become an increasingly important risk assessment and management tool. But while it is easy to imagine a stress scenario and to estimate its impact on the firm’s financial condition, it is not so obvious how to select the most meaningful scenarios in the first place, either to get reasonable coverage of the space of stressful possibilities or even to focus on those that are most probable. In this article, the authors approach the problem from the reverse direction. They begin with a specified level of loss and pick the most likely scenario that generates that loss. They then use principal components to construct a set of alternative scenarios that produce the same level of loss but in (maximally) different ways. This provides much greater insight into which sources of risk are the most important and the most stable across scenarios.

An Empirical Investigation of Variance Reduction Through non-Delta Neutral Hedging

This special issue of The Journal of Investing presents a set of articles that addresses some facets of the complex issues around liability-driven investment (LDI). Given the complexity of LDI, the authors recognize that many issues have been left unaddressed in this special issue. However, the researchers in this special issue have considerable experience in managing fixed-income portfolios and in designing custom solutions for client needs in the LDI arena. The articles address important issues, such as the role of risk factors in LDI solutions, distinguishing between liability-hedging and risk-seeking assets in designing LDI programs, the importance of diversification for LDI, and, importantly, the role of glide paths in dynamic de-risking.

On the existence of an optimal tick size

The option delta plays a key role in eliminating the risk in continuously rebalanced hedges. This need not be the case in discretely rebalanced hedges. Robins and Schachter (Management Science, June 1994) show how the minimum variance discretely rebalanced (European call) option hedge ratio deviates from the option delta. In this paper, we use the binomial option pricing framework to calculate an analytical approximation to the return variance of a discretely rebalanced (American call) option hedge. We then conduct an empirical examination of the differences between delta-neutral and non-delta neutral (i.e., minimum variance) hedges. We use OEX option quotes for the first 42 trading days of 1986. We construct hedges (buying options at the ask and selling at the bid) with rebalancing intervals of from 1 to 10 days. We categorize realized hedge returns according to rebalancing interval, option time to expiration and degree to which the option is in- or out-of- the-money. Using the heteroskedasticity and autocorrelation consistent test statistic developed by Mizrach (1992) we document statistically significantly smaller hedge variances for the non-delta-based hedges across most categories.