Stephen Figlewski

Theory says an American call should never be exercised early, except possibly just before an ex-dividend date. But the best market bid is regularly lower than the intrinsic value for in-the-money short-maturity options. An American option can always be exercised to recover the intrinsic value, whereas selling a European call in the market may return considerably less. The article derives the liquidity value of American exercise in closed form as a function of the bid–ask spread and shows empirically that it is of comparable magnitude to, and often greater than, the theoretical value of American exercise to collect a dividend.

The authors dispute the theory that American call options should not be exercised early. By looking at intraday pricing, they find that the best bid price available is often lower than the option’s intrinsic value and early exercise can be rational. Our empirical results overturn the well-known textbook theory that American options should not be exercised early except in very limited conditions. In the real world, the best bid available in the market is frequently below the option’s intrinsic value (e.g., nearly half of all quotes for in-the- money calls), which we call an “exercise boundary violation” (EBV). In an EBV, early exercise can be the right strategy and the “American” option characteristic has economic value. Some option holders do exercise, some sell at EBV bid prices, and our results suggest that actual exercise behavior is much less irrational than earlier studies concluded. Disclosure: The authors report no conflicts of interest. Editor’s Notes Submitted 26 May 2019 Accepted 4 November 2019 by Stephen J. Brown. This article was externally reviewed using our double-blind peer-review process. When the article was accepted for publication, the authors thanked the reviewers in their acknowledgments. Giuseppe Ballocchi, CFA, and William Fung were the reviewers for this article.

Peter Christoffersen passed away on June 22, 2018 at an early age. He was one of the most prolific and gifted researchers in the area of derivatives, combining a very strong theoretical background with an appreciation for the practical details of real world markets. He will be sorely missed. This memorial article provides a few general comments about his work and offers reminiscences about Peter as a researcher, a teacher, and a mentor from three of his coauthors and former students.

In theoretical models, derivatives are redundant securities, which is what gives rise to pricing models based on arbitrage. Real world derivatives are not redundant, which gives rise to multiple related instruments and trading venues all based on the same underlying asset and all moving along somewhat together through time. Derivatives tied to the S&P 500 index offer a prime example. The index has spawned literally dozens of interrelated futures, options, and ETFs on both its level and also its volatility. When many S&P-related securities are trading simultaneously, the question arises of where new information enters this collection of markets first: which instruments lead and which follow along behind as prices adjust to new levels? In this article, Figlewski and Frommherz look at information leadership in both price levels and (implied) volatilities across the S&P 500 (SPX) spot market, E-mini (ES) futures and futures options, and the SPDR (SPY) exchange-traded fund and its options. For both price level and volatility, the SPY appears to be the dominant market for information flow. Among the different options, where volatility information is embedded, out of the money puts seem to contain the most new information, although out of the money calls were more important for SPX options. Contracts with very short maturity are found to lose their informational role, which passes to contracts with more time remaining to expiration.

Under Black–Scholes (BS) assumptions, empirical volatility and risk-neutral volatility are given by a single parameter that captures all aspects of risk. Inverting the model to extract implied volatility from an option’s market price gives the market’s forecast of future empirical volatility. But real world returns are not lognormal, volatility is stochastic, and arbitrage is limited; thus, option prices embed both the market’s estimate of the empirical returns distribution and also investors’ risk attitudes, including possibly distinct preferences over different volatility-related aspects of the returns process, such as tail risk. All these influences are reflected in the risk-neutral density (RND), which can be extracted from option prices without requiring restrictive assumptions from a pricing model. The author computes daily RNDs for the S&P 500 Index over 15 years and finds that risk-neutral volatility is strongly influenced both by investors’ projections of future realized volatility and by the ris...