Relative Implied-Volatility Arbitrage with Index Options

In the study reported here, we investigated the efficiency of markets as to the relative pricing of similar risk by using implied volatilities of options on highly correlated indexes and a statistical arbitrage strategy to profit from potential mispricings. We first analyzed the interrelationships over time of the three most highly correlated and liquid pairs of U.S. stock indexes. Based on this analysis, we derived a relative relationship between implied volatilities for each pair. If this relationship was violated (i.e., if we detected a relative implied-volatility deviation), we suspected a relative mispricing. We used a simple no-arbitrage barrier to identify significant deviations and implemented a statistical arbitrage trade each time such a deviation was recorded. We found that, although many deviations can be observed, only some of them are large enough to be exploited profitably in the presence of bid–ask spreads and transaction costs. The question of market efficiency is of great interest to practitioners and academics. For derivatives markets, many tests of market efficiency have examined arbitrage relationships; none, however, has attempted to test the efficiency of options markets regarding relative implied volatilities of highly correlated underlying assets—or, in other words, the relative pricing of similar risk. This aspect of option market efficiency cannot be tested in a conventional way through the use of exact arbitrage relationships. Rather, a statistical arbitrage rule has to be applied. In the study reported here, we followed this approach to investigate the relative pricing of U.S. stock index options.First, from end-of-month March 1995–December 1999 data, we ranked all pairs of U.S. stock indexes for which exchange-listed options are available. Of 11 indexes, we found that the S&P 500 Index (SPX), S&P 100 Index (OEX), and NYSE Composite Index (NYA) exhibited the highest correlations and the best liquidity. We thus selected these three for the analysis.Second, we analyzed the interrelationship over time of the SPX–OEX, SPX–NYA, and OEX–NYA pairs by using rolling ordinary least-squares regressions of index returns and a robust minimum–maximum approach to determine high and low boundaries for regression coefficients. Based on this analysis, we show that relative implied-volatility relationships can be derived for options with matching maturities. That is, given an observed implied volatility, we obtained low and high boundaries for the implied volatility of the option on the other index. If such a relationship was violated (i.e., if relative implied volatility of the one option was higher than the high boundary or lower than the low boundary of the other option), we considered that we had observed a deviation and we suspected relative mispricing.We measured implied volatility deviations from 3 January 1995 through 10 February 2000 and found that, although many deviations were observable, only a few of them were large enough to be used profitably in the presence of bid–ask spreads and transaction costs. Therefore, to test an arbitrage trading strategy, we used an additional margin as a no-arbitrage barrier to account for spreads and trading costs. The deviations that remained after we imposed this barrier we considered significant enough to allow the implementation of a potentially profitable statistical arbitrage trade.We implemented a statistical arbitrage trade for the period January 1995 through February 2000 every time such a deviation was newly recorded. Such situations occurred on 10 days for the SPX–OEX pair, on 37 days for the SPX–NYA pair, and on 28 days for the OEX–NYA pair. The average holding time before relative volatilities were back within the no-arbitrage boundaries and the position was closed out was 2.9 days for the SPX–OEX pair, 9.6 days for the SPX–NYA pair, and 3.6 days for the OEX–NYA pair.On average, the trades were profitable for all index pairs after deduction of transaction costs. One losing trade occurred for the SPX–OEX pair, and six losing trades occurred for each of the other pairs. For all pairs, the average profit was larger than the average loss. With respect to profitability, however, because we used daily closing prices and assumed that trading would be possible at the signal price, not the tick following the signal price, we could not conclude with certainty that such a trading strategy would have generated exactly the same results in practice.Overall, although the arbitrage strategy was profitable, we found too few potential arbitrage transactions to conclude that option markets are inefficient.