Index Options-Futures Arbitrage: A Comparative Study with Bid/Ask and Transaction Data

We can infer from bid/ask quotations and transaction prices that where options contracts are traded under a competitive open‐outcry market‐making system, the options and futures markets are dynamically efficient. Ex‐ante analysis shows that potential arbitrage opportunities disappear within five minutes. Transaction price data understate both the frequency and magnitude of arbitrage opportunities that are signaled by bid/ask quotes. Quotes stale fast, so opportunities are short‐lived and some of the arbitrage opportunities are deceptive. Nonetheless, the evidence suggests that bid/ask quotes provide valuable trading signals to arbitrageurs. Profitability from exploiting the quotes is negatively related to execution delay and execution risk.     

Arbitrage violations and implied valuations: the option market

The ideas presented in this paper are those of the authors and not necessarily reflect the views of the National bank of Canada. Both authors thank the National Bank of Canada and the SSHRC of Canada for their help. Thanks are also due to Professor Y. Tian for his comments, and for participating, together with students of the Financial Engineering program at York University, in the data preparation and the execution of the Matlab programs. In this paper, we propose a necessary and sufficient condition for bid and ask prices of European options to be free of arbitrage, and derive from it an efficient numerical methodology to determine its satisfaction by a given set of prices. If the bid and ask prices satisfy the no-arbitrage (NA) condition, our methodology produces a vector of NA prices that lie between the bid and ask prices. Otherwise, our methodology generates a vector of arbitrage-free prices that is as close as possible, in some sense, to the bid–ask strip. The arbitrage-free prices detected by our methodology render the commonly used practice of using mid-points and then ‘cleaning’ arbitrage from them as unnecessary. Moreover, a vector of ‘cleaned’ prices obtained from mid-point prices may be outside the bid–ask spread even in an arbitrage-free market and, hence, in this case will not be representative of the current market. A new procedure of estimating implied valuation operators is also suggested here. This procedure is rooted in the economic properties of put and call prices and is based on Phillips and Taylor's approximation of a convex function. This approach is superior to common estimation techniques in that it produces an analytical expression for the implied valuation operator and is not data intensive as some other studies. Empirical findings for the new methods are documented and their economic implications are discussed.     

Are Options on Index Futures Profitable for Risk‐Averse Investors? Empirical Evidence

American options on the S&P 500 index futures that violate the stochastic dominance bounds of Constantinides and Perrakis (2009) from 1983 to 2006 are identified as potentially profitable trades. Call bid prices more frequently violate their upper bound than put bid prices do, while violations of the lower bounds by ask prices are infrequent. In out‐of‐sample tests of stochastic dominance, the writing of options that violate the upper bound increases the expected utility of any risk‐averse investor holding the market and cash, net of transaction costs and bid‐ask spreads. The results are economically significant and robust.     

Why Option Prices Lag Stock Prices: A Trading-based Explanation

While many studies find that option prices lead stock prices, Stephan and Whaley (1990) find that stocks lead options. We find no evidence that options, even deep out-of-the-money options, lead stocks. After confirming Stephan and Whaley's results, we show their results can be explained as spurious leads induced by infrequent trading of options. We show that the stock lead disappears when the average of the bid and ask prices is used instead of transaction prices. Hence, we find no evidence of arbitrage opportunities associated with the stock lead.     

Transactions data tests of efficiency of the Chicago board options exchange

This paper performs lower boundary condition tests based on rational pricing of call options and an implied standard deviation test based on the bid/ask prices of options. These efficiency tests attempt to closely approximate conditions in the option markets to avoid the pitfalls indicated by Phillips and Smith (1980). The tests use transactions data and account for the effects of stock and option bid/ask prices, simultaneity of stock and option prices, depth of market, execution lag and transaction costs. The small and relatively infrequent profits due to market mispricing disappear in the lower boundary tests when transaction costs are taken into account. Frequent violations of the tighter boundary conditions in the implied standard deviation test are reported, but the estimated profits cannot be unambiguously attributed to option market inefficiency.     

The liquidity effects associated with addition of a stock to the S&P 500 index: evidence from bid/ask spreads

This study examines changes in stock liquidity, as measured by the bid/ask spread, when a stock is added to the S&P 500 Index. The paper presents evidence of a significant decrease in the bid/ask spread upon S&P 500 addition, however, this effect is limited to only those stocks that were not trading listed options. Further, the decrease in the bid/ask spread for nonoptioned stocks is accompanied by a significant and permanent increase in share price and trading volume. While optioned stocks experience a permanent increase in trading volume, they experience only a temporary increase in share price. The findings for optioned stocks support the hypothesis that the price and volume effects associated with S&P 500 addition derive from temporary price pressure. Findings pertaining to the nonoptioned stocks indicate that the price and volume effects associated with S&P addition reflect enhanced stock liquidity. The decrease in the bid/ask spread for nonoptioned stocks is attributed to informational efficiencies achieved via index arbitrage trading, and it is argued that this effect is mitigated for optioned stocks due to the pre-existence of arbitrage trading between the option and the underlying stock.     

The Price of Immediacy

This paper models transaction costs as the rents that a monopolistic market maker extracts from impatient investors who trade via limit orders. We show that limit orders are American options. The limit prices inducing immediate execution of the order are functionally equivalent to bid and ask prices and can be solved for various transaction sizes to characterize the market maker's entire supply curve. We find considerable empirical support for the model's predictions in the cross-section of NYSE firms. The model produces unbiased, out-of-sample forecasts of abnormal returns for firms added to the S&P 500 index.     

The relevance of information and trading costs in explaining momentum profits: Evidence from optioned and non-optioned stocks

Considerable evidence from many countries suggests momentum strategies generate profits. These have been difficult to rationalise and evidence on the sources of such profitability is inconclusive. We utilise a sample of optioned stocks, characterised by high liquidity, high market capitalisation and fewer short sales constraints and compare results with control samples of non optioned stocks chosen on the basis of market value, turnover and bid–ask spread. The sample characteristics, and the fact that derivatives improve the impounding of information into prices, enable us to draw conclusions about the causes of momentum profits. While we find that short sales constraints are not the major driver of profitability and that most momentum profits disappear using two transactions costs measures of the bid–ask spread, one not previously used, the persistence of some momentum profits indicates that the market underreacts even to the most publicly available information.     

On the Distribution of CBOE Option Trade Prices Occurring Between Consecutive Stock Trades

This paper examines the volume distribution of option trade prices that occurs when the underlying stock price remains constant. The width of these option trade price bands provides direct evidence on the law of one price and the redundancy of options assumed in many option models. We find that index option bands are narrower than equity option bands. Furthermore, for both equity and index options, puts have narrower bandwidths than calls. In general, option price bandwidth is narrow and can be explained by the minimum price movement allowed by the Chicago Board Options Exchanges (CBOE). This supports the single price law and the redundancy assumption. The existence of bid/ask quotes on the option does not materially affect the above results although it does alter the frequency of multiple option trade prices for a given underlying stock price. We note that over 53% of option trading volume occurs without bid/ask quotes on the CBOE compared to less than 15% a decade ago. Our results suggest that the effective bid/ask spread on options is probably no larger than the minimum price movements allowed by the CBOE. Furthermore, the need for the liquidity services of market makers may be declining if the decline in quoting activity stems from cross trading (i.e. trades not involving market makers).     

EMPIRICAL TESTS OF THE EFFICIENCY OF THE CURRENCY OPTION MARKET

Tests of a hedge and a rational boundary of the efficiency of the currency option market are conducted in this study. These tests use transactions data and account for the effects of currency and option bid/ask spreads, synchronization of option prices and underlying exchange rates, market depth, execution lags, and transaction costs. Currency options, unlike domestic stock options, exhibit continuous dividends. The nature of the option and of the data set employed makes the immediate exercise lower bound test one of the purest tests of market efficiency to date. Results reported here indicate no ability to earn abnormal economic or riskless arbitrage profit for the period when these tests are conducted.     

Pricing Efficiency in a Thin Market with Competitive Market Makers: Box Spread Strategies in the Hang Seng Index Options Market

Using a box spread arbitrage strategy, we examine the pricing efficiency of the emerging, thinly traded Hang Seng Index options market in Hong Kong, where market makers operate under a competitive open outcry system. In 20 months of tick‐by‐tick bid‐ask and transaction quotes we find very few arbitrage opportunities. Our examination of the reporting time of quotes shows that in effect, all the apparent mispricings are deceptive and could be explained by stale quotes. The absence of real arbitrage opportunities supports the pricing rationality hypothesis in the Hong Kong options market.     

A multiple-curve Lévy forward rate model in a two-price economy

An advanced Heath–Jarrow–Morton forward rate model driven by time-inhomogeneous Lévy processes is presented which is able to handle the recent development to multiple curves and negative interest rates. It is also able to exploit bid and ask price data. In this approach in order to model spreads between curves for different tenors, credit as well as liquidity risk is taken into account. Deterministic conditions are derived to ensure the positivity of spreads and thus the monotonicity of the curves for the various tenors. Valuation formulas for standard interest rate derivatives such as caps, floors, swaptions and digital options are established. These formulas can be evaluated numerically very fast using Fourier-based valuation methods. In order to exploit bid and ask prices we develop this approach in the context of a two-price economy. Explicit formulas for bid as well as ask prices of the derivatives are stated. A specific model framework based on normal inverse Gaussian and Gamma processes is proposed which allows for calibration to market data. Calibration results are presented based on multiple-curve bootstrapping and cap market quotes. We use data from September 2013 as well as September 2016. The latter is of particular interest since rates were deep in negative territory at that time.     

A quasi-analytical interpolation method for pricing American options under general multi-dimensional diffusion processes

We present a quasi-analytical method for pricing multi-dimensional American options based on interpolating two arbitrage bounds, along the lines of Johnson in J Financ Quant Anal 18(1):141–148 (1983). Our method allows for the close examination of the interpolation parameter on a rigorous theoretical footing instead of empirical regression. The method can be adapted to general diffusion processes as long as quick and accurate pricing methods exist for the corresponding European and perpetual American options. The American option price is shown to be approximately equal to an interpolation of two European option prices with the interpolation weight proportional to a perpetual American option. In the Black-Scholes model, our method achieves the same efficiency as the quadratic approximation of Barone-Adesi and Whaley in J Financ 42:301–320 (1987), with our method being generally more accurate for out-of-the-money and long-maturity options. When applied to Heston’s stochastic volatility model, our method is shown to be extremely efficient and fairly accurate.     

A new methodology for studying intraday dynamics of Nikkei index futures using Markov chains

The empirical study of intraday patterns of stock trading volatilities and bid–ask spreads in the literature depends on assumptions of specific price generating process and may therefore not be robust to distributional assumptions. By creating discrete states that conform more naturally to the way prices are actually quoted in the derivatives and assets markets, we employ a new methodology of Markov chains for studying the intraday dynamics of derivative prices. We apply the method to study the intraday behavior of the Nikkei index futures prices, trading volumes, and spreads. We find some interesting results such as higher probabilities of transitions between larger volatilities at the opening and closing times. The volatility at lunch break is strikingly low. Contrary to most of the literature, the Nikkei intraday bid–ask spread does not show a U-shaped pattern. We offer some explanations.     

Conditions on option prices for absence of arbitrage and exact calibration

Under the assumption of absence of arbitrage, European option quotes on a given asset must satisfy well-known inequalities, which have been described in the landmark paper of Merton [Merton, R., 1973. Theory of rational option pricing. Bell Journal of Economics and Management Science 4 (1), 141–183]. If we further assume that there is no interest rate volatility and that the underlying asset continuously pays deterministic dividends, cross-maturity inequalities must also be satisfied by the bid and ask option prices.     

Asset pricing theory for two price economies

We show that nonlinearly discounted nonlinear martingales are related to no arbitrage in two price economies as linearly discounted martingales were related to no arbitrage in economies satisfying the law of one price. Furthermore, assuming risk acceptability requires a positive physical expectation, we demonstrate that expected rates of return on ask prices should be dominated by expected rates of return on bid prices. A preliminary investigation conducted here, supports this hypothesis. In general we observe that asset pricing theory in two price economies leads to asset pricing inequalities. A model incorporating both nonlinear discounting and nonlinear martingales is developed for the valuation of contingent claims in two price economies. Examples illustrate the interactions present between the severity of measure changes and their associated discount rates. As a consequence arbitrage free two price economies can involve unique discount curves and measure changes that are however specific to both the product being priced and the trade direction. Furthermore the developed valuation operators call into question the current practice of Debt Valuation Adjustments.     

Stochastic Volatilities and Correlations of Bond Yields

I develop an interest rate model with separate factors driving innovations in bond yields and their covariances. It features a flexible and tractable affine structure for bond covariances. Maximum likelihood estimation of the model with panel data on swaptions and discount bonds implies pricing errors for swaptions that are almost always lower than half of the bid–ask spread. Furthermore, market prices of interest rate caps do not deviate significantly from their no‐arbitrage values implied by the swaptions under the model. These findings support the conjectures of Collin‐Dufresne and Goldstein (2003) , Dai and Singleton (2003) , and Jagnnathan, Kaplin, and Sun (2003) .     

Relevance of the disposition effect on the options market: New evidence

A moneyness‐based propensity to sell (MPS) measure, at the aggregate level, determines the propensity of option holders to exercise their winning relative to losing positions. Using data on individual stock and S&P 500 Index options, we find that the MPS measure has significant predictive power over the cross section of delta‐hedged option returns. We test the disposition effect in the options market based on a long–short strategy that exploits price distortions induced by the disposition bias. More pronounced evidence of the disposition bias is found for individual at‐the‐money call options than put options where the significance of abnormal returns remains robust across different subsamples even after we control for the portfolio option greeks and market‐based risk factors. The profitability of the long–short strategy is related to limit‐to‐arbitrage proxies suggesting that behavioral explanations help explain the positive relation between the MPS measure and delta‐hedged option returns.     

Efficiency in index options markets and trading in stock baskets

Researchers have reported mispricing in index options markets. This study further examines the efficiency of the S&P 500 index options market by testing theoretical pricing relationships implied by no-arbitrage conditions. The effect of a traded stock basket, Standard and Poor’s Depository Receipts (SPDRs), on the link between index and options markets is also examined. We find that pricing efficiency within option markets improves but there is little evidence to support the hypothesis that a stock basket enhances arbitrage across markets. When transactions costs and short sales constraints are included, very few violations of inter-market pricing relationships such as put–call parity are reported. However, violations of within market pricing relationships such as the box spread remain frequent. Extensive analysis suggests that the results are robust.     

Relative performance of bid–ask spread estimators: Futures market evidence

The issue of transaction costs is the mainstay of the equity market microstructure. Research in the microstructure of futures markets has lagged behind. A primary reason is that futures exchanges in the U.S. do not record bid–ask quotes, requiring these costs to be imputed from transaction price data. A reliable estimator of bid–ask spreads would significantly enhance microstructure research in futures markets. Unique intraday data from the Sydney Futures Exchange (SFE) that include both transaction prices and bid–ask spreads allow us to compare bid–ask spread estimation techniques proposed in the literature against the benchmark of actual spreads in a futures market, and thus identify the best-performing estimator. To maximize relevance, we impose all the constraints that apply in U.S. futures data to perform our estimations. We find that the four bid–ask spread estimators considered significantly underestimate the actual spreads. However, simple moments-based estimators perform better in predicting spreads.