Are Mispricings Long-Lasting or Short-Lived? Evidence from S & P 500 Index ETF Options

A rapidly 
growing literature has documented evidences suggesting the mispricing of 
options. Building on recent results of option pricing bounds imposed by 
stochastic dominance, this paper examines the time-series proprieties of such 
mispricing. In an application to high-frequency bid/ask quotes of S & P 500 index ETF options, this 
paper provides evidences that most violations of the stochastic dominance bounds last 
no more than 10 trading hours. The typical duration of mispricing is even 
shorter for near to maturity options. The results imply that the observed 
widespread mispricing in options might be the result of temporary inefficiency 
(e.g. transaction costs, overreaction, liquidity etc.) rather than a model 
misspecification, such as estimation biases of the parameters, or an overlooked 
persistent risk factor.


Introduction
Finance literature documents a number of evidences suggesting the mispricing of options. Given a steep smile in the implied volatility of S & P 500 index option, the out-of-the-money (OTM) options seem to be expensive [1] [2]. For example, shorting the zero beta straddles/strangles offered a return of 3.15 percent per week [3] [4]. Also, widespread violations of stochastic dominance by 1-month S & P 500 index call options imply that any risk-averse trader can improve expected utility by writing call options net of transaction costs and bid-ask spread [5] [6] [7]. Santa-Clara and Saretto [8] find that strategies involving short positions in options generally compensate the investor with Sharpe ratios as high as 1.69.  Answering these questions may shed light on the underlying mechanism of such mispricing. If the "mispricings" were the results of model mis-specifications, e.g. unknown risk factors, or errors in estimating parameters, they should be present for a long duration, as the flawed pricing models generate systematic pricing biases. If such "mispricings" were simply market temporary inefficiency due to market frictions or overreactions, they should be short-lived as arbitragers can take advantages of such opportunities quickly. The answer of long-lasting or short-lived mispricing is also crucial to practitioners since it determines how soon the arbitrage strategies will pay off. If prices converge towards fair value slowly, it may take too long to realize any profit.
This paper investigates the time series properties of option mispricings using high frequency bid-ask prices. After constructing option pricing bounds based on stochastic dominance, this paper provides evidences that most violations of the stochastic dominance upper bounds of Constantinides and Perrakis [7] last no more than 10 trading hours. This study also identifies that options move in and out of the pricing bounds frequently during the last few days before maturi-

Option Pricing Bounds Imposed by Stochastic Dominance
Constantinides and Perrakis [7] investigate the restrictions on option prices imposed by stochastic dominance. They conclude that options prices should stay within a set of bound in equilibrium. Otherwise, any trader can increase expected utility by trading in the options, the index, and the risk-free bond -hence violates the conditions of equilibrium.
In this study, the term "mispricing" is defined as the option prices which violate the restrictions in Constantinides and Perrakis [7].

Assumptions to Derive the Option Pricing Bounds
To derive the option pricing bounds, Constantinides and Perrakis [7] assumes

Pricing Bounds on Call Options
This section presents the pricing bounds on call options without proof. At any time t prior to expiration T, the upper bound on the price of a call option is given by where t S is the underlying price at time t, K is the strike, k is the proportional transaction costs, and S R is the expected return of the underlying per period.
For the lower bound, where f R is the gross risk-free return per period, and δ is the dividend yield.
Constantinides and Perrakis [9] also derived the option pricing bounds imposed by stochastic dominance on put options. However, empirical evidences suggest that the violations of bounds on puts are sparse [6]. Consequently, this

Data and Methodology
This research intends to find the mispricings implied by upper/lower bounds of option prices, and to examine the time series properties of such violations. This section first presents the methodology and dataset to construct the option pricing bounds. Then, we discuss about the criteria to identify the mispricings when both ask and bid price are present.

Estimation of Option Pricing Bounds
There are three steps involved to implement the empirical test: estimating input parameters and distribution; feed into Equation (1) and (2)  The first approach is to bootstrap from one month (22 trading days) overlapping index returns with a rolling window of six months (132 trading days), such that each day is the beginning of another one-month return. This is a widely accepted approach in financial investment industry.
In the second way, the conditional distribution comes from the forecast of ARMA(1,1)-GARCH(1,1) model with error terms distributed as skewed student's t. An ARMA(m, n)-GARCH(p, q) process model the index return as a stationary ARMA(m, n) process, and the conditional volatility as a GARCH(p, q) process. The deviation from traditional assumptions of normal distributed error terms allows for the negative skewness and excessive kurtosis observed in actual index return.
The third estimation approach parallels the bootstraps approach, but rescales the distribution such that the volatility of expected returns matches the adjusted VIX index. This method adopts the VIX index as the benchmark volatility because VIX reflects the market expectation of one-month ahead volatility. The rationales of the adjustment of VIX index are that VIX generally overpredicts the realized volatility. As a result, the last approach sets the adjusted VIX as the fitted values of regressing VIX index on realized volatility 1 .
The input parameters used to calculating the option pricing bounds are summaries as follows ( at 0.3%. This is based on the best estimation of two senior derivatives traders with more than 10 years of experiences in an assets management firm. Calculation of Pricing Bounds. Finally, after estimating the statistical distribution of index return, the calculation of the term (1) and (2)  To avoid any vagueness, this paper claims an option to be reasonably priced when both its bid and ask prices stand within the bounds. The duration of mispricings, as a result, is simply the span between the time when an option becomes mispriced for the first time, and the time when it subsequently turns back to be reasonably priced.

Data
The dataset in this study comes from Interactive Broker trading platform, which contains high frequency option bid-ask prices in a realistic trading environment

Caveats
Obviously, there are many other possible ways of estimating the statistical distribution of the S & P 500 index returns other than these three models listed here. One caveat of the empirical results of "mispricings", as a result, is simply that the options market is priced with a different probability distribution than any of the three estimated probability distributions. Nevertheless, this may not be a major concern. I argue that if "mispricings" would result from inappropriate estimated return distribution, the identified "mispricings" should be quite frequent and persistent. Yet, the final results indicate the opposite. The last section will conduct further robustness checks on the issue.
Another noticeable concern is that the option pricing bounds stated in section 2 were derived specifically to price European options. Yet, the options on Index ETFs are American style. Although our option pricing bounds would underestimate the true price, several empirical designs may ease such concern. Note that an American call is identical to a European one if there is no dividend. Fortunately, the S & P 500 index ETF has a schedule of dividend payment, i.e. approximately every three months. In order to eliminate the effect of dividend, the sample used in this study remove all options covering an ex-dividend date. Consequently, the resulted sample contains options with a maximum maturity of three months.

Empirical Results
This section describes the pattern of observed violations for the pricing bounds.
The first part of the exercise identifies that more than one third of the options are mispriced, especially for out-of-money options. The next exercise find that the average duration of mispricing lasts around 5.5 trading hours, and the prices move in and out of the bounds frequently as maturity approaches.  (1) and (2).    improve expected utility by writing those "mispriced" call options net of transaction costs and bid-ask spread.

The Frequency of Violations
An experiment not shown in Table 3 indicates that the typical violation size of the bounds is between $0.01 to $0.05 for 65% of the total violations. The violations are widespread, with a proportion of approximately 30%, when maturity approaches (less than one week).

The Duration of Violations
This section further investigates the time series properties of violations of pricing bounds. Specifically, Table 4 illustrates the average duration of the violations under different estimation approaches for options with different time to maturi- On average, the duration of a mispricing persists less than two trading days.
This implies that the majority of the mispricings disappear in a short period of time, which refutes the prediction of a persistent model misspecification. The bootstrapping method identifies a longer duration than others as it usually produces wider bounds.  the bounds. These agree with the arguments pinpointing the irrationality of investors shortly before maturity [10].
As argued before, the "mispricings" may be either the results of model miss-

Discussion
This section conducts several robustness checks which may undermine the results.
In the first place, the short-lived violations may purely results from overestimation of upper bounds or underestimate the lower bounds. For example, the upper bounds could be so high that only sporadic extreme market fluctuations are documented. To ease this concern, this paper manually adjusts the pricing bounds downwards. The results show that only when decreasing the bounds by as much as $0.15 could we observe an average mispricing duration of five trading days.
In addition, the Monte Carlo simulation to calculate the pricing bounds may also lead to frequent short term violations of the bounds, as the bounds fluctuate across time simply because of the sampling errors. To address this issue, the simulation in this paper employs a fixed seed in the random number generator.
Thus, this procedure could only bias the duration of the violations upward, as the sampling errors persist through time.

Conclusions
A number of literatures have documented evidences suggesting the mispricing of options. Since it is hard to believe the markets are inefficient for a long term, the observed "mispricing" might either result from transitory market inefficiency or from model misspecifications.
After constructing option pricing bounds based on stochastic dominance, this paper examines the time series properties of option mispricings using high frequency bid-ask quotes. This study contributes to the literature by showing that most violations of the stochastic dominance upper bounds of Constant inides and Perrakis [7] last no more than 10 trading hours. The results imply that the observed widespread mispricing in options might be the result of temporary inefficiency (e.g. transaction costs, overreaction, liquidity etc.) rather than a model misspecification, such as estimation biases of the parameters, or an overlooked persistent risk factor.
Possible extension that could substantiate the results obtained in this paper could be establishing a high-frequency trading rule and testing its profitability.
As the results of this paper suggest that the mispricings are mostly short-lived, traders could profit from fast convergence of mispriced options. Another possible future improvement could be including high-frequency quotes for a longer sample period. I leave these extensions to future endeavors.