Information Content of Right Option Tails: Evidence from S&P 500 Index Options

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1 Information Content of Right Option Tails: Evidence from S&P 500 Index Options Greg Orosi* October 17, 2015 Abstract In this study, we investigate how useful the information content of out-of-the-money S&P 500 index call options is to predict the size and direction of the underlying index for the period First, based on the results of Orosi (2015), we demonstrate that behavior of the right tail of the option implied risk-neutral distribution can be characterized by a single parameter. Subsequently, we find that weekly changes in the tail parameter can be used to devise trading strategies that outperform the underlying index on a risk adjusted basis. In particular, two strategies that have comparable drawdowns to the S&P 500 index, have cumulative returns of % and %, while the index has a cumulative return of 60.16%. Keywords: Trading strategy, Option-implied information, Performance evaluation JEL Classification: C13, G10, G11, G13 1 Introduction The idea that option prices contain information about future density of the underlying asset has been understood since the work of Black and Scholes (1973). It is now well established that option-implied information can be used to construct portfolios with improved risk-return characteristics (see for example DeMiguel et al. (2013)). Moreover, Audrino et al. (2014) find that profitable trading strategies can be devised by estimating the physical density from the risk-neutral one. In this study, we propose a trading strategy that relies on the information content of out-of-the-money S&P 500 index call options to predict the future change in the underlying index. First, based on the results of Orosi (2015), 1 Assistant Professor, Department of Mathematics and Statistics, American University of Sharjah, Offi ce: Nab 254, P.O. Box 26666, Sharjah, UAE, Tel: , 1 Electronic copy available at:

2 we demonstrate that a single parameter characterizes the behavior of the right tail of the option implied risk-neutral distribution implied by option prices. Subsequently, we find that this parameter contains predictive information about the S&P 500 index and that it can be used to devise trading strategies with improved risk-return characteristics. The remainder of the paper is organized as follows. In Section 2, we summarize the findings of Orosi (2015), and introduce the trading strategies. In Section 3, we present the results of our empirical study and discuss our findings. Finally, Section 4 presents our conclusions. 2 The Trading Strategy In this section, we review the work of Orosi (2015) and point out that the behavior of the right tail of the distribution is determined by a single parameter. Moreover, we illustrate how trading strategies can be developed using these results. 2.1 Extracting Information from Option Prices Orosi (2015) demonstrates that call option prices that provide a good fit S&P 500 index call options can be generated using the following procedure. First, the following transformations are applied to the call option prices, C(K, T ), with strike price K and maturity T : c = C(K, T ) Se dt x = Ke rt Se dt. where S is the price of the underlying, r is the interest rate, and d is the continuous dividend yield. Then, the following the model is fitted to all available call option prices: (1 c)β x = 1 c + G c α, (1) where G, β, and α are three-parameters of the model with the following restrictions: G > 0, α > 0, β > 2. The behavior of the call prices for large strikes can be determined by first observing that 1 c + G c (1 c) β c α c α+1 + G (1 c) β lim α c G = lim c 0 c α + G c α = 1. Then, the above implies that the call option prices given by (1) are asymptoti- 2 Electronic copy available at:

3 cally equivalent to call prices given by (1 c)β x = 1 c + G c α. Therefore, as c approaches zero (or equivalently as x approaches infinity), the transformed strikes in (1) are approximately given by x 1 + G c α. From the above, the transformed call options are approximately given by c ( ) 1 G α 1 G α (x 1) 1 α. x 1 Assuming equality in the above relation, the call option prices can be expressed as C (K, T ) = G 1 α S ( Ke rt S ) 1 α 1, (2) and the binary call option price and the survival function (complementary cumulative distribution function) are B(K, T ) = G 1 α P (S T > K) = e rt B (K, T ) = G 1 α ( ) 1 Ke rt 1 α 1 α Se dt 1 (3) e rt α ( ) Ke rt 1 α 1 Se dt 1. (4) It can be easily shown that the above distribution is a type of generalized Pareto distribution (see, e.g., Orosi (2015)). Therefore, since the shape parameter determines the weight of the tail of the distribution, the behavior of the right tail of the option implied distribution is determined by α. Specifically, the shape parameter of the risk-neutral probability distribution, ξ S, is given by Moreover, note that ξ S = α α + 1. (5) α 2 α α 1 α = α 2 α 1 (α 1 + 1)(α 2 + 1). Therefore, if α increases or decreases, then the shape parameter increases or decreases accordingly. 3

4 2.2 Trading Strategies It is well known there is a difference between real-world parameters and riskneutral ones that are derived from option prices. Although studies attempting to recover the physical density or moments from the risk-neutral ones rely on restrictive assumptions (see for example Aït-Sahalia and Lo (2000), Ross (2013), and Ghysels and Wang (2014)), some empirical evidence indicates that these approaches can be useful to practitioners. For example, Audrino et al. (2014), using the recovery theorem of Ross (2013), extract the market s forecast of the physical density and use this to develop trading strategies that outperform the S&P 500 index. Based on the above results, we consider two trading strategies. The first strategy (referred to as Strategy 1 hereafter) takes both long and short positions and is based on the following rule: if the value in parameter α is larger than it was the week before, we go long the S&P 500 index for one week; otherwise we short the index for one week. The idea behind the strategy is that when the right tail of the risk-neutral density (and hence distribution) gets fatter or thinner, investors expect the market to rise or fall, respectively. Therefore, it takes advantage of the information content of option prices by extracting the market s current expectation about the change in the underlying. The second strategy (referred to as Strategy 2 hereafter) only takes on long positions and is based on the following rule: if the value in the parameter α is larger than it was the week before, we go long the S&P 500 for one week; otherwise we do not take on a position. Note Strategy 2 is similar to Strategy 1, and it avoids shorting the index because the historical weekly change in the S&P 500 index is positive more often than negative. 3 Empirical Analysis In this section, we outline our methodology. Moreover, we evaluate the performance of the two trading strategies by comparing these with the performance of the S&P 500 index. 3.1 Data To assess the performance of the proposed trading strategies, for our empirical work, we use S&P 500 Index call option prices obtained from the IVolatility 1 database. Moreover, we use daily closing prices for each Wednesday between January 1, 2004 to December 31, To filter the data for possible biases, the following criteria are applied: (i) following Bakshi, Cao, and Chen (1997), we exclude all options that violate at least one of a number of basic, no-arbitrage conditions and contracts that cost less than $1; (ii) we consider only data for 4

5 contracts with more than six trading days to maturity and less than a year to maturity, and exclude maturities with fewer than ten strike prices; (iii) we exclude options with absolute moneyness in excess of 20%, where moneyness is given by the following: M = K S 1, S representing the stock price and K the strike price. Table 1 summarizes the remaining data set consisting of 137,654 contracts. 3.2 Methodology For each of the 518 trading days in the sample, following Orosi (2012), a thin-plate spline-based interpolant was fitted to all available implied volatilities. Then, using this interpolant, 21 uniformly distributed implied volatilities with one month maturity were generated such that the range of the generated strike prices corresponded to the range of the quoted option prices. Finally, the implied volatilities were converted to call option prices. To calibrate the option pricing model in (1), the procedure presented in Orosi (2015) is used: 1. Transform all of the available option prices {C} n i=1 and strike prices {X}n i=1 using c i = C i Se dt x i = X ie rt Se dt. 2. Determine the model parameters that minimize the following objective: n (c (x i ) c i ) 2, i=1 where c (x i ) has to be numerically determined from the equation x i = 1 c (x i ) + G (1 c (x i)) β c (x i ) α. During this step, the call option prices have to be numerically retrieved from these equations by employing golden section search (or some other technique). 3. During Step 2, ensure that the parameters are restricted to the intervals G > 0, α > 0, β > 2, to guarantee arbitrage-free call option prices. In our study, the above procedure was carried out by using the Matlab programming language. To calculate the parameters for the option pricing model, the built-in fminsearch function was used. 5

6 3.3 Performance of the Strategies The performances, based on weekly returns 2, of the S&P 500 index and the two trading strategies over the 10-year period are reported in Table 1. It can be observed that the S&P 500 index that has a total return of 60.16% and average annual return of 4.71%. Consequently, Strategy 2 significantly outperforms the S&P 500 index with a total and average annual return of % and 15.49%. Although Strategy 2 also outperforms the S&P 500 index with a total and average annual return of % and 7.91%, the returns are lower than that of Strategy 1. Moreover, the Sharpe ratios are more than double than that of the index that indicates that the excess returns are not generated by taking in excess risk. The cumulative returns are plotted in Figure 1. It is shown that both the index and Strategy 1 experienced similar and significant drawdowns during the financial crisis of In addition, it can be observed that Strategy 1 does not consistently outperform the index every year. For example, the strategy did not generate any profit from the middle of 2012 to the middle of 2013, during which period the S&P 500 index made modest gains. However, Strategy 1 made up for this loss during the remaining part of 2013 and our results suggest that it significantly outperforms the index over long periods. Finally, the weekly returns of Strategy 2 and the index have a correlation of , which make it attractive for diversification purposes. Although Figure 1 suggests that the excess return of Strategy 1 is achieved by the strategy avoiding the large drawdown of the index during the financial crisis of 2008, we demonstrate that this is not the case. To this effect, we examine the performance of two strategies that leverage the returns of Strategy 1 by a factor of 1.5 and a factor of 2. The cumulative returns of these strategies are plotted in Figure 2 and the performances are reported in Table 3. Although the strategies based on Strategy 2 do not always outperform the index, their performance is superior over longer periods. Moreover, Strategy 2, and consequently the leveraged strategies, have a win-loss ratio of 1.536, while the S&P has a weekly win-loss ratio of It is also noteworthy that Strategy 2 leveraged by a factor of 1.5 has a significantly lower drawdown than the index and has a significantly higher cumulative return of %. Furthermore, Strategy 2 leveraged by a factor of 2 has a similar drawdown than the index, and it has a cumulative return of %. Although executing the strategies is more costly than replicating the S&P 500 index, the trading fees are minimal compared to the excess returns the strategies generate. In particular, CME Group (2015) reports that the transaction cost of purchasing an ES future is 1.5 bps (including execution fees and market impact), and the annual holding cost is 20 bps. Therefore, even with frequent rebalancing, the strategies are highly profitable. 6

7 4 Conclusion and Future Research In this study, we present evidence that the thickening or thinning of the right tail of the risk-neutral distribution contains information about the future weekly returns of the S&P 500 index. Moreover, we demonstrate that weekly changes in the tail parameter can be used to devise profitable trading strategies that substantially outperform the S&P 500 index. The profitability of the strategies indicates that the thickening of the right tail is more likely to be followed by a positive weekly change in the S&P 500 index, and vice versa. Therefore, our results are in line with current literature and confirm that option prices contain forward looking information about the direction of the underlying index. Although we do expect the market to eventually incorporate this information such that the profitability of the strategies is eliminated, the option implied information can still be beneficial for portfolio and risk managers. Finally, in our future research, we wish to explore if our findings can be generalized to other markets. Moreover, besides examining the information content of equity index options, we wish to explore whether an improved portfolio consisting of individual equities could be constructed by incorporating information from the options written on these stocks. References [1] Aït-Sahalia, Y. and Lo. A.W. (2000) Nonparametric Risk Management and Implied Risk Aversion, Journal of Econometrics, 94, 1-2, [2] Audrino, F., Huitema, R., and Ludwig M. (2014) An Empirical Analysis of the Ross Recovery Theorem, Available at SSRN, [3] Bakshi, G., Cao, C., and Chen, Z. Empirical Performance of Alternative Option Pricing Models. Journal of Finance, 52 (1997), pp [4] Black, F. and Scholes, M. S. (1973) The Pricing of Options and Corporate Liabilities, Journal of Political Economy, 81, [5] CME Group (2015), The Big Picture: A Cost Comparison of Futures and ETFs, Available at: [6] DeMiguel, V., Plyakha, Y., Uppal, R., and Vilkov, G. (2013) Improving Portfolio Selection Using Option-Implied Volatility and Skewness, Journal of Financial and Quantitative Analysis, 48, 6, [7] Ghysels, E. and Wang, F. (2014) Moment-Implied Densities: Properties and Applications, Journal of Business & Economic Statistics, 32, [8] Orosi, G. (2012) Empirical Performance of a Spline-Based Implied Volatility Surface, Journal of Derivatives & Hedge Funds, 18, 4,

8 [9] Orosi G. (2015) Estimating Option-Implied Risk-Neutral Densities: A Novel Parametric Approach, Journal of Derivatives, 23, 1, [10] Ross, S.A. (2015) The Recovery Theorem, Journal of Finance, 70, 2, Endnotes Although we refer to the changes as weekly, one set of changes is biweekly because July 4, 2007 was a holiday. 8

9 D<30 30<D<90 D>90 D>180 All -0.1>M> >M> >M> >M> >M> >M> All Table 1: S&P 500 Index call option data. The table shows the number of call option contracts broken down by moneyness (M) and days to maturity (D). 9

10 Parameter G α β Mean Standard Deviation Table 2: Distribution of the parameters of the option pricing model fitted to generated call option prices with one month maturity. 10

11 S&P 500 Strategy 1 Strategy 2 Positive Weekly Change 295 (57.2%) 273 (52.9%) 149 (28.9%) No Weekly Change 0 (0%) 0 (0%) 270 (52.3%) Negative Weekly Change 221 (42.8%) 243 (47.1%) 97 (18.8%) Average Arithmetic Return per Week 0.119% 0.356% 0.166% Total Return 60.16% % % Average Geometric Annual Return 4.71% 15.49% 7.91% Annualized Sharpe Ratio Correlation of Weekly Returns with S&P Table 3: The table presents the performance of the S&P 500 Index and the two strategies. 11

12 S&P 500 Strategy 2 Strategy 2 Leveraged 1.5X Leveraged 2X Total Return 60.16% % % Average Geometric Annual Return 4.71% 11.35% 14.46% Table 4: The table presents the returns of the S&P 500 Index and the two leveraged strategies based on Strategy 2. 12

13 Figure 1. Plotted are the cumulative returns for the S&P 500 index and the two strategies. 13

14 Figure 2. Plotted are the cumulative returns for the S&P 500 index and the two leveraged strategies based on Strategy 2. 14

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