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Monotonicity of the stochastic discount factor and expected option ...

Monotonicity of the stochastic discount factor and expected option ...

eturns,

eturns, and the sample reduces to 1,643,925 after the restrictions have been applied. Themean (median) number of unique stocks on each buying date is 1,380 (1,421) and the mean(median) number of strikes for each stock on each buying date is 3.43 (3.00).The thintrading in deep in-the-money (ITM) and deep out-of-the-money (OTM) options increasesthe possibility of stale quotes.Therefore as a robustness test we repeat our analysis withonly those option contracts which have a positive volume on the buying date.The numberof calls in our sample that have a positive volume on the buying date falls to 598,826. Theresults however remain qualitatively similar.The initial sample for put options consists of 2,224,293 return observations, with 1,982,501observations satisfying our sample restrictions, and 450,931 returns having a positive volumeon the buying date. The mean (median) number of unique stocks on each buying date is1,423 (1,461) and the mean (median) number of strikes for each stock on each buying dateis 3.72 (3.00).III.Empirical ResultsWe first test weak monotonicity by testing whether call and put expected returns are increasingin the strike price. We then test strict monotonicity by testing the monotonicity ofexpected returns in strike for additional option strategies with log-concave payout functions.Each pairwise test, presented in SubsectionA, assigns strategies into strike groups, computesreturn differences between consecutive strike groups (higher strike return minus lower strikereturn), and then tests whether the average return differences between groups are positive.Subsection B compliments the pairwise tests by conducting a joint test of monotonicity ofexpected returns in strike using the Page test for ordered alternatives, which generates asingle test statistic.17

A. Pairwise TestsWe first assign returns into strike groups.For any underlying stock on any buying date,we assign to strike group 3 the option-strategy return with strike closest to at the money.The next two higher strikes are assigned to strike groups 4 and 5, respectively, and thenext two lower strikes are assigned to strike groups 2 and 1, respectively. Note that thereturns are first divided into strike groups before any filters are applied to the data. Averagereturn differences are reported in Table II with, for example, the column “ 3 − 2 ” presentingaverage strike-group-3 minus strike-group-2 returns. For a given strategy (e.g., long calls),each difference used to compute the average is obtained by subtracting the strike-group-3and strike-group-2 returns for the same stock on the same week.Under the hypothesis ofweak monotonicity, each call and put return difference has a positive expected value. Underthe hypothesis of strict monotonicity, all return differences have positive expected values.The return differences are averaged across different stocks to form a weekly time series ofaverage return differences.The average of the weekly time series and the correspondingt-statistics are reported in Table II.[Insert T able II here]For vanilla calls and puts and binary calls we compute an additional difference.Forvanilla call options, the column “ 1 − R S” presents the average difference between eachstrike-group-1 return and the return of the underlying stock (which represents a zero-strikecall).For put options, the column “ R f − 5 ” presents the average difference between therisk-free rate (the limiting return of a put as the strike price increases towards infinity) andthe strike-group-5 return.And for binary calls, the column “1 − R f ” presents the averagedifference between the strike-group-1 return and the risk-free rate (the return on a binarycall with zero strike).Table II also provides a skewness-adjusted t-statistic for the average return difference.Central-limit approximations are problematic because of the small sample sizes and the heavy18

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