- Text
- Returns,
- Monotonicity,
- Strict,
- Increasing,
- Binary,
- Strategies,
- Statistics,
- Stocks,
- Prices,
- Underlying,
- Stochastic,
- Discount,
- Factor

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

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) **option**s increases**the** 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 number**of** 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 **option**s 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 whe**the**r call **and** put **expected** returns are increasingin **the** strike price. We **the**n test strict monotonicity by testing **the** monotonicity **of****expected** 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** **the**n tests whe**the**r **the** average return differences between groups are positive.Subsection B compliments **the** pairwise tests by conducting a joint test **of** monotonicity **of****expected** 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** **the**next two lower strikes are assigned to strike groups 2 **and** 1, respectively. Note that **the**returns 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-3**and** strike-group-2 returns for **the** same stock on **the** same week.Under **the** hypo**the**sis **of**weak monotonicity, each call **and** put return difference has a positive **expected** value. Under**the** hypo**the**sis **of** strict monotonicity, all return differences have positive **expected** values.The return differences are averaged across different stocks to form a weekly time series **of**average 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 **option**s, **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 **option**s, **the** column “ R f − 5 ” presents **the** average difference between **the**risk-free rate (**the** limiting return **of** a put as **the** strike price increases towards infinity) **and****the** 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

- Page 1: Monotonicity of the stochastic disc
- Page 4 and 5: 500 results, our empirical analysis
- Page 6 and 7: I. Characterization of SDF Monotoni
- Page 8 and 9: The intuition for the equivalence o
- Page 10 and 11: consumption in the two middle state
- Page 12 and 13: The first two examples below presen
- Page 14 and 15: tional to the slope of m (), as is
- Page 16 and 17: strict monotonicity because it requ
- Page 20 and 21: skewness of option returns (most OT
- Page 22 and 23: money.Assuming only differentiabili
- Page 24 and 25: against the alternative hypothesis
- Page 26 and 27: in an increasing across strike grou
- Page 28 and 29: Further compounding the heterogenei
- Page 30 and 31: is equivalent to0 > E ( mS T 1 {ST
- Page 32 and 33: The inverse of the expected return
- Page 35 and 36: Using integration by parts:Cov (Y,
- Page 37 and 38: Equation B3 is the skewness adjuste
- Page 39 and 40: where we have used, for any k ∈ {
- Page 41 and 42: Brown, David, and Jens Carsten Jack
- Page 43 and 44: Huang, C., and R. Litzenberger, 198
- Page 45 and 46: Shive, S., and T. Shumway, 2006, Is
- Page 47 and 48: Table IIAverage Return Differences
- Page 49 and 50: Table IVOption Elasticities for AIG