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- Returns,
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Monotonicity of the stochastic discount factor and expected option ...

in an increasing across strike groups.For example, **the** statistic for call **option**s is 9.73 for**the** classical Page test **and** 6.11 for **the** restricted sample. These high statistic values resultbecause, in **the** call-**option** samples, for example if we look at a stock-by-stock case, 72%**of** **the** stocks have average-return ranks that are monotonically increasing in strike (thatis, rank vectors (1, 2, . . . , k i )).The results are even stronger if we fur**the**r increase **the**required minimum number **of** weekly observations . For example, restricting **the** number **of**minimum weekly observations to 150 (250) increases **the** number **of** stocks that have averagecall returns strictly increasing in strike to 81%( 90%). Note also that although **the** pairwisetest results for puts from Table II Panel B were somewhat weak, **the** joint test providesconvincing evidence **of** strict monotonicity.C. Heterogeneity **and** Test DesignThe previous two subsections provide strong evidence in support **of** strict monotonicity **of****the** SDF.These results contradict **the** results **of** Ni (2007), who finds violations **of** weakmonotonicity. Ni (2007) sorts call **option** contracts on each buying date into five portfoliosbased on moneyness **and** shows that **the** portfolio with OTM **option** contracts earns significantlylower returns than **the** portfolio with ITM **option** contracts. She suggests thatinvestors are sometimes risk-seeking **and** a preference for idiosyncratic skewness leads to apremium for deep OTM **option**s **and** could be a possible explanation for **the** puzzling callreturns. We will show that such a testing procedure can bias her results because it fails tocontrol for **the** stocks underlying **the** **option**s in each portfolio, unlike our pairwise tests, inwhich matched pairs **of** returns (same underlying stock **and** same time period) for differentstrikes are subtracted before averaging.[Insert T able IV here]To illustrate **the** problem, consider **the** Black-Scholes setting, in which **the** excess (over**the** riskless rate) **option** return equals **the** product **of** **the** **option** elasticity **and** **the** excess25

stock return.Elasticities depend on **the** **option** moneyness **and** stock volatility (as well as**the** short rate **and** time to expiration). Table IV presents elasticities for AIG **and** USRX call**option**s for a particular date. The Black-Scholes elasticity **of** **the** lowest strike call on AIG isalmost equal to **the** elasticity **of** **the** highest strike call on USRX. The heterogeneity between**the**se two stocks would not bias **the** results if AIG **and** USRX each had one **option** in each**of** **the** five strike groups each day.However, if both stocks have calls in strike group 4, butUSRX calls are overrepresented in strike group 5, this may induce a downward bias in strikegroup5 returns relative to strike-group 4 returns. Many stocks **of**fer only a limited number**of** strikes, especially after applying reasonable filtering rules (e.g., eliminating no-arbitrageviolations). The problem is likely to be most severe in **the** deep ITM **and** deep OTM groups,where strike listings are thinnest, **and** where thin trading is most likely to result in morefrequent filter violations.The combination **of** **the** underlying stock heterogeneity **and** **the**heterogeneity **of** strike-group representation would have **the** most severe impact in **the** deepOTM group because **of** **the** much higher elasticities in this group compared to **the** deep ITMgroup.[Insert T able V here]Table V reports **the** average Black-Scholes elasticities **of** calls sorted into strike groupsbased on **the** **option**’s moneyness. The sample **and** moneyness cut**of**fs 21 used are from Ni(2007). The mean **and** median elasticities **of** deep OTM calls are in fact smaller than that **of**ATMs, despite **the** fact that, ceteris paribus, elasticities are increasing in strike. Assuming,for illustration, identical stock betas, positive risk premia, **and** **the** validity **of** CAPM, **the****expected** return **of** **the** ATM group would **the**refore exceed that **of** **the** OTM group, despite**the** fact that **expected** returns **of** calls would be increasing in strike in **the** B-S setting. Alsonote that **the** st**and**ard deviation **of** **the** elasticities are enormous suggesting huge variationswithin strike groups.21The moneyness cut**of**fs used are as follows: K/S ≤ 0.85, 0.85 < K/S ≤ 0.95, 0.95 1.15.26

- 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 18 and 19: eturns, and the sample reduces to 1
- 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 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