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

Using integration by parts:Cov (Y, g (Y )| A) =∫ b= −=a−∫ ba−∫ ba−(y − E (Y | A)) 1 {y∈A} g (y)g (y) dh (y; A)h (y−; A) dg (y) .dF (y)P (A)If g is strictly decreasing **the**n **the** left h**and** side must be strictly negative.Part b) The sufficiency **of** g strictly decreasing follows from Part a.To prove necessity,suppose (A7) holds, but g is not strictly decreasing.Then **the**re must exist points c < esuch that g (c) ≤ g (e), **and** **the**refore a point d ∈ (c, e) such that g ′ (d) > 0 (we can rule outg ′ (x) = 0 for all x ∈ (c, e), because this contradicts Cov ( )Y, g (Y ) 1 {Y ∈[c,e]} < 0). Continuity**of** g ′ implies g ′ (x) > 0 for x in a neighborhood **of** d, implying Cov ( )Y, g (Y ) 1 {Y ∈[d−ε,d+ε]} > 0for some ε > 0, contradicting (A7).Pro**of** **of** Proposition 8The pro**of** **of** (i) ⇐⇒ (ii) follows from Lemma A1 above.The pro**of** **of** **the** equivalence**of** (ii), (iii) **and** (iv) is **the** same as in Proposition 3 (**the** weak monotonicity case). Thesufficiency part **of** (ii) follows from Proposition 7 above. The necessity pro**of** **of** (ii) follows.Suppose m () is not strictly decreasing. Then **the**re must exist points c < e such thatm (c) ≤ m (e).If m () is constant in **the** interval [c, e], **the**n fix some ¯K **and** define some Gsatisfying Condition 6 so that G ( s − ¯K ) is positive only for s within some interval within**the** interior **of** [c, e].Then it is easy to show that R (G (S T − K)) will be constant for Kwithin some neighborhood **of** ¯K.If m () is not constant in **the** interval [c, e], **the**re mustexist a point d ∈ (c, e) such that g ′ (d) > 0.Continuity **of** g ′ implies g ′ (x) > 0 for x in aninterval [d 1 , d 2 ] containing d.Fix some ¯K **and** construct a G that satisfies Condition 6 sothat G ( s − ¯K ) is positive only for s ∈ [d 1 , d 2 ]. It is easy to show from **the** sufficiency part **of****the** pro**of** **of** Proposition 7 that R (G (S T − K)) is decreasing in K within some neighborhood34

- 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 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 36 and 37: of ¯K.Appendix BSkewness-adjusted
- Page 38 and 39: To compute the variance, we first s
- Page 40 and 41: ReferencesAit-Sahalia, Yacine, and
- Page 42 and 43: Fama, Eugene F., and Kenneth R. Fre
- Page 44 and 45: Newey, Whitney K., and Kenneth D. W
- Page 46 and 47: Table IAverage Returns for option s
- Page 48 and 49: Table IIIPage Test for Ordered Alte
- Page 50: Table VAverage Deltas and Elasticit