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

money.Assuming only differentiability, it is well known thatddK C (K) =ddK E { }m · (S T − K) 1 {ST >K} = −B (K) .Because strikes are insufficiently dense, a better way to computed C (K) (ra**the**r th**and**Kdifferencing market premiums at different strikes) is to estimate a smooth Black-Scholes (B-S)implied volatility curve σ (K) **and** **the**n estimate B (K) from **the** B-S model in **the** followingway.Letting r denote **the** annualized continuously compounded zero-coupon-bond yieldfrom 0 to T , **the**n (assuming a smooth B-S implied volatility curve σ ()):B (K) = − d{dK CBS (S, K, σ (K)) = e −rT N (d 2 ) − KN ′ (d 2 ) √ }T σ ′ (K) .We apply two obvious no-arbitrage restrictions:B (K) ∈ ( 0, e −rT ) ,dB (K) ≤ 0.dKThe second can be checked usingd 2 C= e −rT N ′ (d 2 ){1 + ddK 2 2 K √ } { }T σ ′ 1(K)σ (K)K √ T + d 1σ ′ (K)+e −rT KN ′ (d 2 ) √ T σ ′′ (K) .To estimate a smooth B-S implied volatility curve, implied volatilities for all calls on**the** same stock **and** same buying date are used to fit a smooth curve using cubic splineinterpolation. To ensure better estimation **of** **the** implied volatility curve, we restrict oursample to only those calls whose underlying stocks have at least four strikes on any tradingday. 18Finally, to reduce **the** possibility **of** unrealistic returns, we omit binary calls wi**the**stimated prices below $0.01, giving us a total **of** 94,009 binary call returns that satisfies all18We get similar results when we require at least three strikes instead **of** four.21

**the** restrictions. Results from Table II Panel D indicate that all average return differencesare positive, with three out **of** five significant at **the** 1% level. Binary calls for strike group 1earn on average 4.9% higher weekly return than **the** risk free whereas binary calls for strikegroup 4 are 5.9% higher than those for strike group 3.Modified bullish call spreadsOnce we have **the** prices **of** **the** cash-or-nothing binary call**option**, computing **the** prices **of** **the** modified bullish call spread is straightforward. Recallthat we defined a modified bullish call spread as a portfolio that is long a call with strike K,short a call with strike K + ∆K, where ∆K > 0, **and** short a cash-or-nothing binary callwith pay**of**f ∆K · 1 {ST >K+∆K} (all on **the** same stock **and** same expiration T ). We impose noo**the**r restriction on **the** bullish call spreads o**the**r than requiring a minimum price **of** 0.125,which results in a total **of** 63,291 bullish call return observations. Table II Panel E showsthat **the** return differences are all positive **and** significant at **the** 1% level **of** significance. Forexample, bullish call spread returns for strike group 2 earn on average 9.7% higher weeklyreturn than strike group 2, **and** returns for strike group 3 are 4.7% higher than those forstrike group 2.B. Page test for ordered alternativesThe results in Table II are overall consistent with strict SDF monotonicity.A problemwith **the** pairwise tests, however, is that four or five separate tests statistics are providedfor each strategy.If all are positive **and** significant, this clearly supports **the** hypo**the**sis **of**monotonicity, but mixed results can be difficult to interpret statistically.Letting ¯r i denote**the** mean strike-group-i return, **the** Page test (or Page’s L test) for ordered alternativesprovides a single statistic to test **the** null hypo**the**sis **of** equal means returns,H 0 : ¯r 1 = ¯r 2 = · · · = ¯r k ,22

- 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 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