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

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

## money.Assuming only

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) (rather thandKdifferencing market premiums at different strikes) is to estimate a smooth Black-Scholes (B-S)implied volatility curve σ (K) and then 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 , then (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 onthe 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 withestimated 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 calloption, 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 payoff ∆K · 1 {ST >K+∆K} (all on the same stock and same expiration T ). We impose noother restriction on the bullish call spreads other 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 hypothesis ofmonotonicity, but mixed results can be difficult to interpret statistically.Letting ¯r i denotethe mean strike-group-i return, the Page test (or Page’s L test) for ordered alternativesprovides a single statistic to test the null hypothesis of equal means returns,H 0 : ¯r 1 = ¯r 2 = · · · = ¯r k ,22

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